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ED  383  538 

SE  055  811 






Geeslin,  William,  Ed.;  Graham,  Karen,  Ed. 
Proceedings  of  the  Conference  of  the  International 
Group  for  the  Psychology  of  Mathematics  Education 
(PME)    (16th,  Durham,  NH,  August  6-11,   1992).  Volumes 

International  Group  for  the  Psychology  of  Mathematics 


Aug  92 


Collected  Works  -  Conference  Proceedings  (021) 
MF06/PC39  Plus  Postage. 

Action  Research;  Affective  Behavior;  Algebra; 
Arithmetic;  Cognitive  Development;  College 
Mathematics;  Constructivism  (Learning);  Content  Area 
Writing;  Context  Effect;  Cooperative  Learning; 
"Cultural  Influences;  Discussion  (Teaching 
Technique);  Elementary  Secondary  Education;  Equations 
(Mathematics);  ^Geometry;  Imagery;  Language; 
Mathematics  Achievement;  Mathematics  Education; 
Mathematics  Instruction;  Metacogni t i on ; 
Misconceptions;  Number  Systems;  Problem  Solving; 
Proof   (Mathematics);  Ratios   (Mathematics);  Sex 
Differences;  Social  Psychology;  Spatial  Ability; 
Student  Attitudes;  *Teacher  Education;  Thinking 
Skills;  Visualization;  Writing  Across  the 

Advanced  Mathematics;  LOGO  Programming  Language; 
Mathematical  Communications;  Mathematical  Thinking; 
"Mathematics  Education  Research;  '^Psychology  of 
Mathematics  Education;  Representations  (Mathematics); 
Teacher  Candidates;  Teacher  Change;  Teacher 




The  Proceedings  of  PME-XVI  has  been  published  in 
three  volumes  because  of  the  large  number  of  papers  presented  at  the 
conference.  Volume  1  contains:   (1)  brief  reports  from  each  of  the  11 
standing  Working  Groups  on  their  respective  roles  in  organizing 
PME-XVI;   (2)  brief  reports  from  6  Discussion  Groups;  and  (3)  35 
research  reports  covering  authors  with  last  names  beginning  A~K. 
Volume  II  contains  42  research  reports  covering  authors  with  last 
names  beginning  K~S.  Volume  III  contains  (1)   15  research  reports 
(authors  S~W) ;    (2)  31  short  oral  presentations;   (3)  AO  poster 
presentations;    (4)  9  Featured  Discussion  Groups  reports;    (5)   1  brief 
Plenary  Panel  report  and  4  Plenary  Address  reports.  In  summary,  the 
three  volumes  contain  95  full-scale  research  reports,  4  full-scale 
plenary  reports,  and  96  briefer  reports.  Conference  subject  content 
can  be  conveyed  through  a  listing  of  Work  Group  topics,  Discussion 
Group  topics,  and  Plenary  Panels/Addresses,  as  follows.  Working 
Groups:  Advanced  Mathematical  Thinking;  Algebraic  Processes  and 
Structure;  Classroom  Research;  Cultural  Aspects  in  Mathematics 
Learning;  Geometry;  Psychology  of  Inservice  Education  of  Mathematics 
Teachers;  Ratio  and  Proportion;  Representations;  Research  on  the 
Psychology  of  Mathematics  Teacher  Development;  Social  Psychology  of 
Mathematics  Education;  Teachers  as  Researchers  in  Mathematics 
Education.  Discussion  Groups:  Dilemmas  of  Constructivist  Mathematics 
Teaching;  Meaningful  Contexts  for  School  Mathematics;  Paradigms  Lost 
-  What  Can  Mathematics  Education  Learn  From  Research  in  Othe- 
Disciplines?;  Philosophy  of  Mathematics  Education;  Research  in  the 
Teaching  and  Learning  of  Undergraduate  Mathematics;  Visualization  in 
Problem  Solving  and  Learning.  Plenary  Panels/Addresses:  Visualization 
and  Imagistic  Thinking;  "The  Importance  and  Limits  of  Epistemological 
Work  in  Didactics"  (M.  Artigue) ;  "Mathematics  as  a  Foreign  Language" 
(G.  Ervynck) ;  "On  Developing  a  Unified  Model  for  the  Psychology  of 
Mathematical  Learning  and  Problem  Solving"  (G.  Goldin) ; 
"Illuminations  and  Reflections — Teachers,  Methodologies,  and 
Mathematics"  (C.  Hovles).  (MKR) 

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University  of  New  Hampshire 
Durham,  NH  (USA) 
August  6- 11,  1992 

Volume  I 

Published  by  the  Program  Committee  of  the  16th  PME  Conference,  USA. 

All  rights  reserved. 

William  Geeslin  and  Karen  Graham 
Department  of  Mathematics 
University  of  New  Hampshire 
Durham,  NH  03824 


1  -i 

Edited  by  WiUiam  Geeslin  and  Karen  Graham 
Mathematics  Department 
University  of  New  Hampshire 
Durham  NH 


The  first  meeting  of  PME  took  place  in  Karlsruhe,  Germany  in  1976.  Thereafter  different 
countries  (Netherlands,  Germany,  U.K.,  U.S.A.,  France,  Belgium,  Israel,  Australia,  Canada, 
Hungary,  Mexico,  Italy)  hosted  the  conference.  In  1992,  the  U.S.A.  will  again  pjay  host  to  PME. 
The  conference  will  take  place  at  the  University  of  New  Hampshire  in  Durham,  NH.  The 
University  was  founded  in  1866  as  the  New  Hampshire  College  of  Agriculture  and  the  Mechanic 
Arts.  The  state  legislature  granted  it  a  new  charter  as  the  University  of  New  Hampshire  in  1923. 
The  University  now  has  about  800  faculty  members  and  more  than  10,000  students  enrolled  in  100 
undergraduate  and  75  graduate  programs.  The  University's  Mathematics  Department  has  a  strong 
history  of  commitment  to  research  and  service  in  mathematics  education.  We  are  pleased  to  be  the 
host  site  for  PME  XVI. 

The  academic  program  of  PME  XVI  includes: 

•  92  research  reports 

•  4  plenary  addresses 

•  1  plenary  panel 

•  11  working  groups 

•  6  discussion  groups 

•  2  featured  discussion  groups 

•  31  short  oral  presentations 

•  40  poster  presentations. 

The  short  oral  presentations  represent  a  new  format  for  sessions  at  PME. 

The  review  process 

The  Program  Committee  received  a  total  of  181  research  proposals  that  encompassed  a 
wide  variety  of  themes  and  approaches.  Each  proposal  was  submitted  to  three  outside  reviewers 
who  were  knowledgeable  in  the  specific  research  area.  In  addition,  one  or  more  program 
committee  members  read  each  paper.  Based  on  these  reviews  each  paper  was  accepted,  rejected, 
or  accepted  as  a  short  oral  presentation  or  poster.  If  a  reviewer  submitted  written  comments,  they 
were  forwarded  to  the  authors)  along  with  trie  Program  Committee's  decision. 


1  -  ii 

List  of  PME  XVI  Reviewers 

The  Program  Committee  wishes  to  thank  the  following  people  for  their  help  during  the  review 

Alice  Alston,  USA 

MicWle  Arn'gue,  France 

Arthur  Baroody,  USA 

MarioUna  Bartollni  Bussi,  Italy 

Thomas  Bassarear,  USA 

Michael  Battista,  USA 

Merlyn  Behr,  USA 

Alan  Bell,  United  Kingdom 

Jacques  Bergeron,  Canada 

Alan  Bishop,  United  Kingdom 

Cathy  Brown,  USA 

Deborah  Carey,  USA 

Thomas  Carpenter,  USA 

Randall  Charles,  USA 

Daniel  Chazan,  USA 

Doug  Clements,  USA 

Paul  Cobb.  USA 

Jere  Confrey,  USA 

Beatriz  D'Ambrosio,  USA 

Linda  Deguire,  USA 

Barbara  Dougherty,  USA 

EdDubinsky,  USA 

Sharon  Dugdale,  USA 

Laurie  Edwards,  USA 

Theodore  Eisenberg,  Israel 

Nerida  Ellerton,  Australia 

Joan  Ferrini-Mundy,  USA 

Olimpia  Figueras,  Mexico 

Eugenio  FUloy,  Mexico 

Joaquim  Giminez,  Spain 

Gerry  Goldin,  USA 

David  Green,  United  Kingdom 

Brian  Greer,  United  Kingdom 

Angel  Gutierrez,  Spain 

Gila  Hanna,  Canada 

Guershon  Harel,  USA 

Lynn  Hart,  USA 

Rina  Hershkowitz,  Israel 

Jim  Hiebert,  USA 

Celia  Hoyles,  United  Kingdom 

Robert  Hunting,  Australia 

Claude  Janvier,  Canada 

Barbara  Jaworski,  United  Kingdom 

James  Kaput,  USA 

Carolyn  Kieran,  Canada. 

David  Kirshner,  USA 

Cliff  Konold.  USA 

Colette  Laborde,  France, 

Sue  Lamon,  USA 

Marsha  Landau,  USA 

Gilah  Leder,  Australia 

Miriam  Leiva,  USA 

Fou-Lai  Lin,  Taiwan 

Wendy  Mansfield,  USA 

Zvia  Markovits,  Israel 

Doug  McLeod,  USA 

Jamce  Mokros,  USA 

Steve  Monk,  USA 

Jim  Moser,  USA 

Ricardo  Nemkovsky,  USA 

Pearia  Nesher,  Israel 

Nobujiko  Nohda,  Japan 

Terezinha  Nunes,  United  Kingdom 

Tony  Orton,  United  Kingdom 

John  Pace,  USA 

Jo3o  Pedro  Ponte,  Portugal 

David  Pimm,  Canada 

Thomas  Post,  USA 

Ferd  Prevost,  USA 

Ian  Putt,  Australia 

Sid  Rachlin,  USA 

John  Richards,  USA 

Andee  Rubin,  USA 

Susan  Jo  Russell,  USA 

Deborah  Schifter,  USA 

Thomas  Schroeder,  Canada 

Karen  Schultz,  USA 

Mike  Shaughnessy,  USA 

Yoshinori  Shimizu,  Japan 

Edward  Silver,  USA 

Larry  Sowder,  USA 

Judith  Sowder,  USA 

Leslie  Steffe,  USA 

Kevin  Sullivan,  USA 

Lindsay  Tartre,  USA 

Dina  Tirosh,  Israel 

Shlomo  Vinner,  Israel 

Terry  Wood.  USA 


Michal  Yerushalmy,  Israel 



1  -iii 



-  President  KathHart  (United  Kingdom) 

-  Vice-President  Gilah  Leder  (Australia) 

-  Secretary  Martin  Cooper  (Australia) 

-  Treasurer  Angel  Gutierrez  (Spain) 


Mich61e  Artigue  (France)  -  Frank  Lester  (USA) 

Mariolina  Bartolini-Bussi  (Italy)  -  -  Fou-Lai  Lin  (Taiwan) 

BernadetteDenys  (France)  -  Carolyn  Maher  (USA) 

Claude  Gaulin  (Canada)  -  Nobuhiko  Nohda  (Japan) 

Gila  Hanna  (Canada)  -  Joao  Ponte  (Portugal) 

Barbara  Jaworski  (U.K.)  -  Dina  Tirosh  (Israel) 
Chronis  Kynigos  (Greece) 


-  Paul  Cobb  (USA)  -  Frank  Lester  (USA) 

-  Claude  Gaulin  (Canada)  -  Carolyn  Maher  (USA) 

-  William  EGeeslin  (USA)  -  Nobuhiko  Nohda  (Japan) 

-  Karen  Graham  (USA)  -  Barbara  Pence  (USA) 

-  Kathleen  Hart  (UK)  -  David  Pimm  (UK) 


-  Joan  Ferrini-Mundy  -  Karen  Graham 

-  William  E.  Geeslin  -  Lizabeth  Yost 

-William  EGeeslin 



1  -iv 


At  the  Third  International  Congress  on  Mathematical  Education  (ICME  3,  Karlsruhe,  1976) 
Professor  E.  Fischbein  of  the  Tel  Aviv  University,  Israel,  instituted  a  study  group  bringing 
together  people  working  in  the  area  of  the  psychology  of  mathematics  education.  PME  is  affiliated 
with  the  International  Commission  for  Mathematical  Instruction  (ICMI).  Its  past  presidents  have 
been  Prof.  Efraim  Fischbein,  Prof.  Richard  R.  Skemp  of  the  University  of  Warwick,  Dr.  Gerard 
Vergnaud  of  the  Centre  National  de  la  Recherche  Scientifique  (C.N.R.S.)  in  Paris,  Prof.  Kevin  F. 
Collis  of  the  University  of  Tasmania,  Prof.  Pear  la  Nesher  of  the  University  of  Haifa,  Dr.  Nicolas 
Balacheff,  C.N.R.S.  -  Lyon. 

The  major  goals  of  the  Group  are: 

•  To  promote  international  contacts  and  the  exchange  of  scientific  information  in  the  psychology 
of  mathematics  education; 

•  To  promote  and  stimulate  interdisciplinary  research  in  the  aforesaid  area  with  the  cooperation  of 
psychologists,  mathematicians  and  mathematics  teachers; 

•  To  further  a  deeper  and  better  understanding  of  the  psychological  aspects  of  teaching  and 
learning  mathematics  and  the  implications  thereof. 


Membership  is  open  to  people  involved  in  active  research  consistent  with  the  Group's  aims,  or 
professionally  interested  in  the  results  of  such  research. 

Membership  is  open  on  an  annual  basis  and  depends  on  payment  of  the  subscription  for  the  current 
year  (January  to  December). 

The  subscription  can  be  paid  together  with  the  conference  fee. 

1  -V 

Addresses  of  Authors  Presenting  Research  Reports  at  PME  XVI 

Vcma  Adams 

Dept.  of  Elementary/Secondary  Education 
Washington  State  University 
Pullman.  WA  99164-2122 


Department  of  Education 
University  of  Witwatersrand 
PO  Wits  2050,  Johannesburg 

M.C.  Batanero 

Escuela  Universitaria  del  Profesorado 
Campus  de  Cartuja 
18071,  Granada 

Michael  Battista 
404  White  Hall 
Kent  State  University 
Kent.  OH  44242 

Nadine  Bednarz 

CP.  8888  -  Sue  a  -  Montreal 

P.Quebec  H3C3P8 


David  Ben-Chaim 
Weizmann  Institute  of  Science 
Rehovot,  76100 

Janette  Bobis 

University  of  New  South  Wales 

Kensington,  NSW,  2033 


Dipartimento  Matematica  Umversita 


16132,  Genova 


Card  Brekke 
Telemark  Laercrhugskole 

Lynne  Cannon 

Faculty  of  Education 

Memorial  University  of  Newfoundland 

St  John's,  Newfoundland  A1B  3X8 


Olive  Chapman 

Dept  of  Curr.  &  Instruction,  U.  Calgary 
2500  University  Drive,  NW 
Calgary,  AB 

Giampaolo  Chiappini 
ViaL.  B.  Alberti,  4 
16132  Genova 

David  Clarke 

Australian  Catholic  University 
Oakleigh,  Victoria,  3166 

M.A.  (Ken)  Clements 
Faculty  of  Education 
Deakin  University 
Geelong,  Victoria  3217 


Dept  of  Education,  Kennedy  Hall 
Cornell  University 
Ithaca,  NY  14853 

Kathryn  Crawford 
Faculty  of  Education 
The  University  of  Sydney 
NSW  2006 

Linda  Davenport 
P.O.  Box  751 
Portland,  OR  97207 

Gary  Davis 

Institute  of  Mathematics  Education 
La  Trobe  University 
Bundoora,  Victoria  3083 



1  -vi 

Guida  de  Abreu 

Dept.  of  Education,  Trumpington  St. 
Cambridge  University 
Cambridge,  CB2 1QA 

Linda  DeGuire 
Mathematics  Department 
California  State  Univerisity 
Long  Beach,  CA  90840 

M.  Ann  Dirkes 

School  of  Education 

Purdue  University  at  Fort  Wayne 

Fort  Wayne,  IN  46805-1499 


Barbara  Dougherty 
University  of  Hawaii 
1776  University  Avenue 
Honolulu,  HI  96822 

Laurie  Edwards 
Crown  College 
University  of  California 
Santa  Cruz,  CA  95064 

Pier  Luigi  Ferrari 
Dipattimento  di  Matematica 
via  L.B.  Alberti 
4-16132  Genova 

Rossella  Garuti 

Dipattimento  Matematica  University 
via  L.B.  Alberti,  4 
16132,  Genova 

Linda  Gattuso 

College  du  Vieux  Montreal 

3417  Ave.  de  Vendome 

Montreal,  Quebec  H4A  3M6 


J.D.  Godino  Escuela 
Universitaria  del  Profesorado 
Campus  de  Cartuja 
18071,  Granada 

Susie  Groves 

Deakin  University  •  Burwood  Campus 
221  Burwood  Highway 
Burwood,  Victoria,  3125 

Elfriede  Guttenberger 
Avenida  Universidad  3000 
Maestria  en  Education  Matematica 
Mexico,  D.F.,  Of  Adm.  2, 1  piso 

Lynn  Hart 

Atlanta  Math  Project 

Georgia  State  University 

Atlanta,  GA  30303 


James  Hiebert 
College  of  Education 
University  of  Delaware 
Newark,  DE  19716 

Robert  Hunting 
La  Trobe  University 
Bundoora,  Victoria,  3083 

Barbara  Jaworski 
University  of  Birmingham 
Edgbaston  Birmingham  B15  2TT 

Clivc  Kanes 
Division  of  Education 
Griffith  University 
Nathan,  4111 

TE.  Kieran 

Dept.  of  Secondary  Education 
University  of  Alberta 
Edmonton  T6G2G5 

Cliff  Konold 

Hasbrouck  Laboratory 

University  of  Massachusetts  -  Amherst 

Amherst,  MA  01003 



1  -  vii 

Masataka  Koyarna 

Faculty  of  Education 

Hiroshima  University,  3-101 

2-365  Kagamiyama  Higashi-Hiroshima  City 


Konrad  Krainer 
IFF/Universitat  Klagenfurt 
Sterneckstrasse  IS 

Gilah  Lcder 
Monash  University 
Clayton,  Victoria  3168 

Stephen  Lerman 
103  Borough  Road 
London  SE1  OAA 

Liora  Linchevski 

School  of  Education 

Hebrew  University 

Mount  Scoups,  Jerusalem  91-905 


R.C.  Lins 

Shell  Centre  for  Math  Education 
University  Park 
Nottingham,  NG7  2QR 

Susan  Magidson 

EMST  -  4533  Tolman  Hall,  School  of  Ed. 
University  of  California 
Berkeley,  CA  94720 

Enrique  Castro 

Departamemo  Didactica  de  la  Matemaanca 
Campus  de  Cartuja  s/n 
18071  Granada 

Amy  Martinet 

Center  for  Math,  Science,  &  Computer  Ed. 

192  College  Avenue 

New  Brunswick,  NJ  08903-5062 


Joanna  Masingila 
Education  309 
Indiana  University 
Bloomington,  IN  47405 


University  Montpellier  U 

Place  Eugene  Bataillon 



Luciano  Meira 

Mestrado  em  Psicologia  Cognitiva 
CFCH  -  8"  andar,  Recife  50739  PE 

A.L  Mesquita 
R.  Marie  Brown,  7/8c 
1500  Lisbon 

Saburo  Minato 
College  of  Education 
Akita  University 
Gakuencho,  Tegata,  Akita  City 

Michael  Mitchelmore 
School  of  Education 
Macquarie  University 
NSW  2109 

Judit  Moschkovich 
4533  Tolman  Hall 
University  of  California 
Berkeley,  CA  94720 

Judith  Mousley 
Faculty  of  Education 
Deakin  University  * 
Geelong,  Victoria,  3217 

Joanne  Mulligan 

27  King  William  Street 

Greenwich  2065 




Hanlie  Murray 
Faculty  of  Education 
University  of  Stellenbosch 

Mitchell  Nathan 

University  of  Pittsburgh 
Pittsburgh,  PA  15260 

Dagmar  Neuman 
Box  1010 

University  of  GSteburg 
S-43126  MOlndal 

F.A.  Norman 
Dept.  of  Mathematics 
University  of  North  Carolina 
Charlotte,  NC  28223 

Richard  Noss 

Institute  of  Education,  U.  of  London 
20  Bedford  Way 
London  WC1H0AL 

School  of  Education 
Macquarie  University 
NSW  2109 

Kay  Owens 
P.O.  Box  555 

University  of  Western  Sydney,  Macarthur 
Campbelltown,  NSW  2560 

Marcela  Perlwitz 


Purdue  University 

West  Lafayette,  IN  47907-1442 


JoSo  Pedro  Ponte 

Av.  2y  de  Julmo,  134-4" 

1300  Lisboa,  PORTUGAL 


Matthias  Reiss 
Stedingerstr.  40 
7000  Stuttgart  31 

Joe  Relich 
PO  Box  10 

c/o  Faculty  of  Education 



Anne  Reynolds 

Math  Education,  219  Carothers,  B-182 
Florida  State  University 
Tallahassee,  FL  32306 

Mary  Rice 
Deakin  University 
Geelong,  Victoria  3217 

Naomi  Robinson 

Department  of  Science  Teaching 

Weizmann  Institute 

Rehovoth,  76100 


Adaiira  Sienz-Ludlow 

Dept  of  Mathematical  Sciences 

Northern  Illinois  University 

DcKalb,  IL  60115 


Victoria  Sanchez 
Avdo.  Ciudad  Jardin,  22 
41005  Sevilla 

Vinia  Maria  Santos 

School  of  Education,  Room  309 

Indiana  University 

Bloomington,  IN  47405 


Manvel  Joaquim  Saraiva 
Universidade  da  Beira  Interior 
Rue  Ferreira  de  Castro,  5-3 



1  -ix 

Analucia  Schliemann 
Mestrado  em  Psicologia 
8"  andar,  CFCH-UFPE 
50739  Recife 

Thomas  Schroeder 

Faculty  of  Education,  212S  Main  Mall 
University  of  British  Columbia 
Vancouver,  BC,  V6T  1Z4 

Yasuhiro  Sekiguchi 
Institute  of  Education 
University  of  Tsukuba 
Tsukuba-shi,  Ibaraki,  305 

Keiichi  Shigematsu  Takabatake 
Nara  University  of  Education 

Yoshinori  Shimizu 
4- 1  - 1 ,  Nukuikita-Machi 
Tokyo,  184 

Dianne  Seimon 

School  of  Education 

Phillip  Institute  of  Technology 

Alva  Grove,  Coburg  3058 


Martin  Simon 

176  Chambers  Building 

Department  of  Curriculum  and  Instruction 

Univesity  Park,  PA  16802 


Jack  Smith 

ColL  of  Education,  436  Erickson  Hall 
Michigan  State  University 
East  Lansing,  MI  48824 

Judith  Sowder 

Ctr.  for  Research  in  Math  and  Science  Ed. 
5475  Alvarado  Road,  Suite  206 
San  Diego,  CA  92120 

Kaye  Stacey 

School  of  Science  &  Math  Education 
University  of  Melbourne 
Parkville,  Victoria  3 142 

Rudolf  StrSsser 

Institut  fur  Didaktik  dcr  Mathematik 
UniversitSt  Bielefeld 
4800  Bielefeld 

L.  Streefland 
3561  GG,  Utrecht 

Susan  Taber 
717  Harvard  Lane 
Newark.  DE  19711 

Cornelia  Tierney 

2067  Massachusetts  Avenue 
Cambridge,  MA  02140 

School  of  Education 
Tel  Aviv  University 
Tel  Aviv,  69978 

Maria  Trigueros 
Rio  Hondo  Num  1 
ColoniaTizapan  San  Angel 
03100,  Mexico  D.F. 

Pessia  Tsamir 
School  of  Education 
Tel- Aviv  University 
Tel  Aviv,  69978 

Diana  Underwood 

Purdue  University 


West  Lafayette,  IN 




1  -X 

Marjory  Witte 

OCT  -  University  of  Amsterdam 
Grote  Bidiersstnut  72 
1013  KS,  Amsterdam 

Addresses  of  Presenters  of  Plenary  Sessions  at  PME  XVI 

Michele  Artigue 
IREM,  Universe  Paris  7 
2  Place  Jussieu 
75251  Paris  Cedex  5 

M.A.  (Ken)  Cleme-its 
Faculty  of  Education 
Deakin  University 
Geelong,  Victoria,  3166 

Tommy  Dreyfus 

Center  for  Technological  Education 
PO  Box  305 
Holon  58102 

Gontran  Ervynck 
Kath.Univ.  Leuven 
Campus  Kortrijk 

Gerald  A.  Goldin 

Center  for  Math,  Science,  &  Computer  Ed. 
Rugers  University 
Piscataway,  NJ  08855-1179 


Institute  of  Education,  Math 
University  of  London 
20  Bedford  Way 
London  WC1HOAL 

John  Mason 
Open  University 
Walton  Hall 

Milton  Keynes  MK7  6AA 

Bernard  Parzysz 

IUFMde  Lorraine 

Departement  de  mathematiques 

University  deMetz 

lie  du  Suilcy 

F  57000  Metz 


Norma  Presmeg 

219  Corothers  Hall,  B-182 

Florida  State  University 

Tallahassee,  FL  32306-3032 




1  -  xi 


Preface  P-  l-» 

International  Group  for  the  Psychology  of  Mathematics  Education  p.  1-i" 

History  and  Aims  of  the  PME  Group  p.  Mv 

Addresses  of  Authors  Presenting  Research  Reports  at  PME  XVI  p.  1-v 

Addresses  of  Presenters  of  Plenary  Sessions  at  PME  XVI  p.  1-x 

Working  Groups 

Advanced  mathematical  thinking  P-  1-3 

Organizers:  G.  Ervynck  &  D.  Tall 

Algebraic  processes  and  structure  P-  1-4 

Organizer  R.  Sutherland 

Classroom  Research  P-  1-5 

Organizer  F.J.  van  den  Brink 

Cultural  aspects  in  mathematics  learning  P-  1-7 
Organizer  B.  Denys 

Geometry  P-  1-8 
Organizer:  H.  Mansfield 

Psychology  of  inservice  education  of mathematics  teachers:  A  research  perspective  p.  1-9 

Organizers:  S.  Dawson,  T.  Wood,  B.  Dougherty,  &  B.  Jaworski 

Ratio  and  proportion  P- 
Organizers:  F.L.  Lin,  K.M.  Hart,  &  J.C.  Bergeron 

Representations  P-  1-11 
Organizer  G.  Goldin 

Research  on  the  psychology  of  mathematics  teacher  development  P-  1-12 
Organizer  N.EUerton 

Social  psychology  of  mathematics  education  P-  1-13 
Organizer:  A.J.  Bishop 

Teachers  as  researchers  in  mathematics  education  p.  1-14 
Organizers:  S.  Lerman  &  J.  Mousley 


x  o 

1  -  xii 

Discussion  Groups 

Dilemmas  ofconstructivist  mathematics  teaching:  Instances  from  classroom  practice 
Organizers:  R.  Carter  &  J.  Richards 

Meaningful  contexts  for  school  mathematics 
Organizers:  L.  Bazzini  &  L.  Grugnctti 

Paradigms  lost:  What  can  mathematics  education  learn  from  research  in  other  disciplii 
Organizer  B  A.  Doig 

Philosophy  of  mathematics  education 
Organizer  P.Ernest 

Research  in  the  teaching  and  learning  of  undergraduate  mathematics: 
Where  are  we?  Where  do  we  go  from  here? 

Organizers:  J.  Ferrini-Mundy,  E.  Dubinsky.  &  S.  Monk 

Visualization  in  problem  solving  and  learning 
Organizers:  M.A.  Mariotti  &  A.  Pesci 

Research  Reports 
de  Abreu,  Guida 

Approaches  to  research  into  cultural  conflicts  in  mathematics  learning 
Adams,  V.M. 

Rhetorical  problems  and  mathematical  problem  solving:  An  exploratory  study 

Action  research  and  the  theory-practice  dialectic:  Insights  from  a  small  post 
graduate  project  inspired  by  activity  theory 

Batanero,  M.C.,  Vallecillos.  M.A.  &  Godino,  J.D. 

Students'  understanding  of  the  significance  level  in  statistical  tests 

Battista.  M.T.  &  Clements.  D.H. 

Students'  cognitive  construction  of  squares  and  rectangles  in  Logo  Geometry 

Bcdnarz,  N.,  Radford,  L.,  Janvier,  B.  &  Lepage,  A. 
Arithmetical  and  algebraic  thinking  in  problem-solving 

Ben-Chaim,  D.,  Carmeii,  M.  &  Fresko,  B. 

Consultant  as  co-teacher:  Perceptions  of  an  intervention  for  improving 
mathematics  instruction 

Bobis,  J.,  Cooper,  M.  &  Sweller,  J. 

The  redundancy  effect  in  a  simple  elementary-school  geometry  task:  An  extension 
of  cognitive-load  theory  and  implications  for  teaching 

p.  1-20 
p.  1-21 

p.  1-22 

p.  1-25 
p.  1-33 
p.  1-41 

p.  1-49 
p.  1-57 
p.  1-65 
p.  1-73 

1  -  xiii 

Boero,       Shapiro,  L.  P>  !'89 

On  some  factors  influencing  student!'  solutions  in  multiple  operations  problems: 
Results  and  interpretations 

Brckke,G.&B*H,A.  P- 1-97 

Multiplicative  structures  at  ages  seven  to  eleven 

Cannon,  P.L.  P-  1-105 

Middle  grade  students'  representations  of  linear  units 

Castro,  E.M.  LJ       p.  1-113 

Choice  of  structure  and  interpretation  of  relation  in  multiplicative  compare  problems 

Chapman,  O.  p.  1-121 

Personal  experience  in  mathematics  learning  and  problem  solving 

Chiappini,  G.  &  Lemut,  E.  p.  1-129 

Interpretation  and  construction  of  computer-mediated  graphic  representations 
for  the  development  of  spatial  geometry  skills 

Clarke,  D.J.  &  Sullivan,  P.A.  P-  I-I37 

Responses  to  open-ended  tasks  in  mathematics:  Characteristics  and  implications 

Clements,  M.A.  &  EUerton,  N.F.  P-  1-145 

Over-emphasising  process  skills  in  school  mathematics:  Newman  error 
.  analysis  data  from  five  countries 

Confrey,  J.  &  Smith,  E.  ,  P-  1-153. 

Revised  accounts  of  the  function  concept  using  multi-representational  software, 
contextual  problems  and  student  paths 

Crawford,  K.  P-  1-161 

Applying  theory  in  teacher  education:  Changing  practice  in  mathematics  education 

Davenport,  L.  &  Narode,  R.  P- 1-169 

School  math  to  inquiry  math:  moving  from  here  to  there 

Davis,  G.  p.  1-177 

Cutting  through  Chaos:  A  case  study  in  mathematical  problem  solving 

DeGuire,L.J.  p.  1-185 

The  development  of  problem-solving  abilities:  its  influence  on  classroom  teaching 

Dirkes,  M.A.  P-  M93 

Self-directed  problem  solving:  Idea  production  in  mathematics 

Dougherty,  B.J.  P-  1-201 

Project  DELTA:  Teacher  change  in  secondary  classrooms 

Edwards,  L.D.  P- 1'209 

Reasoning  and  representation  in  first  year  high  school  students 

Ferrari,  P.L.  L        ^         J  p.  1-217 

Problem-solving  in  geometrical  setting:  Interactions  between  figure  and  strategy 




1  -  xiv 

Garuti,  R.  &  Boero,  P.  p.  1-225 

A  sequence  cf  proportionality  problems:  An  exploratory  study 

Gattuso,  L.  p.  1-233 

Discrepancies  between  conceptions  and  practice:  A  case  study 

Godino,  J.D.,  Navarro-Pelayo,  V.  &  Batanero,  M.C  p.  1-241 

Analysis  of  students'  errors  and  difficulties  in  solving  combinatorial  problems 

Groves,  S.  p.  1-249 

Processes  and  strategies  of  third  and  fourth  graders  tackling  a  real  world 
problem  amenable  to  division 

Hart,  L.C.  &  Najce-ullah,  D.H.  p.  1-257 

Pictures  in  an  exhibition:  Snapshots  of  a  teacher  in  the  process  of  change 

Heiscovics,  N.  &  Linchevski,  L.  p.  1-265 

"Cancellation  within-the-equation"  as  a  solution  procedure 

Hiebert,  J.  &  Weame,  D.  p.  1-273 
Emerging  relationships  between  teaching  and  learning  arithmetic  during 
the  primary  grades 

Hunting,  R.P.,  Pepper,  K.L.  &  Gibson,  SJ.  p.  1-281 
Preschoolers'  schemes  for  solving  partitioning  tasks 

Jaworski,  B.  p.  1-289 

The  emancipatory  nature  of  reflective  mathematics  teaching 

Kanes,  C.  p.  1-297 

Reference,  structure  and  action:  Eliminating  paradoxes  in  learning  and 
teaching  mathematics 

Research  Reports  (continued) 

Kieren,  T.  &  Pine,  S.  p.  2- 1 

The  answer  determines  the  question.  Interventions  and  the  growth  of  mathematical 

Konold,  C.  &  Falk,  R.  p.  2-9 

Encoding  difficulty:  A  psychological  basis  for  'misperceptions' of  randomness 

Koyama,  M.  p.  2-17 

Exploring  basic  components  of  the  process  model  of  understanding  mathematics 
for  building  a  two  axes  process  model 

Krainer,  K.  p.  2-25 

Powerful  tasks:  Constructive  handling  of  a  didactical  dilemma 


1  -  XV 

Letter,  G.C.  P-  2-33 

Measuring  attitudes  to  mathematics 

Lerman,  S.  P-  2-40 

The  Junction  of  language  in  radical  constructivism:  A  Vygotskian  perspective 

Linchevsky,  L.,  Vinner,  S..  &  Karscnty,  R.  P-  2-48 

To  be  or  not  to  be  minimal?  Student  teachers'  views  about  definitions  in  geometry 

Lins,  R.  P-  2-56 

Algebraic  and  non-algebraic  algebra 

Magidson.  S.  ^  P- 164 

What's  in  a  problem?  Exploring  slope  using  computer  graphing  software 

Martino,  A.M.  &  Maher,  C.A.  P- 2-72 
Individual  thinking  and  the  integration  of  the  ideas  cf  others  in  problem 
solving  situations 

Masingila.J.  P- 2-80 
Mathematics  practice  in  carpet  laying 

Maury,  S.,  Lerouge,  A.,  &  Bailie,  J.  P- 2-88 

Solving  procedures  and  type  of  rationality  in  problems  involving  Cartesian 
graphics,  at  the  high  school  level  (9th  grade) 

Meira,L.  P-  *96 

The  microevolution  of  mathematical  representations  in  children  s  activity 

Mesquita,  AJL.  P-  2-104 

Les  types  d'apprehension  en  geometrie  spatiale:  une  etude  clinique  sur  le 
developpement-plan  du  cube 

Minato,S.AKamada,T.  J    .  J        p.  2-112 

Results  of  researches  on  causal  predominance  between  achievement  and  attitude 
in  junior  high  school  mathematics  of  Japan 

Mitchelmore,  M.  P-  2"120 

Children's  concepts  of  perpendiculars 

Moschkovich,  J.  P-  2-128 

Students'  use  of  the  x-intercept:  An  instance  of  a  transitional  conception 

Mousley,  J.  ^  P- 2-136 

Teachers  as  researchers:  Dialectics  cf  action  and  reflection 

Mulligan,  J.  „  P- 2-144 

Children's  solutions  to  multiplication  and  division  word  problems: 
A  longitudinal  study 

Murray,  H.,  Olivier,  A.,  &  Human,  P.  P-  2-152 

The  development  of  young  students'  division  strategies 



1  -xvi 

Nathan,  M.J.  p.  "160 

Interactive  depictions  of  mathematical  constraints  can  increase  students' 
levels  cf  competence  for  word  algebra  problem  solving 

Neuman,  D.  p.  2- no 

The  influence  cf  numerical  factors  in  solving  simple  subtraction  problems 

Norman,  F.A.  &  Prichard,  M.K.  p.  2-178 

AKrutetskiianframeworkfor  the  interpretation  of  cognitive  obstacles: 
An  example  from  the  calculus 

Noss,  R.  &  Hoyles  C.  p.  2-186 

Logo  mathematics  and  boxer  mathematics:  Some  preliminary  comparisons 

Outhred,  L.  &  Mitchelmore,  M.  p.  2- 194 

Representation  of  area:  A  pictorial  perspective 

Owens,  K.  p.  2-202 

Spatial  thinking  takes  shape  through  primary-school  experiences 

Perlwitz,  MD.  p.  2-2 10 

The  interactive  constitution  of  an  instructional  activity:  A  case  study 

Ponte,  J.P.,  Matos,  J.  F.,  Guimaries,  H.M.,  Leal,  L.C.,  &  Canavarro,  A.P.  p.  2-218 

Students'  views  and  attitudes  towards  mathematics  teaching  and  learning: 
A  case  study  of  a  curriculum  experience 

Reiss,  M.  &  Reiss,  K.  p.  2-226 

Kasimir:  A  simulation  of  learning  iterative  structures 

Relich,  J.  p.  2-234 
Self-concept  profiles  and  teachers  of  mathematics:  Implications  for  teachers 
as  role  models 

Reynolds,  A.  &  Wheatley,  G.H.  p.  2-242 
The  elaboration  of  images  n  the  process  of  mathematics  meaning  making 

Rice,  M.  p.  2-250 

Teacher  change:  A  constructivist  approach  to  professional  development 

Robinson,  N.,  Even,  R.,  &  Tirosh,  D.  p.  2-258 

Connectedness  in  teaching  algebra:  A  novice-expert  contrast 

Saenz-Ludlow,  A.  p.  2-266 

Ann's  strategies  to  add  fractions 

Sanchez,  V.  &  Llinares,  S.  p.  2-274 
Prospective  elementary  teachers' pedagogical  content  knowledge  about 
equivalent  fractions 

Santos,  V.  &  Kroll,  D.L.  p.  2-282 
Empowering  prospective  elementary  teachers  through  social  interaction, 
reflection,  arid  communication 


1  -  xvii 

Saraiva,  M  J.  P-  2-290 
Students'  understanding  of  proof  in  a  computer  environment 

Schliemann,  A.,  Avel«r,  A.P.,  &  Santiago,  M.  p.  2-298 
Understanding  equivalences  through  balance  scales 

Schroeder.  T.L.  P-  2-306 
Knowing  and  using  the  Pythagorean  theorem  in  grade  10 

Sekiguchi,  Y.  P-  2-314 
Social  dimensions  of  proof  in  presentation:  From  an  ethnographic  inquiry 
in  a  high  school  geometry  classroom 

Shigcmatsu.  K.  P-  2-322 
Metacognidon:  The  role  of  the  "inner  teacher" 

Shimizu.  Y.  P-  2-330 
Metacognition  in  cooperative  mathematical  problem  solving:  An  analysis 
focusing  on  problem  transformation 

Research  Reports  (continued) 

Seimon,  D.  P-  3-3 

Children's  approaches  to  mathematical  problem  solving 

Simon.  MA.  &  Blumc.  G.W.  P-  3-H 

Understanding  multiplicative  structures:  A  study  of  prospective  elementary  teachers 

Smith.  J.P.  P-  3-19 

Misconceptions  and  the  construction  of  mathematical  knowledge 

Sowder,  J..  Philipp,  R.,  &  Flores,  A.  P-  3-27 

The  act  of  teaching  mathematics:  A  case  study 

Stacey.  K.  &  del  Bcato.  C  P-  3"35 

Sources  ofcertainy  and  uncertainty  in  mathematical  problem  solving 

SuHBer,  R.  &  Bromine,  R.  P-  S"43 

The  description  of  solids  in  technical  drawing  -  Results  from  interviews  of 
experienced  draughtsmen 

Streefland,  L.  &  van  den  Heuvel-Panhuizen,  M.  P-  3-51 

Evoking  pupils'  informal  knowledge  on  percents 

Taber  S.B.  P-  3"'^ 

The  "multiplier  effect"  and  sixth-grade  students' performance  on  multiplication 
word  problems  with  unit-fraction  factors 

Tiemey,  C.C.,  Weinberg,  A.S.,  &  Nemirovsky,  R.  P-  3-66 

Telling  stories  about  plant  growth:  Fourth  grade  students  interpret  graphs 



1  -  xviii 

Tirosh,  D.  &  Sttvy,  R.  p.  3-74 
Overgeneralization  between  science  end  mathematics:  The  case  of  successive 
division  problems 

Trigueros,  M.  &  Cantoral,  R.  p.  3-82 
Exploring  understanding  and  its  relationship  with  teaching:  Variation  and  movement 

Tsarnir,  P.  &  Tiiosh,  D.  p.  3-90 
Students'  awareness  of  inconsistent  ideas  about  actual  infinity 

Underwood,  D.L.  p.  3-98 
Mathematics  and  gender:  An  interactional  analysis 

Wenzelburger,  E.  p.  3-106 
The  learning  of  trigonometric  functions  in  a  graphical  computer  environment 

Witte.M.  p.  3-114 
Euclidian  constraints  in  mathematics  education 

Short  Oral  Presentations 

Abcle,  A.  .  p.  3-125 

The  concept  of  speed:  Two  case  studies  in  the  primary  school 

Albert,  J.  &  Friedlander,  A.  p.  3- 1 26 

Achievement  and  thinking  strategies  on  "reversed  items" 

Boulton-Lewis,  G.M.  p.  3-126 

The  processing  loads  and  relations  between  counting  and  place  value 

Cantoral-Uriza,  R.  p.  3-127 

From  research  to  teaching:  An  analysis  of  students' performance  on  calculus 

Ernest,  P.  p.  3-127 

Metaphors  for  the  mind  and  the  world  in  the  psychology  of  mathematics  education 

Oelfman,  E.,  Demidovt,  L.,  Kholodnaja,  M.,  Lobanenko,  N.,  &  Wolfengaut,  J.  p.  3-128 

The  psychology  of  pupil's  intellect  development  in  the  process  of  teaching  mathematics 

Kaplan,  R.G.,  Jani,  M,  &  Schmidt,  A.  p.  3-128 

Implementing  the  NCTM  Standards:  Reconciling  the  planned  impact  with  the 
experienced  reality  in  an  urban  school  district 

Kynigos,  C.  p.  3-129 

Children  using  the  turtle  metaphor  to  construct  a  computational  tool  in  a 
geometrical  Logo  microworld 

Lo,  J.  &  Wheatley,  G.H.  p.  3-129 

Understanding  mathematics  class  discussions 

MacGregor,  M.  &  Sttcey,  K.  p.  3-130 

Cognitive  origins  of  students'  errors  in  writing  equations 


1  -  xix 

Malara,  N.A.,  Pellegrino.  C.  &  Iaderosa,  R.  P-  3-130 

Towards  applied  problem  solving 

Markovits,  Z.  &  Hershkowitz,  R.  P-  3-131 
Visual  estimation 

Morgan.  C.  P-  3-131 
Written  mathematical  communication:  The  child's  perspective 

Nantais,  N.,  Francavilla,  M.,  &  Boulet,  G.  P-  3-132 

Young  pupils'  logico-physical  concept  of  multiplication:  15  case  studies 

Nevile.  L.  P-  3-132 

Teaching  recursion  as  shifts  of  attention 

Nunes,  T.  &  Bryant,  P.  P-  3-133 

Rotating  candy  bars  and  rearranging  oranges:  A  study  of  children's  understanding 
of  commutativity 

Pegg.,  J.  &  Davey,  G.  p.  3-133 

Interpreting  children's  understanding  of  geometric  concepts:  A  comparison  of 
the  Van  Hiele  theory  and  the  solo  taxonomy 

Pchkonen.  E.  &  Tompa.  K.  .  p.  3-134 

Are  there  any  differences  in  pupils'  conceptions  about  mathematics  teaching  in 
different  countries?  The  case  of  Finland  and  Hungary 

Perks.  P.  P-  3-134 

Introducing  calculators  to  six-year  olds:  Views  on  support  for  teachers 

Quintal,  C.  P-  3-135 

Hierarchies  of  cognitive  difficulty  in  early  algebra 

Rojano,  T.  &  Sutherland,  R.  P-  3-135 

Pupil  strategies  for  solving  algebra  word  problems  with  a  spreadsheet 

Rubin.  A.  &  Russell.  S.J.  P-  3-136 

Children's  developing  concepts  of  landmarks  in  the  number  system 

Sfard.  A.  &  Linchevsky.  L.  P-  3-136 

Equations  and  inequalities  •  Processes  without  objects? 

Shiu.  C.  p.  3-137 

Assumptions  and  intentions  in  distance  learning  materials  for  mathematics 

Swinson.  K.V.  &  Partridge.  B.D.  P-  3-137 

Writing  in  mathematics:  Is  it  always  beneficial? 

Teppo.A.R.  „  P-3'138 

The  impact  of  understanding  and  expectations  of  performance  on  college 
students'  self-conjidence 




1  -  XX 

Experiences  and  effects  of realistic  mathematics  education:  The  case  of 
exponential  growth 

Watson,  J.M.,  Collis,  K.F.,  &  Campbell,  KJ. 

Ikonic  and  early  concrete  symbolic  responses  to  two  fraction  problems 

Williams,  S.R.  &  Walen.  S.B. 

Conceptual  splatter  and  metaphorical  noise:  The  case  of  graph  continuity 

Yoshida,  M. 

Trying  the  theory  on  the  determination  to  study  -  Applying  mathematical 
activities  based  on  varied  problem  solving 

Zazkis,  R. 

Inverse  of  a  product  -  A  theorem  out  of  action 

Poster  Presentations 

Barocio  Quijano,  R.  &  Brefia  Sanchez,  J. 

Teaching  mathematics  in  the  first  years  of  elementary  education:  Kama's 
proposal  in  action 

Becker,  G. 

Analogical  reasoning:  Basic  component  in  problem  solving  activities 

Bell,  A.,  Crust,  R.,  Shannon,  A.  &  Swan,  M. 
Pupils'  evaluations  of  learning  activities 

Berenson,  S.B. 

Race  and  gender  interactions  and  constructivist  teaching 

Bergsten,  C. 

Schematic  structures  of  mathematical  form 

Borba,  M.  &  Confrcy,  J. 

Transformations  of  functions  using  multi-representational  software: 
Visualization  and  discrete  points 

Carraher,  D.W. 

Relational  thinking  and  rational  numbers 

Chazan,  D. 

F(x)  =  G(x)?:  An  approach  to  modeling  with  algebra 

Clarke,  D.,  Wallbridge,  M.,  &  Fraser,  S. 

The  other  consequences  of  a  problem-based  mathematics  curriculum 

Clements,  D.H.,  Meredith,  J.S.,  &  Battista,  M.T. 

Design  of  a  Logo  environment  for  elementary  geometry 

p.  3-138 

p.  3-139 
p.  3-140 
p.  3-141 

p.  3-141 

p.  3-145 

p.  3-146 
p.  3-147 
p.  3-14S 
p.  3-149 
p.  3-149 

p.  3-150 
p.  3-150 
p.  3-151 
p.  3-152 

1  -  xxi 

Collis,  K.F.,  Watson,  J.M.,  &  Campbell,  KJ. 

Multimodal  functioning  in  mathematLal  problem  solving 

Coidero-Osorio,  F. 

The  idea  of  variation  and  the  concept  of  the  integral  in  engineering  students: 
Situations  and  strategies 

DeFranco,  T.C. 

The  role  ofmetacognition  in  mathematical  problem  solving  among  PhD. 

dc  Villiers,  M. 

Childrens'  acceptance  of  theorems  in  geometry 

Doig,  B.A. 

Exploring  mathematical  beliefs 

Ebert,  C.L. 

An  assessment  of  students'  graphing  strategies  in  a  technology-rich  environment 

Ellerton,  N.F.  &  Clements,  M.A. 

Teaching  mathematics  education  at  a  distance:  The  Dealdn  University  experience 

Emori,  H.  &  Nohda,  N. 

Communication  process  in  learning  mathematics 

Farah-Sarkis,  F. 

Problem  familiarity  and  experts:  The  case  of  transitivity 

Gal,  L.  Mahoncy,  P.,  &  Moore,  S. 
Children's  usage  of  statistical  terms 

Gallaido,  A.  &  Rojano,  T. 

The  status  of  negative  numbers  in  the  solving  process  of  algebraic  equations 

Gclfman,  E.,  Demidova,  L.,  Grinshpon,  S.,  Kholodnaja,  M.,  &  Wolfengaut,  J. 
Study  of  identities  in  the  school  course  of  algebra 

Gimcnez,  J. 

Some  wrong  strategies  to  determine  probabilities  in  8th  graders  -  Report 
of  a  preliminary  study 

Goldberg,  M.D.  Sc.  Hershkowitz,  R. 
From  Concept  to  proof:  A  first  step 

vjuuya,  *«. 

Metacognitive  strategies  in  the  classroom:  Possibilities  and  Imitations 

Irwin,  K.  &  Britt,  M. 

A  two  year  project  for  improving  the  mathematics  teaching  for  11-13  year-olds 

Ito-Hino,  K. 

An  assessment  of  mathematics  learning  through  students'  intra-  and  inter- 
communication processes 

p.  3-153 
p.  3-153 

p.  3-154 

p.  3-155 
p.  3-156 
p.  3-157 
p.  3-158 
p.  3-159 
p.  3-160 
p.  3-160 
p.  3-161 
p.  3-162 
p.  3-163 

p.  3-163 
p.  3-164 
p.  3-165 
p.  3-166 

1  -  xxii 

Jones,  G.A.,  Bidwell,  J.K.,  &  Ziukelis,  R.  p.  3-166 

The  effect  of  different  school  environ,,ients  on  mathematics  learning  across 
the  elementary-secondary  interface 

Kaufman  Fainguelemt,  E.  p.  3-167 

The  importance  of  teaching  practice  in  mathematics  teacher  courses 

Konold,  C.  p.  3-168 

Prob  Sim  and  Datascope:  Interactive  software  tools  for  introductory  courses 
in  probability  and  data  analysis 

Krainer,  K.  p.  3-169 

PFL-Mathematics:  An  in-service  education  university  course  for  teachers 

Lawson,  M.  &  Chinnappan.  M.  p.  3-170 

The  effects  of  training  in  use  of  generation  and  management  strategies 
on  geometry  problem  solving 

LeBlanc,  M.D.  p.  3-171 

When  more  is  less  ■  Interactive  tools  for  relational  language 

Long,  E.  p.  3-172 

Teachers'  questioning  and  students'  responses  in  classroom  mathematics 

Ojeda,A.M.  p.  3-173 

Students' problems  in  understanding  the  idea  of  conditional  probability 

Putt,  I.,  Annesley,  F.,  &  Clark,  J.  p.  3-174 

Development  of  an  instrument  for  teacher  and  student  use  in  the  measurement 
of  affective  development  in  school  students 

Rhodes,  S.  p.  3-175 

Research  and  psychological  factors  influencing  materials  development  in 
mathematics:  Imagery 

Schwarz,  B.  &  Resnick,  L.  p.  3-175 

Acquisition  of  meaning  for  pre-algebraic  structures  with  "the  Planner" 

Shier,  G.B.  p.  3-176 

Correlates  of  direct  proportional  reasoning  among  adolescents  in  the  Philippines 

Zellweger,  S.  p.  3-177 

Cards,  mirrors,  and  hand-held  models  that  lead  into  elementary  logic 

Featured  Discussion  Group  I 
Chair,  Kath  Hart  (United  Kingdom) 

Fischbein,  E.  p.  3-181 

The  three  facets  of  mathematics:  The  formal,  the  human,  and  the  instrumental 
educational  implications 


1  -  xxiii 

Martin,  W.G.  P-  3"182 
Research-based  curriculum  development  in  high  school  geometry: 
A  construcdvist  model 

Pace,  J.P.  .        J  p.  3-182 

Needing  conscious  conceptions  of  human  nature  and  values  to  inform  and 
develop  pedagogy 

Featured  Discussion  Group  II 
Chair,  Eugenio  RUoy  (Mexico) 

Bechara  Sanchez,  L.  „  ,    ,.      P- 3-185 

An  analysis  of  the  development  of  the  notion  of  similarity  in  confluence:  Multiplying 
structures;  spatial  properties  and  mechanisms  of  logic  and  formal  framework 

Graciosa  Velosa,  M.  P-  3-185 

Appropriation  and  cognitive  empowerment:  Cultural  artifacts  and  educational 

Gutierrez,  A.  &  Jaime,  A.  p.  3- 186 

Exploring  students'  mental  activity  when  solving  3 -dimensional  tasks 

Hitt,  F.  P-  3" 186 

Visual  images,  availability  and  anchoring,  related  to  the  polynomial  numbers 
and  the  use  of  microcomputers 

Nasser,  L.  P-  3"187 

A  Van-Hlele-bascd  experiment  on  the  teaching  of  congruence 

Orozco  Hormaza,  M.  p.  3-187 

Modes  of  use  of  the  scalar  and  functional  operators  when  solving  multiplicative 

Plenary  Sessions 
Plenary  Panel 

Dreyfus,  T.  (organizer),  Clements,  K.,  Mason,  J.,  Parzysz,  B.,  &  Presmeg,  N.  p.  3-191 

Visualization  and  imagistic  thinking 

Plenary  Addresses 

Artique.M.  ...    .  p.  3-195 

The  importance  and  limits  of  epistemological  work  in  didactics 

Ervynck,  G.  P-  3"217 

Mathematics  as  a  foreign  language 



1  -  xxiv 

Goldin,  G.  p.  3-235 
On  developing  a  unified  model  for  the  psychology  of  mathematical  learning 
and  problem  solving 

Hoyles,  C.  p.  3-263 
Illuminations  and  reflections  -  Teachers,  methodologies  and  mathematics 



1  -  1 

Working  Groups 




1  -3 


Organisers:  Gontran  Ervynck,  David  Tall 

Four  initiators  will  present  different  approaches  to  what  seems  to  be  basically  the  same 

-  Michele  Artigue  (France):  Tool  and  Object  status  of  mathematical  concepts;  the 
case  of  complex  numbers. 

-  Anna  Sfard  (Israel):  On  Operational-structural  Duality  of  Mathematical  Concep- 

-  Ed  Dubinsky  (U.S.A.):  A  Theoretical  Perspective  for  Research  in  Learning  Math- 
ematics Concepts:  Gtnetic  Dtcomposition  and  Groups. 

-  David  Tall  (U.K.):  The  Construction  of  Objects  through  Definition  and  Proof,  with 
emphasis  on  Vector  Spaces  and  Group  Theory. 


Discussion  of  the  contribution  of  the  initiators.  All  discussion  has  to  come  from  reflec- 
tion on  the  content  of  the  presentations.  Disagreement  is  to  be  seen  as  a  vehicle  not  for 
attempting  to  convince  others  of  one's  own  view  but  of  trying  to  find  out  the  source  of 
the  disagreement. 

The  initiators  are: 

-  Dick  Shumway  (U.S.A.):  The  Role  of  Proof  and  Definition  in  Concept  Learning. 

.  The  intention  is  that  a  link  should  be  established  with  the  subjects  discussed  in 
Sessions  I  and  II. 

-  John  Selden  (U.S.A.):  Continuation  of  the  work  on  rigor  in  mathematics  . 
Three  aspects  of  rigor  in  undergraduate  mathematics  will  be  discussed:  (1)  Is  it 
possible  to  construct  objects  through  definition  and  proofs  ?  (2)  What  can  be  said 
about  understanding  the  concept  of  proof  itself  ?  (3)  How  can  students  learn  to 
construct  proofs  and  what  kind  of  background  knowledge  is  needed  ? 


Discussion  of  the  work  of  the  AMT  Group  at  PME-17,  Japan  1993. 

%j  0 


Working  Group  on  Algebraic  Processes  and  Structure 
Coordinator:  Rosamund  Sutherland 
Institute  of  Education  University  of  London 

During  the  Assisi  meeting,  the  group  aimed  to  characterize  the  multiple  "jumps'Vshifts  that  appear 
to,be  involved  in  developing  an  algebraic  mode  of  thinking  and  to  investigate  the  role  of 
symbolizing  in  this  development  Other  concerns  of  the  group  are  the  role  of  meaning  in  algebraic 
processing,  the  potential  of  computer-based  environments  and  implications  for  classroom  practice. 
Key  issues  were  discussed  and  worked  on  in  small  groups  with  die  aim  of  producing  a  set  of 
questions  and  working  hypotheses  for  future  collaboration. 

1  -5 

Classroom  Research  Working  Group 
Problems,  standpoints  and  purposes  of  the  working  group 

A .  Problems 

1 .  In  their  research  all  the  participants  of  this  working  group  have  encountered  similar 
methodological  problems  arising  from  their  developmental  approach  to  classroom  research. 
One  of  the  problems  is  how  to  collect  and  to  analyze  the  classroom  data  within  the  working 

An  important  aspect  of  our  research  is  the  defining  of  new  variables  for  each  new  set  of  data. 
Different  standpoints  for  'good'  research  arise:  doing  classroom  research  for  the  data 
themselves,  for  the  research  methods  involved,  for  enrichment  mathematics  education,  for 
psychologic  phenomena,  for  theoretical  considerations. 

2.  Our  research  has  raised  many  questions  including: 
-ways  of  collecting  data  (video  tapes) 




-qualitative/quantitative  aspects 

-reduction  of  collected  data 

•developing  new  research  methods  and  techniques 

B .  Standpoints 

The  following  perspectives  are  implicit  in  our  research: 

1.  Classroom  Descriptors 

As  well  as  describing  the  data  collected  from  various  activities  presented  to  and/or  undertaken 
by  the  children,  we  record  key  classroom  descriptors.  In  particular,  details  of  actual  instruction 
given  by  the  teachers  is  noted.  This  is  not  common  in  research  where  any  instruction  involved 
is  mostly  described  in  global  terms.  We  have  found,  however,  in  all  our  work  that  the  type  of 
instruction  given  by  the  classroom  teacher  can  be  a  distinguishing  feature  in  the  data  collection 
from  the  children. 


2.  Mood  conditions 

One  of  the  key  issues  to  be  considered  by  the  group  will  involve  searching  for  education 
conditions  which  produced  a  suitable  mental  climate  for  the  children  to  work  towards  their  own 
productions.  They  have  to  bring  the  children  into  the  mood  to  do  so.  These  conditions  are 
mostly  of  a  social  character  and  help  to  legitimize  the  particular  children's  activities;  the 
activities  make  sense  to  the  children. 

We  consider  the  educational  setting  in  the  classroom  (the  manner  of  teaching,  and  so  on)  to  be 
a  source  of  techniques  and  methods  for  the  researcher. 

4.  Mutual  nature  of  tlx  research 

Our  focus  is  on  mutual  research  situations  in  which  the  children  can  recognize  themselves 
(e.g.  as  a  writer,  as  an  author,  etc.).  This  is  seen  as  important  as  it  helps  to  justify  the  research 
objects  (children). 

C.  Purposes  of  the  working  group  Classroom  Research 

1 .  To  become  aware  of  methods  we  use  in  the  classroom,  their  possibilities  and  their  constraints 
(watching  video  tapes) 

2 .  To  collect  and  develop  methods  and  techniques  which  can  be  categorized  under  one  of  the 
standpoints  mentioned  above. 

3.  To  collect  and  develop  different  mood  conditions. 

4 .  To  criticize  these  methods,  techniques  and  conditions  and  indicate  their  constraints. 

5 .  To  prepare  a  booklet  to  support  researchers  working  in  the  area  of  classroom  research  and 
closely  connected  with  the  practice  of  teaching. 

D.  Contact  person: 

Dr.  F.  Jan  van  den  Brink, 
Freudenthal  Institute 
State  University  of  Utrecht, 
3561 GG  Utrecht 
The  Netherlands 
Tel.  31-30-611611 
Fax.  31-30-660430 

E.  Language:  English 




Working  group  on 


Although  the  process  of  learning  mathematics  takes  place  in  the  school 
environment,  this  educational  process  cannot  be  isolated  from  the  effects  of 
the  child's  cultural  context. 

In  other  words,  mathematical  knowledge  is  a  product  of  schooling  filtered 
through  culturally  conditioned  individual  characteristics. 

Our  new  Working  Group  has  grown  out  of  the  Discussion  Group  Learning 
Mathematics  and  cultural  context,  which  has  been  active  since  PME  13. 
At  PME  13,  we  explored  the  main  interests  of  the  participants  in  this  area. 
The  following  themes  were  touched  on:  minorities  in  mathematics 
education,  social  pressures  in  mathematics  education,  the  role  of  language 
in  the  acquisition  of  a  given  mathematical  concept,  learning  and  teaching  in 
multicultural  classrooms,  teachers'  conceptions  of  mathematics,  problems 
of  cooperative  research  in  math  education  using  a  comparative  approach  for 
different  cultures. 

PME  14  and  PME  15  represented  our  first  attempt  to  focus  on  the  following 
question:  "What  is  the  meaning  of  culture  in  the  learning  of  mathematics?" 
Presentations  of  some  research  results  in  various  areas  led  to  a  discussion 
about  whether  these  cultural  aspects  are  to  be  considered  a  starting  point  or 
an  end-product. 

At  this  point,  we  have  identified  and  are  working  on  several  types  of  studies 
related  to  the  cultural  field: 

1.  Informal  education  and  formal  mathematical  knowledge. 

2.  The  effects  of  language  and  cultural  environment  on  the  mental 
representa-tions  of  students  and  teachers. 

3.  Cognitive  processes  in  learning  mathematics,  using  a  comparative 
approach  for  different  cultures. 

The  objectives  of  our  working  group  consist  of  the  following: 

1.  To  exchange  views  on  the  impact  of  cultural  context  on  the  learning  of 

2.  To  ensure  that  contact  between  conferences  is  maintained  through  the  ex- 
change of  information  about  relevant  research. 

3.  To  contribute  to  the  formulation  of  a  methodological  and  theoretical 
framework  by  presenting  original  research  at  PME  conferences.  These 
contributions  may  be  interdisciplinary  in  nature,  possibly  by  making  use  of 
the  fields  of  psychology,  mathematics  education,  art  education,  and  didactics 
of  geography.  . 

4.  To  identify  the  areas  relevant  to  this  approach  where  further  research  is 


At  PME  16,  more  specifically,  we  will  invite  participants  to  discuss  the 
problematique  of  our  working  group  as  they  are  reflected  in  research 
situations  presented  by  selected  group  members.  At  the  request  of  last  year's 
participants,  a  combined  session  of  this  working  group  and  the  Social 
Psychology  of  Math  Education  working  group  will  be  held  at  PME  16. 

Bernadette  Denys 

ERJC  ° 1 



Helen  Mansfield 
Curtin  University  of  Technology,  Western  Australia 

At  the  Assisi  meeting,  the  Geometry  Working  Group  had  the  overall  theme  Learning  and 
Teaching  Geometry:  a  Constructivist  Point  of  View.  This  theme  was  chosen  because  the 
committee  of  the  Geometry  Working  Group  believes  that  it  is  timely  to  examine  constructivism  as  a 
theoretical  framework  for  research  into  aspects  of  teaching  and  learning  geometry. 

Within  this  overall  theme,  two  sub-themes  were  discussed.  These  were  What  Constructivism 
has  to  say  about  Learning  and  How  Teaching  can  Promote  Learning  in  Geometry; 
and  Helping  Students  to  Construct  Knowledge  in  the  Geometry  Classroom. 

In  the  first  session  of  the  Geometry  Working  Group,  there  were  introductory  presentations 
concerning  the  first  subtheme  followed  by  a  discussion.  The  focus  in  the  second  session  was  on 
the  role  of  computer  environments  in  the  learning  and  teaching  of  geometry. 

The  third  session  provided  opportunities  for  group  members  to  present  brief  papers  on  their  current 
research.  These  papers  did  not  have  to  report  on  work  that  was  completed,  but  provided 
opportunities  for  the  presenters  to  discuss  work  in  progress,  to  seek  feedback  from  other 
participants,  and  to  discuss  with  colleagues  collaborative  research  projects. 

1  -9 

PME  XVI  Working  Group 

Psychology  of  Inservice-education 
with  Mathematics  Teachers: 
a  Research  Perspective 

Group  Organizers:  Sandy  Dawson-Simon  Fraser  University,  Canada 
Terry  Wood-Purdue  University,  USA 
Barbara  Dougherty-University  of  Hawaii,  USA 
Barbara  Jaworski-University  of  Birmingham,  UK 

This  is  the  fourth  year  the  group  has  been  studying  the  role  of  the  teacher 
educator  in  doing  inservice  with  mathematics  teachers.  Last  year  at  PME  XV  in 
Italy,  the  group  critically  examined  a  proposed  conceptualization  of  a  framework 
for  inservice  education  of  mathematics  teachers.  These  discussions  gave  rise 
to  a  revised  draft  of  the  framework.  This  draft  was  circulated  and  reactions  to  it 
were  sought  during  the  spring  of  1992. 

The  organizers  of  the  working  group  see  the  revised  framework  as  a  basis  for 
the  creation  of  a  working  manuscript  (and  thence  a  book)  about  INSET. 

The  preparation  of  a  book  is  in  line  with  the  aim  of  the  working  group  which  is: 

to  extend  knowledge  regarding  the  psychology  of  mathematics  teacher 
inservice  education,  in  order  to  broaden  arid  deepen  understanding  of 
the  interactions  among  teachers  and  teacher  educators. 

The  group  will  meet  for  four  sessions  during  PME  XVI.  The  first  two  sessions 
will  centre  on  collecting  reactions  to  the  revised  framework,  examining  issues 
arising  from  this  discussion,  and  to  hearing  accounts  of  how  others  on  the 
international  scene  have  attempted  to  handle  the  issues  raised.  The  first  half  of 
the  third  session  will  be  devoted  to  laying  out  the  chapters  for  the  manuscript. 
The  latter  portion  of  the  third  session  will  address  the  group's  presentation  at 
ICME7,  the  detailed  planning  for  which  will  take  place  during  the  fourth  session, 
a  joint  meeting  with  the  other  two  Teacher  Education  working  groups. 

Though  the  work  of  the  group  is  a  carry  over  from  discussions  at  the  Italy 
meeting,  new  members  are  most  welcome  to  join.  The  preparation  of  the 
manuscript  will  require  input  from  participants  representing  a  broad  spectrum  of 
the  international  mathematics  education  community  served  by  PME.  Hence, 
new  participants  are  not  only  welcomed  but  are  needed  if  the  manuscript  is  to 
truly  cover  the  spectrum  of  experiences  lived  by  PME  members. 

When  participants  leave  PME  XVI  they  will  have  contributed  to  the  preparation 
of  an  outline  for  a  manscript  on  INSET,  and  will  have  made  a  commitment  to 
write  a  chapter  for  the  book  based  on  their  taken-as-shared  experiences. 


1  - 10 

Ratio  and  Proportion 
F.L.  Lin,  K.M.  Hart  and  J.C.  Bergeron 

Is  the  research  topic  'ratio  and  proportion'  dead  or  alive? 

To  tackle  this  question,  this  group  have  tried  to  define  what  is  proportional  reasoning  abilities 
and  found  that  what  we  have  known  is  incomplete. 

In  the  Mexico  meeting,  the  group  worked  on  some  very  fundamental  questions,  such  as 

(1)  what  is  fractions?  Is  n/2  a  fraction? 

(2)  what  is  ratio?  Is  a:b  =  3:4  a  ratio? 

(3)  what  is  the  relation  between  ratio  and  fraction? 

In  the  Assisi  meeting,  the  group  addressed  these  questions  during  one  of  their  time  slots. 

The  group  also  worked  on  advanced  proportional  reasoning.  Discussion  on  some  recent 
developments  and  problems  for  further  investigation  occurred. 

In  the  remaining  time  slot,  the  group  worked  on  questions  such  as, 

(1)  what  is  the  origin  of  fraction/ratio  concepts? 

(2)  how  to  develop  a  diagnostic  teaching  module  on  beginning  fractions/ratio?  ...  etc. 
Welcome  to  ratio  and  proportion  group. 

1  - 11 

The  PME  Working  Group  on  Representations 

Representations  are  key  theoretical  constructs  in  the  psychology  of  mathematics  education.  For 
the  purposes  of  our  working  group,  the  meaning  of  this  term  is  quite  broad.  It  includes: 

*  External,  structured  physical  situations  or  sets  of  situations,  that  can  be  described 
mathematically  or  seen  as  embodying  mathematical  ideas.  External  physical 
representations  range  from  peg-boards  to  microworlds. 

*  External,  structured  symbolic  systems.  These  can  include  linguistic  systems,  formal 
mathematical  notations  and  constructs,  or  symbolic  aspects  of  computer  environments. 

*  Internal  representations  and  systems  of  representation.  These  include  individual 
representations  of  mathematical  ideas  (fractions,  proportionality,  functions,  etc.),  as 
well  as  broader  theories  of  cognitive  representation  that  range  from  image  schemata  to 
heuristic  planning. 

Included  in  the  scope  of  the  Working  Group  are  many  kinds  of  issues.  The  following  are  just  a 
few  of  the  questions  we  have  been  addressing: 

*  What  are  appropriate  philosophical  and  epistemological  foundations  of  the  concept  of 

*  How  are  internal  representations  constructed?  How  can  we  best  describe  the 
interaction  between  external  and  internal  representations? 

*  What  are  the  theoretical  and  practical  consequences  for  mathematics  education  of  the 
analysis  of  representations? 

*  How  can  the  creation  and  manipulation  of  external  representations  foster  more 
effective  internal  representations  in  students? 

*  What  are  the  roles  of  visualization,  kinesthetic  encoding,  metaphor,  and  other  kinds  of 
non -prepositional  reasoning  in  effectively  representing  mathematical  ideas? 

*  How  can  linkages  between  representations  be  fully  developed  and  exploited? 

*  Can  individual  differences  be  understood  in  relation  to  different  kinds  of  internal 

Our  group  was  guided  during  its  first  years  by  Frances  Lowenthal  (Mons,  Belgium).  I  began 
coordinating  it  after  the  1980  meeting  in  Parie,  with  the  help  of  Claude  Janvier  (Montreal, 
Canada).  In  Mexico  in  1090  we  had  42  participants,  and  in  Italy  in  1991  we  had  47.  Some 
detailed  notes  from  these  two  meetings  will  be  available  thie  year  to  participants;  those  who  are 
not  at  the  meeting  are  welcome  to  write  for  copies.  We  continue  to  aim  toward  publishing  a 
special  volume  of  the  Journal  of  Mafnematfca/  Bthavhr  devoted  to  the  topic  of  "representation". 

Gerald  A.  Gotdin,  Center  for  Mathematics,  Science,  and  Computer  Education, 
Rutgers  University,  New  Brunswick,  New  Jersey  08903,  USA 

1  -12 


The  Working  Group  Research  on  the  Psychology  of  Mathematics  Teacher  Development  was  first 
convened  as  a  Discussion  Group  at  PME  X  in  London  in  1986,  and  continued  in  this  format  until 
the  Working  Group  was  formed  in  1990.  This  year,  at  PME  XVI,  we  hope  to  build  on  the 
foundation  of  shared  understandings  that  have  developed  over  the  last  five  years. 

Aims  of  the  Working  Group 

-  The  development,  communication  and  examination  of  paradigms  and  frameworks  for  research 
in  the  psychology  of  mathematics  teacher  development. 

-  The  collection,  development,  discussion  and  critiquing  of  tools  and  methodologies  for 
conducting  naturalistic  and  intervention  research  studies  on  the  development  of  mathematics 
teachers'  knowledge,  beliefs,  actions  and  thinking.  * 

-  The  implementation  of  collaborative  research  projects. 

-  The  fostering  and  development  of  communication  between  participants. 

-  The  production  of  a  joint  publication  on  research  frameworks  and  methodological  issues  within 
this  research  domain. 

Research  Questions 

At  the  Working  Group  sessions  in  1991  it  was  decided  that  the  focus  for  the  1992  Working 
Groups  sessions  would  be  the  sharing  of  examples  of  the  practice  of  mathematics  teachers  and 
teacher  educators  that  inform  our  notions  of  what  constitutes  good  pedagogy  in  general,  and  the 
role  of  assessment,  in  particular.  The  format  of  the  sessions  will  include  the  presentation  and 
discussion  of  brief,  anecdotal  vignettes.  The  following  research  questions  may  help  to  mould  the 
thinking  of  researchers  interested  in  contributing  an  anecdotal  vignette  to  one  of  the  sessions. 

-  Should  professional  development  programs  for  teachers  of  mathematics  be  basically  the 
same,  the  world  over? 

-  Do  we  have  examples  of  professional  development  programs  that  help  practising  teachers 
build  confidence  in  their  mathematical  ability  and  in  their  ability  to  teach  mathematics? 

-  Can  the  tension  between  constructivist  ideas  recommended  in  mathematics  teacher 
development  programs  and  assessment  practices  and  pedagogy  in  the  programs  be  reconciled? 

Proposed  Outcomes  of  the  Working  Group  at  PME  XV 

/.  Collaborative  Research  Projects:  Members  of  the  Working  Group  have  overlapping 
research  interests,  and  it  is  hoped  that  collaborative  research  projects  can  be  mounted. 

2.  Publication  of  an  Edited  Collection  of  Research  Papers:  The  Working  Group 
plans  to  publish  a  collection  of  research  papers. 

3.  Preparation  of  the  Working  Group's  Presentation  to  1CME  7:  A  session  at 
ICME  7,  combined  with  the  other  two  Working  Groups  involving  teacher  education,  has  been 
scheduled.  At  least  one  session  at  PME  will  be  devoted  to  planning  this  session. 

Nerida  Ellerton,  Convenor 


1  - 13 

Workiny  flmnp  nn  "Social  Psychology  nf  Mathematics  Education" 
Alan  J.  Bishop,  Organizer 

All  mathematics  learning  takes  place  in  a  social  setting  and  particularly  within  the  PME  community, 
we  need  to  be  able  to  interpret,  and  theorize  about,  mathematics  learning  imopersonally  as  well  as 
intrapersonallv.  Mathematics  learning  in  its  educational  context  cannot  be  fully  interpreted  as  an 
intrapersonal  phenomenon  because  of  the  social  context  in  which  it  occurs.  Equally,  interpersonal 
or  sociological  constructs  will  be  inadequate  alone  since  it  is  always  the  individual  learner  who 
must  make  sense  and  meaning  in  the  mathematics.  Therefore,  it  is  vitally  important  to  research  the 
ways  this  intra-interpersonal  complementarity  influences  the  kind  of  mathematical  knowledge 
acquired  by  pupils  in  classrooms.  In  order  to  pursue  this  research  it  is  therefore  necessary  to 
analyze  and  develop  both  theoretical  constructs  and  methodological  tools. 

This  is  what  the  SPME  working  group  has  concentrated  on.  At  PME  10,  the  first  official  meeting 
of  the  group,  we  tried  out  various  small  group  tasks  amongst  ourselves  and  discussed  their  value 
as  research  'sites'  and  also  teaching  situations.  At  PME  1 1,  we  moved  to  other  social  determinants 
of  mathematics  leaning,  particularly  thinking  about  influences  of  other  pupils  and  of  the  teacher. 
At  PME  1 2,  we  focused  on  the  idea  of  "bringing  society  into  the  classroom"  and  the  issues  of 
justifying  research  which  might  conflict  with  what  "society",  considers  education  should  be  doing. 
At  PME  13,  we  worked  on  two  areas,  firstly  the  ways  in  which  the  construct  "mathematics"  is 
socially  mediated  in  the  classroom,  and  secondly,  the  use  of  videos  of  classroom  interactions,  and 
their  analyses.  At  PME  14,  we  considered  the  situation  of  bi-cultural  learners,  the  social  setting  of 
the  nursery-school,  and  the  learning  values  of  cooperative  games.  At  PME  15,  we  considered  the 
following:  (1)  bi-cultural  learners  -  particularly  ideas  from  the  evidence  of  Guida  de  Abreu  from 
Brazil,  (2)  the  relationships  between  the  social  contexts  of  mathematics  and  the  child  development 
model,  led  by  Leo  Rogers  from  UK,  (3)  social  issues  of  assessment,  led  by  Luciana  Bazzini  and 
Lucia  Grugnetti  from  Italy,  and  (4)  aspects  of  cultural  and  social  'difference*  which  may  be  of 
significance  in  mathematics  learning. 


1  -  14 

Working  Group:  iwi^r.  m  BeMMasben  in  Education 

co-convenors:  Steve  Lerman  and  Judy  Mousley 

The  group  has  been  meeting  annually  since  1988  and  as  a  working 
group  since  1990.  The  aims  of  the  group  are  to  review  the  issues 
surrounding  the  theme  of  teachers  as  researchers  in  mathematics 
education,  and  to  engage  in  collaborative  research. 

The  stimulus  for  the  notion  that  classroom  teachers  can  and  should 
carry  out  research  whilst  concerned  with  the  practice  of  teaching 
mathematics  comes  from  a  number  of  sources,  including:  teachers  as 
reflective  practitioners;  teaching  as  a  continuous  learning  process;  the 
nature  of  the  theory/practice  interface;  the  problems  of  dissemination  of 
research  when  it  is  centred  in  colleges;  research  problems  being 
generated  in  the  classroom,  and  finding  solutions  within  the  context  in 
which  the  questions  arise.  These  themes  are  seen  to  be  equally  relevant 
to  the  teacher  education  situation,  and  provide  a  focus  for  the  reflective 
activities  of  ourselves  as  teacher-educators. 

Since  the  meeting  in  Assissi  in  1991,  we  have  established  a  network 
amongst  members  and  circulated  papers,  ideas  and  questions.  The 
programme  in  New  Hampshire  will  centre  around  the  issue  that  was 
raised  in  Assissi,  namely  what  constitutes  research  in  the  context  of 
teachers  researching  their/our,  own  practice.  We  will  also  review  the 
work  of  present  and  new  members  in  this  field  and  report  on  research 
carried  out  during  the  year. 


Discussion  Groups 


1  - 17 


Richard  Carter  and  .lohn  Richardsl 

Bolt,  Beranek,  and  Newman  10  Mouiton  St. 
Cambridge,  MA  02138 

The  NCTM  Curriculum  and  Professional  Standards  (NCTM  1989)  and  the  California 
Framework  (California  Department  of  Education  1991)  lay  out  a  vision  of  how 
mathematics  learning  and  teaching  should  happen.  This  vision  is  in  strong  contrast  to  what 
one  finds  in  the  vast  majority  of  standard  classrooms.  This  new  vision  is  becoming 
common  in  the  mathematics  education  community.  Researchers  have  written  about  their 
own  attempts  to  transform  the  classrooms  they  work  in  and  the  difficulties  they  have 
encountered  (Lampeit  (1990),  Ball  (1990),  Cobb  (1991)).  The  NCTM's  Professional 
Standards  are  full  of  classroom  vignettes,  and  there  are  even  some  videotapes  that  show 
exemplary  practices.  Yet,  there  has  been  little  written  about  what  it  means  for  regular 
classroom  teachers  to  try  to  make  the  transition  from  traditional  mathematics  teaching  to  the 
vision  of  inquiry  learning  articulated  in  these  documents.  In  this  paper  we  present  some  of 
our  own  efforts  to  help  classroom  teachers  make  this  transition  and  some  of  the  enduring 
dilemmas  these  teachers  have  encountered. 


1  - 18 


Luciana  B  AZZINI,  Dipartimento  di  Matematka,  Universita  di  Pavia,  (Italy) 
Lucia  GRUGNETTI,  Dipartimento  di  Matematica,  Universita  di  Cagliari,  (Italy) 

A  Discussion  Group  explicitly  devoted  to  the  analysis  of  meaningful  contexts  for  school 
mathematics  had  its  fust  meeting  at  PME  IS  in  Assisi  last  year. 

A  primary  reason  leading  to  the  establishment  of  this  group  was  the  growing  interest  in  the 
role  of  contexts  in  mathematics  education ,  as  shown  by  recent  research  (for  a  basic  bibliog- 
raphy, see  the  presentation  of  this  Discussion  Group  in  the  Proceedings  of  PME  IS,  Vol.  1, 
pag.  XXIX). 

The  two  sessions  of  the  Discussion  Group  took  into  account  the  role  of  context  from  a 
general  point  of  view.  It  was  noticed  that  the  word  "context"  can  have  different  meanings, 
related  to  socio-  cultural  or  ethno-anthropological  factors,  or  to  the  conditions  in  which  teach- 
ing-learning processes  take  place.  For  our  purposes,  we  defined  context  as  the  set  of  environ- 
mental conditions  and  experiences  created  to  evoke  thinking  in  the  classroom  in  order  to  give 
meaning  to  mathematical  constructions.  Evidence  suggests  that  the  ability  to  control  and 
organize  cognitive  skills  is  not  an  abstract  context-free  competence  which  may  be  easily 
transfered  across  diverse  domains  but  consists  rather  of  a  cognitive  activity  which  is  specifi- 
cally tied  to  context.  This  is  not  to  say  that  cognitive  activities  are  completely  specific  to  the 
episode  in  which  they  were  originally  learned  or  applied.  However,  it  is  of  vital  importance  to 
be  able  to  generalize  aspects  of  knowledge  and  skills  to  new  situations.  Attention  to  the  role  of 
context  focuses  on  determining  how  generalization  can  be  stimulated  or  blocked.  A  specific 
context  ca  represent  a  powerful  opportunity  for  mathematical  investigation  but  also  a  potential 
obstacle  to  abstraction. 

In  this  perspective,  we  propose  two  main  foci  for  the  two  sessions  of  this  Discussion  Group 
at  PME  16.  They  are: 

-  analysis  of  how  mathematical  activity  can  be  contextualized  in  experiences  taken  from 
children's  extrascolastic  knowledge; 

-  analysis  of  how  school  mathematics  can  be  lir  ied  to  other  school  disciplines,  in  view  of  a 
meaningful  contextualization  of  mathematics  itself. 

In  our  opinion,  special  attention  to  the  interaction  of  school  mathematics  and  the  world 
outside  and  of  school  mathematics  and  other  domains  gives  rise  to  important  questions  related 
to  the  meaningfulness  of  a  given  context:  meaningful  for  children,  for  mathematics  or  other. 

Finally,  questions  related  to  how  mathematical  constructions  can  be  contextualized  and 
de-contextualized  according  to  a  spiral  process  can  be  discussed  and  investigated  hereafter. 

1  - 19 


BRIAN  A.  DOIG  The  Australian  Council  for  Educational  Research 

Many  researchers  appear  to  work  in  isolation  from  their  brethren  in  related  fields. 
Nowhere  is  this  more  true  than  in  educational  research.  Language  research  has  had 
little  to  say  about  the  language  of  mathematics,  yet  the  mathematics  research 
literature  is  replete  with  references  to  'the  language  of  mathematics'.  This  'language 
of  mathematics'  though,  seems  not  connected  to  the  notions  of  language  generally 
used  in  language  research.  There  are  two  questions  here.  First,  why  does  this 
disconnection  occur,  and  second  are  there  indeed  any  benefits  to  be  had  from 
looking  at  other  disciplines?  I  do  not  propose  to  enter  the  debate  regarding  to  the 
former  question,  but  do  in  regard  to  the  latter. 

Mathematics  education  research  may  benefit  from  looking  at  some  related 
disciplines,  but  which?  Let  us  look  at  one  related  discipline,  namely  science. 
Although  science  appears  to  be  similar  to  mathematics  to  the  uninitiated,  and  indeed 
historically  was  so,  the  end  of  the  twentieth  century  sees  two  quite  distinct  research 
areas  defined.  Where  previously  researchers  like  Piaget  investigated  both  science 
and  mathematical  concept  development  we  now  see  separate  studies.  Scanning  the 
relevant  journals  gives  the  impression  that  mathematics  is  about  content  and  how  it 
may  be  best  taught,  while  a  similar  overview  of  science  journals  reveals  an  emphasis 
upon  development  of  concepts.  How  this  divergence  has  occurred  is  of  no 
importance  here,  but  rather  that  it  exists. 

Mathematics  is  supposedly  about  concept  development,  so  can  we  use  the  science 
research  as  a  guide  to  better  research  efforts  in  mathematics?  I  believe  we  can.  An 
example  of  science  research  exploring  children's  conceptual  development  and 
providing  information  for  teachers  to  better  plan  their  students'  further  learning,  is  a 
study  recently  undertaken  in  Australia  (Adams,  Doig  and  Rosier,  1990).  This  survey 
of  children's  science  beliefs  used  novel  assessment  instruments  collectively  entitled 
Tapping  Students'  Science  Beliefs  (TSSB)  units.  Children  were  asked  to  role  play, 
complete  a  short  story  or  comment  upon  the  activities  of  characters  in  a  cartoon  strip. 
By  the  use  of  modern  psychometrics  the  data  was  collated  and  analyzed  to  produce 
continua  describing  the  development  of  concepts  over  a  number  of  scientific  areas. 
Descriptions  of  students'  likely  scientific  beliefs  at  various  points  along  these 
continua  make  the  planning  of  future  experiences  for  these  students  much  simpler 
and  more  likely  to  match  the  students'  needs. 

It  is  my  contention  that  mathematics  education  can  learn  from  these  current  efforts  in 
science.  For  example,  is  it  possible  to  construct  assessment  instruments  that  engage 
students  and  measure  their  underlying  beliefs  about  mathematical  concepts?  The 
answer  must  be  'yes'  if  we  are  to  attempt  to  create  any  sort  of  constructivist 
curriculum  -  one  based  upon  the  student's  needs  and  perceptions,  and  not  solely  on 
the  received  wisdom  of  previous  generations,  which  is  apparently  what  we  have. 
Ask  yourself  'How  different  is  my  curriculum  from  that  of  my  grandparents?' 


Adams,  R.  J.,  Doig,  B.  A.  and  Rosier,  M.  (1991).  Science  Learning  in  Victorian 
Schools:  1990  Melbourne:  The  Australian  Council  for  Educational  Research. 

ERiC  4C- 

ummmmmi-M  "j  %J 

1  -20 

Discussion  Group 
Paul  Ernest 

In  mathematics  education  epistemological  and  philosophical 
issues  are  gaining  in  importance.  Theories  of  learning  are 
becoming  much  more  epistemologically  orientated,  as  in  the  case 
of  constructivism.  A  number  of  areas  of  inquiry  in  the 
psychology  of  mathematics  education,  including  problem  solving, 
teacher  beliefs,  applications  of  the  Perry  Theory,  and 
ethnomatheraatics ,  all  relate  directly  to  the  philosophy  of 
mathematics.  Researchers  in  mathematics  education  are  becoming 
increasingly  aware  of  the  epistemological  assumptions  and 
foundations  of  their  inquiries.  This  is  because  any  inquiry 
into  the  learning  and  teaching  of  mathematics  depends  upon  the 
nature  of  mathematics,  and  teachers'  and  researchers' 
philosophical  assumptions  about  it.  Whilst  many  of  these  issues 
have  been  raised  before  at  PME,  none  have  been  or  can  be 
resolved.  This  suggests  that  a  continuing  discussion  would  be 
useful  and  timely. 

In  fact,  the  most  central  of  the  philosophical  issues,  the 
philosophy  of  mathematics,  has  been  insufficiently  addressed  at 
PME.  Although  reference  has  been  made  to  it  in  a  number  of 
plenary  and  other  presentations,  there  has  not  been  sufficient 
recognition  that  it  is  undergoing  a  revolution,  and  the 
absolutist  paradigm  is  being  abandoned.  Publications  by 
Lakatos,  Davis  and  Hersh,  Kitcher  and  Tymoczko,  for  example, 
are  pointing  towards  a  new  fallibilist  paradigm.  This  has 
profound  implications  for  the  psychology  of  mathematics 
education.  For  if  mathematics  itself  is  no  longer  seen  as  a 
fixed,  hierarchical  body  of  objective  knowledge,  then  what  is 
the  status  of  hierarchical  theories  of  mathematical  learning  or 
of  subjective  knowledge  of  mathematics?  One  outcome  is  sure. 
They  can  no  longer  claim  to  be  representing  the  logical 
structure  of  mathematics. 

The  aim  of  the  group  is  to  provide  a  forum  for  a  discussion 
some  of  these  issues,  including: 

1.  Recent  developments  in  the  philosophy  of  mathematics. 

2.  Implications  of  such  developments  for  the  psychology  of 
mathematics  education. 

3.  The  epistemological  bases  of  research  paradigms  and 
methodologies  in  mathematics  education. 

This  discussion  group  was  first  offered  at  PME-14  in  Mexico. 
This  meeting  will  continue  the  discussion  begun  there,  and 
consider  becoming  a  working  group. 




Organizers:  Joan  Ferrini-Mundy,  University  of  New  Hampshire,  Ed  Dubinsky,  Purdue 
University,  and  Steve  Monk,  University  of  Washington 

There  is  growing  interest,  particularly  in  the  community  of  mathematicians,  in 
questions  about  the  teaching  and  learning  of  mathematics  at  the  undergraduate  level. 
Professional  organizations  such  as  the  Mathematical  Association  of  America  and  the  American 
Mathematical  Society  have  begun  to  encourage  attention  to  this  emerging  research  area  within 
their  conference  and  publication  structures.  This  discussion  session  is  organized  by  members 
of  the  Mathematical  Association  of  America's  Committee  on  Research  in  Undergraduate 
Mathematics,  to  promote  a  more  sustained  focus  on  this  area  of  research.  We  will  address  the 
following  questions: 

Can  we  summarize  major  research  areas  and  methodologies  concerning  the  learning  and 
teaching  of  undergraduate  mathematics,  and  what  are  the  most  appropriate  vehicles  for  sharing 
this  work  with  a  wider  audience? 

How  can  mathematicians  and  researchers  in  mathematics  education  collaborate  to  formulate  and 
investigate  significant  questions  about  the  teaching  and  learning  of  undergraduate  mathematics? 

How  can  we  encourage  more  systematic  and  widespread  interest  in  this  area  of  research,  while 
also  maintaining  high  levels  of  quality  for  audiences  of  mathematicians,  mathematics  education 
researchers,  and  others? 

What  mechanisms  can  be  developed  for  sharing  work  that  has  implications  for  practice,  in 
terms  of  instruction  and  curriculum,  with  the  community  of  college  mathematics  teachers? 

Is  it  viable  to  propose  a  PME  Working  Group  on  the  Teaching  and  Learning  of  Undergraduate 
Mathematics?  What  might  be  the  relationship  with  the  Advanced  Mathematical  Thinking 
Working  Group? 

A  wide  range  of  research  has  been  undertaken  concerning  the  teaching  and  learning  of 
undergraduate  mathematics.  There  are  serious  challenges  in  considering  how  this  work  might 
be  summarized  and  organized  so  that  it  can  be  accessible  and  helpful  to  interested  researchers 
and  practitioners.  Several  working  reference  lists  and  bibliographies  will  be  assembled  for  this 
discussion  session,  and  participants  are  encouraged  to  supply  additional  material.  Certainly  the 
monograph  produced  by  the  PME  Working  Group  on  Advanced  Mathematical  Thinking 
provides  a  very  useful  organization.  Additional  compilations  and  formats  might  be  helpful  to 
various  communities.  ,  , . 

College  and  university  teachers  of  mathematics  often  have  serious  and  important 
Questions  concerning  issues  in  student  learning  and  in  teicbing£ommunic(Uing  Among 
Communities,  the  final  report  of  a  fall,  1991  conference  sponsored  by  the  MAA,  includes  as 
one  of  its  recommendations  that  "those  faculty  whose  professional  work  is  devoted  to  research 
in  mathematics  education,  as  well  all  those  whose  work  centers  on  curriculum  development  or 
educational  practice"  should  be  appropriately  rewarded.  Issues  in  this  area  also  wiU  be  raised. 

»     '  j   .11  A-...\~~.Z  k~4„nf  in  tAvancfA  mathematical  thlnkin 

there  ce.  , .   .  , 

mathematics  learning  and  teaching.  These  include  various  intervention-type  ~  - 

curricular  innovation  or  instructional  strategies,  studies  of  teaching  processes,  and  studies 
about  the  mathematics  preparation  of  preservice  teachers.  We  hope  to  expand  the  discussion  to 
determine  the  ways  that  these  other  lines  of  research,  many  of  which  have  more  profound 
implications  for  practice,  may  be  extended  and  communicated 




Maria  Alessandra  MARIOTTI,  Dipattimento  di  Matematica,  Universita  di  Pisa,  (Italy) 
Angela  PESCI,  Dipattimento  di  Matematica,  Universita  di  Pavia,  (Italy) 

When  we  had  the  idea,  last  year,  to  start  a  discussion  group  on  this  theme  we  did  not  expect 
so  large  a  presence.  There  were  45  participants  from  the  following  countries:  Australia, 
Canada,  Spain,  Finland,  Germany,  Israel,  Italy,  Mexico,  Portugal,  Sweden,  UK  and  the  USA. 

Today  many  people  are  very  interested  in  this  topic  and  the  related  studies  are  multiplying. 
On  this  subject  T.  Dreyfus,  last  PME,  gave  a  lecture  "On  the  status  of  visual  reasoning  in 
mathematics  and  mathematics  education".  Our  theme  also  intersects  some  aspects  that  are 
widly  discussed  by  the  Working  Group  on  Representations,  guided  by  G.  A.  Goldin.  In  the  last 
three  years  the  growing  number  of  participants  has  made  evident  the  growing  interest  in  these 
problems.  On  the  basis  of  last  year's  discussion,  we  think  it  opportune  to  direct  our  work 
along  the  following  lines. 

Since  visualization,  that  is  the  action  "to  see"  mentally,  can  be  the  result  of  different  visual 
stimuli,  among  these  we  plan  to  deal  in  particular  with  graphical  representation;;.  By  graphical 
representation  we  mean  every  graphical  sign  different  from  the  written  word:  from  a  pictorial 
drawing  to  a  schematic  and  symbolic  one,  up  to  the  most  specific  mathematical  signs.  The 
graphical  sign  can  be  produced  by  a  pupil,  a  teacher,  a  textbook,  a  computer  and  so  on.  During 
a  lesson  of  mathematics,  geometrical  figures,  symbols,  schemes,  tables,  tree  diagrams,  arrows, 
...  are  frequently  used.  Often  they  are  not  only  didactic  aids  but  visual  messages  which  are 
crucial  in  building  the  "meaning"  of  a  concept  or  in  schematizing  a  problematic  situation.  In 
several  instances  these  representations  play  a  very  important  role:  perhaps  they  are  able  to 
suggest  mental  images  which  are  very  effective  and  functional  (for  instance,  in  some  memory 
tasks,  in  associations  useful  in  producing  cognitive  acquisitions,  in  partially  new  resolution 
processes  and  so  on).  We  consider  very  important  to  study  the  dialectics  between  graphical 
representations  and  internal  cognitive  processes  and  to  discuss  how  this  study  can  be  faced. 

Therefore  we  consider  important  that  our  group  try  to  discuss  the  following  problems: 
a  -  To  what  extent  and  how  are  internal  images  influenced  by  external  ones  in  arithmetic, 

algebra,  geometry  and  analysis? 
b  -  How  are  internal  images  used  to  generate  external  ones  (diagrams,  pictures,  sketches,...)  for 

example  during  problem  solving  processes? 
c  -  Which  graphical  representations  are  particularly  effective?  In  which  conceptual  contexts? 

For  which  ages?  Which  could  be  the  reasons  of  their  effectiveness? 
d  -  Which  are  the  most  common  misunderstandings  in  using  external  representations?  How 

can  we  find  a  remedy  for  them?  How  can  we  be  sure  that  the  meaning  of  a  graphical 

scheme  is  completely  determined  without  ambiguity? 
e  -  How  to  face  the  analysis  suggested  by  the  previous  points?  By  which  tools  and  methods? 

In  which  theoretical  frames? 


1  -23 

Research  Reports 






Gnida  de  Abreu  -  Department  of  Education  /  Cambridge  University.  UK 
Alan  J.  Bishop  -  Department  of  Education  /  Cambridge  University,  UK 
Oeralrin  Pompeu  Jr.  -  Catholic  University  of  Campinas.  Brazil 

With  the  aim  of  discussing  alternatives  approaches  to  research  into  cultural 
conflicts,  some  results  from  two  research  projects  are  presented.  The  first  is  concerned 
with  clarthitna  the  cultural  conflict  as  experienced  from  the  child's  perspective,  when  her 
home  mathematics  is  substantially  different  from  the  school  mathematics.  The  second 
analyses  changes  in  teachers'  attitudes  in  the  transition  from  a  culture-free  approach  to 
mathematics  teaching,  to  an  approach  that  acknowledges  the  cultural-conflict. 

The  recent  constructivist  framework,  as  exemplified  by  Saxe  (1990).  focuses  on  a 
level  of  mathematics  learning  where  culture  and  cognition  are  constitutive  of  one 
another.  Saxe  developed  his  empirical  studies  in  an  out-of-school  setting,  candy  selling 
on  tiie  street,  and  found  evidence  that  the  children  gradually  interweave  their  school 
mathematics  with  the  mathematics  generated  by  the  participation  in  the  out-of-school 
practices.  A  considerable  amount  of  research  describing  the  mathematical  competence 
of  people  out-of-school.  in  contrast  to  in-school.  is  also  available,  e.g..  (Carraher.  1988; 
Lave  1988).  but  little  is  known  about  the  interactions  occurring  when  children  are 
confronted  by  the  two  sets  of  mathematics  cultural  practices,  in  a  school  setting.  That 
seems  a  crucial  area  to  clarify  when  developing  new  approaches  to  teaching  in 
situations  where  the  school  mathematics  culture  is  markedly  different  from  that 
demonstrated  outside  school. 

A  second  crucial  area  in  such  situations  is  that  of  the  attitudes  of  teachers 
concerning  the  relevance  of  children's  out-of-school  knowledge  for  classroom  teaching. 
There  is  a  body  of  research  on  teachers'  attitudes  in  mathematics  teaching  in  general, 
but  none  which  focuses  on  this  specific  aspect.  This  paper  will  be  a  report  of  ongoing 
research  in  both  of  these  areas,  illustrating  as  well  the  enormous  research  challenges 
facing  mathematics  educators  working  in  cultural  conflict  situations,  where  'cultural 
conflict-  means  the  conflict  the  children  experience  in  terms  of  contradictory 
understandings  generated  through  their  participation  in  two  different  mathematics 
cultures,  one  outside  school,  linked  to  their  everyday  pracUces.  and  the  other  at  school. 

*  -  Guida  de  Abreu  's  research  is  sponsored  by  CNPq  /  Brazil 
Geraldo  Pompeu' s  research  is  sponsored  by  Capes  /  Brazil 




Children's  cultural  conflicts 

The  assumption  that  mathematical  knowledge  is  cultural  implies  that  its  learning 
is  associated  with  values,  '^llefs.  rules  about  its  use.  etc.  Therefore,  the  traditional 
belief  that  school  mathematics  is  a  culture-free  subject  is  questionable,  and  there  is  a 
growing  feeling  that  it  should  be  treated  as  a  specific  school  mathematics  culture  which 
is  not  taking  into  account  the  mathematics  practiced  in  the  out-of-school  culture.  That 
split  between  the  two  mathematics  cultures  is  the  source  of  conflict  for  children.  By  the 
nature  of  human  cognition  they  should  build  their  knowledge  upon  their  previous 
understandings,  but  because  of  the  cultural  gap  they  are  being  faced  with 
contradictions,  which  appear  in  different  ways,  such  as:  (a)  beliefs  ;  (b)  performance; 
(c)  representations  (di  self-concepts.  To  exemplify  these  aspects  some  results  from  an 
empirical  investigation  developed  among  Brazilian  children,  aged  between  8  and  16. 
from  primary  schools  in  a  sugar  cane  farming  community,  will  be  reported.  This  is  a 
development  of  the  research  reported  in  Bishop  and  Abreu  (1991). 

(a)  Beliefs:  When  investigating  children's  beliefs  a  great  imbalance  was  found  in 
terms  of  the  value  that  they  give  to  the  outside  mathematics,  used  in  the  predominant 
activity  of  the  local  economy,  sugar  cane  farming  and  the  in  school  mathematics  (see 
Table  1). 

Table  1:  Percentage  of  children's  answers  related  to  their  beUefs  about 
mathematics  (n»26) 

Children  believe  that: 


.  People  working  in  an  office  use  mathematics 

.  People  working  in  sugar  cane  farming  do  not  use  mathematics 


.  The  pupils,  who  performed  best  in  school  mathematics, 
work  in  offices 

.  The  pupils,  who  performed  worst  in 

school  mathematics,  work  in  sugar  cane  farming 


.  People  working  in  an  office  are  schooled 

.  People  working  in  sugar  cane  farming  are  unschooled 


.  Sugar  cane  workers  can  work  out  sums  without  being  schooled 
.  Sugar  cane  workers  cannot  do  sums  without  being  schooled 


On  the  other  hand  they  also  acknowledged  that  sugar  cane  workers  cope 
successfully  with  their  everyday  sums.  This  seems  to  be  one  area  of  cultural-conflict, 
which  apparently  they  resolved  in  terms  of  contextualization.  that  is  school 

mathematics  is  different  from  sugar  cane  mathematics.  However,  in  practice  they  are 
copl.ig  with  contradictions  such  as:  their  parents  can  do  sums  better  than  them,  but 


1  -27 

they  believe  that  people  who  are  in  sugar  cane  farming  do  not  have  proper  knowledge: 
they  need  to  rely  on  their  parents'  mathematics  to  get  help  to  cope  with  difficulties  in 
their  school  mathematics  homework.  Severlna.  14  years  old.  a  sugar  cane  worker's 
daughter,  described  by  the  teacher  as  an  unsuccessful  pupil,  gives  evidence  of  the 

/  (Interviewer)  -  Why  doesn't  that  man  (in  a  picture]  on  the  tractor  know 

S  (Severina)  -  He  doesn't  know.  He  doesn't  have  a  Job.  He  works  in  sugar  cane. 

I -Is  it  possible  that  some  people  (in  pictures]  had  never  been  to  school? 

S  -  {Among  pictures  with  people  in  offices,  markets,  school  and  sugar  cane  she 

choose  a  man  working  in  sugar  cane.]  Yes.  this.  1  think  that  if  he  has  been  to 

school  he  would  not  be  working  in  that  place. 

I  -  Any  more? 

S  -  These  (again  people  in  sugar  cane). 

I -Why?  , 

S  -  ft  is  the  same.  If  they  had  studied  they  will  not  be  working  in  that  place.  This  is 

an  example  of  those  who  had  never  been  to  school  like  my  father. 

I  -  (...)  You  told  me  that  your  father  doesn't  know  to  write,  but  for  oral  sums  he  is 

the  best  How  does  he  help  you  in  your  mathematics  homework? 

S  -  /  ask  him,  for  example:  how  much  is  3  times  7  or  8  and  he  answers.  How  much 

is  3  phis  12?  He  answers  all 

I  ■  (On  another  interviewl  Can  you  tell  what  you  think  about  the  way  your  father 
did  the  sums,  is  it  the  same  or  different  from  the  way  you  learned  in  school? 
S  -  ft  is  a  different  way.  he  does  it  in  his  head,  mine  is  with  the  pen. 
I  ■  Which  do  you  think  is  the  proper  way? 
S  -  School. 

/  -  Which  do  you  think  gives  a  correct  result? 
S-  My  father. 
I -Why? 

S  -  Because  ljust  think  so. 

(b)  Performance:  Analysing  children's  performance  in  group  tasks,  on  which 
they  were  asked  to  imagine  they  were  farmers,  to  allow  them  to  bring  their  out-of- 
school  knowledge  to  solve  the  task,  no  differences  were  found  related  to  their  school 
mathematical  performance,  but  there  seems  to  be  a  relation  with  gender  (see  Table  2). 

Table  2:  Number  of  answers  according  to  children's  performance  In  school  and 

How  the  child 
understands  the 
inverse  relation 

Only  qualitatively 

Both  qualitatively 
and  quantitatively 

Number  of  answers  given  by  pupils: 

Successful  Unsuccessful 
at  school    at  school 

Boys  Girls 




The  mathematics  concept  involved  was  the  inverse  proportional  relation  between 
the  size  of  the  unit  of  measurement  and  the  total  number  of  units  needed  in  four  tasks: 
halving  and  doubling  the  size  of  a  stick  used  to  measure  length:  halving  and  doubling 
the  size  of  a  square  to  measure  area.  Children  show  two  types  of  understandings:  (a) 
only  qualitatively,  e.g..  if  it  is  half  the  size  of  the  unit,  then  I  will  need  more  units,  but 
they  show  difficulties  in  the  calculations,  (b)  qualitatively  and  quantitatively,  e.g..  this 
is  half,  then  I  need  twice  the  number  I  have. 

Again,  these  results  brought  more  cultural  conflicts  into  question.  One  is  related 
to  the  contradiction  in  the  child's  performance  in  school  and  out.  of  which  they  seem  to 
be  aware,  as  for  example,  a  5th  grade  girl,  unsuccessful  in  school  who  said:  "Sometimes 
my  sister  comes  to  my  house,  brings  the  money  and  I  go  shopping  for  her.  They  give  me 
the  note  (account).  I  check  and  get  it  correct.  But.  in  school  there  is  no  way.  I  cannot 
learn."  Another  is  related  to  the  contradiction  between  child-specific  experiences  and 
the  task  presented,  e.g.  this  community  has  specific  social  roles  for  girls  and  boys, 
allowing  boys  to  have  Jobs  in  agriculture  or  in  the  market,  but  which  are  improper  for 
girls.  Perhaps  that  difference  between  girls  and  boys  could  account  for  their 
engagement  in  different  social  practices  outside  which  lead  to  specific  mathematics 

(c)  Representation*:  When  confronting  the  children  with  school-like  tasks,  the 
first  thing  that  was  obvious  was  the  difficulty  of  the  children  of  that  community  in 
coping  with  written  language,  rhis  make  sense  since  in  the  children's  homes  there  is 
mainly  an  oral  culture.  They  mention  about  their  fathers  that:  only  37%  can  read,  but 
77%  can  do  sums  orally  and  70%  in  writing.  For  the  mothers.  57%  can  read,  but  only 
47%  can  cope  with  sums  orally  or  writing.  Analysing  the  results  of  twenty  pupils  in  four 
school  tasks  they  succeed  on  89%  of  the  sums  when  solving  them  orally,  but  only  39% 
succeed  in  representing  it  in  a  written  way  acceptable  in  school  (these  findings  agree 
wtth  Carraher.  Carraher  and  Schliemann.  1987.  who  described  the  oral  strategies 
accounting  for  success).  However,  in  the  research  presented  here,  the  focus  is  on  the 
process  of  producing  a  written  representation  for  their  oral  solution.  It  seems  as  if  they 
are  mixing  their  oral  system  of  representation  with  the  written  system,  which  is 
being  taught  in  school.  For  example  509b  of  the  third  grade  pupils  in  writing  the  sum  of 
the  sides  of  quadrilaterals,  put  more  than  one  side  in  the  same  line,  as  in  the  following 

Example  1 :  After  measuring  a  square  with  5  (cm)  each  side,  and  answering  orally  that 
the  total  is  20.  the  child  produced  the  following  written  representation 
5  5 


10  10 

Reading  the  written  answer:  "  It  Is  twenty.  Because  S  plus  5  Is  10,  with  S  plus  5  is 


1  -29 

Example  2:  Adding  the  sides  of  a  parallelogram  3  by  4  (cm). 

Reading  the  written  answer: "  It  is  seventy  seven." 

Both  of  these  children  wrote  in  the  same  way.  but  in  the  firs-:  example  the  child 
seems  to  follow  the  oral  reasoning  and  give  a  correct  result,  while  in  the  second  the 
child  reads  the  number  following  the  rules  of  written  numbers  and  giving  an  incorrect 
answer.  This  seems  to  be  another  way  of  experiencing  the  conflict,  that  is.  children 
appear  to  be  very  confused  when  asked  to  choose  which  answer  is  correct.  A  child 
argued  in  one  problem  that  the  written  sum  is  correct  "Because  this  one  here  (written) 
we  did  getting  the  nwnbers  from  here,  working  out,  and  checking",  while  in  fact  the 
correct  result  was  the  one  he  did  orally.  On  another  problem  the  same  child  chose  as 
correct  the  oral  result. 

(d)  Self-concepts:  Comparing  children's  self-Judgements  about  their 
performances  in  mathematics  with  their  teachers' judgements  it  was  found  that  they  do 
not  agree  in  55%  of  the  cases.  That  high  rate  of  disagreement  between  children's  self- 
concepts  and  teachers  seems  to  be  another  source  of  cultural  conflict.  The  majority  of 
the  children  who  disagree  with  their  teachers  are  the  ones  judged  as  low  achievers  by 
the  teachers.  The  child's  self-concept  seems  more  coherent  to  their  mathematics 
abilities  in  general,  than  the  teacher's  judgement  based  on  the  scores  from  school  tests. 

Educational  approaches  to  cultural-conflict 

Cultural-conflict  between  in  and  out-of-school  mathematics  is  being  reflected  in 
different  kinds  of  contradictions  that  affect  children  mathematics  learning,  as 
exemplified  by  research  results  like  those  described  above.  There  appear  to  be,  from 
an  educational  perspective,  two  broad  approaches  to  this  conflict,  one  which  ignores  it 
and  keeps  the  traditional  mathematics  teaching  approach,  and  the  other  which 
acknowledges  it.  Following  the  second,  different  alternatives  are  followed  in  terms  of  the 
extent  to  which  the  home  culture  of  the  child  will  be  taken  into  account  in  the  school 
context.  We  refer  to  these  as:  assimilation:  accommodation:  amalgamation  and 
appropriation.  The  way  the  school  culture  will  interact  with  the  child's  home  culture 
will  vary  according  to  the  four  alternatives,  therefore  raising  different  questions  about 
children'  learning:  teacher's  attitudes:  school  curriculum,  etc.  (see  Table  3) 




Table  3:  Different  approaches  to  culture-conflict 

to  culture 








No  culture 


No  particular 





V,  Li  11 14 1  C 

should  be 
useful  as 

Some  child's  Caring 
cultural  approach 
contexts        !  perhaps  with 
included        !  some  pupils 
■  in  groups 

Official,  j 
plus  | 

contrasts  1 

and  j 


for  second  ! 

language  j 

learners  j 







Currk  !um 
due  to 


modified  as 
preferred  by 

Child's  home  | 
language  • 
accepted  in  \ 
class,  plus  | 



in  education 

organised  by 
teachers  and 

Shared  or 






should  take 


wholly  by 

entirely  by 

Teaching  in 




To  illustrate  one  aspect  of  research  into  those  approaches,  some  results  from  a 
study  of  teachers'  attitudes  are  presented.  This  research,  also  in  Brazil,  investigated 
changes  in  attitudes  occurring  during  the  implementation  of  an  'accommodation' 
approach  -  specifically  in  the  transition  from  traditional  teaching,  called  canonical- 
structuralist',  to  one  called  'ethnomathematlcal',  which  took  into  account  the  children's 
social  and  cultural  knowledge  and  values.  This  work  is  the  culmination  of  the  research 
described  in  a  preliminary  form  in  Bishop  and  Pompeu  (1991). 

The  lesearch  study  was  designed  in  three  main  phases:  In  the  first,  the 
theoretical  background  for  the  Ethnomathematical  approach  was  introduced  to  the 
teachers:  in  the  second  phase,  the  teachers  planned  and  developed  six  Teaching 


1  -31 

Projects  fTPs)'  based  on  the  Ethnomathematical  approach;  finally,  in  the  third  phase, 
the  teachers  applied  the  TPs  with  their  pupils.  Nineteen  teachers  were  involved.  In 
twelve  ordinary  state  schools,  teaching  the  six  projects  lasting  between  three  and  five 
weeks,  to  a  total  of  450  pupils.  In  order  to  monitor  and  assess  the  changes  In  teachers' 
attitudes,  a  questionnaire  was  applied,  as  an  attitude  •thermometer".  In  three  different 
points  in  the  research  study:  at  the  beginning,  after  the  theoretical  phase,  and  after  the 
application  of  the  TPs  (for  details  of  procedure,  see  Bishop  and  Pompeu.  1991).  The 
teachers  also  wrote  about  their  attitudes  and  were  interviewed.  Three  conclusions 
about  the  effects  of  the  different  phases  of  the  implementation  strategy  upon  the 
teachers'  attitudes  are  relevant  here: 

From  the  general  perspective  of  mathematics  as  a  school  subject,  and  In  terms  of 
the  intended,  implemented  and  attained  levels  of  the  mathematics  curriculum,  the 
theoretical  phase  signiflc<_utly  affected  the  teachers*  attitudes  towards  the  first  two  of 
these  perspectives.  That  is.  the  introduction  of  the  Ethnomathematical  theoretical 
background  to  the  teachers,  substantially  changed  their  attitudes  towards  the  general 
perspective  of  mathematics  as  a  school  subject,  and  towards  the  Intended  mathematics 
curriculum  (why  mathematics  occupies  an  important  place  into  the  school  curriculum). 
According  to  the  data,  the  emphasis  on  mathematics  as  a  school  subject  most 
increased  in  relation  to  the  'particular'  and  the  'exploratory  and  explanatory'  features  of 
the  subject,  and  most  decreased  in  relation  to  its  'universal'  and  'logical'  features. 
Similarly,  the  emphasis  on  the  reasons  why  mathematics  occupies  an  important  place 
Into  the  school  curriculum,  most  increased  in  relation  to  the  'social  and  cultural  basis 
of  the  subject",  and  most  decreased  in  relation  to  its  'informative'  aspects.  For  example, 
one  teacher  wrote: 

"What  a  big  mistake  it  was  to  think  inWally  that  the  'cultural  and  social'  basis  of 
mathematics  has  so  little  importance.  Mathematics  is  basically  a  product  of  the 
culture  of  each  race.  It  grows  from  the  needs  of  each  society,  and  the  experience  of 
each  one.  These  are  the  bases  of  its  truth." 

The  planning  and  development  of  the  TPs,  as  well  as  their  application  with  the 
pupils,  most  substantially  changed  the  teachers'  attitudes  towards  the  attained  level  of 
the  mathematics  curriculum.  In  other  words,  the  teachers*  action  as  'designers  of 
curriculum  fTPs  in  this  case),  and  guides  to  learning*,  as  suggested  by  Howson  and 
Wilson  (1986).  most  substantially  changed  their  attitudes  In  relation  to  'what  abilities 
pupils  should  have  after  they  have  learnt  mathematics*.  According  to  the  data,  for 
example,  after  the  application  phase,  the  emphasis  on  pupils'  ability  to  "analyse 
problems'  was  the  one  which  most  increased  its  importance  in  the  teachers*  view. 



Interestingly,  the  attitude  questionnaire  did  not  reveal  any  major  change  In 
teachers'  attitudes  towards  'how  mathematics  should  be  taught'  (the  implemented  level 
of  the  mathematics  curriculum).  However,  by  the  end  of  the  study,  the  emphasis  on  a 
'debatable'  approach  to  the  teaching  of  mathematics  was  the  perspective  which  had 
most  increased  its  importance  from  the  teachers'  point  of  view.  In  contrast  the 
emphasis  on  a  "one-wa/and  a  "reproductive"  approach  to  the  teaching  of  mathematics 
were  the  aspects  which  most  decreased  in  their  Importance  for  these  teachers.  In 
addition,  from  some  teachers  comments,  it  was  also  possible  to  see  other  changes  in 
teachers'  attitudes  at  this  level  of  analysis.  One  of  these  changes  is  related  to  the 
assessment  procedure  adopted  In  mathematics,  and  at  the  end  of  the  research  study,  a 
teacher  wrote  about  this: 

"I  was  not  expecting  the  kind  of  reaction  which  some  of  the  pupils  had  (some  pupils 
manifested  disagreement  about  the  final  results  of  the  assessment  -  researcher 
observation).  On  the  other  hand,  (...)  I  learnt  that  the  assessment  procedure  is  too 
complex  to  be  so  little  discussed.  (...)  After  all  I  believe  that  an  assessment 
procedure  should  take  into  consideration  the  individual  aspects  of  each  pupil 
demanding  from  each  one  of  them  a  proportional  response  to  Ivts/her  earlier 

(More  data  will  be  presented  at  the  conference) 

Cultural  conflicts  are  increasingly  being  recognised  as  a  source  of  mathematical 
conceptual,  and  attltudinal  obstacles  for  pupils  and  teachers  alike.  The  analysis, 
research  approaches  and  findings  reported  In  this  paper  indicate  some  promising 
directions  which  research  in  this  area  could  take,  and  demonstrate  the  educational 
complexity  which  must  be  appreciated  if  progress  is  to  be  made. 


Bishop,  A.J.  and  Abreu.  G.  de.:  1991,  Children's  use  of  outslde-school  knowledge  to 
solve  mathematics  problems  ln-school.  PMEXV.  voLl.  Italy,  128-135. 

Bishop.  A.J.  and  Pompeu  Jr.,  G.:  1991,  Influences  of  an  ethnomathematical  approach 
on  teacher  attitudes  to  mathematics  education.  PMEXV.  vol.1.  Italy.  136-143. 

Carraher,  T.  N.:  1988.  Street  mathematics  and  school  mathematics,  PME  xn.  vol.1. 
Hungary,  1-23. 

Carraher,  T.N.,  Carraher,  D.W.  and  Schllemann,  A.D.:  1987,  Written  and  oral 

1  -33 



Verna  M.  Adams 

Washington  State  University 


This  investigation  examined  ways  in  which  a  theory  of  knowledge  telling 
and  knowledge  transforming  from  written  composition  might  be  relevant  to 
mathematical  problem  solving.    Rhetorical  problems  were  identified  in  problem 
solving  interviews  as  subjects  attempted  to  understand  the  problem  statement. 
These  problems  generally  dealt  with  understanding  language  and  were 
sometimes  resolved  as  a  result  of  expectations  of  text  forms  for  mathematical 
problems.    Revisions  of  text  during  mathematical  problem  solving  occurred 
when  the  problem  solver  modified  diagrams,  charts,  equations,  etc.  Revisions 
often  occurred  at  critical  times  in  the  solution  process. 

Although  solving  a  mathematics  problem  and  writing  a  composition  are 
very  different  activities,  from  one  perspective  they  have  much  in  common.  Like 
skilled  writers,  good  problem  solvers  in  mathematics  must  use  and  exert 
control  over  complex  cognitive  activities  such  as  goal  setting,  planning,  and 
memory  search  and  evaluation.     Some  researchers  on  writing  (Bereiter  & 
Scardamalia,  1987;  Carter,  1988;  Flower  &  Hayes.  1977)  consider  writing  to  be 
a  problem-solving  activity.    This  view  of  writing  has  led  researchers  in 
mathematical  problem  solving  and  in  writing  to  rely  on  some  of  the  same 
sources  in  building  theories  in  the  two  domains.   Bereiter  and  Scardamalia 
developed  a  theory  of  written  composition  that  involves  two  modes  of  mental 
processing  called  knowledge  telling  and  knowledge  transforming. 



1  -34 

In  this  study,  the  investigator  examined  Bereiter's  and  Scardamalia's 
theory  of  written  composition  (1987)  from  the  perspective  of  a  mathematics 
educator  interested  in  mathematical  problem  solving.    The  investigator,  in 
effect,  first  stepped  outside  the  domain  of  mathematics  to  acquire  an 
understanding  of  the  theory  of  knowledge  telling  and  knowledge  transforming  as 
it  applies  to  written  composition.   That  understanding  was  then  brought  back 
into  the  domain  of  mathematics  to  ground  the  theory  in  data  on  mathematical 
problem  solving.   This  paper  reports  on  the  component  of  the  investigation  of 
knowledge  telling  and  knowledge  transforming  in  mathematical  problem  solving 
(Adams,  1991)  that  identified  rhetorical  problems. 

The  idea  of  two  modes  of  mental  processing  has  its  roots  in  theories  from 
cognitive  psychology  (Anderson,  1983).   According  to  Anderson,  one  mode  of 
cognitive  functioning  is  "automatic"  and  "invoked  directly  by  stimulus  input." 
The  other  mode  "requires  conscious  control  .  .  .  and  is  invoked  in  response  to 
internal  goals"  (pp.  126-127).    Bereiter  and  Scardamalia  (1987)  identified  the 
characteristics  of  these  modes  of  processing  within  the  domain  of  written 
composition  and  labeled  the  first  mode  of  processing  knowledge  telling  and  the 
second  mode  knowledge  transforming. 

Bereiter  and  Scardamalia  (1987)  proposed  that,  when  a  writer  is  engaged 
in  knowledge  transforming,  the  writer  creates  a  rhetorical  problem  space  and  a 
content  problem  space.   These  problem  spaces  are  not  created  if  the  writer  is 
engaged  in  knowledge  telling.   The  rhetorical  problem  space  is  tied  to  text 
production  and  contains  mental  representations  of  actual  or  intended  text.  One 
of  its  functions  is  to  put  thoughts  into  a  linear  sequence  for  output  as  written 
text.   The  content  problem  space  is  tied  to  idea  production. 

Theoretical  Background 


1  -35 

To  make  the  transition  from  written  composition  to  mathematical 
problem  solving,  the  investigator  viewed  the  representation  of  a  mathematical 
problem  created  by  the  problem  solver  as  "text."   Examples  of  text  forms  in 
mathematical  problem  solving  include  charts,  tables,  and  proofs.   Knowledge  of 
how  to  structure  a  mathematical  proof  is  an  example  of  knowledge  of  a  way  of 
forming  text.    Because  mathematical  problems  are  often  presented  in  written 
form,  the  function  of  a  rhetorical  problem  space  in  mathematical  problem 
solving  was  assumed  to  include  the  interpretation  of  text  as  well  as  the 
creation  of  text.   This  assumption  has  a  basis  in  literature  on  reading  and 
writing:    Birnbaum  (1986)  suggests  that  reflective  thinking  about  language  is  a 
common  thread  between  reading  and  writing.   Readers  and  writers  share  the 
common  goal  of  constructing  meaning  (Dougherty,  1986;  Spivey,  1990). 

The  origins  of  the  concept  of  problem  spaces  lie  in  theory  related  to 
computer  simulations  of  human  thought  (see  Newell,  1980;  Newell  &  Simon, 
1972;  Simon  &  Newell,  1971).   Simon  and  Newell  (1971)  describe  a  problem 
space  as  "the  way  a  particular  subject  represents  the  task  in  order  to  work  on 
it"  (p.  151).   Newell  (1980)  proposes  that  problem  spaces  are  mental  constructs 
"whicn  humans  have  or  develop  when  they  engage  in  goal-oriented  activity"  (p. 
696).     Whether  or  not  these  mental  constructs  were  created  was  of  interest  in 
this  investigation  as  one  means  of  distinguishing  knowledge  telling  and 
knowledge  transforming.    For  that  purpose,  it  was  useful  to  think  of  a  problem 
space  as  setting  boundaries  on  the  knowledge  structures  used  in  finding  a 
solution  to  the  problem. 

Although  the  concept  of  problem  spaces  is  not  new  to  research  on 
mathematical  problem  solving  (e.g.,  Goldin,  1979;  Jensen,  1984;  Kantowski, 
1974/1975),  the  idea  of  a  rhetorical  problem  space  has  not  been  proposed  for 
mathematical  problem  solving.    Researchers  in  mathematics  education, 
however,  have  been  interested  in  issues  dealing  with  mathematics,  language, 



and  learning  (e.g.,  Cocking  &  Mestre,  1988;  Pimm,  1987).    Pimm  (1987) 
identified  features  of  the  mathematical  writing  system  and  the  complexity  of 
the  syntax  of  written  mathematical  forms.   He  suggested  that  the  same 
difficulties  that  children  have  with  natural  language  are  evident  in  learning 

In  this  investigation,  a  problem  solver  was  described  as  experiencing  a 
problem  if  a  direct  route  to  a  goal  was  blocked.  Because  a  task  that  is  a 
problem  for  one  person  may  not  be  a  problem  for  another  person,  a  mathematical 
task  cannot  be  labeled  a  problem  until  after  the  problem  solver  has  worked  on 
the  task.  Thus,  in  order  to  characterize  the  task  as  a  problem,  an  observer  must 
evaluate  the  mental  activity  of  the  problem  solver  by  making  inferences  about 
that  mental  activity  from  what  the  problem  solver  says  and  does. 

The  mathematical  tasks  used  in  the  investigation  were  written  word 
problems.   Problems  were  identified  as  compositional  problems  if  the  problem 
solver  suspended  attention  to  the  problem  goal  identified  in  the  problem 
statement  in  order  to  create  his  or  her  own  goals  for  understanding  and  solving 
the  problem.   Mathematical  tasks  may  be  problems  without  being  compositional 

As  problem  solvers  work  on  mathematical  tasks,  they  construct  mental 
representations  of  the  tasks.  The  mental  representation  may  be  thought  of  as  a 
cognitive  structure  constructed  on  the  basis  of  the  problem  solvers  domain- 
related  knowledge  and  the  organization  of  that  knowledge  in  memory  (Yackel, 
1984/1985).    As  the  problem  solver  develops  the  mental  representation,  the 
problem  solver's  mathematical  concepts  and  problem-solving  processes  may 
undergo  change  (see  Lesh,  1985).  Kintsch  (1986)  suggested  that  the  problem 

Research  Perspective  on  Mathematical  Problems 


solver  builds  a  mental  model  of  the  text  (problem  statement)  and  a  mental 
model  of  the  situation  described  in  the  text.   The  mental  model  of  the  text  "is 
built  from  propositions  and  expresses  the  semantic  content  of  the  text  at  both  a 
local  and  a  global  level"  (p.  88). 

For  this  investigation,  I  assumed  that  the  mental  model  of  the  text  is  part 
of  the  rhetorical  problem  space  and  the  mental  model  of  the  problem  situation 
is  part  of  the  content  space,  which  I  refer  to  as  the  main  problem  space.  The 
contents  of  each  problem  space  must  be  inferred  from  analysis  of  the  problem 
solver's  written  representation  of  the  problem  and  what  the  problem  solver 
does  and  says  while  solving  the  problem. 

A  theory  developed  in  one  domain  cannot  be  expected  to  manifest  itself  in 
exactly  the  same  ways  in  a  second  domain.    Thus,  the  main  purpose  of  the  the 
study  was  to  generate  theory  for  mathematical  problem  solving  and  was  not 
approached  as  a  verification  study.   A  general  method  of  comparative  analysis 
(Glaser  &  Strauss,  1967)  was  used  throughout  the  investigation,   in  order  to 
examine  potentially  different  situations  in  which  a  grounded  theory  might 
manifest  itself,  purposeful  sampling  (Bogdan  &  Biklen,  1982)  was  used  to  select 
data  from  the  mathematics  education  literature  and  to  collect  additional  data. 
This  procedure  made  it  possible  to  compare  "novices"  and  "experts"  solving  a 
variety  of  mathematical  tasks. 

A  total  of  21  interviews  of  subjects  solving  problems  were  coded  and 
analyzed.   Three  levels  cf  protocol  analysis  were  used:  (a)  coding  and 
categorizing  of  idea  units  in  order  to  identify  characteristics  of  the  theory  as  it 
related  to  mathematical  problem  solving  (b)  identification  of  problem  spaces 
created  by  the  problem  solver,  and  (c)  characterization  of  the  problem  solving 

Design  and  Procedures 



episodes  as  primarily  knowledge  telling  or  knowledge  transforming. 
Consistency  between  the  levels  of  analysis  was  monitored. 

The  identification  of  rhetorical  problems  proceeded  in  two  ways:  (a) 
Situations  involving  difficulties  with  language  were  examined.    If  the  problem 
solver  appeared  to  be  pursuing  a  particular  goal,  explicitly  or  implicitly,  and 
appeared  to  be  purposefully  working  toward  the  goal,  the  activity  was 
identified  as  taking  place  in  a  rhetorical  problem  space,    (b)  Situations  in  which 
text  was  modified  were  examined. 


The  first  problematic  situation  for  many  problem  solvers  occurred  as  they 
attempted  to  understand  the  problem  statement.   For  example  an  eighth-grade 
student  read  a  problem  and  stated:  "Okay  .  .  .  You  have  1 9  coins  worth  .  .  .  worth 
a  dollar.  How  many  of  each  type  of  coin  can  you  have?  ...  19  coins  worth  a 
dollar.  Well  I  ...  I  don't  know  if  all  19  coins  are  worth  a  dollar  each  or  just  one 
dollar."  The  student  resolved  the  problem  in  this  way:   "but  I  guess  if  it  was 
worth  a  dollar  each  they'd  tell  you."  In  this  case,  the  student  relied  on  her 
expectations  of  text  forms  for  problem  statements  to  make  the  problem 
situation  meaningful. 

Revisions  of  text  were  identified  in  the  problem  solving  process  and 
appeared  to  be  undertaken  for  different  reasons.   For  example,  one  subject, 
solving  the  coin  problem  above,  modified  a  chart  he  was  creating:  "Urn.  I  am 
going  to  put  a  column  over  here  for  the  total  so  that  I  can  keep  a  running  total." 
Modifications  of  this  type  seemed  replace  creating  a  plan  before  beginning  the 
problem  solving  process.   One  problem  solver  working  a  problem  involving 
similar  triangles  modified  a  drawing  in  order  to  make  the  drawing  more 
consistent  with  the  problem  solver's  knowledge  of  similar  triangles.  The 



revision,  in  this  case,  seemed  to  be  part  of  a  process  of  reformulating  the 
problem.   Revisions  also  occurred  after  the  problem  solver  observed  an  error  in 
earlier  work. 

In  summary,  rhetorical  problems  were  more  easily  identified  during  the 
process  of  understanding  the  problem  statement  than  at  other  times  during 
individual  problem  solving.    Although  revisions  of  text  did  not  seem  to  indicate 
the  same  types  of  problematic  situations  that  language  presented  in 
understanding  the  problem  statement,  revisions  of  text  often  occurred  at 
critical  points  in  the  problem  solving  process. 


Adams,  V.  M.  (1991).  Knowledge  tailing  and  knowledge  transforming  in 
mathematical  problem  solving.  Doctoral  dissertation,  University  of 

Anderson,  J.  R.  (1983).  The  architecture  of  cognition.  Cambridge,  MA:  Harvard 
University  Press. 

Bereiter,  C,  &  Scardamalia,  M.   (1987).  The  psychology  of  written 
composition.  Hillsdale,  NJ:  Erlbaum. 

Birnbaum,  J.  C.   (1986).   Reflective  thought:  The  connection  between  reading 
and  writing.  In  B.  T.  Petersen  (Ed.),  Convergences:   Transactions  in  reading 
and  writing  (pp.  30-45).  Urbana,  IL:  National  Council  of  Teachers  of 

Bodgan,  R.  C,  &  Biklen,  S.  K.  (1982).  Qualitative  research  for  education:  An 
introduction  to  theory  and  methods.    Boston,  MA:  Allyn  and  Bacon. 

Carter,  M.   (1988).   Problem  solving  reconsidered:   A  pluralistic  theory  of 
problems.   Collage  English.  50.  551  -565. 

Cocking,  R.  R.  &  Mestre,  J.  (Eds.)  (1988).    Linguistic  and  cultural  influences  on 
learning  mathematics.   Hillsdale,  NJ:  Erlbaum. 




Dougherty,  B.  N.  (1986).  Writing  plans  as  strategies  for  reading,  writing,  and 
revising.  In  B.  T.  Petersen  (Ed.),  Convergences:  Transactions  in  reading  and 
writing  (pp.  82-96).  Urbana,  IL:  National  Council  of  Teachers  of  English. 

Flower,  L.  S.,  &  Hayes,  J.  R.  (1977).  Problem-solving  strategies  and  the 
writing  process.    College  English,  39,  449-461. 

Goldin,  G.  A.  (1979).  Structure  variables  in  problem  solving.  In  G.  A.  Goldin  & 
C.  E.  McClintock  (Eds.),  Task  variables  in  mathematical  problem  splving  (pp. 
103-169).  Columbus,  OH:  ERIC  Clearinghouse  for  Science,  Mathematics  and 
Environmental  Education. 

Jensen,  R.  J.  (1984).  A  multifaceted  instructional  approach  for  developing 
subgoal  generation  skills.    Doctoral  dissertation,  University  of  Georgia. 

Kantowski,  E.  L.   (1975).   Processes  involved  in  mathematical  problem  solving. 
(Doctoral  dissertation,  University  of  Georgia,  1974).  Dissertation 
Abstracts  International.  36.  2734A. 

Kintsch,  W.  (1986).  Learning  from  text.  Cognition  and  Instruction.  3(21.  87- 

Lesh,  R.  (1985).  Conceptual  analyses  of  problem-solving  performance.  In  E.  A. 
Silver  (Ed.),  Teaching  and  learning  mathematical  pmhlem  solving:  Multiple 
research  perspectives  (pp.  309-329).   Hillsdale,  NJ:  Erlbaum. 

Newell,  A.  (1980).  Reasoning,  problem  solving,  and  decision  processes:  The 
problem  space  as  a  fundamental  category.  In  R.  S.  Nickerson  (Ed.), 
Attention  and  performance.  VIII  (pp.  693-718).   Hillsdale,  NJ:  Erlbaum. 

Newell,  A.,  &  Simon,  H.  A.  (1972).  Human  problem  solving.  Englewood  Cliffs, 
NJ:  Prentice-Hall. 

Pimm,  D.  (1987)  Speaking  mathematically.  London:  Routledge  &  Kegan  Paul. 

Simon,  H.  A.,  &  Newell,  A.  (1971).  Human  problem  solving:  The  state  of  the 
theory  in  1970.   American  Psychologist.  26.  145-159. 

Spivey,  N.  N.  (1990).  Transforming  texts:  Constructive  processes  in  reading 
and  writing.   Written  Communication.  7_,  256-287. 

Yackel,  E.  B.  S.  (1985).  Characteristics  of  problem  representation  indicative 
of  understanding  in  mathematics  problem  solving.  (Doctoral  dissertation, 
Purdue  University,  1984).    Dissertation  Abstracts  International  ££,  2021  A. 




Jill  Adlar.  University  of  the  Wltwatarsrand.  S  Africa. 

Action  research  it  a  context  and  a  process  by  which  pnctiting  mathematics  teachers  enrolled  *» 
post  graduate  study  can  explore  end  explain  the  relationship  between  theory  and  practice.  From 
this  starting  point  I  develop  the  argument  that  action  rataarch  It  enhanced  by  teachers'  prior 
angagamant  In  theoretical  debate  on  learning  and  teaching.  In  particular,  key  concepts  in  activity 
theory'  provide  teacher  researcher*  with  useful  tool*  to  explore,  change  and  reflect  on  their 
practice.  Thete  arguments  are  explored  in  thit  paper  through  the  work  of  one  particular 
mathematics  teacher  during  hit  pott  graduate  ttudy.  I  w*  describe  how  hit  understandings  and 
ute  of  activity  theory  both  shape,  and  are  shaped  by.  hit  classroom  practices.  Thit  view  from  the 
teacher-researcher  at  pott  graduate  student  It  complemented  by  my  own  reflections  at  the 
supervisor  of  hit  work.  Through  the  latter,  a  third  thrust  emerges  at  I  become  aware  of  the 
complexity  of  the  questions  provoked  by  the  project  the  limitations  of  activity  theory  and  the 
constraints  on  such  research  within  pott  graduate  ttudy. 


As  an  educational  method  concerned  to  break  the  research-practice  divide,  action 
research  has  spawned  many  different  interpretations  and  practices.  Differences 
can  be  linked  to  the  assumptions  and  interests  underlying  the  projects  (Grundy, 
1987),  and  to  whether  projects  focus  inwards  towards  the  classroom  or  also 
outwards  towards  the  broader  social  structure  (Liston  and  Zeichner,  1990). 
Common  to  all  projects  is  a  concern  with  improvement  of  classroom  practice 
through  the  involvement  of  teechers-as-feseerchers  in  their  own  classrooms. 

Action  research  is  distinguished  from  flood  practice  (what  many  teachers  do 
anyway)  in  that  it  is  systematic,  deliberate  and  open  to  public  scrutiny  (McNiff, 
1988;  Davidoff  and  Van  den  Berg,  1990;  Walker,  1991)  and,  while  enhancing 
reflective  teaching  (Liston  and  Zeichner,  1990),  is  distinct  from  it  in  that  reflective 
teaching,  is  not  always  conscious  (Lerman  and  Scott-Hodges,  1991).  Action 
research  involves  a  continuous  cycle  of  planning,  acting,  observing,  reflecting  and 
replanning  instances  of  classroom  practice.  Through  critical  reflections,  teachers 
not  only  develop  their  practice  but  also  their  theoretical  understanding  of  that 
practice.  Action  research  thus  provides  for  e  constant  interplay  between  theory, 
research  and  practice. 

'    as  developed  in  the  Soviet  school  of  thought. 



Recently  I  have  encouraged  mathematics  teachers  enrolled  for  post  graduate  study 
at  the  Bachelor  of  Education  {B  Ed)  level  at  the  University  of  the  Witwatersrand 
(Wits)  to  turn  the  research  requirements  of  their  degree  towards  small  scale  action 
research  projects.  There  is  no  particular  innovation  in  this:  in-service  education, 
directed  as  it  is  towards  curriculum  change,  has  long  drawn  on  the  methods  of 
action  research.  Currently,  action  research  is  being  explored  for  its  possibilities 
within  pre-service  teacher  education  (Listen  and  Zeichner,  1991)  and  M  Sc  study 
(Lerman  and  Scott-Hodges,  1991).  Their  motivation  is  not  dissimilar  to  the  two 
inter-related  reasons  for  my  advocating  action  research  within  the  B  ED:  (!)  The  B 
Ed  degree  has  a  predominantly  theoretical  thrust.  While  students  are  generally 
excited  by  new  ideas  and  ways  of  looking  at  education,  these  often  have  little 
impact  on  their  educational  practice.  Action  research  can  ground  theory  in 
practice.  <ii)  More  broadly,  practising  teachers  in  South  Africa,  by  and  large, 
remain  alienated  from  educational  research  and  educational  theory.  They  tend  to 
perceive  themselves  as  users  and  not  producers  of  knowledge.  In  particular, 
mathematics  teaching  is  characterised  by  a  rather  slavish  adherence  to  a  prescribed 
syllabus  and  its  related  prescribed  text  book  (Adler,  1991 ).  Action  research  offers 
possibilities  for  shifting  such  curriculum  processes. 


Embedded  in  action  research  is  a  view  of  learning  as  an  active  process.  Teacher- 
researchers  learn  through  action,  and  reflection  on  that  action.  Action  research 
thus  shares  assumptions  about  learning  with  constructivist  theories,  be  they  naive, 
radical  or  'socio -constructionist'  (i.e.  drawing  on  activity  theory)  (Bussi,  1991). 
All  see  knowledge,  not  as  given,  but  as  constructed  through  activity.  It  is  thus 
not  surprising  that  current  research  into  mathematics  teaching  and  learning  from 
constructivist  perspectives  (eg  Bussi,  1991;  Jaworsky,  1991;  Lerman  and  Scott- 
Hodges,  1991)  are  within  an  action  research  tradition  .  While  radical 
constructivists  and  activity  theorists  share  their  rejection  of  knowledge  as 
transmitted,  they  differ  crucially  in  the  importance  they  attach  to  the  social 



mediation  of  learning.  I  do  not  wish  to  debate  the  pros  and  cons  of  these  theories. 
In  fact,  like  Confrey  {1991),  1  would  look  to  'steering  a  course  between  Vygotsky 
and  Piaget".  However,  activity  theory,  with  its  emphasis  on  social  mediation, 
provides  useful  concepts  like  the  Zone  of  Proximal  Development  (ZPD)  and  the 
interpersonal  becoming  the  intrapersonal  with  which  mathematics  teachers  can 
examine  social  interaction  and  its  impact  on  learning  in  their  classrooms. 
The  focus  of  the  rest  of  this  paper  is  the  work  of  one  such  teacher,  Mark2.  Mark's 
action  research  projact(i)  illuminates  the  theory-practice  dialectic  in  the  context  of 
the  B  Ed  course  and  (ii)  provides  particular  insights  into  how  key  concepts  in 
activity  theory  can  be  useful  tools  for  a  teacher  attempting  to  reflect  on, 
understand,  and  change  his  teaching  practices. 


While  enrolled  in  the  B  Ed,  part-time,  during  1990  and  1991,  Mark  was  teaching 

mathematics  in  senior  classes  in  a  middle-class  state  (still  segregated,  whites-only) 

school,  and  bogged  down  yet  again  by  'word  problems'  in  Std  9  (Grade  11). 

During  his  studies,  he  was  inspired  by  the  Piaget  •  Vygotsky  debate  on  cognition. 

After  engaging  with  some  theoretical  extrapolations  from  activity  theory  to 

mathematics  teaching  and  learning,  he  wrote: 

Various  authors,  including  Christiansen  and  Walther  (C  &  W)  (1985),  and 
Mellin-Olsen  (1986)  have  provided  an  elaboration  of  this  theoretical 
framework  (activity  theory)  into  mathematical  learning.  The  relationship 
between  educational  'task  and  activity'  is  analysed  in  detail  by  C  &  W, 
whereas  Mellin-Olsen  locates  activity  within  a  broadar  socio-political 
context.  Givan  that  the  focus  of  this  project  is  specifically  syllabus-related 
in  terms  of  teaching  word-problems  with  less  emphasis  on  socio-political 
problems,  I  will  draw  primarily  on  the  work  of  C  &  W. 

There  were  thus  two  interacting  starting  points  for  Mark's  project:  (i)  a  problem 
identified  in  his  own  classroom,  and,  (ii)  a  desire  to  develop  his  interest  in  activity 
theory  and  knowledge  es  socially  constructed.  He  acknowledged  and  then  rejected 
socio-political  concerns  as  outside  the  scope  of  his  project.  His  selection  from 
elements  of  activity  theory  is  pragmatic  and  clearly  shaped  by  his  understandings 

'    See  Phillips,  M  (1991) 


of  tha  scopa  of  his  project  and  the  constraints  of  his  practice.  Right  from  the 
start,  Mark's  project  is  an  interaction  of  theoretical  and  practical  concerns. 

C  &  W's  product-process  framework  and  their  characterisation  of  'drilling  of 
problem-types'  resonated  with  what  Mark  perceived  as  inadequacies  in  his  (and 
others')  practice.  He  translated  C  &  W's  contrast  of  typical  and  novel  problems 
and  their  argument  for  internalisation  through  learning  in  two  dimensions  (action 
and  reflection)  into  a  series  of  tasks  structured  around  the  solving  of  'word 
problems'  related  to  quadratic  equations.  These  were  to  be  tackled  by  Std  9  pupils 
in  a  socially  interactive  setting  so  as  to  incorporate  key  activity  theory  concepts 
such  as  'mediation  in  the  ZPD'  and  the  'interpsy etiological  becoming  the 
intrapsy  etiological'. 

To  facilitate  both  his  own  and  others'  critical  reflection  and  interpretation  of  his 
strategies  for  changed  practice,  and  because  of  his  focus  on  mediation  and 
interaction,  Mark  tape-recorded  a  group  of  learners,  his  interactions  with  them,  and 
their  interactions  with  each  other.  Mark's  detailed  self-critical  reflections  are  not 
possible  to  reproduce  in  full  here.  Some  of  the  most  significant  are  captured  in  the 
following  extracts  and  descriptions. 

Despite  his  intent  at  establishing  both  pupil-pupii  and  teacher-pupii  interaction 
Mark's  transcript  revealed  that:  'I  did  most  of  the  talking  ...  each  pupil  tended  to 
interact  predominantly  with  me'. 

On  the  question  of  establishing  inter-subjective  meaning  for  the  tasks,  he  noticed 

I  often  asked  the  question  "do  you  all  agree?"  without  actually 
confirming  whether  they  did  ...  ' 


1  -45 

As  regards  his  mediation  of  activity,  close  scrutiny  of  the  transcript  revealed  that: 

'...I  provided  a  great  deal  of  help,  but  my  domination  prevented  pupils  from 
collectively  or  independently  solving  the  problems  without  my  Interference 

More  specifically, 

I  effectively  mediated  Grant's  and  Robert's  activity...  I  failed  to 
effectively  draw  others  into  the  process  of  interaction  ...  I  concentrated  far 
more  on  the  boys  than  on  the  girls  ...' 

And  reflecting  most  critically  on  the  last  point  he  says: 

'A  hidden  assumption  that  the  boys  are  automatically  more  successful  than 
girls  at  solving  word  problems  seemed  to  prevail.' 

Page  4 

Anastasia:  I  don't  understand 

Teacher:  OK,  Robert,  see  if  you  can  explain  to  her 

Page  6 

Teacher:  ...  good  Robert  ...  (and  later) 
Teacher:  Help  Anastasia,  Grant. 

Page  8 

Teacher:  ...that's  great.  Look  at  Grant's  attempt  everybody. 

Anastasia:  Gee  wiz,  kit  hey. 

Teacher:  ...  show  the  others  what  you  did.' 

Examining  who  entered  tasks,  how,  why,  he  observed  that  many  different  methods 
emerged,  revealing  a  'virtue'  of  his  new  approach  in  encouraging  pupils  to  use  their 
own  methods  rather  than  simply  adopting  a  'correct-method'  mentality  and  that: 

"...  Grant  and  Robert  were  the  most  actively  involved  on  the  tasks  and  came 
closest  to  solving  all  the  problems...  this  seems  to  have  a  bearing  on  their 
acceptance  of  the  tasks  and  their  willingness  to  communicate  in  the  group 
setting.. .The  girls  tended  to  be  demotlvated  ...  none  of  the  girls  were  able 
to  effectively  solve  any  of  the  problems  ...  A  reason  for  the  girls' 
demotlvatlon... might  well  be  related  to  my  ineffectual  mediation  ...Another, 
more  subtle  reason,  which  C  &  W  overlook  and  which  Mellin-Olsen 
addresses  more  adequately,  is  that  "iris'  failure  to  solve  the  problems  could 
be  ...  that  they  did  not  adequately  accept  the  tasks  as  part  of  their  own 

ERIC  70 


In  addition,  ha  concludes  that  mora  tima  was  naadad  for  all  to  gat  at  aach  task, 
and  that  in  their  construction,  the  tesks  did  not  fecilitate  or  provoke  sufficient 
pupil-pupil  interaction.  He  draws  his  insights  into  his  conclusions: 

'Although  there  were  flaws  in  the  tasks  ...  the  approach  adopted  was  by  far 
an  improvement ...  Pupils  were  more  positive  ...  and  I  was  provided  some 
interesting  "revelations"  about  my  own  teaching  of  which  I  was  not  aware. 
My  mediation  was  unwittingly  sexist  ...  I  dominated  the  activities  ...  With 
these  insights  ...  I  can  further  improve  my  teaching  approach  so  that  more 
pupils  will  benefit  in  the  future 

and  finally, 

'By  researching  my  own  methods  of  teaching  word  problems  in  the  context 
of  'task  and  activity',  I  have  been  able  to  provide  an  example  of  how 
research  and  practice  can  be  integrated  so  teaching  is  enhanced  ...  the 
teaching  process  is  truly  a  continual  research  one.' 


There  is  no  doubt  that  Mark  gained  tremendous  insight  into  his  practice.  In  his 
conscious  attempt  at  preparing  for  socially  interactive  learning,  he  came  to  see  that 
his  practices  were  such  that  not  only  did  he  dominate  classroom  interaction,  but 
his  mediation  was  exclusive  (focused  on  only  two  pupils)  in  general  and  gendered 
in  particular.  Without  some  systematic  method  of  observing  his  mediating 
processes,  -  he  would  still  be  unaware  of  how  much  he  actually  talks  in  class  and 
to  whom.  This  specific  project,  therefore,  speaks  volumes  of  the  powerful  impact 
of  action  research  with  an  activity  theory  framework.  However,  the  weakness  of 
activity  theory  in  relation  to  the  'process  of  internalisation  of  collective  activity  and 
the  conditions  of  its  functioning  within  the  ZPD'  (Bussi,  1991)  is  reflected  In 
Mark's  focus.  His  emphasis  on  the  gender  dimensions  of  the  process  of 
internalisation  of  collective  activity  fails  to  open  up  the  difficulties  attached  to 
mediating  a  whole  group  all  at  once.  What  does  the  ZPD  mean  in  whole  class 
interaction?  Commenting  on  this,  Bussi  (1991)  notes  that  this  is  still  an  'open 


1  -47 

problem  in  activity  theory.. .(and)  mathematics  classrooms  are  suitable  settings  for 
further  research.' 

Mark's  observations  and  reflections  are  explicitly  theory-laden.  The  structure  of 
his  analysis  fits  the  framework  he  established  for  his  research.  His  discussion 
foregrounds  tasks,  mediation  and  interaction  and,  through  this  focus,  Mark  is  able 
to  develop  and  share  detailed  insight  into  with  whom  he  interacted,  how  and  to 
what  effect.  However,  his  failure  to  mediate  all  pupils,  and  in  particular  the 
demotivation  of  the  girls  in  the  class,  required  a  reconsideration  of  his  theoretical 
framework.  C  &  W  do  discuss  goal-directed  activity,  but  (as  Mark  says  above) 
this  does  not  address  the  gendered  outcome  of  his  teaching.  Gendered  practices 
need  to  be  interpreted  in  relation  to  wider  social  practices  and  Mark  is  pushed  to 
reconsider  the  worth  of  Mellin-Olsen's  location  of  activity  in  a  socio-political 
context  i.e.  to  alter  and  expand  his  initial  theoretical  frame.  The  interaction  of 
theory  and  practice  evident  at  the  beginning  is  thus  just  as  evident  at  the  end  of 
Mark's  project. 

An  important  question  at  this  point,  both  for  Mark  and  for  teacher-educators,  is 
how  to  sustain  the  'symbiotic  relationship  between  teacher  as  theory  maker  and 
teacher  as  developer  of  practice'  (Jaworsky,  1991)  outside  of  the  supporting 
structure  of  the  B  Ed  degree?  The  need  for  strategies  such  as  support  networks 
for  past  B  Ed  students  becomes  important  if  the  gains  made  by  Mark  during  his 
formal  study  are  to  be  consolidated  and  developed.  Within  such  networks,  action 
research  as  the  structured  and  rigorous  activity  described  in  my  paper,  can  become 
a  continuous  part  of  a  teacher's  reflection  on  their  practice. 

The  more  serious  challenge,  however,  lies  in  linking  issues  such  as  gender  bias  in 
the  maths  classroom  to  deeply  rooted  social  practices.  Once  this  link  becomes 
clearer,  as  it  did  to  Mark,  the  solution  to  the  problem  becomes  less  obvious.  Mark 
will  be  able  to  draw  from  a  large  and  growing  body  of  literature  on  gender  in 
mathematics  education.  The  experience  gained  and  the  analytical  and 
methodological  tools  developed  in  the  research  component  of  his  B  Ed  should 
enable  him  to  act,  reflect  and  deal  creatively  with  these  issues.  But  whether  and 


how  this  rational  method  of  noticing,  analysing  and  acting  will  gat  at  deeply  seated 
social  practices,  e.g.  gender,  remains  a  question. 


Adler,  J  (1991)  'Vision  and  constraint:  politics  and  mathematics  national  curricula 
in  a  changing  South  Africa'  in  Pimm,  D  and  Love,  E  (eds)  The  Tucking 
and  Learning  of  School  Mathematics  Hodder  and  Stoughton.  London 

Bussi,  M  B  (1991)  'Social  interaction  and  mathematical  knowledge'  in 
Proceedings  of  the  Fifteenth  Conference  of  the  International  Group  for  the 
Psychology  of  Mathematics  Education  (PME15).  Assisi.  Vol  1.  1-16. 

Christiansen,  B  and  Walther,  6  (Eds)  (1985)  Perspectives  on  Mathematics 
Education.  Reidel.  Dordrecht.  Chapter  7. 

Confrey,  J  (1991)  'Steering  a  course  between  Vygotsky  and  Piaget'  in 
Educational  Researcher.  November. 

Davidoff,  S  and  Van  den  berg,  0  (1990  Changing  your  teaching:  the  challenge 
of  the  classroom.  Centaur  with  UWC.  Cape  Town. 

Grundy,  S  (1987)  Curriculum:  Product  or  Praxis.  Falmer  Press.  Lewes. 

Jaworsky,  B  (1991)  'Some  implications  of  a  constructivist  philosophy  for  the 
teacher  of  mathematics'  in  Proceedings  of  PME15  Assisi.  Vol  II.  213  - 

Lerman,  S  and  Scott-Hodges,  R  (1991)  'Critical  incidents  in  classroom  learning  - 

their  role  in  developing  reflective  practitioners'  in  Proceedings  of  PME15. 

Assisi.  Vol  II.  293  -  299. 
Liston,  D  P  and  Zeichner,  K  M  (1990)  'Reflective  teaching  and  action  research 

in  preservice  teacher  education'  in  Journal  of  Education  for  Teaching.  16. 


McNiff,  J  (1988)  Action  Research:  Principles  and  Practice.  Macmillan.  London. 
Mellin-Olsen,  S  (1986)  The  Politics  of  Mathematics  Education.  Reidel. 

Phillips,  M  (1991)  Teaching  word  problems  through  'task  and  activity'  •  an 

integration  of  research  and  practice.  Unpublished  B  Ed  Special  Project. 

University  of  the  Witwatersrand.  Johannesburg. 
Walker,  M  (1991)  Reflective  Practitioners  -  A  Case-Study  in  Facilitating  Teacher 

Development  in  Four  African  Primary  Schools  in  Cape  Town. 

Unpublished  PhD  thesis.  University  of  Cape  Town. 

o  73  - 


1  -49 


M.A.  VallecMos;  M.  C  Batanero  and  J.  D.  Godino 
University  of  Granada  (Spain) 


In  this  paper  the  initial  results  of  a  theoretical-  experimental  study 
of  university  students'  errors  on  the  level  of  significance  of 
statistical  test  are  presented.  The  "a  priori"  analysis  of  the  concept 
serves  as  the  base  to  elaborate  a  questionnaire  that  has  permitted  the 
detection  of  faults  in  the  understanding  of  the  same  in  university 
students,  and  to  categorize  these  errors,  as  a  first  step  In 
determining  the  acts  of  understanding  relative  to  this  concept. 


One  of  the  key  aspects  in  the  learning  of  the  test  of  hypothesis,  is  the  concept 
of  the  level  of  significance,  which  is  defined  as  the  "probability  of  rejecting  a 
null  hypothesis,  when  it  is  true".  Falk  (1986)  points  out  the  change  of  the 
conditional  and  the  conditioned  as  a  frequent  error  in  this  definition  and  the 
mistaken  interpretation  of  the  level  of  significance  as  "the  probability  that  the 
null  hypothesis  is  true,  once  the  decision  to  reject  it  has  been  taken".  Likewise, 
White  (1980)  describes  several  errors  related  to  the  belief  of  conservation  of  the 
significance  level  value  a,  when  successive  tests  of  hypothesis  are  carried  out  on 
the  same  set  of  data,  that  is,  relative  to  the  so  called  "problem  of  the  multiple 

In  this  paper  the  concept  of  level  of  significance  in  a  test  of  hypothesis  is 
analyzed,  determining  different  aspects  related  to  its  correct  understanding.  The 
analysis  of  the  components  of  the  meaning  of  mathematical  concepts  and  procedures 
should  constitute  a  previous  phase  to  the  experimental  study  of  students' 
difficulties  and  errors  on  the  said  objects.  The  study  of  the  interconnections 
between  the  concepts  enables  us  to  know  their  degree  of  complexity  and  to  determine 
the  essential  aspects  that  should  be  pointed  out  to  achieve  a  relational  learning  and 
not  merely  an  instrumental  learning  of  the  same  (Skemp,  1976). 

Thl.  report  formi  p«rt  of  the  Project  PS°0-0Z4b.  DCICYT.  kUdrld 

er|c  74 


Likewise,  we  describe  the  results  of  an  exploratory  study  carried  out  on  a 
sample  of  35  students,  that  shows  the  existence  of  misconceptions  related  to  each  one 
of  the  aspects  identified  In  the  conceptual  analysis.  The  errors  of  the  students  when 
faced  with  specific  tasks  indicate  faults  in  the  understanding  of  the  concepts  and 
procedures,  and  so,  this  analysis  "should  be  considered  a  promising  researching 
strategy  for  clarifying  some  fundamental  questions  of  mathematics  learning"  (Radatz, 
1980,  p.  16).  •  • 


In  the  classical  theory  (see  for  example,  Zacks,   (1981))  a  parametric  test  of 

hypothesis     is    a    statistical    procedure    of    decision    between    one    of    the  two 

■com;. 'ementary  hypothesis  Ho  and  H  ,  hypothesis  that  refer  to  the  unknown  value  of  a 

population  parameter,  starting  from  the  observation  of  a  sample.  To  carry  this  out,  a 

statistic  Vtx)  whose  distribution  is  known     in  terms  of  the  value  of  the  parameter, 

is  used.   The  set  of  possible  values  of  the  statistic,   supposing  that  the  hypothesis 

Ho  is  verified,   is  divided  into  two  complementary  regions,  acceptance  region  A  and 

critical  region  C,   in  such  a  way  that  having  observed  the  particular  value  of  the 

statistic  in  the  sample  we  decide  to  accept  the  HQ  hypothesis  if  this  value  belongs 

to  the  region  A  and  reject  it  if  it  belongs  to  C.  We  will  only  consider  the  case  H  : 


0c8o  of  simple  null  hypothesis,  to  facilitate  the  discussion. 

The  application  of  a  test  can  give  rise  to  two  different  types  of  errors:  to 
reject  the  hypothesis  Ho  when  it  is  true  (type  I  error)  and  to  accept  it  when  it  is 
false  (type  II  error).  Although  we  cannot  know  whether  we  have  committed  one  of 
these  errors  in  a  particular  case,  we  can  determine  the  probability  of  type  I  error 
as  a  function  of  the  value  of  the  parameter,  that  is  called  the  power  function  of  the 

Power  («)  =  P  (Rejecting  HJ0) 

In  the  case  of  0=0q  ,  we  obtain  the  probability  of  rejecting  HQ  with  the  chosen 
criteria,  supposing  that  is  true,  the  so-  called  probability  of  type  I  error,  or 
level  of  significance  a  of  the  test: 

a  =  P  (Rejecting  HQ         =  P  (Rejecting  H  |H    is  true) 

The  contrary  event  of  rejecting  the  hypothesis  HQ  consists  of  accepting  it  and 
its  probability  can  also  be  expressed  as  a  functloi  of  the  parameter: 




0(0)    =  P  (Accepting  HjO) 

In  this  case  and  whenever  «  is  different  from  the  supposed  value  <>o,  a  type  II 
error  is  being  committed.  As  we  can  see.  in  the  case  of  a  simple  null  hypothesis, 
while  the  type  I  error  has  a  constant  probability,  the  probability  of  type  11  error 
is  a  function  of  the  unknown  parameter.  Finally,  and  taking  into  account  that  the 
events  to  accept  and  to  reject  the  null  hypothesis  are  complementary,  we  see  that  the 
relationships  between  these  probabilities  are  given  by  the  following  expression: 

o=  l-f3(«o) 

In  the  understanding  of  the  idea  of  level  of  significance,  we  can  as  a  result  of 
this,  distinguish  four  differentiated  aspects,  that  we  have  used  in  the  elaboration 
of  a  questionnaire  that  enables  us  to  identify  and  classify  the  misconceptions 
related  to  this  understanding.  This  classification  constitutes  a  first  step  in  the 
categorization  of  the  acts  of  understanding  of  synthesis  of  the  said  concept 
(Sierpinska,  1990),  that  would  be  added  to  the  acts  of  identification,  discrimination 
and  generalisation  of  the  objects  that  intervene  in  its  definition.  The;,e  aspects  are 
the  following: 

a)  The  test  of  hypothesis  as  aproblem  of  decision:  between  two  excluding  and 
complementary  hypothesis,  with  the  possible  consequences  of  committing  or  not  one  of 
the  types  of  error  that  are  incompatible  but  not  complementary  events. 

b)  Probabilities  of  error  and  relation  between  them:  understanding  of  the  conditional 
probabilities  that  intervene  in  the  definition  of  o  and  0,  of  the  dependence  of  f3  in 
terms  of  the  unknown  value  «  of  the  parameter,  and  of  the  relation  between  o  and  f3. 

c)  Level  of  significance  as  the  risk  of  the  decision  maker:, 

The  values  o  and  |3  determine  the  risks  that  the  decision  maker  is  willing  to 
assume  in  his  decision  and  will  serve,  along  with  the  hypothesis,  for  the  adoption  of 
the  decision  criteria. 

d)  Level  of  significance  and  distribution  of  the  statistic^  interpretation,  of  a 
significant  result: 


1  -52 

The  level  of  significance;  is  the  probability  that  the  statistic  chosen  as  a 
decision  function  takes  a  value  in  the  critical  region,  in  the  case  that  the  null 
hypothesis  is  true.  Obtaining  a  significant  result  leads  to  the  rejection  of  the  null 
hypothesis,  although  this  does  not  necessarily  imply  the  practical  relevance  of  this 


Description  of  the  sample 

Tht  study  was  carried  out  on  a  group  of  35  students  studying  Statistics  in  their 
2nd  year  of  Civil  Engineering  in  the  University  of  Granada.  Seventy  five  per  cent  of 
these  students  had  not  studied  Statistics  or  Probability  before  and  the  rest  had  only 
studied  it  in  some  previous  course.  These  students  had  studied  Infinitesimal  Calculus 
and  Algebra  in  their  first  year  of  studies,  so  they  can  be  considered  to  have  an 
excellent  previous  mathematical  base.  The  subject  of  Statistics,  which  includes  the 
basis  of  descriptive  statistics,  probability  theory  and  inference  has  been  given 
three  hours  per  week  throughout  a  whole  course,  and  the  test  having  been  carried  out 
at  the  end  of  the  same. 

Questionnaire  used. 

The  questionnaire  used  consists   of  20  questions,   and   was   elaborated   by  the 

authors    to    study    conceptual    difficulties  of    the    test    of    hypothesis.    Due    to  the 

limitations  of  space  we  will  only  present  the  results  obtained  in  four  of  the  items, 

whose   distracters    have   been   chosen    by  trying   to    detect   errors    in   the   acts  of 

understanding  of  synthesis  referred  to  the  level  of  significance.  These  items  are  the 

This  item  asks  about  the  possibility  that  the  two  types  of  error  can  occur 
simultaneously.  Since  by  carrying  out  a  test  of  hypothesis  we  have  a  problem  of 
decision,  the  null  and  alternative  hypothesis  are  complementary  like  the  events  of 
accepting  and  rejecting  the  null  hypothesis.  However,  the  events  of  committing  type  I 

ITEM  1: 

A:  I; 

The  probability  of  committing  both  type  I  and  type  II  errors  In  a  test  of  hypothesis  It: 
B:  0;         C:  a  ♦  b;  D:  ths  product  ab  ,  since  th«  errors  are  Indspsndsnt 

1  -53 

error  or  type  II  error  are  incompatible  but  not  complementary. 

ITEM  ?: 

A  scientist  always  chooses  to  use  0.05  as  the  level  of  significance  in  his  experiments. 
This  means  that  in  the  lone  run: 

A:  S  X  of  the  times  he  will  reject  the  null  hypothesis. 

B:  5  X  of  tha  times  that  he  rejects  the  null  hypothesis  he  will  have  made  a  mistake. 
C:  He  will  have  mistakenly  rejected  the  null  hypothesis  only  5  X  of  his  experiments. 
D:  He  will  have  accepted  a  false  null  hypothesis  95  X  of  the  times. 

In  this  item  the  definition  of  the  level  of  significance  appears  as  a 
conditional  probability  and  the  distracters  refer  to  the  incorrect  interpretation  of 
the  same.  In  particular,  in  the  classical  inference,  it  is  not  possible  to  know  the 
probability  of  having  committed  one  of  the  types  of  error  once  the  decision  has  been 
taken,  although  we  can  know  the  probabilities  of  type  I  or  I)  error  "a  priori".  That 
is,  although  we  cannot  perform  an  inductive  inference  about  the  probability  of  the 
hypothesis  referring  to  the  population,  once  the  particular  sample  has  been  observed, 
we  are  able  to  make  a  deducive  inference  from  the  population  of  possible  samples  to 
the  sample  that  is  going  to  be  obtained  before  having  extracted  it  (Rivadulla,  1991). 

When  we  chance  from  a  level  of  sltnlflcance  of  0.01  to  one  of  0.05  we  have: 
A:  Less  risk  of  type  I  error. 
B:  More  risk  of  type  I  error. 
C:  Less  risk  of  type  II  error. 
D:  Both  B  and  C. 

In  this  item  we  study  the  interpretation  of  the  level  of  significance  as  a  risk 
of  error  as  well  as  the  relationship  between  the  probabilities  a  and  ft,  which  implies 
that  it  is  not  possible  to  simultaneously  reduce  the  two  risks,  when  the  sample  size 
has  been  fixed. 

ITEM  4: 

What  can  be  concluded  If  the  result  In  a  test  of  hypothesis  Is  significant?: 
A:  The  result  Is  very  Interesting,  from  the  practical  point  of  view. 
B:  A  mistake  has  been  made. 

C:  The  alternative  hypotheslt  Is  probably  correct. 
D:  The  null  hypothesis  Is  probably  correct. 

The  level  of  significance  determines  the  critical  and  the  acceptance  regions  of 
a  test,  together  with  the  null  and  alternative  hypothesis  and  the  test  statistic.  The 
problem  of   carrying  out   a   test   of  hypothesis  has  been  transformed   into  that  of 

ITEM  3: 



1  -54 

dividing  the  population  of  possible  samples  in  two  complementary  subsets:  those  who 
provide  evidence  in  favour  or  against  the  null  hypothesis.  So.  the  level  of 
significance  is  the  probability  that  the  statistic  take  a  value  in  the  critical 
region.  One  statistically  significant  result  does  not  necessarily  imply  the 
significance  (relevance)  from  a  practical  point  of  view. 


The  frequencies  and  percentages  of  responses  to  the  different  items  are 
presented  in  Table  1.  The  relative  difficulty  of  the  same  have  been  quite 
homogeneous,  although  somewhat  higher  in  item  4  which  refers  to  the  interpretation  of 
results  and  the  difference  between  statistical  and  practical  significance.  From  the 
analysis  of  the  distracters  that  have  been  chosen  by  the  students  in  the  different 
items,  we  obtain  a  first  information  about  the  conceptual  errors,  that  we  classify  in 
accordance  with  the  previous  conceptual  analysis,  in  four  sections: 

Table  I 

Frequencies  (and  percentages)  of  responses  in  the  items 

Item  A(7.)  B(7.)  C(7.)         D(7.)     R.  Correct 

1  5  (14.3)  *16  (45.7)  5  (14.3)  6  (17.1)  16  (45.7) 

2  4  (11.4)  9  (25.7)  «17  (48.6)  4  (11.4)  17  (48.6) 

3  4  (11.4)  12  (34.3)  I  (  2.9)  "16  (  45.7)  16  (  45.7) 

4  6  (17.1)  2  (  5.7)  "12  (34.3)  14  (40.0)  12  (  34.3) 

•  Correct  option. 

Misconceptions  in  the  identification  of  a  test  of  hypothesis  as  a  problem  of 

-  Consideration  of  the  type  I  and  II  errors  as  complementary  events  that  are 
shown  in  the  four  responses  to  distracter  D  of  item  2  and  in  the  5  responses  to 
distracter  A  of  item  1. 

-  Errors  type  1  and  II  are  not  perceived  as  incompatible  events.  (5  responses  to 
distracter  C  and  6  to  D  of  item  1). 


Misconceptions    in    the    interpretation    fif    the    probabilities    of    error.    ajjd.  tfiejr 

-  Confusion  of  the  two  following  conditional  probabilities  in  the  definition  of 
the  level  of  significance 

a  =  P  (reject  H  |H0true)  and   a  =  P  (HQ  true  |  HQ  has  been  rejected) 

shown  by  the  9  responses  to  dlstracter  B  of  item  2,  that  is  the  error  mentioned  in 

Falk's  research  (1986). 

-  Interpretation  of  a  as  P  (reject  in,  that  is  to  say  ,  the  suppression  of  the 
condition  in  the  conditional  probability,  in  the  4  responses  to  distracter  A  of  item 

-  Not  to  take  into  account  the  relationship  between  the  probabilities  of  type  i 
and  II  error  (12  responses  to  distracter  B  of  item  3). 

Misconceptions  in  the  interpretation  of  the  level  of  significance  as  the  rjsjc  of  the 
decision  maker: 

-  A  higher  level  of  significance  gives  less  probability  of  type  I  error.  (4 
responses  to  distracter  A  of  item  3). 

-  By  changing  the  level  of  significance  the  risk  of  type  1  error  does  not  change 
(1  case,  in  distracter  C  of  item  3). 

Misconceptions  in  the  interpretation  of  a  significant  result: 

-  A  statistical  significant  result  is  also  significant  from  a  practical  point  of 
view,  (6  responses  to  distracter    A  of  item  4). 

-  Since  the  level  of  significance  is  a  very  small  value  of  a  probability,  it  is 
associated  with  an  incorrect  result  (2  responses  to  distracter  B  of  item  4). 

-  Confusion  of  the  significant  result  as  one  that  corroborates  the  null 
hypothesis,  this  is  confusion  of  the  critical  and  acceptance  regions  (14  responses  to 
distracter  D  of  item  4). 


In  the  analysis  of  the  responses  to  the  questions  put  forward,  the  existence  of 

o  SO 



a  great  diversity  of  misconceptions  has  been  shown  in  the  interpretation  given  by  the 
students  of  the  sample  to  the  concept  of  the  level  of  significance,  thus  completing 
the  results  of  Falk  (1986)  and  White's  (1980)  research.  Although  in  an  exploratory 
way,  this  study  constitutes  a  first  step  towards  the  search  for  the  structure  of  the 
components  of  the  meaning  of  the  test  of  hypothesis  and  the  identification  of 
obstacles  in  the  learning  (Brousseau  (1983),  that  without  doubt  can  contribute  to  an 
improvement  in  the  teaching  and  application  of  statistical  methods. 


BROUSSEAU,    G.    (1983).    Les   obstacles    epistemologiques     et     les     problemes  en 
mathematiques.  Recherches  en  didactiques  des  mathematiques.  Vol  4,  n.2, 

FALK,  R.  (1986).  Misconceptions  of  Statistical  Significance.    Journal    of  Structural 
Learning.  Vol.  9, .  pp.  83-96. 

RADATZ,  H.  (1980).  Students'  errors  in  the  mathematical  learning  process:    a  survey. 
For  the  learning  of  mathematics,  I,  t,  p.  16-20. 

RIVADULLA,  A.  (1991).  Probabilidad  e  inferencia  cientlfica.  Barcelona:  Anthropos. 

SIERPINSKA,  A.  (1990).  Some  remarks  on  understandig  in  mathematics.  For  the  learning 
of  mathematics,  10,  3,  p.  24-36. 

SKEMP,  R.  (1976).  Relational  and  instrumental  understandig.  Mathematics  teaching,  77. 

WHITE,  A.  L.  (1980).  Avoiding  Errors  in  Educational  Research.  En  Richard  J. 

Shu'mmway  (Ed.).  Research  in  Mathematics  Education,  pp.  47-65.  The  National  Council  of 

Teachers  of  Mathematics,  Inc. 

ZACKS,  S.  (1981).  Parametric  statistical  inference.  Oxford,  U.K.:  Pergamon  Press. 


Students'  Cognitive  Construction  of  Squares  and  Rectangles  in  Logo  Geometry- 

VMrtT.  Battista  Potiflas  H.  ClcmcPis 

Kent  State  University  State  University  of  New  York  at  Buffalo 

It  has  been  argued  that  appropriate  Logo  activities  can  help  students  attain  higher  levels  of  geometric 
thought.  The  argument  suggests  that  as  students  construct  figures  such  as  quadrilaterals  in  Logo,  they  will 
analyze  the  visual  aspects  of  these  figures  and  how  their  component  parts  are  put  together,  encouraging  the 
transition  from  thinking  of  figures  as  visual  wholes  to  thinking  of  them  in  terms  of  their  properties. 
Research  has  demonstrated  that  this  theoretical  prediction  is  sound;  appropriate  use  of  Logo  helps  students 
begin  to  make  the  transition  from  van  Hiele's  visual  to  the  descriptive/analytic  level  of  thought  (Battista  & 
Clements.  1988b;  1990;  Clements  &  Battista,  1989;  1990;  in  press).  The  current  report  will  extend  the 
previous  findings  by  giving  a  detailed  account  of  how  students'  Logo  explorations  can  encourage  them  to 
construct  the  concepts  of  squares  and  rectangles  and"  the  relationship  between  these  two. 

The  Instructional  Setting 
Students  (n  =  656)  worked  with  activities  from  Logo  Geometry  (Battista  &  Clements,  1988a; 
Battista  &  Clements,  1991a;  Clements  &  Battista.  1991).  which  was  designed  to  help  students  construct 
geometric  ideas  out  of  their  spatial  intuitions.  Control  students  (644)  worked  with  their  regular  geometry 
curriculum.  After  introductory  path  activities  (e.g..  walking  paths,  creating  Logo  paths),  students  engaged 
in  off-  and  on-computer  activities  exploring  squares  and  rectangles,  including  identifying  them  in  the 
environment,  writing  Logo  procedures  to  draw  them,  ard  drawing  figures  with  these  procedures  (Hg.  1). 

Figure  1.  "Rectangle:  What  can  you  draw?" 
Students  are  asked  to  determine  if  each  figure  could 
or  could  not  be  drawn  with  a  Logo  rectangle 
procedure  with  inputs  and  to  explain  their  findings. 
They  are  permitted  to  turn  the  turtle  before  they 
draw  a  figure.  From  Logo  Geometry. 

Data  came  from  two  sources-case  studies  and  relevant  items  from  the  Geometry  Achievement 
Test  that  was  administered  to  all  students  involved  in  the  Logo  Geometry  project  (Clements  &  Battista. 
1991 )  None  of  the  items  from  this  test  were  related  to  Logo.  The  case  studies  were  conducted  by  the 
authors,  who  observed  and  videotaped  four  pairs  of  students  from  grades  K,  2.  and  5  (two  pairs)  every 
day  they  worked  on  the  materials. 

do  nw  necessity  reflect  the  views  of  the  National  Science  Foundation. 

o  82 


1  -58 


Paper-and-Pencil  Items 

The  first  item,  adapted  from  (Burger  &  Shaughnessy,  1986), 
asked  students  to  identify  rectangles.  One  point  was  given  for  each 
correct  identification.  Note  that  good  performance  on  this  item  requires 
knowledge  not  only  of  the  properties  of  rectangles,  but  of  the  fact  that 
squares  are  rectangles. 

Directions:  Write  the  numbers  of  all  the  figures  below  that  are  rectangles. 

There  was  a  significant  treatment  by  time  interaction  (E(l.  1030)  =  21.83,  e  <  .001).  The  Logo 
posttest  score  was  higher  than  all  other  scores  (Logo  pretest,  control  pretest  and  posttest)  and  the  control 
posttest  scores  were  higher  than  both  pretest  scores  (p.  <  .01).  It  is  noteworthy  that  Logo  students 
showed  dramatic  growth  between  the  pre-  and  post-tests  for  the  squares  (shapes  2  &  7).  Control  groups 
also  showed  growth  on  these  items,  but  nowhere  near  as  strong  as  did  the  Logo  groups.  For  both  of  these 
shapes  the  most  striking  growth  occurred  in  grades  4, 5,  and  6.  This  may  be  due  to  students'  increased 
knowledge  of  the  properties  of  shapes  or  to  the  thinking  engendered  by  the  "Rectangles:  What  Can  You 
Draw?"  activity  and  class  discussion.  It  is  also  relevant  that  the  Logo  group  outperformed  the  control 
group  on  the  parallelogram  items;  therefore,  there  was  little  indication  that  the  students  were  simply 
overgeneralizing  all  quadrilaterals  as  rectangles.  In  a  similar  vein,  students  were  asked  on  a  separate  item 
to  identify  all  the  squares  in  the  same  group  of  figures.  Logo  students  significantly  outperformed  control 
students.  There  was  no  indication  of  an  overgeneralization  that  "all  rectangles  are  squares." 

On  another  paper-and-pencil  item,  students  were  asked  which  geometric  properties  applied  to 
squares  and  rectangles.  Logo  Geometry  students  improved  more  than  control  students.  Thus,  students- 
knowledge  ot  properties  was  increased  by  work  with  Logo  Geometry.  According  to  the  van  Hiele  theory, 
this  lays  groundwork  for  later  hierarchical  classification. 

The  increased  attention  that  Logo  students  gave  to  properties,  however,  sometimes  made  it  seem 
like  their  performance  declined  compared  to  control  students.  First,  Logo  more  than  control  students 
claimed  that  rectangles  have  two  long  sides  and  two  short  sides.  While  not  mathematically  correct,  this 
response  indicates  an  increased  attention  to  properties  of  figures— students  consider  it  to  be  a  property  of 
rectangles.  Logo  students  also  claimed  more  often  that  rectangles  had  "four  equal  sides,"  possibly  an 
overgeneralization  from  squares  to  rectangles  or  a  misinterpretation  of  what  the  property  states. 

On  the  other  hand,  Logo  students  learned  to  apply  the  property  "opposite  sides  equal"  to  the  class 
of  squares  in  much  greater  numbers  than  control  students.  Logo  instruction  may  have  helped  students 
understand  that  the  property  "opposite  sides  equal"  is  not  inconsistent  with  the  property  "all  sides  equal  in  I 
length.''  Most  students  could  apply  both  properties  to  the  class  of  squares,  demonstrating  flexible 

er|c  83 

1  -59 

consideration  of  multiple  properties  that  may  help  lay  the  groundwork  for  hierarchical  classification. 

In  conclusion,  Logo  explorations  helped  students  move  toward  van  Hiele  level  2  by  focusing  their 
attention  on  properties  of  figures.  Explorations  of  relationships  between  shapes  might  have  provided 
important  precursors  for  hierarchical  classification.  Indeed,  on  the  paper-and-pencil  task  in  which  students 
identified  rectangles,  the  Logo  Geometry  group  showed  a  strong  increase  in  the  frequency  of  identifying 
squares  as  rectangles,  compared  to  the  control  group.  This  effect  was  particularly  strong  in  the 
intermediate  grades.  No  evidence  of  overgeneralization  was  found  (e.g.,  that  "all  rectangles  arc  squares"). 

There  are  several  possible  reasons  why  Logo  Geometry  instruction  helped  substantial  numbers  of 
students  to  identify  figures  consistent  with  the  hierarchical  relationship.  First,  when  Logo  Geometry 
students  succeeded  in  identifying  squares  as  rectangles  on  the  rectangles  item,  they  could  have  done  so  by 
asking  themselves  if  a  "Rectangle"  procedure  could  have  drawn  each  of  ihe  given  shapes.  Second, 
increased  knowledge  of  properties  of  shapes  and  movement  toward  level  2  thinking  may  have  enabled 
students  to  see  squares  as  rectangles  because  squares  have  all  the  properties  of  rectangles.  Class 
discussions  of  the  classification  issue  may  have  suggested  to  students  that  squares  should  be  classified  as 
rectangles.  Some  students  may  have  simply  accepted  this  as  a  fact  to  be  remembered.  As  we  will  see 
below,  others  made  sense  of  this  notion  by  using  visual  transformations. 

Case  Studies 

What  is  the  basis  for  students'  classifications  of  shapes? 

Most  of  the  students  fell  into  either  level  1  (visual),  level  2  (property-based),  or  the  transition 
between  the  two  levels  in  the  van  Hiele  hierarchy.  The  two  student  responses  below  illustrate  these  levels 
of  thinking  when  classifying  figures  as  squares.  A  second  grader  was  examining  her  attempt  to  draw  a 
tilted  square  in  Logo.  Although  not  really  a  square,  she  reasoned  as  follows: 
Int:  How  do  you  know  its  a  square  for 

M:  It's  in  a  tilt.  But  it's  a  square  because  if  you  turned  it  this  way  it  would  be  a  square. 

M  does  not  refer  to  properties  in  making  her  decision;  it  is  sufficient  that  it  looks  like  a  square. 

Contrast  this  visual  response  with  that  of  two  fifth  graders  who  had  drawn  a  tilted  square. 

Im:  Is  it  a  square? 

Ss:  Yes.  a  sideways  square.  [InL  How  do  you  know?)  It  has  equal  edges  and  equal  turns. 

So,  what  criteria  do  students  use  to  judge  whether  a  figure  is  a  square?  M  was  operating  at  the 
visual  level:  a  figure  is  a  square  if  it  looked  like  or  could  be  made  to  look  like  a  square.  The  5th  graders 
required  a  figure  to  possess  the  properties  of  a  square,  demonstrating  level  2  thinking  in  this  instance. 
Squares  as  rectangles 

Students  also  dealt  with  the  relationship  between  squares  and  rectangles  in  different  ways.  The 
first  example  of  a  kindergarten  student  illustrates  an  unsophisticated  visual  approach  to  judging  the  identity 


1  -60 

of  shapes.  Chris  is  using  the  Logo  Geometry  "Shape"  command  to  draw  Figures  of  various  sires.  He 
types  S  (for  Shape),  then  types  the  first  letter  of  the  shape  he  wants  (e.g.,  S  for  Square,  from  a  menu), 
and  finally  receives  a  prompt  to  type  a  number  for  the  length  of  each  side  of  the  shape.  After  first  being 
puzzled  that  pressing  R  for  rectangle  required  two  numbers  as  inputs,  Chris  enters  two  5s. 

Inl:  Now  what  do  the  two  5s  mean  for  the  rectangle? 

Chris:  1  don't  know,  now!  Maybe  I'll  name  this  a  square  rectangle! 

Inu  That  looks  like  a  square. 

Chris:  It's  both. 

Int:  How  can  it  be  both? 

Chris:  'Cause  S  and  5  will  make  a  square. 

Inl:  But  how  do  you  know  it  is  still  a  rectangle  then? 

Chris:  'Cause  these  look  a  little  longer  and  these  look  a  little  shorter. 

Int:  Would  this  square  (drawing  a  square  with  Logo)  also  be  a  rectangle,  or  not? 

Chris:  No. 

Int:  Even  though  I  made  it  with  the  rectangle  command? 

Chris:  It  would  be  a  rectangle  square. 

Even  though  Chris  uses  a  terminology  ("square  rectangle")  that  suggests  that  he  might  be  thinking 
of  a  square  as  a  special  kind  of  rectangle,  his  response  of  "No"  indicates  that  he  is  not  making  a 
hierarchical  classification.  He  also  judges  the  figure  to  be  a  rectangle,  not  because  it  was  made  by  the 
rectangle  procedure,  but  because  of  the  way  the  sides  "look." 

Int:  So  is  a  square  a  special  kind  of  rectangle? 

Chris:  Yeah,  if  you  pushed  both  numbers  the  same. 

Int:  How  about  10  on  two  sides  and  9  on  the  oiher  two?  Would  that  make  a  square?  Or  a  reclanglc?  Or  both? 

Chris:  It's  both  [a  square  and  a  rectangle]. 

Int:  Is  it  a  square? 

Chris:  Yes. 

Int:  How  come  it's  a  square? 

Chris:  'Cause  9  is  close  to  10. 

Again,  we  see  the  strength  of  visual  thinking  in  Chris'  judgments.  He  is  willing  to  call  the 
rectangle  with  side  lengths  of  9  and  10  a  square,  presumably  because  his  visual  thinking  causes  him  to 
judge  9  close  enough  to  10  as  side  lengths.  Contrast  this  with  the  second  grader  M's  thinking  about 
squares.  It  too  was  visual,  but  it  was  more  sophisticated  because  of  her  use  of  visual  transformations. 

Robbie,  another  kindergartner,  already  indicated  that  he  understood  why  two  numbers  must  be 
input  for  a  rectangle  and  only  one  for  a  square. 

Int:  What  about  this?  What  if  I  put  in  S  R  5  5. 

Robbie:  That  would  be  a  rectangle  for  R. 

Inl:  Right,  and  then  I  tell  it  S  and  S. 

Robbie:  R  draws  on  paper  what  he  thinks  it  would  be  (a  square)  and  calls  it  a  square. 

Inl:  How  did  that  happen? 

Robbie:  Because  if  I  goofed. ..and  I  think  1  put  some  number  the  same,  I  got  a  square,  and  I  wanted  a  rectangle. 

Inl:  Why  is  that? 

Robbie:  I  can  go  wrong  on  the  rectangles.  Because  the  rectangle  is  likea  square,  except  that  squares  aren't  long. 

Inl:  What  else  do  you  know  about  a  rectangle?  What  docs  a  shape  need  to  be  to  be  a  rectangle? 

Robhic-  All  of  the  sides  aren't  equal.  These  two  lopposile)  and  these  two  lothcr  opposite!  sides  have  to  be  equal. 

Int:  How  about  10  on  two  sides  and  9  on  the  other  two?  Would  that  make  a  square? 

Robbie:  Kind  of  likea  rectangle. 



1  -61 

Inl:  Would  it  be  i  square  loo? 

Robbie:      [Pause.]  I  think  may... .  [Shaking  head  negatively.)  It's  not  a  square.  'Cause  if  you  make  a  square,  you 

wouldn't  go  10  up.  then  you  turn  and  it  would  be  9  this  way,  and  turn  and  10  this  way.  That's  not  a  square. 

Robbie  too  is  not  using  any  type  of  hierarchical  classification.  He  thinks  of  squares  and  rectangles 
in  terms  of  visual  prototypes—  "the  rectangle  is  like  a  square,  except  that  squares  aren't  long."  And 
according  to  his  past  experiences,  he,  like  most  students,  decides  that  rectangles  have  opposite  sides  equal, 
but  not  all  sides  equal. 

In  conclusion,  tlie  Logo  microworlds  proved  to  be  evocative  in  generating  thinking  about  squares 
and  rectangles  for  these  Idndergartners.  Their  constructions  were  strongly  visual  in  nature,  and  no  logical 
classification  per  se,  such  as  class  inclusion  processes,  should  be  inferred.  Squares  were  squares,  and 
rectangles  rectangles,  unless— for  some  students— they  formed  a  square  with  a  Logo  rectangle  procedure 
or  they  intended  to  sketch  a  rectangle,  in  which  case  the  figure  might  be  described  as  a  "square  rectangle." 

The  comments  of  another  second  grader  illustrate  how  visual  thinking  is  used  by  some  students  to 
make  sense  of  relationships  between  figures  (Battista  &  Clements,  1991b).  This  student,  who  had 
previously  discovered  that  she  needed  90°  turns  to  draw  a  square,  used  90s  on  her  first  attempt  at  making  a 
tilted  rectangle,  reasoning  as  follows: 

C:  Because  a  rectangle  is  just  like  a  square  but  just  longer,  and  all  the  sides  are  straight  Well,  not  straight,  but  not 

lilted  like  that  (makes  an  acute  angle  with  her  hands).  They're  all  like  that  (shows  a  right  angle  with  her  hands) 
and  so  are  the  squares. 

Int:  And  that's  90  [showing  hands  put  together  at  a  90°)? 

C:  Yes. 

She  then  stated  that  a  square  is  a  rectangle. 
Int:  Does  that  make  sense  to  you? 

C:  It  wouldn't  to  my  [4  year  old]  sister  but  it  sort  of  does  to  me. 

Int:  How  would  you  explain  it  to  her? 

C:  We  have  these  stretchy  square  bathroom  things.  And  I'd  tell  her  to  stretch  it  out  and  it  would  be  a  rectangle. 

It  "sort  of  made  sense"  that  a  square  is  a  rectangle  because  a  square  could  be  stretched  into  a 
rectangle.  This  response  may  be  more  sophisticated  than  one  might  initially  think,  for  C  had  already 
demonstrated  her  knowledge  that  squares  and  rectangles  are  similar  in  having  angles  made  by  90°  turns. 
Thus,  she  may  have  understood  at  an  intuitive  level  that  all  rectangles  could  be  generated  from  one  another 
by  certain  "legal"  transformations,  that  is,  ones  that  preserve  90°  angles. 

A  fifth  grader  was  working  on  the  square  in  the  "Rectangle:  What  Can  You  Draw?"  activity. 

Jon:  This  one  Ipoinling  to  the  square]  is  not  a  rectangle.  It's  a  square.  It  has  equal  sides. 

Inl:  Can  you  do  il  with  youi  rectangle  procedure? 

Jon:  No.  because  the  sides  are  equal.  So  that  would  be  a  "no." 

Inl:  So,  no  matter  what  you  iricd.  you  couldn't  make  il  with  your  rectangle  procedure? 

Jon:  You  couldn't  no,  because  the  sides  are  equal. 

Int:  On  your  rectangle  procedure,  what  decs  this  first  input  stand  for? 

Jon:  The  20?  These  sides. 

Int:  What  docs  ihc  40  stand  for? 

Jon-  Yea.  you  could  do  it.  If  you  put  like  40, 40.  40.  and  40.  |ag;un.  motionsl 

Int:  Ok.  try  it. 


1  -62 

Jon:  So  that  would  be  a  square? 

Im:  Can  you  draw  a  square  with  your  rectangle  procedure? 

Jon:  You  could  draw  it,  but  it  wouldn't  be  a  rectangle. 

Even  with  prompting,  Jon  is  resistant  to  calling  the  square  a  rectangle.  In  his  conceptualization,  one  can 
draw  a  square  with  the  rectangle  procedure,  but  that  does  not  "make  it"  a  rectangle. 
Here  is  another  fifth  grader  discussing  the  issue. 

Teacher.      Why  do  you  think  a  square  is  not  a  recuuigle,  Jane? 

Jane:  Each  side  is  equal  to  each  other.  But  in  a  rectangle  there  are  two  longer  sides  thai  equal  each  other  and  the  other 

two  sides  equal  each  other  but  they're  short 

This  response  is  typical.  Jane  has  simply  elaborated  the  essential  visual  characteristics  of  the  set  of  figures 

she  thinks  of  as  rectangles.  So,  because  almost  all  of  the  figures  that  she  has  seen  labeled  as  rectangles 

have  two  long  sides  and  two  short  sides,  she  includes  this  characteristic  in  her  list  of  characteristics  or 

properties.  The  teacher  asked  how  she  could  make  u  square  with  the  RECT  procedure. 

Jane:  Because  you  put  in  two  equal  numbers.  And  that's  the  distance  (length]  and  the  width.  If  they  are  the  same 

amount,  then  it  will  come  out  to  be  a  square. 
Teacher      So  it  did  come  out  to  be  a  square?  Thai  is  a  square  you're  telling  me? 
Jane:  Yes,  and  a  rectangle.  But  it's  more  a  square,  because  we  know  it  more  as  a  square. 

The  second  grader  below  tries  to  deal  with  the  problem  by  inventing  new  language,  much  like  one 

of  the  kindergartners  that  we  discussed. 

Im:  Is  everything  that  RECT  draws  a  rectangle? 

Bob:  Thai's  (points  to  square  on  the  screen)  not  a  rectangle. 

Int:  How  come? 

Bob:  Because  the  sides  are  the  same  size? 

Int:  So ...  this  square  (pointing  to  the  square  on  the  sheet]  is  not  a  rectangle? 

Bob:  I  think  it's  a  special  kind  of  rectangle. 

Int:  So  is  this  (pointing  to  the  square  on  the  screen)  a  rectangle? 

Bob:  It's  a  special  kind  of  rectangle. 

So  Bob  dealt  with  the  conflict  of  a  square  being  drawn  by  a  rectangle  procedure  by  inventing  a  language 
that  allowed  him  to  avoid  the  uncomfortable  statement  that  a  square  is  a  rectangle  by  saying  that  a  square  is 
a  special  kind  of  rectangle  but  not  a  rectangle. 

Other  fifth  graders  trying  to  come  to  grips  with  the  same  question  in  a  class  discussion. 

Lisa:  I  have  a  different  question.  Why  can't  we  call  squares  equilateral  rectangles? 

Keith:  A  square  classifies  as  a  bunch  of  things.  Equilateral  rectangle  doesn't  classify  as  all  the  things  that  arc  square. 

Teacher.  Give  me  an  example  of  a  square  that  isn't  an  equilateral  rectangle. 

Keith:  Well,  like  a  diamond. 

Teacher  (Draws  one  and  has  Keith  clarify  that  he  means  a  diamond  with  90°  turns.  Keith  still  maintains  that  the  drawing 

is  not  an  equilateral  rectangle.) 

Lisa:  All  you  have  to  do  is  turn  it  and  it  would  be  both  a  square  and  an  equilateral  rectangle  in  my  definition. 

Interestingly,  and  illustrating  his  lack  of  hierarchical  classification,  Keith  does  not  think  a  square 
and  an  equilateral  rectangle  arc  the  same.  Lisa,  who  does,  still  uses  visual  thinking  to  support  her 
argument.  In  the  episode  below,  the  teacher  has  asked  the  students  whether  a  variable  square  procedure 
(SQUARE  :X)  can  be  used  to  make  a  variable  rectangle  procedure. 

K:  No.  There  are  two  longer  lines  on  a  rectangle.  They  are  longer  than  a  square.  All  the  lines  are  not  equal  in  u 



rectangle;  they  are  in  a  square.  So  if  you  think  that,  you  can't  draw  a  rectangle  with  a  square  procedure. 
P:  In  the  sense  that  the  10  or  whatever  yc„  put  down  for  the  square  represents  all  the  side*,  which  wouldn't  work 

because  alt  the  sides  would  be  equal.  So  you'd  have  to  make  a  new  procedure  for  it. 
J:  You  have  mentioned  that  opotBiie  sides  arc  parallel  and  equal.  It's  the  same  way  with  a  square  except  that  all 

sides  are  equal.  So  that  the  two  sides  that  are  parallel  are  still  equal  So  a  square  in  the  sense  that  you're  saying 

is  a  still  a  rectangle,  but  a  rectangle  is  not  a  square. 
Teacher      Can  we  build  any  rectangle  with  the  square  procedure? 
J:  Yes  you  can. 

Teacher      Can  I  build  a  rectangle  with  sides  of  20  and  40? 

J:  No.  sorry.  You  can't  build  every  single  rectangle  with  the  square  procedure,  but  you  can  build  one  rectangle 

with  the  square  procedure. 

In  pairs,  students  now  move  on  to  the  Rectangle:  What  Can  You  Draw?  activity.  As  they  get  to  the  square 
on  the  sheet,  J  says  "It's  a  square."  P  illustrates  his  confusion  over  classification,  saying  "A  square  can  be 
a  rectangle,  wait  A  rectangle  can  be  a  square  but  a  square  can't  be  a  rectangle."  J  starts  to  correct  him  "A 
square  can  be  a  rectangle.  P  interrupts,  "Oh  yeah  [laughs]." 

In  this  episode,  all  of  these  students  see  that  the  square  procedure  cannot  be  used  to  make 
rectangles.  J,  however,  is  the  only  student  who  seems  capable  of  comprehending  the  mathematical 
perspective  of  classifying  squares  and  rectangles.  However,  her  comment  "in  the  sense  that  you're 
saying"  suggests,  that  she  has  not  yet  accepted  this  organization  as  her  own.  The  episode  below  further 
illustrates  that  she  has  not  yet  adopted  a  mathematical  organization  in  her  classification  of  shapes. 

Int:  If  I  typed  in  RECT  50  5 1 .  what  would  it  be  (before  hitting  return)? 

P:  Probably  about  a  square. 

J:  A  rectangle  but  it  wouldn't — 

P:  It  would  be  a  rectangle  but  sorta  like— 

J:  It  would  be  a  rectangle,  but  it  wouldn't  be  a  perfect  square.  [They  hit  return.] 

J:  You  see  it's  not  a  perfect  square. 

P:  (Measures  the  top  side  (the  longer)  with  his  fingers.)  It's  only  one  step  off. 

Even  though  P  and  J  say  that  the  50  5 1  rectangle  is  a  rectangle  and  not  a  square,  their  language 
seems  to  indicate  their  belief  in  such  a  thing  as  an  "imperfect  square" — that  is  we  presume,  a  figure  that 
looks  like  a  square  but  does  not  have  all  sides  equal.  They  cling  to  an  informal  rather  than  logical 
classification  system,  one  that  still  contains  remnants  of  their  visual  thinking. 

Finally,  we  examine  the  comments  of  a  6th  grader  during  a  class  discussion  of  the  square/rectangle 
issue  raised  by  trying  to  draw  the  square  with  the  rectangle  procedure.  Kelly  asked  "Why  don't  you  call  a 
rectangle  a  square  with  unequal  sides?"  After  the  teacher  defined  a  rectangle  as  a  shape  that  has  four  right 
turns  and  opposite  sides  parallel,  however,  Kelly  stated  "If  you  use  your  definition,  then  the  square  is  a 
rectangle"  (Lewellen,  in  press).  Kelly's  comments,  like  those  of  the  5th  grader  J,  clearly  indicate  an 
ability  to  follow  the  logic  in  the  mathematical  classification  of  squares  and  rectangles.  But  neither  student 
has  yet  made  that  logical  network  her  own — each  still  clings  to  the  personal  network  constructed  from 
previous  experiences.  As  van  Hiele  says,  "Only  if  the  usual  las  taught  in  the  classroom]  network  of 
relations  of  the  third  level  has  been  accepted  does  the  square  have  to  be  understood  as  belonging  to  the  set 
of  rhombuses.  This  acceptance  must  be  voluntary;  it  is  not  possible  to  force  a  network  of  relations  on 



1  -64 

someone"  (van  Hiele,  1 986,  p.  50).  For  Kelly  or  J  to  move  to  the  next  level  requires  them  to  reorganize 
their  definitions  of  shapes  in  a  way  that  permits  a  total  classification  scheme  to  be  constructed.  That  is,  the 
attainment  of  level  3  does  not  automatically  result  from  the  ability  to  follow  and  make  logical  deductions; 
the  student  must  utilize  this  ability  to  reorganize  her  or  his  knowledge  into  a  new  network  of  relations.  In 
this  network,  "One  property  can  signal  other  properties,  so  definitions  can  be  seen  not  merely  as 
descriptions  but  as  a  way  of  logically  organizing  properties"  (Clements  &  Battista,  in  press).  Normally 
this  entails  making  sense  of  and  accepting  the  common  definitions  and  resulting  hierarchies  given  in  the 


Logo  environments  can  promote  students'  movement  from  the  visual  van  Hiele  level  to  the  next 
level  in  which  students  think  of  shapes  in  terms  of  their  properties.  Logo  explorations  of  relationships 
between  shapes  such  as  squares  and  rectangles  differentially  affect  students  at  different  levels  of  thinking. 
For  some  students  such  as  second-grader  C,  such  explorations  cause  their  visual  thinking  to  become  more 
sophisticated,  incorporating  visual  transformations  that  express  their  knowledge  of  these  relationships. 
For  several  of  the  fifth  graders,  the  explorations  engendered  analysis  and  refinement  of  their  definitions  for 
shapes  in  terms  of  properties,  further  promoting  the  attainment  of  level  2  thinking.  And  finally,  for  some, 
such  explorations  promoted  the  transition  to  level  3  thinking — first  they  understand  a  logical  organization 
of  properties,  and  finally  they  adopt  it. 


Battista,  M.  T.,  &  Clements,  D.  H.  (1988a).  A  case  for  a  Logo-based  elementary  school  geometry 

curriculum.  Arithmetic  Teacher,  36,  1 1-17. 
Battista,  M.  T.,  &  Clements,  D.  H.  (1988b).  Using  Logo  pseudoprimitives  for  geometric  investigations. 

Mathematics  Teacher,  81, 166-174. 
Battista,  M.  T.,  &  Clements,  D.  H.  (1990).  Constructing  geometric  concepts  in  Logo.  Arithmetic  Teacher, 

35(3),  15-17. 

Battista,  M.  T.,  &  Clements,  D.  H.  (1991a).  Logo  geometry.  Morristown,  NJ:  Silver  Burden  &  Ginn. 
Battista,  M.  T.,  &  Clements,  D.  H.  (1991b).  Using  spatial  imagery  in  geometric  reasoning.  Arithmetic 
Teacher,  39(3),  18-21. 

Burger,  W.,  &  Shaughnessy,  J.  M.  (1986).  Characterizing  the  van  Hiele  levels  of  development  in 

geometry.  Journal  for  Research  in  Mathematics  Education,  1 7, 3 1  -48. 
Clements,  D.  H.,  &  Battista,  M.  T.  (1989).  Learning  of  geometric  concepts  in  a  Logo  environment. 

Journal  for  Research  in  Mathematics  Education,  20, 450-467. 
Clements,  D.  H.,&  Battista,  M.  T.  (1990).  The  effects  of  Logo  on  children's  conceptualizations  of  angle 

and  polygons.  Journal  for  Research  in  Mathematics  Education,  21. 356-37 1 . 
Clements,  D.  H.,  &  Battista,  M.  T.  (1991).  The  development  of  a  Logo-based  elementary  school 

geometry  curriculum  (Final  Report:  NSF  Grant  No.:  MDR-8651668).  Buffalo,  NY/Kent,  OH:  State 

University  of  New  York  at  Buffalo/Kent  State  University. 
Clements,  D.  H.,  &  Battista,  M.  T.  (in  press).  Geometry  and  spatial  reasoning.  In  D.  A.  Grouws  (Ed.), 

Handbook  of  research  on  mathematics  teaching.  Reston,  VA:  National  Council  of  Teachers  of 


Ixwcllen,  H.  (in  press).  Conceptualizations  of geometric  motions  in  elementary 'school  children:  An 

extension  of  the  van  Hiele  model.  Doctoral  dissertation,  Kent  State  University, 
van  Hiele,  P.  M.  (1986).  Structure  and  insight.  Orlando:  Academic  Press. 



1  -65 


Nadlne  Bednarx,  Luis  Radford,  Bernadette  Janvier,  Andr<  Lepage 
GRADE,  University  du  Quebec  a  Montreal 

The  fact  that  students  have  difficulty  acquiring  and  developing  algebraic  procedures  in  problem-solving, 
considering  the  arithmetical  experience  that  they  have  acquired  overyears,  calls  for  a  didactic  rtpectton  on 
the  nature  of  the  conceptual  changes  which  mark  the  transition  from  one  mode  of  treatment  to  the  other.  In 
this  perspective,  our  study  seeks  to  characterize  the  spontaneous  problem-solving  strategies  used  by 
Secondary  III  level  students  (14  -and  -15-year-olds),  who  have  already  taken  one  algebra  course,  when 
solving  different  problems.  The  analysis  of  the  problem-solving  procedures  developed  by  these  students 
reveals  the  differences  between  the  conceptual  basis  which  underlie  the  two  modes  of  thought. 

The  difficulties  experienced  by  students  learning  algebra  have  been  the  subject  of  many  studies  which  have 
shown  that  certain  conceptual  changes  are  necessary  to  make  the  transition  from  arithmetic  to  algebra 
(Booth,  1984;  Collis,  1974,  Kieran,  1981;  Filloy  and  Rojano,  1984;  Hercovicz  and  Linchevski,  1991; 
Arzarello,  1991  ...)•  I»  the  area  of  problem-solving,  which  is  one  of  the  important  heuristic  functions  of 
algebra  (Kieran,  1989)  and  which  proves  very  difficult  for  students  (Lochead,  1988;  Kaput,  1983; 
Clement,  1982;  Mayer  1982),  the  analyses  examining  the  passage  to  an  algebraic  mode  of  thinking  have 
either  focused  on  a  certain  dialectic  between  procedural  and  relational  thought  (Kieran,  1991;  Arzarello, 
1991),  or  on  the  symbolism  and/or  the  solving  of  equations.  With  regard  to  the  latter,  the  history  of 
mathematics  shows  that  algebra  began  to  develop  well  before  symbols  were  used  to  represent  unknown 
quantities.  Rhetoric  was  an  important  stage  in  this  development  among  the  Arabs,  for  whom  language  was 
the  natural  means  to  represent  the  (known  and  unknown)  quantities  of  a  problem  to  be  solved  and  to 
express  the  solution  process.  The  studies  carried  out  among  students  also  show  that  most  of  them,  from 
high  school  to  university,  solve  algebraic  problems  in  an  "abridged"  style  (natural  syncopated  language) 
rather  than  in  a  symbolic  style  (Kieran,  1989;  Harper,  1979).  Few  studies,  however,  have  focused  on  the 
students'  reasoning  in  solving  the  problems. 

1  This  study  is  pan  of  a  larger  project  undertaken  by  a  group  from  CIRADE.  subsidized  by 

the  FCAR  (Quebec),  which  is  researching  the  conditions  for  the  construction  of  algebraic 

reasoning  and    representations,    with    regard   to   the   situations   which   allow   for  their 

emergence  and  development. 

1  -66 

In  our  didactic  perspective,  the  main  objective  of  our  analysis  was  to  gain  a  better  understanding  of  the 
conceptual  basis  which  underlie  the  arithmetical  mode  of  thought  on  one  hand,  and  the  algebraic  mode  of 
thought  on  the  other,  as  well  as  the  possible  articulation-conflicts  which  are  possible  in  the  transition  from 
one  mode  of  treatment  to  the  other. 

From  a  didactic  point  of  view,  because  of  the  previous  experience  acquired  by  the  students,  problem- 
solving  appears  to  be  an  interesting  terrain  for  examining  the  two  modes  of  thought  and  the  conceptual 
changes  which  mark  the  passage'  from  the  arithmetical  to  the  algebraic  thinking.  Moreover,  from  the 
historical  point  of  view,  the  solving  of  problems  played  an  important  role  in  the  development  of  algebra.  It 
is  at  the  heart  of  the  algebra  of  Diophantus  and  of  the  Arabs,  and  is  explicit  in  Vieta's  objective  of 
developing  a  method  that  could  solve  every  problem.  Thus,  problem-solving  is  a  doubly  interesting  terrain 
for  the  examination  of  the  emergence  of  the  algebraic  mode  of  thought  and  its  characteristics.  This 
historical  analysis  is  now  being  carried  out,  and  is  the  object  of  investigation  of  some  of  the  members  of 
our  team  (Charbonneau,  1992;  Lefebvre,  1992;  Radford,  1992). 
Objective  of  the  Study 

By  examining  the  ways  in  which  secondary  school  students  (Sec.  Ill,  14-and- 15-year-olds  who  had 
already  taken  an  algebra  course)  spontaneously  solved  different  types  of  problems,  this  exploratory 
research  project,  carried  out  with  a  small  group  of  students,  aimed  to  analyse  the  solution  processes  of  the 
students.  In  the  characterization  of  the  arithmetical  and  algebraic  procedures  used,  the  accent  was  placed 
not  on  the  use  of  symbolism,  but  rather  on  the  students'  capacity  to  grasp  the  known  and  unknown 
quantities  in  the  problem,  and  their  way  of  solving  it. 

In  order  to  delineate,  on  an  exploratory  basis,  the  procedures  used  by  students,  and,  through  these,  to 
better  elucidate  the  differences  between  the  conceptual  basis  which  underlie  the  arithmetical  and  the 
algebraic  thinking,  54  students  from  two  regular  classes  in  a  Montreal  area  public  high  school  (Secondary 
III,  14-and- 15-year-olds)  were  given  a  paper-and-pencil  test  with  five  different  written  problems  to 
solve?-  The  choice  of  the  students'  level  (they  had  taken  an  introductory  course  in  algebra)  made  it 

2  The  different  problems  presented,  involving  complex  relations,  could  all  be  solved  a 
priori  by  either  arithmetical  or  algebraic  reasoning,  even  if  some  of  the  methods  appear 
more  complicated  than  others. 

Er|c  91 


possible  for  us  to  show  the  conflicts  that  can  arise  for  students  at  this  sage  when  facing  two  possible 
modes  of  solving  the  problems. 
Analysis  of  the  Results 

Our  analysis  centered  on  one  of  the  problems,  which  read  as  follows:  "588  passengers  must  travel  from 
one  city  to  another.  Two  trains  are  available.  One  train  consists  only  of  1 2-seat  cars,  and  the  other  only  of 
16- seat  cars.  Supposing  that  the  train  with  16-seat  cars  will  have  eight  cars  more  than  the  other  train,  how 
many  cars  must  be  attached  to  the  locomotives  of  each  train  ?" 

In  this  problem,  different  solution  processes  were  possible.  These  took  into  account  a  certain  implicit 
mental  representation  of  the  data  and  the  relations  which  linked  the  elements  involved,  a  representation 
which  evolved  during  the  solution  process.  How  can  we  distinguish  between  the  arithmetical  and  the 
algebraic  procedures  in  the  ways  that  this  data  and  these  relations  were  dealt  with  ? 
A  preliminary  analysis  of  the  above  problem  brought  out  the  key  elements  around  which  the  solution  will 
be  organized:  "the  total  number  of  passengers:  588",  the  existence  of  "two  trains",  of  "  1 6-seat  cars",  "1 2- 
seat  cars",  and  the  "eight  cars  more"  that  one  kind  of  train  had  in  relation  to  the  other. 
However,  the  resolution  of  the  problem  required  the  use  of  other  elements  which  made  it  possible  to  "build 
bridges"  between  the  different  data,  elements  which  were  not  at  all  explicit  in  the  problem:  the  number  of 
16-seat  cars  and  12-seat  cars,  the  relation  between  the  two  types  of  quantities  involved:  the  number  of  cars 
and  the  number  of  passengers,  which  must  be  built  from  the  rates  given  in  the  problem,  the  number  of 
passengers  in  each  train...  This  a  priori  analysis  brought  to  light  important  reference  points  which  guided 
the  subsequent  analysis  of  the  students'  ways  of  solving  the  problem. 


It  was  easily  observed  that  certain  elements  were  retained  by  the  students,  and  that  these  were  used  as  a 
kind  of  point  of  entry ,  or  engagement,  in  the  organization  of  their  solution  procedures:  a)  the  two  trains; 
b)  the  whole:  the  588  passengers;  c)  the  difference  between  the  number  of  cars  of  one  type  and  those  of 
other  type;  d)  the  data:  "16-seat  cars"  and  "12-seat  cars". 

In  general,  the  arithmetical  procedures  were  organized  around  these  four  known  elements,  in  attempts 
to  build  bridges  between  them  to  be  able  to  work  with  known  data.  The  unknown  quantity  therefore 


appeared  at  the  end  of  the  process.  Two  types  of  entry  points,  or  engagements,  were  distinguished.  In  the 
first  case,  the  first  two  elements  (a  and  b)  frequently  gave  rise  to  a  numerical  strategy  which  we  call 
equitable  partition,  which  consisted  in  dividing  the  number  of  passengers  by  the  number  of  trains  (in 
this  case,  two)  to  obtain  the  number  of  passengers  in  each  train  (see  Procedure  3).  Another  less  frequent 
type  of  engagement  was  the  adjustment  of  the  difference  between  the  two  trains  (c)  at  the  beginning, 
to  obtain  two  trains  having  the  same  number  of  cars  (see  Procedures  1  and  2). 

1.  Procedure  taking  the  difference  into  account  at  the  beginning: 
Student:  Comments: 

8  x  16  =  128  passengers  Numbers  of  passengers  in  the  8  extra  cars 

588  -  128  »  460  passengers  By  eliminating  the  extra  cars,  the  number  of  cars  in  each  train  is  equal 

460  +  28=  16.4  12  seats +  16  seats*  28  seats 

(one  28-seat  car  train) 


17,  12-seat  cars  and  25,  16-seat  cars 

This  strategy  clearly  showed  the  modifications  which  occurred  in  the  representation  of  the  problem  during 
the  solution  process:  this  representation  was  not  at  all  static.  The  problem,  and  the  relations  linking  the  (lata 
had  to  be  transformed  by  the  students  into  a  new  configuration  of  the  whole,  which  made  it  possible  for 
the  calculations  to  progress.  The  arithmetical  procedure  used  here,  which  only  dealt  with  the  known 
elements,  could  not  advance  without  those  necessary  modifications,  because  at  the  beginning  there  was  no 
relation  directly  linking  the  known  quantities  provided  in  the  problem. 

2.  Another  procedure  taking  the  difference  into  account  at  the  beginning, 
followed  by  partition: 

Student:  Comments: 

16  x  8  =  1 28  Numbers  of  passengers  in  the  eight  extra  cars 

588- 1 28  *  460  Modification  of  the  initial  representation  into  a  new  configuration  of 

equality  of  cars  (see  previous  strategy) 
460  +  2  -  230  The  equitable  sharing  strategy 

(230+  12-  19,230+  16=  14)      Number  of  cars  of  each  type 
1 2-scat  car  train  ->  19  cars 

16-seat  car  train  ->  14  +  8  =  22  cars  Re-utilization  of  the  difference 

Just  as  in  the  first  procedure,  the  representation  of  the  problem  was  modified  during  the  solving  process. 
The  ch  inge  from  the  initial  representation  of  inequality  to  one  of  equality  authorized  the  use  of  equitable 
partition  schema. 




1  -69 

3.  Procedure  with  partition  at  the  beginning: 

Student:  Comments: 

588  +  2-294  Division  by  two  of  the  given  toul 

294  +  16-19  Calculation  of  the  number  of  16-scatcars 

19  +  8-27  The  use  of  the  difference 

27x  16  -  432  Number  of  passengers  in  the  16-seat  car  train 

588  -  432  -  156  Calculation  of  the  number  of  passengers  in  the  12-seat  car  train 

156  +  12  -  1 3  Number  of  12-seat  cars 

The  more  frequent  recourse  to  the  equitable  partition  schema  at  the  beginning  suggested  a  less  complicated 
representation  than  the  preceding  one,  in  which  the  inequality  of  the  number  of  cars  had  to  be  taken  into 
consideration.  The  students'  errors  in  all  of  the  arithmetical  procedures  occurred  precisely  in  the 
coordination  of  the  equitable  partition  schema  and  the  inequality  of  the  number  of  cars. 

PROCEDURES  BETWEEN  ALGEBRA  AND  ARITHMETIC  (revealing  a  process  in  formation) 
In  the  following  strategy  (see  Procedure  4),  after  undertaking  an  arithmetical  method,  the  student 
subsequently  abandoned  it,  and  wrote  an  equation.  The  solution  of  the  equation  was  used  immediately 
afterward  in  a  step  which  went  back  to  an  arithmetical  procedure. 
4.  Student:  Comments: 

Arithmetical  trial:  16  x  8  =128        previous  procedure  which  took  the  difference  into  account  at  the 
588-128  =  460  beginning 

Algebraic  step:  x  +  8x  =  588;  9x  =  588;  x  =  65 

Arithmetical  procedure: 

65  +  2  =  32  equitable  partition  schema:  the  number  of  cars  is  divided  by  two 

32  -  8  =  24  use  of  the  difference 

24  x  12  -  288  passengers  travelling  in  the  12-seat  cars 

In  this  example,  the  student  began  by  adjusting  the  number  of  passengers  to  arrive  at  two  trains  having  an 
equal  number  of  cars.  In  this  arithmetical  engagement,  there  is  a  semantic  control  of  the  situation  and  the 
relations  which  link  the  elements  involved.  When  the  student  left  this  procedure  in  favour  of  an  algebraic 
one,  the  continuation  shows  that  there  was  no  longer  any  control  over  the  rates  (which  appear  to  be 
completely  ignored)  or  the  difference,  although  the  algebraic  treatment  of  the  equation  is  correct.  There 
was  a  complete  loss  of  control  over  the  situation.  But  as  soon  as  the  student  returned  to  arithmetic,  the 
control  was  regained.  This  and  the  following  examples  clearly  show  the  distinctions  effected  by  the 
student  in  the  transition  from  one  mode  to  the  other.  The  passage  to  algebra  requires  the  construction  of  a 


1  -70 

more  global  representation  of  the  problem,  which  is  in  opposition  to  the  sequence  of  'dynamic 
representations  which  are  the  basis  of  the  arithmetical  reasuninf . 

5.  Student: 

Arithmetical  trial:      16  x  8  -128;  588  -  128  -  460;  460  +  2  -  230       (Control  of  the  situation) 

Algebraic  trial:         588  +  2  +  12  +  16  «  16  + 8x 

578  = '6+ 8x 
562  *  8x 

Note  that  the  order  of  the  terms  of  the  equation  followed  that  of  the  presentation  of  the  numbers  in  the  text 
(loss  of  control  -  the  student  did  not  take  into  account  the  meaning  of  the  quantities  and  of  the  problem). 

6.  Arithmetico-algebraic  strategy: 

8  x  16«  128;  588  •  128  «  460        (difference  taken  into  account) 

Then  the  student  switched  to  an  algebraic  mode,  with  the  equation:  12x  +  16x  -  460,  and  ended  by 
solving  the  problem. 


In  contrast  to  the  arithmetical  procedures,  in  the  algebraic  procedures,  the  representation  of  the  problem 
and  the  calculations  do  not  generally  undergo  a  parallel  development.  The  solution  process  -  which  in 
arithmetic  is  based  on  a  necessary  transformation  of  the  representation  of  the  problem,  in  relation  to 
meaning  of  the  numbers  obtained  in  successive  calculate  is  -  needs  ai  the  beginning  a  representation  of  the 
relations  between  the  daw.  It  requires  then  for  the  stude.ii global  representation  of  the  problem,  from  the 
start  of  the  procedure,  to  infer  an  external  symbolic  representation  modeling  these  relations,  in  the  form 
here  of  an  equation.  Once  the  equation  is  expressed,  the  algebraic  calculations  often  proceed  independently 
of  this  representation  of  the  situation.  If  the  semantic  control  of  the  problem  is  re-established,  it  only 
happens  at  the  end  of  the  process.  This  type  of  engagement,  totally  different  in  its  management  of  the  data, 
is  based  on  an  element  which  is  not  present  in  arithmetic,  that  is,  the  introduction  of  precisely  that  quantity 
which  is  sought,  the  unknown  quantity.  We  find  there  the  analytical  character  of  algebra  so  important  to 




1  -  71 


1st  x  12 

2nd  (x+8)12    588  .  x.12  +  (x+8)I6 
588  »  12x  +  16x  +  128 
-12x  -  16x  -  128  -  588 
-28x  -  -460 
x  =  16.42 

1st  =>  16.42  x  12  -197.04  2nd  ->  (16.42  +  8)  16  =  390.72 

1*197.04  2nd  390.72 

The  equality  -28x  =  -460,  for  example,  cannot  be  interpreted  in  the  context  of  the  problem.  This  distance 
from  the  problem,  necessary  to  proceed  with  the  algebraic  operations,  makes  it  impossible,  at  this  point,  to 
verify  if  the  results  obtained  concur  with  what  is  sought  in  the  problem.  A  further  effort  must  be  expended 
to  reinterpret  the  results  from  the  symbolic  operations. 

The  analysis  of  the  students'  errors  in  constructing  their  equations,  throughout  their  procedures,  showed 
that  they  did  not  take  certain  elements,  such  as  rates,  into  account.  Their  symbolizations  only  retained 
certain  aspects  of  what  had  to  be  represented. 


This  analysis  brought  out  differences  between  the  conceptual  basis  that  underlie  the  arithmetical  and  the 
algebraic  modes  of  thought. 

Arithmetical  reasoning  is  based  on  representations  which  are  particular  to  it,  and  involves  a  particular 
relational  process.  The  successive  calculations  which  work  with  known  quantities  are  effectively  based 
upon  the  necessary  transformation  of  the  relations  which  link  the  elements  present,  requiring  a  constant 
semantic  control  of  the  quantities  involved  and  of  the  situation. 

In  algebraic  reasoning  on  the  contrary,  the  rclitions  expressed  in  the  problem  are  integrated  from  the 
beginning  into  a  global  "static"  representation  of  the  problem,  nevertheless  requiring  specific  necessary 
representations  for  this.  This  engagement,  which  is  quite  different  in  its  management  of  the  data,  is  based 
on  the  introduction  of  the  unknown  quantity  at  the  very  beginning  of  the  ptocess,  and  requires  a 
detachment  from  the  meaning  of  both  the  quantities  and  the  problem  to  solve  it. 

Our  results  suggest  that  the  difficulty  experienced  in  the  transition  from  arithmetic  to  algebra  occurs 
precisely  in  the  construction  of  the  representation  of  the  problem. 



1  -72 


Arxtrcllo,  F.  (1991).  Procedural  and  Relational  Aspects  of  Algebraic  Thinking.  Proceeding  ofPMEXV, 

Assisi,  Italy,  I,  pp.  80-87. 
Booth,  L.R.  (1984).  Algebra:  Children's  strategies  and  errors.  Windsor,  U.K.:  NFR-Nelson. 
Charbonneau,  L.  (a  paraitre).  Du  raisonnement  Uits*  a  lui-meme  au  raisonnement  outill*:  l'algebre  depuis 

Baby  lone  jusqu'a  VitocJSulietin  de  VAssociation  Mathtmatique  du  Qutbec. 
Clement.  J.  (1982).  Algebra  word  problem  solutions:  Thought  processes  underlying  a  common 

misconception.  Journal  for  Research  in  Mathematics  Education,  14,  pp.  16-30. 
Collis,  K.F.  (1974).  Cognitive  development  and  mathematics  learning.  Paper  presented  at  the  Psychology 

of  Mathematics  Workshop.  Center  for  Science  Education.  Chelsea  College.  London. 
Filloy,  E..  Rojano.  T.  (1984).  From  an  arithmetical  to  an  algebraic  thought.  Proceedings  of  the  sixth 

annual  meeting  of  PME-NA,  Madison,  pp.  51  -56 . 
Harper,  E.W.  (1979).  The  child's  interpretation  of  a  numerical  variable.  University  of  Bath,  Ph.  D. 

Thesis,  400  pages. 

Hercovicz,  N..  Linchevski.  L.  (1991).  Pre-algebraic  thinking:  range  of  equations  and  informal  solution 
processes  used  by  seventh  graders  prior  to  any  msavc&onJ'roceedings  of  PME  XV,  Assisi.  Italy.  D, 
pp.  173-180. 

Kaput.  J.  Sims-Knight,  J.  (1983).  Etrors  in  translations  to  algebraic  equations:  Roots  and  implications. 

Focus  on  Learning  Problems  in  Mathematics,  5,  pp.  63-78. 
Kieran,  C.  (1981).  Concepts  associated  with  the  equality  symbol.  Educational  Studies  in  Mathematics,  12, 

pp.  317-326. 

Kieran,  C.  (1989).  A  perspective  on  algebraic  thinking.  Actes  de  la  lie  confirence  intemationale 

"Psychology  of  Mathematics  Education*,  2,  pp.  163-171. 
Kieran,  C.  (1991).  A  procedural-structural  perspective  on  algebra  research.  Proceedings  of  PME  XV. 

Assisi,  Italy,  2,  pp.  245-253. 
Lefebvre,  J.  (a  paraitre).  Qu'est  l'algebre  devenue?  De  Viete  (1591)  a  aujourd'hui  (1991),  quelques 

changements  cleft.  Bulletin  de  f  Association  Mathtmatique  du  Qutbec. 
Lochead,  J.,  Mesne.  J.  (1988).  From  words  to  algebra:  Mending  misconceptions.  The  Ideas  of  Algebra, 

K-12,  NCTM  Yearbook. 

Mayer,  R.E.  (1982).  Memory  for  algebra  story  problems.  Journal  of  Educational  Psychology.  74(2).  pp. 

Radford,  L.  (a  panutre).  Diophante  et  l'algebre  pnf-symbolique.  Bulletin  de  VAssociation  Mathtmatique 
du  Qutbec. 



David  Ben-Chaim.  Miriam  Carmcli,  &  Barbara  Frcsko 
The  Wcizmann  Institute  of  Science 

Abstract.  A  form  of  co-teaching  was  utilized  as  one  mode  of  intervention  in 
a  project  to  improve  mathematics  instruction  in  Israeli  secondary  schools. 
Initial  reactions  of  pupils,  teachers,  school  principals,  and  co-teaching 
consultants  suggest  that,  on  the  whole,  this  is  a  viable  in-service  approach  for 
demonstrating  instructional  strategies  to  teachers  and  for  increasing  their 
involvement  in  reflection  and  planned  instruction. 


For  the  past  two  years,  a  project  for  improving  mathematics  instruction 
has  been  on-going  in  six  comprehensive  secondary  schools  in  the  Northern 
Negev  region  of  Israel.  The  project,  which  will  continue  for  at  least  another 
year,  was  undertaken  following  a  needs  assessment  survey  which  revealed 
that  many  teachers  lacked  proper  teaching  credentials  and  that  few  pupils 
were  taking  and  passing  national  matriculation  examinations  in  mathematics 
and  the  sciences  at  the  end  of  Grade  12  (Ben-Chaim  &  Carmeli,  1990).  The 
project  is  concerned  in  its  entirety  with  improving  mathematics  and  science 
teaching  and  learning  in  Grades  7  through  12.  Attacking  the  problem  from  a 
holistic  perspective,  different  forms  of  activity  are  being  carried  out  at  the 
various  levels  of  instruction  which  include:  1)  weekly  workshops  and 
individual  consultation  for  Grades  7-9  teachers,  2)  individual  assistance  for 
Grades  10-12  teachers,  and  3)  co-teaching  of  some  Grades  10-11  classes.  All 
modes  of  activity  are  explicitly  geared  towards  helping  teachers  with  average 
and  above-average  pupils,  i.e.  those  with  the  ability  to  matriculate.  Project 
consultants  have  extensive  prior  experience  as  teachers  and  as  consultants  in 
their  subject  area. 




The  co-teaching  mode  was  undertaken  primarily  in  mathematics  classes. 
It  has  been  selected  as  the  focus  of  the  present  paper  insofar  as,  compared  to 
workshops  and  individual  consultation  which  are  somewhat  common  teacher 
in-service  activities,  co-teaching  as  an  intervention  mode  is  generally 
unknown.  This  form  of  activity  was  adopted  in  eight  Grade  10  mathematics 
classes  in  1990-91  and  in  nine  Grades  10  and  11  classes  in  1991-92.  Four 
teachers  have  been  involved  in  this  activity  for  two  years. 


Co-teaching  may  be  viewed  as  a  form  of  team  teaching  in  which  two 
teachers  are  responsible  for  the  educational  advancement  of  a  single  class. 
As  reported  by  Goodlad  (1984),  team  teaching  was  extensively  tried  out  in 
different  schoois  in  the  United  States  during  the  60's  as  one  solution  to  the 
teacher  shortage  problem.  Accordingly,  qualified  and  experienced  teachers 
were  expected  to  work  together  with  new  and  under-qualified  teachers,  thus 
ensuring  both  maximal  use  of  personnel  resources  and  the  supervision  of  the 

Co-teaching  as  a  means  for  altering  teaching  behaviors  in  the  context  of  a 
project  is  uncommon.  However,  the  rationale  behind  such  an  approach  is 
similar  to  that  described  by  Goodlad.  By  pairing  a  project  consultant  with  a 
particular  teacher,  expertise  knowledge  can  be  shared  as  both  take 
responsibility  for  the  instruction  of  a  single  class.  In  such  manner,  teachers 
are  provided  with  an  intensive,  site-based,  in-service  experience:  they  are 
thus  offered  the  opportunity  to  directly  view  expert  teachers  in  action  and  to 
learn  their  strategies  and  approaches  through  joint- planning  and  coordination 
of  lessons. 

The  co-teaching  mode  has  been  used  in  the  project  schools  in  the 
following  manner.  Throughout  the  course  of  the  school  year,  on  one  set  day 
every  week,  the  consultant  conies  to  give  a  regular  classroom  lesson  in  the 


1  -75 

co-teacher's  class.  The  class  teacher  observes  the  lesson  and  often  assists  the 
consultant.  Following  the  consultant's  weekly  lesson,  co-teaching  pairs  meet 
to  discuss  the  lesson  and  to  plan  the  next  week's  teaching  schedule.  During 
these  discussions,  consultants  endeavor  to  raise  pedagogical  and  didactical 
issues  relevant  to  the  mathematical  topic  being  taught.  Since  topics  taught  in 
the  consultant's  lessons  are  an  integral  part  of  the  regular  instructional 
curriculum,  careful  coordination  must  be  made  with  the  classroom  teacher. 
Teachers  and  consultants  try  to  plan  their  lessons  and  adjust  their  pace  of 
instruction  so  that  the  consultant's  lesson  can  be  a  natural  continuation  of  the 
material  taught  by  the  teacher  during  the  week.  Accordingly,  the  consultants 
make  suggestions  to  the  teachers  as  to  how  to  continue  the  teaching  of  the 
material  and  try  to  define  for  them  what  students  need  to  accomplish  in 
order  to  enable  their  own  next  planned  lesson  to  be  carried  out  smoothly. 

By  teaching  actual  classes,  consultants  are  able  to  directly  demonstrate 
different  methods  of  instruction,  to  show  how  they  cope  with  learning 
problems,  and  to  demonstrate  how  to  integrate  material.  They  also  become 
familiar  with  the  needs  of  the  specific  class  which  enable  them  to  give  better 
advice  to  the  teacher  concerning  appropriate  materials,  level  of  instruction, 
and  pacing.  Their  intimate  knowledge  of  the  pupils  and  their  demonstrated 
teaching  skills  are  furthermore  intended  to  enhance  their  credibility  in  the 
eyes  of  the  teachers. 

Classroom  teachers  are  exposed  to  new  ways  of  dealing  with  the 
curricular  material  and  are  able  to  view  these  methods  in  action  in  the 
natural  environment  of  the  class.  In  addition,  they  are  given  the  opportunity 
to  participate  in  collective  efforts  to  plan  instruction  and  to  learn  about 
teamwork.  It  should  be  noted  that  this  mode  of  activity  has  involved  only 
those  teachers  who  are  relatively  new  to  teaching  the  grade  level  in  question 
or  who  lack  experience  teaching  it  using  up-dated  materials. 

o  100 



On  occasion,  a  change  is  made  in  this  co-teaching  schedule  such  that  the 
regular  teachers  conduct  the  class  on  the  day  of  the  consultant's  visit  and  the 
consultant  observes  the  lesson.  Observation  enables  the  consultant  to  view 
the  teacher  in  action,  to  diagnose  the  teacher's  weaknesses  and  strengths  in 
the  classroom,  and  to  concentrate  activity  with  the  teacher  in  the  areas 
particularly  requiring  assistance. 

Pupil  progress  in  these  classes  is  monitored  through  periodic 
examinations,  some  of  which  are  specific  to  the  class,  prepared  by  the  teacher 
and  consultant  together,  and  some  of  which  are  general,  prepared  by  the 
Weizmann  staff  for  all  participating  schools.  Dates  for  the  latter  tests  are  set 
in  advance  which  is  intended  as  an  external  incentive  to  the  co-teachers  in 
their  preparation  of  the  pupils. 

Operational  Difficulties 

Intervention  of  this  kind  inevitably  encounters  numerous  organizational 
problems  along  the  way.  One  of  the  major  difficulties  is  the  problem  of 
adjusting  teachers'  schedules  to  fit  project  activities.  It  means  that  each 
participating  class  must  be  studying  mathematics  on  the  day  the  consultant 
comes  to  teach  and  that  their  teachers  have  at  least  one  free  period  for 
discussion  and  ,  'inning  after  viewing  the  consultant's  lesson.  Difficulties  are 
also  encountered  regarding  the  coordination  of  teacher  and  consultant 
lessons.  Classroom  teachers  are  not  always  able  to  accomplish  all  that  was 
planned  for  the  week  (often  due  to  the  cancellation  of  classes  for  school 
purposes)  and  the  consultant  is  forced  to  change  his/her  own  planned  lesson 
accordingly.  The  co-teaching  pair  maintains  telephone  contact  during  the 
week  so  that  the  consultant  is  kept  abreast  of  class  progress  and  can  make 
alterations  as  required. 

The  type  of  difficulties  which  particularly  interested  project  directors 
were  those  whose  source  was     psychological  rather  than  organizational  in 

1  -77 

nature.  With  regard  to  co-teaching,  three  questions  were  of  special  interest: 
1)  How  do  the  teachers  accept  the  consultant  as  a  co-teacher?  2)  How  do  the 
pupils  respond  towards  having  two  teachers,  one  of  whom  is  external  to  the 
school?  and  3)  How  do  the  consultants  themselves  feel  about  their  intensive 
involvement  in  someone  else's  classes?  The  central  issue  is  whether  or  not 
the  teacher's  status  in  the  classroom  is  undermined  by  the  fact  that  an 
outside  expert  shares  with  him/her  the  teaching  responsibilities  for  the  class. 

As  the  project  progresses,  information  is  being  gathered  on  the  reactions 
of  the  different  parties  to  co-teaching  as  a  form  of  intervention.  This 
information  is  being  collected  through  questionnaires  to  pupils,  consultants, 
and  teachers  as  well  as  by  means  of  interviews  with  teachers,  consultants, 
and  school  principals.  Results  from  the  first  U  years  of  project  operation  are 
cited  below;  results  from  the  full  two  years  will  be  presented  at  the 

Reactions  to  Co-teaching 

Pupil  reactions.  At  the  start  of  the  school  year,  it  was  carefully  explained 
to  pupils  in  the  designated  classes  that  both  co-teachers  would  be  responsible 
for  their  mathematics  learning  and  that  the  teacher  from  the  Weizmann 
Institute  would  be  teaching  them  once  a  week  on  a  regular  basis.  The 
general  impression  obtained  from  teachers  and  consultants  was  that  pupils 
easily  accepted  this  situation.  Open-ended  questionnaire  responses  from 
pupils  in  three  classes  indicated  that,  in  two  of  the  three,  reactions  were  very 
positive  and  many  pupils  showed  great  enthusiasm,  commenting  that  having 
two  teachers  was  more  interesting,  made  the  material  easier  to  understand, 
and  reflected  a  more  serious  attitude  in  the  school  towards  the  importance  of 
learning  mathematics.  In  the  third  class,  pupils  also  had  positive  comments 
to  make  but  many  of  them  complained  that  the  pace  of  instruction  was  too 
quick  for  them  and  expressed  a  preference  for  their  own  teacher  who  they 


felt  was  sufficiently  capable  to  teach  them  on  his  own  (a  comment  supported 
by  the  consultant  herself). 

Teacher  reactions."  Teachers  who  were  interviewed  towards  the  end  of 
the  first  year  expressed  satisfaction  with  the  arrangement.  They  commented 
that  working  together  increased  creativity  and  resulted  in  better  worksheets 
and  examination  forms.  They  felt  that  by  working  with  a  co-teaching 
consultant  they  had  learned  to  better  apportion  instructional  time  among 
curricular  topics. 

Teachers  felt  that  observing  the  consultant  in  the  classroom  was 
particularly  useful.  On  a  questionnaire  administered  to  all  project  teachers, 
the  observation  of  a  lesson  given  by  a  consultant  was  the  highest  rated 
project  activity,  receiving  an  average  rating  of  4.22  out  of  5  on  usefulness. 
Teachers  commented  that  observing  the  pupils  from  the  side  made  them  see 
the  class  differently  and  gave  them  greater  insight  into  classroom  dynamics 
and  individual  pupil  difficulties.  In  addition,  many  of  the  teachers,  after 
watching  tne  consultant  give  a  lesson,  expressed  amazement  at  seeing  their 
pupils  achieve  higher  levels  of  comprehension  than  they  had  previously 
thought  them  capable  of  reaching. 

It  is  particularly  significant  that  all  teachers  who  were  asked  to 
participate  for  a  second  year  raised  no  objections;  rather  they  expressed 
satisfaction  with  the  idea.  One  teacher,  who  co-taught  with  a  consultant  last 
year  in  Grade  10  and  this  year  in  Grade  11,  has  already  requested  to 
continue  next  year  with  a  consultant  in  Grade  12.  She  feels  that  if  she  co- 
teaches  once  at  each  grade  level,  then  she  will  be  prepared  to  work  on  her 
own  in  these  classes  in  the  future. 

Consultant  reactions.  On  the  whole,  consultants  felt  comfortable  with  the 
co-teaching  approach  and  were  ✓ery  satisfied  with  their  close  involvement  in 
classroom  instruction.  Only  one  consultant  expressed  some  discomfort 
insofar  as   she   felt  that  the   teachers  she  was  helping  were  already  good 




teachers  and  really  did  not  require  such  intensive  assistance.  Talks  with 
consultants  revealed  that  they  strongly  believed  in  this  form  of  intervention, 
noting  that  teachers  with  whom  they  had  co-taught  during  the  first  year  had 
made  significant  improvement  which  carried  over  into  the  following  year. 
Changes  occurred  particularly  with  respect  to  lesson  planning  (greater 
thought  given  to  goals,  structure,  and  pacing)  and  the  ability  to  design  better 

Consultants  felt  that  several  factors  made  their  entry  into  the  classes  as 
co-teachers  acceptable  to  both  teachers  and  pupils.  First  of  all,  explanations 
given  to  both  groups  at  the  start  of  the  school  year  emphasized  the  joint 
responsibility  of  the  co-teachers  for  the  class.  The  pupils  easily  accepted  this 
situation  as  natural  and  teachers  did  not  feel  that  their  self-esteem  had  been 
harmed.  Secondly,  most  of  these  classes  were  plagued  by  severe  discipline 
problems  and  the  addition  of  another  teacher  was  generally  viewed  with 
relief  by  most  regular  teachers  who  were  only  too  happy  to  share  their 
problems  with  someone  else. 

Although  teachers  were  presented  as  equal,  pupils  however  sometimes 
perceived  the  consultant  as  the  more  expert  and  saved  up  questions  to  be 
asked  during  the  consultant's  lesson.  The  consultants  did  not  feel,  however, 
that  this  was  a  problem  for  the  teachers.  They  reported  that  since  many  of 
these  teachers  were  relatively  new  to  the  profession  or  to  teaching  these 
grade  levels  with  an  up-dated  curriculum,  they  tended  to  feel  unsure  of 
themselves  and  help  from  the  consultants  was  welcomed. 

As  noted  by  Fullan  (1982),  Sarason  (1982)  and  others  concerned  with 
educational  change,  resistance  to  change  efforts-  is  to  be  expected  and 
perfectly  natural  in  the  transition  to  new  modes  of  behavior.  Moreover, 
teachers  who  arc  normally  left  alone  in  their  classrooms  do  not  usually  take 
favorably  to  direct  interference  in  the  management  of  their  "territory". 
Under  the  circumstances,  it  is  almost  surprising  that  the  responses  have  been 

1  -80 

thus  far  so  positive  to  the  co-teaching  form  of  intervention  which  entails 
intensive  "meddling"  in  the  teachers"  territory. 


Ben-Chaim,  D.  &  Carmeli,  M.  (1990).  A  survey  of  mathematics  and  science 
Teaching  in  the  northern  Negcv.  Rehovot:  Department  of  Science  Teaching, 
The  Weizmann  Institute  of  Science. 

Fullan,  M.  (1982).  The  meaning  of  educational  change.  New  York:  Teachers 
College  Press. 

Goodlad,  J.I.  (1984).  A  place  called  school.   N.Y.:  McGraw-Hill. 
Sarason,  S.B.    (1982).    The  culture  of  the  school  and  the  problem  of  change. 
Boston:  Allyn  &  Bacon. 


1  -81 


Janette  Bobls,  Ma  '.ln  Coopar,  John  Sweller 
Un1vars1ty  of  New  South  Weles,  Sydnay,  Australia 

The  results  of  three  experiments  indicate  the  Inadequacy  of  some 
conventionally  formatted  instructional  material  and  emphasize  tha 
debilitating  effect  redundant  material  can  have  during  Initial  Instruc- 
tion.   Elementary-school  children  learning  a  simple  paper-folding  taak 
learned  more  effectively  from  instructional  material  using  diagrams 
alone  than  from  material  containing  redundant  verbal  material,  and 
self-explanatory  diagrams  with  redundant  material  eliminated  were 
superior  both  to  instructions  containing  Informational ly  equivalent 
text  and  to  instructions  consisting  of  redundant  diagrams  end  text. 
This  redundancy  effect  was  evident  not  only  when  text  was  redundant 
to  diagrams,  but  also  when  information  was  conveyed  solely  by  meant  of 
diagrams.    These  findings  extend  the  generality  of  the  redundancy 
effect  and  have  important  implications  for  teaching. 

Printed  Instructional  material  1n  subject  areas  such  as  mathematics  and  physics 
typically  use  text  and  diagrams.    Traditionally,  eapeclally  1n  textbooks,  tha  text 
and  the  Illustrations  are  presented  1n  a  separated  format,  usually  s1de-by-»1de.  It 
has  been  shown  1n  a  number  of  subject-areas   that  Integration  of  text  and  diagrams 
enhances  learning.  (Chendler  and  Sweller,  1991;  Sweller,  Chandler,  Tlerney  end 
Cooper,  1990;  TarmUi  and  Sweller,  1988;  Ward  and  Sweller,  1990).    In  terms  of 
cognitive  load  theory,  the  act  of  splitting  attention  between  and  then  mantel ly 
Integrating  textual  and  diagrammatical  material  preaented  1n  the  traditional  format 
Imposes  an  unnecessary  cognitive  load  and  reducea  cognitive  resources  avelleble  for 
learning.    Comparison  with  modified  Instructional  material  Incorporating  phyalcelly 
Integrated  text  and  Illustrations  generetes  the  split-attention  effect. 

The  effect  occurs  only  when  the  text  end  1llustrat1ona  are  unintelligible  1n 
Isolation.    Both  text  and  diagram  are  necessary  for  the  Information  to  be 
understood.    Sometimes,  however,  a  procedure  can  be  learned  from  diagrams  alone,  any 
accompanying  text  being  Irrelevant,  or  "redundant".    In  such  caaas,  atudenta  tend  to 
Ignore  the  redundant  text,  and  pay  attention  aolely  to  the  dlegrams.  Through 



Integration  of  text  with  diagrams,  however,  studants  can  ba  forced  to  pay  attantlon 
to  the  redundant  text  even  though  diagrammatic  material  1s  sufficient  by  Itself.  It 
has  been  shown  1n  several  contexts  (Chandler  and  Sweller,  1991)  that  1n  such  cases 
learning  1s  less  efficient  than  when  students  are  able  to  Ignore  the  text.    This  1s 
referred  to  as  the  redundancy  affect. 

In  the  three  experiments  reported  here,  the  effect  on  learning  of  both  redundant 
text  and  redundant  diagrams  (each  with  respect  to  diagrams)  was  examined  1n  the 
context  of  a  paper-folding  task.    This  task  1s  found  1n  the  "space"  strand  of  many 
elementary-school  curricula  and  consists  of  folding  a  circular  paper  disk  according 
to  a  sequence  of  Instructions  until  a  triangular  shape  1s  obtained.    Each  experiment 
consisted  of  two  phases:  an  acquisition  phase  1n  which  subjects  learned  the  task  by 
means  of  the  instructional  material  provided,  and  a  testing  phis*  1n  which  they 
carried  out  the  task  without  aids.    The  only  difference  between  the  treatment  groups 
was  the  format  of  the  Instructional  material  used  1n  the  acquisition  phasa. 

In  each  experiment,  children  were  treated  singly.    Each  subject  was  asked  to  use 
the  Instructional  material  as  an  aid  to  learning  the  task.    The  time  needed  for  this 
acquisition  was  recorded.    The  subject  was  then  given  a  paper  disk  and  asked  to 
carry  out  the  task  without  aids,  tha  time  taken  to  complete  the  task  and  the 
accuracy  with  which  1t  was  performed  being  recorded.    In  each  phasa,  a  time  of  ten 
minutes  was  recorded  for  subjects  who  over-ran  this  time. 

Experiment  1     Two  sets  of  Instructional  materials  were  used  in  the  acquisition 
phase:  a  sequence  of  diagrams  Intelligible  by  themselves  ("diagrams-only"  format), 
and  the  same  sequence  of  diagrams  accompanied  by  written  Instructions  that  referred 
to  the  diagrams  ("redundant"  format)  [see  Figure  1j.    These  written  instructions 
were  redundent  to  the  diagrams  but,  unlike  the  diagrams,  were  unintelligible  1n 
Isolation.    Because  LeFevre  and  Dixon  (1988)  have  Indicated  that  subjets  are 
Inclined  to  rely  on  example  Information  (espedelly  1f  1t  1s  1n  diagrammatic  form) 
and  to  Ignore  written  Instructions,  1t  was  stressed  to  each  subject  1n  tha 



"redundant"  group  that  the  written  materials  must  be  read.    It  was  thought  that 
children  using  Integrated  instructional  Material  with  extraneous  Information 
eliminated  (r«o"t)  would  leern  more  effectively  than  those  using  a  format  that 
Includes  redundant,  but  not  self-sufficient,  written  material  (non-rd). 

The  mean  times  in  seconds,  and  the  percentages  of  subjects  correctly  completing 
the  task,  are  presented  for  both  phases  1n  the  following  table. 

acquisition  phase 

testing  phase 

N=15  for 



X  correct 



X  correct 

*ach  group 

















The  values  of  t  for  the  comparisons  of  mean  times  were  t= 1 . 23  (ns)  for  the 
acquisition  phase  and  £=1.90  (p<0.05)  for  the  testing  phase.    Comparison  of 
percentages  correct  by  means  of  Fisher's  exact  test  with  Overall's  correction 
yielded  a  s1gn1f1cent  difference  (p:0.03)  for  the  acquisition  phase,  but  not  for  the 
testing  phase.    The  superiority  of  the  non-redundant  group  provides  evidence  for  the 
redundancy  effect. 

Experiment  2     In  this  experiment,  the  effect  of  self-sufficient  diagrams  alone  and 
self-sufficient  verbal  Instructions  alone  were  compared.    In  addition,  Instructions 
for  a  "redundant"  third  group  were  constructed  by  presentation  to  students  of  both 
diagrammatic  and  textual  material.    Since  the  third  step  1n  Experiment  1  had  proved 
difficult  for  many  children,  1t  was  subdivided  Into  three  steps  for  Experiment  2, 
the  accuracy  of  the  representation  being  Important  1f  children  are  to  construct  an 
accurate  mental  Image  (Johnson-Laird,  1963).    Thus,  three  experimental  groups  were 
used,  each  using  a  different  set  of  instructions:  diagrams-only  format,  text-only 
format,  «nd  d1agrams-and-text  format  [see  Figure  2],  the  latter  being  a  "redundant 
format"  because  the  parts  wore  Intelligible  alone.    Cognitive  load  theory  postulates 
that  the  use  of  the  redundant  format  will  have  a  debilitating  effect  on  learning  in 
comparison  with  the  use  of  a  diagrams-only  format.    Both  textual  Information  and  the 


1  -84 

redundant  format  are  difficult  to  process  for  different  reasons  and  no  expectation 
could  be  stated  with  regard  to  their  relative  efficiency.    It  was  expected  that 
children  using  diagrams  only  (dlag)  would  learn  more  effectively  than  those  using 
1nformat1onally-equ1valent  written  instructions  (text)  and  those  using  a  format  1n 
which  the  same  diagrams  are  accompanied  by  redundant,  Informatlonally-equlvelent 
written  Instructions  (red),  and  that  children  using  text  only  would  learn  better 
than  those  having  redundant  Information. 

The  mean  times  1n  seconds,  and  the  percentages  of  subjects  correctly  completing 
the  task,  are  presented  for  both  phases  1n  the  following  table.    There  were  fifteen 

acquisition  phase 

testing  phase 

mean  time 
dlag     red  text 
387.3  462.6  518.5 

X  correct 
dlag  red  text 
66.7  33.3  33.3 

mean  time 
dlag     red  text 
253.3  455.1  424.5 

%  correct 
dlag  red  text 
66.7  26.7  33.3 

subjects  1n  each  group.    The  data  were  analyzed  by  means  of  planned  orthogonal 
contrast  tests  using  P-tests  1n  the  cese  of  the  mean  times  and  a  test  for 
homogeneity  of  binomial  proportions  (Harascullo,  1975)  In  the  case  of  the 
percentages  correct.    For  the  acquisition  phase,  the  values  of  F  were  3.07  (ns)  for 
the  dlag  vs  combined  red+text  contrast  and  0.03  (ns)  for  the  red  vs  text  contrast; 
for  the  testing  phase,  the  respective  ^-values  were  5.29  (p<0.05)  and  0.11  (ns). 
The  values  of  the  test  statistic  for  the  same  contrasts  based  on  percentages  correct 
were  respectively  5.00  (p<0.05)  and  0.00  (ns)  for  the  acquisition  phase,  and 
respectively  6.10  (p<0.05)  and  0.16  (ns)  for  the  testing  phase.    These  results 
Indicate  that  the  diagrams-only  format  was  superior  to  the  other  formats. 
Experiment  3     To  date,  Investigations  of  the  redundancy  effect  have  concentrated  on 
redundant  text.    Experiment  3  wes  designed  to  examine  the  effect  of  redundant 
diagrams.    The  diagrams-only  Instructional  materials  of  Experiment  2  were  compared 
with  a  modified  version  of  these  materials,  in  which  "back  views"  were  provided  for 
some  steps  [see  Figure  3].    Although  It  might  be  thought  Intuitively  thet  this  extra 



information  would  enhance  learning,  it  My  be  hypothesize  from  cognitive  led 
theory  that  the  Inclusion  of  these  extraneouo,  redundant  diagrams  Imposts  extra 
cognitive  load  a,id  has  a  Militating  effect  on  learning.    It  was  thought  that 
children  using  diagram,  only  Wv's)  would  Tarn  more  effectively  than  those  using 
a  format  In  which  the  same  diagrams  are  accompanied  by  redundant  diagrams  (r*'nt>. 
The  following  table  shows  mean  times  1n  seconds,  and  percentages 

acquisition  phase 

mean  time 



%  correct 




testing  phase 

mean  time 



%  correct 


red 'lit 

of  subjects  correctly  completing  the  task  for  both  phases.    The  values  of  t  for  com- 
parisons of  mean  tlmas  were  t=1.92  (p<0.05)  for  the  acquisition  and  t=2.12  (p<0.05) 
for  the  testing    Thus,  the  diagrams-only  format  was  superior  to  the  format  1n 
which  the  same  diagrams  were  accompanied  by  redundant  diagrams.    Comparison  of 
percentages  correct  using  Fisher's  exact  test  with  Overall's  correction  yield*  no 
significant  difference  for  either  phase. 


The  results  indicate  the  Inadequacy  of  conventionally  formatted  Instructional 
material  and  emphasize  the  debilitating  effect  redundant  material  can  have  during 
initial  instruction.    Experiment  1  demonstrated  the  advantage  of  presenting 
information  with  redundant  material  removed.    The  additional  text  provided  for  the 
group  studying  the  redundant  format  had  an  inhibitory  effect  rather  than  assisting 
comprehension,  as  1s  the  normal  Intention. 

The  findings  of  Experiment  2  suggest  that  self-explanatory  diagrams  with 
redundant  material  eliminated  are  superior  to  Instructions  containing 
informational ly  equivalent  text  and  to  Instructions  consisting  of  redundant  diagrams 
and  text. 


1  -86 

Exp.r1-.nt  3  d«on,tr.t.d  thrt  th.  rrtund.ncy  .ff.ct  1,  evident  not  only  when 
text  1s  redund.nt  to  dlMr—.  but  .Iso  wh.n  Information  Is  conveyed  ,0,.,y  by  m..n, 
of  diagram,;  this  .xt.nds  the  generality  of  th.  redundancy  effect.  Providing 
addition,,  diagrams  dat.nmg  a  persp^tlv.  necessary  for  the  successful  completion 
of  the  task  proved  to  Inhibit  rather  that  facilitate  learning. 

For  teaching  and  learning,  the  Implication,  of  these  findings  bear  on  the  manner 
in  which  initi.!  Instruction.,  materia,  1,  pr,„nt,d.  ,hou,d  .x.rds. 
extreme  care  when  providing  .tudents  with  .ddltlon.,  and  seemingly  u.efu, 
1nform.t1on.    if  the  proce.slng  of  .ddltlon.,  Information  (whether  it  be  textual  or  with  .ss.ntl.l  Information  Impos.s  .n  .xtr.neou,  cognitive  led,  n 
may  a  detrimental  rather  than  beneficial  effect  on  learning.    For  optimum 
effect,  the  usefulness  of  additional  Information  must  outweigh  the  consequence,  of 
processing  It. 


J0$=c.:^  seas si  ^^^f^s-^r — 

Vt*r.',~£'m:  °°  Wr1U,n  "~  -ample.,  common  .nd 

M%T280-290  (1M6>:  L*r«-""»'«  comp.ri.on,.  Psychological  BuU.tin, 

HTX\%s"  Srai-^7):  *»»™tric  .no  Oistrmtlon-rr.. 

«*  ,o.d  theory. 

Psychology:  Central,  119,  176-192  or  txpenmntal 




0.  you  »rt  glvin  a  circli. 

5  Vou  now  hovt  a  triangle 

•i  fold  the  top  Inar.glfe  bock  as  sho*r, 
so  that  bolh  Inanlqes  lie  flat  on  (he 



3  Fold  back  the  edges  of 
the  circle  along  the 
broken  lines 

I  Fold  along  th>  broken  lint 

2.  Fold  along  tht  broktn  Itnt  so 
thai  tht  Itf  t  tldt  of  tht  shape 
(Its  txactly  on  top  of  the  right 

Figure  1:    Redundant  format  Instructional  material  for  Experiment  1 





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1  -89 


Paolo  Booo.Lora  Shapiro  -  Dipartiroento  di  Matemaoca .  Universe  diGenova 

The  report  concerns  the  outcomes  of 1023  pupils ,  aged  9-13  from  two  different  instructional  settings, 
on  a  problem  involving  several  variables  and  solution  strategies.  Quantitative  and  qualitative  analyses 
have  been  performed  to  assess  the  dependence  of  the  strategies  produced  by  pupils  on  age  x 
instruction  and  numerical  data.  The  findings  from  this  study  have  led  to  some  interesting  interpretations 
regarding  the  students'  transition  to  pre-algebtaic  strategies  and  (he  associated  mental  processes. 

1.  Introduction 

In  recent  years  there  has  been  an  extensive  array  of  studies  which  have  investigated  students'  problem  solving 
strategies  in  considerable  depth  (see  Harel&  C,  1991;  Hershkovitz&  Nesher,  1991;  Lesh,  1985  ;  Nesher& 
Hershkovitz.  1991:  Reusser.  i990:  Vergnaud,  1988) 

In  an  attempt  to  add  to  this  body  of  research  ,  the  study  to  be  reported  on  has  explored  the  mental  processes 
underlying  the  strategies  produced  by  students  aged  9-13  when  solving  a  non-  standard  contextually  realistic 
problem  involving  multiple  variables,  operations  and  solution  strategies  ("trial  and  error"  strategies,  mental 
calculation  strategies,  "prc-algebraic  strategies"  •  fora  definition,  see  par.4-.... ). 

The  analysis  of  data  focuses  on  quantitative  and  qualitative  aspects  of  the  evolution  of  strategies  on  the  same 
problem,  with  respect  to  age  and  instruction,  and  the  dependence  of  strategies  on  the  numerical  data. 
A  preliminary  review  of  data  suggests  that  pupils  who  are  encouraged  to  perform  a  variety  of  strategies  ("trial  and 
error",  hypothetical  reasoning...)  without  rigid  formalization  and  schematization  requests  reach  the  point  of  transition 
to  "pre-algebratc"  strategies  earlier  than  those  following  more  traditional  instruction. 

These  results,  along  with  some  additional  qualitative  analyses  of  the  protocols,  bring  to  light  some  understanding  of 
the  roots  of  "prc-algebraic"  strategies  (with  connections  with  research  findings  in  the  domain  of  pre-algebratc 
thinking:  see  par.S). 

2.  The  research  problem 

The  purpose  of  this  study  was  to  better  understand  the  mental  processes  ( i.e.  planning  activities,  management  of 
memory ...).  underlying  students'  problem  solving  strategies  in  a  "complex"  situation.  Towards  this  end  the 
following  problem  was  administered: 

"With  T  liras  for  stamps  one  may  mail  a  letter  weighing  no  more  than  M  grams.  Maria  has  an  envelop  weighing  E 
grams:  how  many  drawing  sheets .  weighing  S  grams  each,  may  she  put  in  the  envelop  in  order  not  to  sumwunt 
(with  the  envelop)  the  weight  ofM  grams  ?" 
Various  numerical  versions  have  been  proposed  to  different  classes: 

needed  (C) 


admissible  weight(M) 

weight  of  the 



weight  of  each 
sheet  of  paper  (S) 











This  problem  was  chosen  because  it  represented  a  realistic  situation  for  most  of  the  students  in  this  age  range 
(interviews  with  a  sample  of  Vth  graders  showed  that  over  75%  of  them  think  that  the  "cost"  to  send  a  letter  must 
depend  on  its  weight).  In  addition,  it  was  possible  to  choose  numerical  values  which  kept  into  account  the  feasibility' 
of  mental  calculations  and  the  number  of  iterations  needed  to  reach  the  result  through  progressive  approximation  from 
below.  For  the  transition  from  (50.7,8  )  to  (100.7,8)  there  is  an  increase  of  iterations  needed  in  an 
"approximation  from  below"  strategy  ,  but  mental  calculations  are  still  easy  .In  the  transition  from  (50,7,8)  to 
( 100, 14, 16 )  the  mental  calculations  become  more  difficult,  but  the  number  of  iterations  needed  remains  the  same.  In 
the  version  with  values  (250. 14. 16)  the  mental  calculations  are  yet  more  difficult  and  the  number  of  iterations  is 
increased.  Finally,  the  problem  format  was  similar  to  "evaluation  problems"  proposed  in  the  Italian  primary  and 
comprehensive  schools  (multiple  choices  tests  are  not  frequently  utilized),  but  no  such  problem  had  ever  been 
proposed  to  the  students  before. 

A  pilot  study  was  conducted  at  the  end  of  1 990  with  two  classes  of  students  in  grades  IV ,  V  and  VIII.  The  research 
problem  with  different  numerical  versions  was  administered  and  the  findings  were  utilized  to  make  subsequent 
choices,  regarding  the  appropriatedncss  of  the  numerical  versions  with  respect  to  grade  level.  In  particular,  the 
(250. 14, 16)  case  was  excluded  for  the  IVth-graders ,  due  to  the  difficulties  encountered  by  many  subjects:  and  the 
(50,7,8)  case  was  not  given  to  the  Vlll-graders,  due  to  a  great  number  of  solutions  written  without  any  indications 
of  the  strategies  performed . 

For  this  study  63  IV.V.VI  and  VIII  grade  classes  were  chosen  from  schools  in  the  north -west  region  of  Italy,  and 
this  resultcs  in  a  total  of  1 023  participants.  The  study  was  conducted  in  October  and  early  November  1 99 1  (after 
about  4/6  weeks  from  the  beginning  of  the  school  year) .  The  classes  were  divided  into  two  groups  and  were  chosen 
in  older  to  assure  a  similar  sociocultural  environment  between  them  and  suitable  conditions  for  the  experiment  .In 
addition,  in  these  classes  age  corresponded  well  with  the  grade  level. 

The  classes  in  Group  I,  hereafter  called  the  "Project"  classes,  included  grades  IV  and  V  which  are  currently 
involved  in  a  long  term  instructional  innovation  in  the  Genoa  Group  for  primary  school  .The  characteristics  of  this 
project  which  arc  relevant  for  this  research,  are  presented  in  Boero(  1 989)and  in  Boero  &  Ferrari  &  Ferrero  ( 1989) 
and  are  summarized  below: 

-  the  written  calculation  techniques  are  progressively  costructed  ,  under  the  guidance  of  the  teacher,  starting  from 
the  strategies  spontaneously  produced  by  pupils.This  allows  a  great  deal  of  "trial  and  error"  numerical  strategies  to 
be  performed  by  pupils,  especially  at  grade  II  and  grade  III 

-  smdcnts.fromthecrKlofgradel.arerequiredtoprovide  verbal  written  representation  of  their  strategies 

-  algebraic  notation  for  an  arithmetic  operation  is  introduced  only  when  the  meaning  of  the  particular  arithmetic 
operation  is  mastered  by  majority  of thcclass.Nodirect  explicit  pressure  is  exerced  by  teachers  for  the  students  to 
give  formal  representation  of  operations  with  algebraic  signs  ("  words  and  numbers"  resolutions  are  admitted  up  to 
the  end  of  the  primary  school ) 

-  in  situations  in  which  the  students  perform  different  strategies,  comparisons  of  strategies  (and  formalizations)  are 
organized  and  discussed  (see  Bondcsan  &  Ferrari,  1991) ;  and. 

•  problems  involving  more  than  one  operation  are  proposed  without  intermediate  questions. 

3.  Method 

1  -91 

The  classes  in  the  second  group  were  composed  of  grades  VI  and  VIII  where  the  Vlth  graders  come  from 
"traditional"  primary  school  classes .  Here  "traditional"  instruction  means: 

-  multiple  operations  problems  guided  by  intermediate  questions  are  widely  proposed  from  the  III  to  the  V  grade, 

-  "trial  and  error  "or  other  case-by-case  strategies  are  not  encouraged, 

-  standard  written  calculations  techniques  are  introduced  early,  and 

-  early  formalization  of  arithmetic  operations  (with  +  ,-,x,  :  signs)  is  introduced  and  rapidly  demanded  as  a 
standard  code  in  problem  solving  (for  single  operations). 

In  Group  II  we  have  decided  to  select  only  VI  and  VIII  grade  classes  with  mathematics  teachersaffiliated  to  the 
Genoa  group  and  working  on  a  parallel ,  similar  research  project  for  the  comprehensive  school  forthe  following 
reasons,  emerging  from  our  pilot  study: 

-  difficulty  to  propose  our  problems  at  the  beginning  of  the  IV  grade  in  "traditional"  classes  (because  the 
"subtraction  and  division"  problems  are  normally  proposed,  during  the  grade  III .  only  as  two-steps  problems) 
-difficulty  to  entera  "foreign"  class  and  get  completely  verbally  expliciled  resolutions(thisisnot  frequent  in  Italy: 
it  is  requested  only  to  indicate  the  most  important  calculations  performed). 

-for  the  comparison  between  VI  graders'  and  VIII  graders'  performances,  it  was  suitable  to  keep  sociocullural 
variables  unchanged 

-the  VHIth  graders  of  the  chosen  classes  had  not  yet  explicitely  been  involved  with  equations 
The  following  table  shows  the  distribution  of  the  population  involved  in  the  study: 


n.  of  classes       n.  of  pupils  n.  of  classes  n.  of  pupils 

grade  IV  9  145  -  - 

grade  V  24  396 

grade  VI  -  -  26  406 

grade  VIII  4  76 

A  sample  analysis  of  the  primary  school  copybooks  of  V  grade  and  VI  grade  pupils  belonging  to  the  classes 
involved  in  this  research  showed  these  relevant  differences 

-  the  written  calculation  techniques  were  introduced  6-8  months  in  the  primary  school  "Project"  classes  later  than  in 
parallel  "traditional"  classes 

-  algebraic  notations  were  introduced  in  the  "Project"  classes  10-15  months  later  than  in  parallel  "traditional  "classes 
(for  instance,  the  sign  "  -  "was  not  introduced  in  the  project  classes  before  the  second  term  of  grade  II ;  the  sign  ":" 
was  not  introduced  before  the  second  term  of  grade  111 ,  and  after  at  least  one  year  of  work  on  division  problems) 

-  20  to  25  "subtraction- division  problems"  were  proposed  in  the  "project"  classes  before  the  V  grade  (generally 
without  an  intermediate  question  )  while  40  to  52  subtraction/division  problems  were  proposed  in  "traditional" 
classes  from  grade  III  to  grade  V  (  18  to  34  problems  with  the  two  step  structure  :  first  question  asking  for  a 
subtraction,  second  question  asking  for  a  division). 

The  problem  was  proposed  by  the  teacher,  with  an  "observer"  present  in  the  classroom  Up  to  the  day  of  the  study, 
the  problem  was  unknown  to  the  teacher .  In  order  to  avoid  any  difficulties  in  the  classroom,  if  a  pupil  met  with 
serious  problems  and  insistently  asked  for  help  .  he  was  helped  by  the  teacher  who  provided  w  riilcn  suggestions  on 




the  paper  (this  fact  remained  registered  on  the  pupil's  sheet  of  paper  and  these  protocols  were  excluded  from  the 
following  analyses).Pupils  were  asked  to  use  only  the  sheet  of  paper  on  which  the  text  of  the  problem  was  written. 
Some  short  interviews  were  performed  (after  the  resolution  of  the  problem)  by  the  observer  with  pupils  who  had 
written  very  concise  text,  and  with  pupils  who  had  adopted  the  "pre-algebraic"  strategy  in  the  cases  (50.7 .8-)  and 
(100,7,8),  in  order  to  understand  their  reasoning  and  motivation. 

4.  Results 

The  students'  strategics  resolutions  have  been  analysed  according  to  a  classification  scheme  suggested  by  the  data 
from  pilot  study,  and  corresponding  to  the  aim  of  exploring  mental  processes  underlying  strategies. 
Strategies  were  coded  in  the  following  manners: 

"Pre -algebraic  "  strategies  (PRE-ALC).  In  this  category  the  strategies  involved  taking  the  maximum  admissible 
weight  and  subtract  the  weight  of  the  envelop  from  it.The  number  of  sheets  is  then  found  multiplying  the  weight  of 
one  sheet  and  comparing  the  product  with  the  remaining  weight,  or  dividing  the  remaining  weight  by  the  weight 
of  a  sheet  of  paper ,  or  through  mental  estimates. If  the  problem  would  be  represented  in  algebraic  form,  these 
strategies  would  correspond  to  transformations  from  : 

Sx  +  E  s  M  to:SxsM-E,  up  to:  x-(M-EyS 

For  the  purposes  of  this  research,  we  have  adopted  the  denomination  "pre-algebraic"  in  order  to  put  into  evidence 
two  important ,  strictly  connected  aspects  of  algebraic  reasoning,  namely  the  transformation  of  the  mathematical 
structure  of  the  problem  ("reducing''  it  to  a  problem  of  division  by  performing  a  prior  subtraction) ;  and  the  discharge 
of  information  from  memory  in  order  to  simplify  mental  work  .  This  point  of  view  is  connected  to  researches 
performed  in  recent  years  in  the  domain  of  pre-algebraic  thinking  (sec  par.5) 

"Envelop  and  sheets"  strategies  (ENV&SH).This  "situational"  denomination  was  chosen  by  us  because  it  best 
represented  students'  production  of  a  solution  where  the  weight  of  the  envelop  and  the  weight  of  the  sheet  arc 
managed  together .  These  strategies  include  "mental  calculation  strategies",  in  which  the  result  is  reached  by 
immediate  simultaneous  intuition  of  the  maximum  admissible  number  of  sheets  with  respect  to  the  added  weight  of 
the  envr'op;  "trial  and  error"  strategies  in  which  the  solution  is  reached  by  a  succession  of  numerical  trials ,  keeping 
into  account  the  results  of  the  preceding  trials  ( for  instance ,  one  works  on  the  weight  of  some  number  of  sheets  and 
adds  the  weight  of  the  envelope ,  checking  for  the  compatibility  with  the  maximum  allowable  weight ) :  "systematic 
approximation  from  below  strategies",in  which  the  result  is  reached  progressively  incrementing  the  number  of 
sheets,  adding  the  weight  of  the  envelop  and  checking  if  the  maximum  allowable  weight  is  reached  or  not: 
"hypothetical  strategies",  in  which  one  keeps  into  account  the  fact  that  the  weight  of  one  sheet  is  near  to  the  weight 
of  the  envelop,  and  thus  hypothesizes  that  the  maximum  allowable  weight  is  filled  by  sheets,  and  then  decreases  the 
number  of  sheets  by  one  and  so  on 

UNCLASSIFIED  .While  almost  all  of  the  strategies  fit  well  in  the  above  two  categories,  there  were  some  which 
were  difficult  to  interpret  .For  example,  the  transition  from  ENV&SH.  to  PRE-ALG.  strategies  during  the 
resolution  process  (especially  with  the  numerical  versions  (100,14. 16)  and  (250. 14.16)  (example:  "16  +  14*30. 

30+16-46    .   100-14-86,  86:16  '5  ":  mixed  strategies  :  "16+14-30  : 100-30- 70:  70:16-4  4+1-? 

sIkcis"  .  etc.  These  "ambiguous"  cases  were  thus  coded  as  "unclassified"(  the  whole  number  of  unclassified  proofs 
was  about  3%  of  the  whole  group). 


1  -93 

INVALIDAnotherprobkm  concerned  the  classification  of  incorrect  resolutions 

in  which  solutions  were  lacking,  or  completely  incoirect(  for  instance,  stnuegies  involving  the  amount  of  money 
...  divided  by  the  weight  of  a  sheet  of  paper)  ;  forgetting  the  weight  of  the  envelope;  or  when  numerical  mistakes 
affected  the  final  result  in  a  relevant  manner  ( more  or  less  than  10  times  the  correct  result ). 
In  the  case  of  numerical  mistakes  affecting  only  the  final  result  in  a  "reasonable"way .  or  acritical  presentations  of 
the  results  (for  instance,  under  the  form :  43:8=5.375  sheets)  protocols  were  classified  (on  the  basis  of  the  adopted 
strategy) . 

The  following  tables  represent  a  breakdown  of  the  data: 

GROUP  1  /GR.  IV 
GROUP  1  /GR.V 

TABLE  2:(M.E.SM  1 





71  (49%) 
35  (  38%) 
33  (34%) 

26  (18%) 
35  (38%) 
24  (25%) 

6  (4%) 
5  (5%) 
2  (2%) 

42  (  29%) 
18  (19%) 
37  (39%) 

00.7.8  ) 

GROUP  1  /GR.V 





52  (39%) 
43  (34%) 

50  (38%) 
34  (27%) 

4  (3%) 
3  (2%) 

26  (20%) 
46  (37%) 





GROUP  1  /GR.V 

24  (28%) 

36  (42%) 

4  (5%) 

21  (25%) 


25  (25%) 

32  (32%) 

2  (2%) 

42  (41%) 





GROUP  1  /GR.V 

13  (15%* 

42  (49%) 

6  (7%) 

25  (29%) 


11  (13%) 

34  (41%) 

1  (1%) 

37  (45%) 


9  (12%) 

57  (75%) 


10  (13%) 



1  -94 

It  is  interesting  to  note  from  the  analysis  .of  the  protocols  thai  the  students  in  the  ENV&SH.  strategics  group  who 
probably  derived  their  solutions  from  a  mental  global  estimation  of  the  situation  represent  (in  grades  V  and  VI)  about 
1 7%  of  the  strategies  performed  in  the  (50,7,8)  case,  13%  in  the  ( 100,7,8 )  case  .while  they  are  less  than  3%  in  the 
( 1 00, 14 . 1 6)  case  is  not  easy  to  distinguish  these  strategies  from  the  others  ( for  instance  in  a  protocol  like 
this:  "5x8*40.  40+7*47  ").  especially  for  the  students  in  the  GROUP  2,  who  might  not  have  derived  their 
solutions  from  mental  evaluation ,  but  from  a  succession  of  mental  trials  not  reported  on  the  sheet.  Some  interviews 
confirm  the  ambiguous  character  of  these  kinds  of  protocols .  It  is  also  interesting  to  observe  that  many  usually 
successful  problem  solvers  in  each  age  group  applied  these  strategies  for  (50,7,8 )  and  (100,7,8) ,  while  almost 
all  of them  were  categorized  as  PRE-ALC.  in  the  version  with  the  numerical  dau(250,l4,l6). 

A  qualitative  analysis  of  the  data  suggested  the  following: 

-  generally,  the  text  of  the  PRE-ALG.  strategies  is  linear,  with  subsequent  declarations  about  the  subtraction  and  the 
division  and  the  result  .For  example : 

"  /  subtract  the  weight  of  the  envelop :  250  -14'  236 ,  and  I  find  the  weight  that  may  be  filled  with  drawing  sheets: 
then  I  divide :  236: 1 6 '(calculations)*  14,75.  and  I  gel  the  number  of  sheets:  14" 

-  frequently,  the  text  of  the  ENV&SH.  strategies  is  involved,  especially  in  the  ( 100. 14,16)  and  (250, 14.16)  cases 
(where  a  global,  mental  estimation  of  the  result  is  more  difficult)  .These  are  typical  texts : 

7  multiply  the  weight  of a  sheet  ( 16  grams)  for  a  number  chosen  by  chance,  but  not  surmounting  10.  and  according 
to  the  result  I  multiply  16  for  a  lower  or  a  greater  number.  When  I  get  a  number  which  works  well  I  add  the  weight  ot 
the  envelop  (that  is  14  gmmsl.  if  the  result  exceeds  100 1  make  other  trials,  if  the  number  does  not  exceed  100 1 
have  solved  the  problem...  (■••trials)": 

"/  multiply  the  weight  of one  sheet  by  a  number  of times  such  that  their  weight  is  contained  in  250 g, but  I  cannot 
exceed  the  weight  of 250 g  if  I  add  the  weight  of one  en  velop  to  the  weight  of  the  sheets  admissible  with  the  3800 
liras  fare,  and  sol  must  add  a  certain  number  of  sheets  to  the  weight  of  the  envelop(+\nals)": 
-many  ENV&SH.  protocols  from  the  students  in  the  "Project"  group  reveal  the  students  are  in  close  proximity  to  a 
transition  to  a  PRE-ALG.  strategy  .For  example: 

"I  must  find  the  number ofsheets  which  can  be  set  in  the  envelop  in  order  not  to  overcome 250 g .  but  with  my 
sheets  I  must  not  arrive  to  250 g.  in  order  to  be  able  to  add  the  weight  of  the  envelop  and  not  to  overcome  250. 1 
must  stop  before.  (+trials  up  to  224) 

The  same  kind  of  protocols  is  infrequent  with  students  in  the  "traditional"  classes 

-there  were  3  solutions  in  which  students  began  with  PRE-ALG  and  moved  to  ENV&SH.  strategies .  while 
transitions  in  the  contrary  direction  were  observed  in  21  solutions 

It  should  be  noted,  however.that  it  was  not  easy  to  evaluate  and  compare  the  protocols  from  different  ctasses.bccausc 
of  the  influences  of  different  teaching  styles  .  both  past  and  present  .This  is  especially  true  in  the  case  of  Vlth 
gradcrs.who  had  only  for  weeks  of  instruction  with  the  same  teacher. 

5.  Conclusions  and  discussion 

A  preliminary  review  of  the  results  shows  that  there  is  a  clear  evolution  with  respect  to  age  x  instruction  from 
ENV&SH. .  strategics  towards  PRE-ALG.strategies  within  and  between  numerical  versions  (this  is  found  in 
homogeneous  groups  of  pupils:  transition  from  IV  grade  to  V  grade:  and  from  VI  grade  to  VIII  grade  .  both  with 



classes  of  the  same  schools,  and  teachers  working  in  analogous  marine.  Another  interesting  finding  relates  to  the 
(50,7,8)  case,  where  we  find  strong  increase  of  PRE-ALG.  strategies  forstudents  from  grades  IV  to  V  in  Group 
I ;  this  increase  has  occurred  despite  the  fact  that  a  PRE-ALG.  resolution  may  be  more  expensive  (in  terms  of 
calculations  to  perform)  than  some  convenient  ENV&SH.  strategies.Some  interviews  performed  with  pupils  in 
different  classes  (after  the  test)  showed  plausible  reasons  for  their  passage  to  PRE-ALG.  strategies  (also  in  cases 
asking  for  easy  mental  calculations),  and  these  included  they  felt  more  secure,  it  helped  avoid  confusion  .and  it  gave 
greater  evidence:  "security,  "avoiding  confusion",  "greater evidence".....  are  expressions  frequently  utilized  by 
pupils  to  explain  the  motivation  of  their  choices. 

If  we  also  look  at  the  protocols  of  pupils  who  appear  to  be  ready  to  make  the  transition  towards  a  pre-algebraic 
strategy  in  a  more  difficult  problem ,  for  example  the  students  who  write : 

"I  repeat  1 6  grams  (which  is  the  weight  of  a  drawing sheet)  rill  I reach  100 .  and then  I subtract  14  grams  (the  weight 
of  the  envelop)  and  so  I  must  consider  one  sheet  less"; 

"treason:  16  x .... "  about  86  because  86+14-100 ;  I  count:  16x4-64,  toolittle:  16x6  •96,toomuch:  16.x  5 
"80:  let  us  try  80+14-  94  ": 

"  I  subtract  1 6  grams  from  100  many  times,  each  time  checking  if  it  remains  14  grams  for  the  envelop :  100-  lb 
-84.  yes:  84- 16-68.  yes;  68- 16-52.  yes:  52- 16-  36.  yes:  36- 16-20.  yes  -  and  I  stop,  because  the  envelop  weighs 
14  grams ..." . 

we  see  that  the  motivations  and  access  to  prc-algebraic  strategies  may  be  different:  but  in  all  of  them  there  is  a 
form  of  reasoning  that  may  derive  from  a  wide  experience  involving  production  of  "anticipatory  thinking"  (see 
also  Boero.  1 990) .  That  is  to  say ,  under  the  need  of  economizing  efforts,  pupils  plan  operations  which  reduce  the 
complexity  of  mental  work .  This  interpretation  provides  a  coherence  amongst  different  results,  concerning  the 
evolution  towards  PRE-ALG.  strategics  with  respect  to  age,  as  sho.wn  in  the  solutions  produced  in  grade  IV  to 
grade  V  and  in  grade  VI  to  grade  VIII,  as  well  as  with  respect  to  the  results  involving  more  difficult  numerical  data. 
Indeed,  in  the  ( 1 00, 14, 1 6)  and  (250, 1 4, 1 6)  cases  we  saw  the  difficulties  encountered  by  students  when  attempting 
to  manage  the  weights  of  the  sheets  and  the  envelop  together  (see  par.4) , 

All  this  may  explain  also  why  the  large  experience  of  subtraction/division  problems  presented  as  two  steps 
problems  (with  an  intermediate  question)  in  "traditional"  classes  does  not  seem  to  produce  all  the  desired  effects: 
experiencing  time  separation  of  tasks,  according  to  the  suggestions  contained  in  the  text  of  the  problem,  may  not 
effectively  develop  planning  skills  in  the  same  direction. 

Concerning  research  findings  in  the  domain  of  pre-algebraic  thinking,  we  may  observe  that  there  is  some  coherence 

-  our  results  .  concerning  an  applied  mathematical  word  problem  (  Lesh.  1 985)  .proposed  to  students  prior  to  any 
experience  of  representation  of  a  word  problem  by  an  equation  and  prior  to  any  instruction  in  the  domain  of 

-  Herscovics  &  Lincbewskis  ( 1 99 1 )  results .  concerning  numerical  equations  proposed  to  seventh  graders  prior  to 
any  instruction  in  the  domain  of  equations.  For  instance,  they  find  that  an  equation  like  4n  +  1 7  -  65  is  solved 
performing  4n- 65- 1 7  and  then  n-48:4  by  4 1  %  of  seventh  graders,  while  an  equation  like  I3n  +  I96»  391  is 
solved  in  a  similar  way  by  77%  of  seventh  graders  .This  dependence  of  strategics  or.  numerical  values  is  similar  to 
that  shown  in  our  tables  (compare  data  concerning  sixth  graders  in  the  cases  (50.7.8)  and  (250. 14  16)). 

equations:  and 

1  -96 

Filloy  &  Rojano  define  (for  nvrmirl  "Mfrcrt^  the  "didactic  cut "  "as  the  moment  when  the  child  faces  for  the  first 
time  linear  equations  with  occurrence  of  the  unknown  on  both  sides  of  the  equal  sign"  :  for  applied  mathematical 
word  problems-  a  "didactic  cut"  might  be  considered  when  the  child  faces  for  the  first  time  a  problem  where  a 
separation  of  tasks  (through  an  inverse  operation)  must  be  performed  in  order  to  simplify  mental  work  and  avoid 
"trial  &  error"  methods .  Our  study  gives  some  indications  about  the  consequences  of  two  different  long  term 
instructional  settings  on  students'  efforts  to  overcome  the  obstacle  represented  by  such  a  "didactic  cut". 

Boero.P. ,  ( 1989),  Mathematical  literacy  for  all:  experiences  and  problems.  Proceedings  PME-XUI.  Vol.1. 62-76 
Boero.P.:  Ferrari,  P.L.:  Ferrero.E.,  (1989),  Division  problems:  meanings  and  procedures  in  the  transition  to  a 

written  algorithm.  For  the  Lcarmng  of  Mathematics,  Vol.9. 3. 17-25 
Boero.P..  ( 1 990).  On  lomg  term  development  of  some  general  problem  solving  skills :  a  longitudinal,  comparative 

study.  Proceedings  PME-XTV.  Oaxtpec.  vot.U,  1 69- 1 76 
Bondesan.  M.G. :  Ferrari,  P.  L.  ( 1 99 1 ),  The  active  comparison  of  strategies  in  problem  sol  ving:  an  exploratory  study. 

Proceedings  PME-XV,  Assist ,  Vol.1. 168-175 
Cortes,  A.:  Vergnaud,  Gerard:  Kavaflan,  Nelly,  (1990) ,  From  arithmetic  to  algebra:  negotiating  a  jump  in  the 

learning  process.  Proceedings  PME-XTV.  Vol.n.  Oaxtpec  .27-34 
Filloy,  E.:  Rojano.  T.,  ( 1 984),  From  an  arithmetical  thought  to  an  algebraic  thought.  Proceedings  PME/NA-V1. 

Columbus.  51-56 

Filloy.E.:  Rojano,T.,(  1989).  Solving  equations:  the  transition  from  arithmetic  to  algebra.  For  the  Learning  of 

Mathematics,  vol.  9.2  .  19  -26 
Kieran.  C.  (1989).  The  early  learning  of  algebra:  a  structural  perspective.  Research  Issues  in  the  Learning  and 

Teaching  of  Algebra,  Wagner.  S.&Kieran.C.  (Eds.).  L.E.A.,  Hillsdale.  33-56 
Harel,G.:Behr.M.:Post,T.:  Lesh,R.  (1991),  Variables  affecting  proportionality:  understanding  of  physical  principles. 

formation  of  quantitative  relations,  and  multiplicative  in  variance,  Proceedings  PME-XV,  Assisi,  Vol.11. 1 25- 

Hershkovitz.  S..  Nesher.P.,  ( 199 1 ).  Two-step  problems  -  The  scheme  approach.  Proceedings  PME-XV.  Assist . 
Vol.U,  189-196 

Herscovics.N.:  Linchewski.L. .  (1991),  Pre-algebnic  thinking:  range  of  equations  and  informal  solution  processes 
used  by  seventh  graders  prior  to  any  instruction.  Proceedings  PME-XV,  Assisi,  Vol.  II,  1 73- 180 

Lesh.  R.,  (1985),  Conceptual  analyses  of  mathematical  ideas  and  problem  solving  processes.  Proceedings  PME-9, 
Noordwijkerhout,  73-96 

Nesher,  P.  ( 1988),  Multiplicative  school  word  problems:  theoretical  approaches  and  empirical  findings .  In  J.Hieben 
&  M.Behr(Eds.).  Number  concepts  and  operations  in  the  Middle  Gntdes.  NCTM,  Reston .  141-161 

Nesher,  P. ,  Hershkovitz,  S.  ( 1 99 1 ),  Two-step  problems  -  Research  findings.  Proceedings  PME-XV,  Assisi . 
Vol.111. 65-71 

Reusser.  K.  ( 1990).  From  text  to  situation  to  equation  .  Cognitive  simulation  of  understanding  and  solving 

mathematical  word  problems.  Learning  and  Instruction,  Vot.2.2. 477-498 
Vergnaud.  G.  ( 1988).  Multiplicative  structures.  In  J.Hieben  &  M.Behr(Eds.).  Number concepts  and operations  in 

the  Middle  Grades.  NCTM.  Reston .  141-161 





Studies  of  children's  conceptual  development  and  diagnostic 
teaching  experiments 

Gard  Brekke  Alan  Bell 

Telemark  Laererhogskole  Shell  Centre  for  Mathematical  Education 

ABSTRACT:  The  conceptions  of  children  aged  7-11  of  various  multiplicative 
problems  were  studied  using  interviews  and  written  tests.  In  four-number 
porportion  problems,  changes  from  easy  integer  ratios  to  3:2  and  5:2,  caused  only 
small  falls  in  facility  in  a  familiar  context  (price),  but  considerable  losses  in  a  less 
familiar  context  ■  xrrency  exchange).  Geometric  enlargement  problems  gave  rise 
to  the  wrong  additive  strategy.  A  diagnostic  teaching  experiment  showed 
successful  use  of  the  method  and  materials  by  10116  teachers  after  2  days' 


This  study  has  two  parts.  One  consists  of  an  analysis  of  primary  school  children's 
conceptions  of  multiplicative  word  problems  in  different  contexts.  The  second 
part  was  a  study  of  the  effectiveness  of  a  diagnostic  responsive  teaching  method 
(Bell  et  al,  1985)  and  associated  teaching  material  developed  from  pilot  studies. 

The  16  teachers  involved  attended  a  two  days  in-service  course.  They  were  then 
free  to  choose  how  to  implement  the  teaching  activities,  and  which  activities  to 
pick  from  the  teaching  material.  The  period  of  teaching  in  each  class  was  two 
weeks.  At  least  two  lessons  of  each  class  were  observed  by  the  researcher  to  assess 
the  teaching  style  being  used. 

Results  concerning  childrens  concepts  and  misconceptions 

This  paper  presents  the  results  of  one  group  of  problems.  For  further  details  see 
Brekke  (1991). 

a  122 



The  four-number  problems  (see  below)  vary  with  respect  to  numerical 
relationship  (3  or  4  to  1  and  1.5  or  2.5  to  1)  and  structural  context,  rate  (price), 
currency  exchange,  measure  conversion  and  geometrical  enlargement: 

AL2:  3  sweets  are  sold  for  9  pence.  How  much  for  12  sweets? 
BL2:  4  sweets  are  sold  for  6  pence.  How  much  for  10  sweets? 
AL5:    German  money  is  called  Mark.  John  changed  £3  and  got  9  Marks. 

Sarah  has  £12  to  change  for  Marks.  How  many  Marks  will  she  get? 
AU6:   John  changed  £4  and  got  6  dollars.  Sarah  has  £10  to  change  for  dollars. 

How  many  will  she  get? 
BL5:    Jane  measured  her  book  using  her  short  pencil.  It  was  3  pencils  long. 

Ian  used  his  rubber  to  measure  the  same  book.  It  was  9  rubbers  long. 

Jane  measured  the  table  with  her  pencil.  It  was  12  pencils  long.  Ian  also 

measured  the  table  with  his  rubber.  How  many  rubbers  long  will  the  table 


(This  text  was  accompanied  by  an  illustration). 
AL12:  A  triangle  is  3cm  wide  and  12cm  high.  A  copy  is  made  of  this  triangle,  it 
should  be  9cm  wide.  How  high  must  the  copy  be  to  have  exactly  the  same 
shape  as  the  triangle?  (This  text  was  accompanied  by  an  illustration). 


Percentage  of  correct  answers  and  wrong  additive  strategies  for  the 
four-number  problems 












Wrong  additive  strategy 

















Wrong  additive  strategy 







1  -99 

There  are  not  big  differences  in  facilities  when  ratios  are  changed  from  3:1  or  4:1 
to  1.5:1  or  2.5:1  for  the  most  familiar  context  of  price,  but  when  this  change  is 
combined  with  a  less  familiar  context  of  currency  exchange  there  is  a  considerable 
drop  in  facility.  Children  regress  to  more  primitive  ideas,  in  this  case  the  wrong 
atiditive  strategy.  (Compare  Hart,  1981;  Karplus  et  al,  1983).  Note  also  the 
widespread  use  of  the  wrong  additive  strategy  for  the  measure  conversion 
problem,  which  maybe  compared  with  Karplus'  "Mr  Short  and  Mr  Tall". 

The  idea  of  geometrical  enlargement  was  not  well  understood.  When  asked  to 
make  a  larger  copy  of  the  same  shape  as  a  given  triangle,  the  children  drew 
triangles  which  were  roughly  the  same  shape  but  without  calculating  or 
measuring.  The  problem  of  making  an  enlargement  involves  more  than  the 
pure  numerical  relationship.  Young  children  lack  the  experience  of  linking 
numerical  relationships  with  geometrical  objects. 

There  are  only  small  variations  in  use  of  a  building  up  strategy  across  problem 
structure  for  the  four-number  problems  for  the  younger  children,  and  correct 
answers  were  equally  distributed  between  multiplicative  and  additive  answers. 
The  exceptions  were  the  problems  with  ratios  1.5  and  2.5  where  building  up 
strategy  using  the  internal  ratio  was  the  most  common  correct  method. 

The  context  influences  the  choice  of  scalar  or  functional  operator,  with  scalar 
procedures  being  dominant  for  rate  and  currency  exchange  problems,  while  the 
majority  of  correct  answers  to  the  measure  and  enlargement  problems  are 
obtained  by  a  functional  operator.  Children  tend  also  to  start  by  considering  the 
relationship  between  the  two  units  used  to  measure  the  book,  and  applying  this 
to  the  table.  They  are  working  within  each  object  (book,  table),  while  they  in  the 
previous  items  were  working  within  the  same  measure  (sweet,  pence  &  £,  $). 
Thus  the  dominant  strategy  of  BU6  might  be  categorised  as  scalar.  In  AU11  the 
starting  point  is  the  relationship  established  between  the  shortest  sides  of  the  two 
triangles,  and  is  thus  a  functional  operator,  though  in  this  case  it  is  also  a  scalar, 
since  the  unit  of  measurement  (cm)  is  the  same  for  the  small  and  the  large 
triangles  (compare  Bell  et  al,  1989).  This  preference  for  the  functional 
relationship  for  geometrical  enlargement  problems  is  also  reported  by 
Friedlander,  Fitzgerald  and  Lappan  (1984).  The  dominance  of  the  scalar  operator 



1  - 100 

is  also  reported  in  other  studies  (e.ft  Vergnaud  1983,  Kurth  1988  and  Karplus  et  al 
1983).  Around  85%  of  the  wrong  additive  answers  used  the  external  difference  as 
a  constant  for  addition. 

New  diagnostic  teaching  tasks 

The  teaching  unit  is  based  on  carefully  chosen  problems  from  different  structural 

The  main  objective  of  the  activities  in  Figure  1  is  to  focus  on  the 
inappropriateness  of  adding  a  constant  difference  in  geometrical  enlargements. 
The  full  set  of  activities  also  exemplifies  the  principle  of  starting  with  a  difficult 
problem  (5L)  to  bring  out  the  expected  misconception  and  following  with  an 
easier  activity  (6LZ)  to  give  practice  in  using  the  correct  strategy. 

The  purpose  of  the  'price-line'  activity 



1  - 101 



i  '  mi  <  »»»* 

d3  *  □  on  ill  d 

□  □  □ 

is  to  emphasise  the  need  to  interpret  multiplication  (or  division)  as  a 
dimensionless  scale  factor  along  the  double  number  line  or  as  a  rate  across  the 
number  line. 

Success  and  communicability  of  the  teaching  method 

The  classroom  observations  formed  the  basis  for  assigning  ths  teachers  to  one  of 
three  categories  according  to  the  types  of  interventions  leading  to  different  levels 
of  reflection  on  key  aspects.  These  were:  1)  the  amount  to  which  discussions  of 
misconceptions  were  generated  to  bring  key  issues  to  the  children's  awareness,  2) 
the  demand  for  explanations  and  justifications  of  statements,  3)  the  amount  to 
which  problem  solving  strategies  were  discussed,  4)  the  amount  of  discussion  of 
problem  structure,  (classifying  and  making  problems  of  the  same  structure)  and  5) 
elements  of  generalisation.  Teaching  style  A  was  described  as  highly  concept 
intensive  with  a  high  level  of  reflection,  where  the  elements  described  above 
were  observed  frequently.  In  a  category  B  style  these  elements  were  observed 
sometimes  and  in  category  C  scarcely.  Of  the  teachers  of  the  lower  primary  school 
classes,  three  were  classed  as  style  A,  three  as  B  and  four  as^of  the  upper  primary 
teachers,  3  were  classed  as  A,  1  as  B,  2  as  C. 





1  - 102 

Thus  we  may  conclude  that  the  2  day  training  course  was  sufficient  to  enable  10  of 
the  16  teachers  to  acquire  the  method  and  to  use  it  at  A  or  B  levels,  and  that  these 
classes  made  clearly  significant  <je»W- 

Further  evidence  of  success  of  diagnostic  teaching  when  done  fairly  well. 
Table  2  shows  mean  gains  from  pre  to  post  test  for  the  lower  primary  classes  (max  22) 























Pre  Mean 











Mean  Gain 











p  value 







(*  indicates  a  p  value  <  0.002) 

A  ONEWAY  test  applied  to  compare  gains  by  teaching  style  showed  significant 
differences  between  group  C  and  the  rest  of  the  sample  and  no  significant 
difference  between  group  A  and  B.  Significant  improvements  from  pre  to  post 
test  for  every  A  and  B  class,  and  maintenance  of  scores  through  to  the  delayed 
post  test,  showed  the  long  term  effect  of  the  teaching  activities. 

As  expected  the  teaching  activities  have  had  various  impact  on  problems  from 
different  categories. 

Some  findings  are 

1)  The  material  has  assisted  a  transition  from  employing  the  wrong  additive 
strategy  to  delivering  correct  answers  (few  children  used  this  strategy  for  the  rate 
items).  The  shift  was  larger  for  style  A  and  B  classes  than  for  style  C. 

2)  The  teaching  activities  have  contributed  to  a  change  from  several  wrong 
categories  to  correct  answers,  but  also  from  naive  answers,  which  do  not  take  into 
consideration  the  structural  relationship  between  all  the  given  numbers,  to  the 
wrong  additive  strategy,  demonstrating  that  this  strategy  is  a  natural  intermediate 
level  of  understanding  of  suchproblems. 




1  -  103 

3)  The  activities  have  not  prevented  children  from  employing  the  wrong  additive 
strategy  as  a  fall-back  strategy  when  the  number  relationship  has  become  more 
difficult  combined  with  an  unfamiliar  context. 

4)  The  errors  children  make  in  four  number  enlargement  problems  are  clearly 
attributed  to  inexperience  of  geometrical  enlargement,  and  secondly  a  failure  in 
relating  this  to  application  of  an  appropriate  number  relationship.  The  activities 
in  the  material  do  not  centre  much  around  the  first  aspect. 

The  progress  is  considerable  for  all  cartesian  product  items,  showing  that  when 
children  are  helped  to  organise  and  represent  the  information  in  a  systematic 
way,  so  that  a  repeated  addition  model  can  be  employed,  such  problems  are  not 
particularly  difficult. 


Bell,  A.,  Greer,  B.,  Grimison,  L.,  Mangan,  C:  1989.  Children's  Performance  on 
Multiplictive  Word  Problems:  Elements  of  a  Descriptive  Theory.  Journal 
for  Research  in  Mathematics  Education  20,  434-449. 

Bell,  A.,  Swan,  M.,  Onslow,  B.,  Pratt,  K  &  Purdy,  D.:  1985.  Diagnostic  Teaching: 
Teaching  for  Long  Term  Learning.  Shell  Centre  for  Mathematical 
Education,  University  of  Nottingham. 

Brekke,  G.:  1991.  Multiplicative  Structures  at  ages  seven  to  eleven.    Studies  of 
children's  conceptual  development  and  diagnostic  teaching  experiment. 
Ph.D  thesis.  Shell  Centre  for  Mathematical  Education,  University  of 

Friedlander,  A.,  Fitzgerald,  W  &  Lappan,  G.:  1984.  The  growth  of  similarity 
concepts  at  sixth  grade  level.  In  J  Moser  (Ed).  Proceedings  of  the  Sixth 
Annual  Meeting  of  PME-NA.  127-132.  Madison  University  of  Wisconsin. 

Hart,  KM  (Ed).:  1981.  Children's  Understanding  of  Mathematics:  U-16.  London: 
John  Murray. 


1  - 104 

Karplus,  R.(  Pulos,  S  &  Stage,  E.:  1983.  Proportional  Reasoning  of  Early 

Adolescents.  In  R  Lesh  &  M  Landau  (eds).  Acquisition  of  Mathematics 
Concepts  and  Processes,  45-90.  New  York:  Academic  Press. 

Kurth,  W.:  1988.  The  Influence  of  Teaching  on  Children's  Strategies  for  Solving 
Proportional  and  Inversely  Proportional  Word  Problems.  In  Proceedings 
of  the  Twelfth  International  Conference  for  the  Psychology  of 
Mathematics  Education.  Vesprem,  441-448. 

Vergnaud,  G.:  1983.  Multiplicative  Structures.  In  R  Lesh  &  M  Landau  (Eds). 

Acquisitions  of  Mathematics  concepts  and  processes,  127-174.  New  York: 
Academic  Press. 




1  -  105 




Students  in  Grades  5  and  7  were  interviewed  with  a  series  of  linear  measurement  tasks.  The  tasks 
were  designed  to  investigate  two  aspects  of  students  thinking  about  units  of  length:  (a)  the 
consistency  with  which  they  identify  or  construct  units  of  length  as  line  segment,  and  (bj  the 
extent  to  which  they  reason  appropriately  about  relationships  between  different  sized  units.  It 
was  found  that  many  students  applied  a  direct,  point  counting  process  to  define  units.  However, 
the  extent  to  which  this  occurred  depended  on  the  task  situation.  Point  counting  occurred  with 
more  students  when  numerals  were  juxtaposed  with  points-  a  phenomena  which  has  instructional 
implications  for  the  use  of  the  number  line  for  representing  mathematical  relationships. 

Children's  conceptions  of  units  initially  develop  in  situations  which  involve  enumerating,  comparing  and 
operating  on  sets  of  discrete  objects.  Regardless  of  variations  in  the  physical  attributes  of  the  objects  In  a  set,  all 
objects  represents  equivalent  units.  Through  years  of  counting  experiences,  children  eventually  leam  to 
establish  a  one-to-one  correspondence  between  their  serial  touching  of  or  attention  to  each  object  and  their 
simultaneous  utterances  of  unique  number  names.  If  the  one-to-one  correspondence  is  not  violated  and  the 
number  names  are  uttered  In  a  standard  order,  then  the  last  number  name  uttered  invariably  determines  the 
cardinal  value  of  the  set  (Gelman  S  Gallistel,  1978).  As  such,  children  eventually  construct  a  schema  in  which 
synchronous  counting  actions  directly  determine  the  number  of  units  represented. 

In  linear  measurement  situations,  the  process  of  iterating  linear  units  still  conforms  to  the  direct  counting 
schema  developed  through  the  experiences  of  counting  discrete  units.  However,  when  partitioning  a  line  to 
represent  linear  units  or  when  interpreting  units  represented  as  line  segments,  a  direct  relationship  between 
synchronous  counting  actions  and  the  number  of  units  represented  by  the  count  is  not  invariant.  Variations 
between  the  results  of  a  counting  process  and  the  number  of  line  segments  implied  by  the  count  depend  on  a 
number  of  factors.  These  factors  include  whether  one  attends  to  line  segments  or  points  as  the  salient  feature  to 
be  counted,  and  the  plan  of  action  followed.  If  line  segments  are  the  salient  feature  to  which  one  attends  then  a 
direct  relationship  between  the  count  and  the  number  of  units  pertains.  If  points  are  the  salient  feature  to  which 
one  attends,  then  the  number  of  line  segments  are  defined  indirectly  through  the  count  of  the  points.  For 
example,  depending  on  whether  one  counts  (1)  all  beginning  and  end-points,  (2)  only  end-points,  or  (3)  only 
internal  points  between  line  segments,  6  line  segments  would  be  represented  by  a  count  of  7,  6,  or  5  points. 
It  is  necessary  to  develop  a  flexible  counting  schema  in  order  to  construct  or  interpret  linear  units  adequately  In 
such  measurement  situations.  Students  must  incorporate  notions  of  the  geometric  relationships  between  points 
and  line  segments,  alternative  plans  of  action  implied  by  these  relationships,  and  a  means  of  evaluating  the 

1  - 106 

number  of  linear  units  implied  by  different  courting  procedures.  Conceptions  of  counting  developed  in  discrete 
unit  situations  are  not  sufficient.  This  paper  explores  the  extent  to  which  students  in  Grades  5  and  7 
accommodate  to  these  different  measurement  situations  and  construct  alternative  views  of  how  units  are 
determined.  The  results  reported  here  are  a  small  part  of  a  larger  study  on  students'  representations  of  units  and 
unit  relationships  in  different  mathematical  domains  (Cannon,  1991). 

Plan  of  the  Study 

Fifteen  students  from  2  schools  participated  in  the  study:  6  in  Grade  5  and  9  in  Grade  7.  These  students 
represented  a  range  in  mathematical  achievement  in  each  grade.  A  Measurement  Concepts  Test  was 
administered,  and  later,  each  student  was  interviewed  individually  on  a  selection  of  the  linear  measurement  tasks 
(See  Figures  2).  The  tasks  were  designed  to  investigate  two  aspects  of  students  thinking  about  units  of  length: 

A.  Ruler  task     (Interview  &  Test  Task) 

This  ancient  ruler  measures  lengths  in  "FLUGS ."  One  "FLUG"  is  the  same  as 
two  centimetres.  Draw  a  line  above  the  ruler  that  is  6  centimetres  long. 

i            i              i             i             i              i  T 
I             1               2              3              4              5  6 

B.  AogrooatB  unit  task  (interview  &  Test  Task) 

The  line  below  is  4  units  long.  Draw  a  line  that  is  12  units  long. 

C.  Partitioning  tasks      (Interview  &  Test  Task) 

This  path  is  5  units  long,    a)  Mark  the  5  units  on  the  path. 

b)  Draw  another  path  lunfls.  long. 

FjQui£_2  Linear  measurement  tasks  with  explicit  reference  to  units  and  number, 
(a)  the  consistency  with  which  they  identify  or  construct  units  of  length  as  line  segment,  and  (b)  the  extent  to 
which  they  reason  appropriately  about  relationships  between  different  sized  units.  Students  were  required  to 
represent  units  in  different  problem  situations.1 

1  In  addition,  tasks  were  used  In  which  students  were  required  to  compared  the  lengths  of  two  "paths"  which 
were  made  up  of  different  sized  line  segments  placed  in  irregular  configurations  These  comparison  tasks  were 
derived  from  Bailey  (1974)  and  Babcock  (1978).  The  nature  of  students  reasoning  strategies  was  Investigated 



1  -  107 

Students'  responses  to  the  tasks  reported  in  this  paper  were  analyzed  in  terms  of  (a)  the  salient  features 
to  which  they  primarily  attended  when  defining  units,  (b)  how  they  interrelated  units  of  different  sizes.  In  the  latter 
set  of  categories,  with  bl-relatibnal  thinking  students  accounted  for  the  simple  ratio  between  different  sized  line 
units,  whereas  with  mono-relational  thinking  they  treated  different  sized  units  as  equivalent.  Not  aH  categories 
were  applicable  to  ail  tasks.  Distinctions  in  students'  reasoning  with  multiple  units  (mono-relational  versus  bl- 
relational  thinking)  did  not  apply  to  the  partitioning  task.  Only  one  unit  was  referenced  in  this  task;  all  of  the 
responses  were  necessarily  mono-relational. 

Tank  Rasponse 

Units     Line  Discrete  Undefined 

Segments      Points  I 

I  I   1  

I  I  I 

peasoning         Numerical       Transformational  Perceptual 

Strategy  |  I 

I  I  I 
 |               Actual  Imagined 

I  I 
Unjl         Bi-  Mono- 
Relations  Relational  Thinking 

Figure  3    Analytical  categories  used  to  classify  student  responses  to  tasks. 


Table  1  is  designed  to  explore  students'  representations  of  units  in  response  to  the  tasks  in  Figure  2  in 
several  ways.  First,  it  permits  us  to  determine  the  general  extent  to  which  students  constructed  units  which  were 
either  discrete  points  or  line  segments.  Second,  it  allows  us  to  compare  each  of  the  tasks  with  regard  to  the  extent 
to  which  discrete  points  or  line  segments  were  constructed  by  students.  And  finally  it  reveals  the  extent  to  which 
their  reasoning  about  the  relationship  between  different  units  was  mono  or  bi-relational.  The  students  have  been 
grouped  in  Table  1  according  to  the  extent  to  which  they  represented  units  as  discrete  points  or  line  segments: 
(a)  predominantly  discrete  points,  (b)  predominantly  line  segments,  and  (c)  consistently  line  segments. 

through  these  tasks  (See  Figure  3).  Students'  thinking  about  units  and  the  comparisons  of  length  in  these 
situations  are  not  reported  here  because  of  limits  of  space. 




1  - 108 

Table  1 

Btiirtoms'  Rupmsantatinns  ot  Units  of  Length  and  Ralatinnshins  Between  Units. 

Ruler  Task 

A<j<ji*tt<3&t6  Unit  Task 

Partitioning  Task 

Test  Interview 

Test  Interview 




U       nl    Li         U      nl  Li 

n    Mi  r       n    Mi  r 

U      nl    Li          U      nl  Li 

D    Mi  L 


Mi  L 

Dominant:  Iv 



M  B 

B  B 


.     ?       .     .  ? 

.     .     ?  M 

M  . 



M  B 

B  B 

.  ? 


Conn  ie 

NR   .      .       M  -->  B 

?  B 

M  . 


B  B 

B  B 

.  ? 


?      .      .  B 

B                         ,  MB 



Dominant lv 

Line  Seoments 

B  B 

.      .     B       .      .  B 

.  M 


Dahl ia 

B  B 

.      .     B             .  MB 




B  B 

.      .     B       .      .  B 




B  B 

.      .     B       .      .  B 




B  B 

.      .     B  B 



Line  seament 


.     B       .      .  B 

.      .     B  B 




.     B      .      .  B 

.      .     B       .      .  B 

.  M 



.     B       .     .  B 

.      .     B       .      .  B 




B       .      .  B 

.      .     B  B 

.  M 



The  names  of  the  students  in  Grade  5  are  underlined. 

iMS  lln»r»tatlnn« 

D  -  discrete  points  M  -  mono-relational 

Mi  -  mixed,  points  &  line  segments  MB  -  mono-  then  M- relational 

L  -  line  segments  B  -  bi-relational 

?  -  no  defined  units 

NR  -  no  response 

As  can  be  seen  in  Table  1 ,  most  students  represented  units  as  discrete  points  In  one  or  more  tasks. 
However,  none  did  so  exclusively  and  a  few  never  did  so.  There  was  a  marked  difference  in  the  patterns  ol 
responses  between  different  tasks.  Students  who  represented  units  predominantly  as  discrete  points,  did  so 
most  consistently  with  the  partitioning  tasks.  With  the  aggregate  unit  and  ruler  tasks,  the  form  of  their  units  was 



1  -  109 

more  variable.  However,  the  ruler  task  was  the  situation  in  which  the  greatest  number  of  students  were 
inconsistent  in  the  nature  ot  their  representation  ot  units.  This  suggests  a  significant  difference  in  the  nature  of 
students'  thinking  strategies  in  different  linear  measurement  situations.  A  closer  examination  of  how  students 
responded  to  the  partitioning  and  ruler  tasks,  in  particular,  reveals  further  differences  in  situational  responses. 
Variations  in  Students-  Representations  of  Units  of  Length-  Partitionino  and  Ruler  Tasks 

In  the  partitioning  tasks,  some  students  appeared  solely  to  attend  to  a  direct  relationship  of  counting 
points  to  determine  the  number  of  units  as  in  Lolande's  case  (See  Figure  4).  Others  incorporated  attributes 
normally  associated  with  linear  units  as  in  the  case  of  Derek  and  James  who  both  appeared  to  consider  equal 
spaces  between  the  points  to  be  important,  but  varied  in  their  attention  to  points  and  line  segments  as  the  salient 
feature  for  determining  the  number  of  units. 

1 .     (Lolande.  Grade  7) 

(Derek.  Grade  5) 

(James.  Grade  5) 

•I— I— I— I- 

-I  1 

-I — I — I- 

Fioura4  Examples  of  students'  use  of  discrete  points  as  units  to  partition  a  line  into  five  units  then 
drawing  a  line  of  three  units. 

When  discrete  units  were  used  with  the  ruler  task  the  reasoning  behind  these  responses  differed  (See 

Figure  5).  In  the  first  example,  the  relationship  between  the  size  of  the  centimetre  and  f lug  was  ignored.  The 

points  with  each  numeral  determined  the  length  of  the  line  drawn.  In  the  second  example,  the  student  attended 

to  the  2:1  relationship  between  centimetres  and  fkigs  but  counted  the  beginning  and  end  points  of  the  line 

segments  as  the  units,  beginning  with  the  point  associated  with  the  1  on  the  ruler.  In  the  third  example,  the 

student  converted  centimetres  to  (tugs  using  mental  arithmetic  and  then  represented  the  3  flugs  to  correspond 

with  the  numerals  on  the  ruler  and  not  the  number  of  line  segments  units. 






1 .     (James,  test) 

I  1         2         3         4  5  6 


2.     (Kasey,  test)  I     I     I     t  { 


I  1  2         3  4  5  6 


3.     (Edwin,  interview) 

I'        I         I         I  I  II 

1  2         3  4  5  6 


Figure  5  Students'  use  of  discrete  points  as  units  with  the  ruler  task. 
Point/Line  Segment  Conflict:  Diffarencns  Among  Tasks  and  Solution  Strateniss 

Ditterent  procedures  for  constructing  units  influenced  some  students'  attention  to  line  segments 
or  discrete  points  as  units.  Students  who  represented  units  predominantly  as  discrete  points  defined 
units  as  points  when  then  used  a  partitioning  process  to  resolve  the  aggregate  units  task.  However, 
those  in  this  group  who  solved  the  aggregate  unit  task  by  iterating  line  segments  faced  no  ambiguity 
about  how  to  determine  the  measure  by  the  counting.  The  partitioning  process  led  students  to  attend  to 
the  points  rather  than  the  line  segments.  With  the  ruler  task  there  was  the  additional  perceptual 
feature  that  points  were  juxtaposed  with  numerals.  This  juxtaposition  further  emphasized  a  counting 
relationship  between  points  and  the  enumeration  of  units.  All  students  who  used  discrete  points  as  units 
and  some  who  used  iine  segments  as  units  interpreted  the  "1"  as  the  beginning  marker  of  their 
representations  of  6  centimetres,  not  as  the  end  marker  of  the  first  "flug"  unit.  The  structure  of  the 
ruler  and  the  normal  meaning  of  the  numerals  did  not  guide  students'  representation  of  6  centimetres.  ■ 


Differences  in  the  representation  of  units  of  length  often  lay  not  in  students'  Initial  responses  to  the  tasks, 
but  in  their  reflection  on  the  consequences  of  their  first  responses.  For  example,  initial  partitions  of  a  line  often 
were  based  on  an  assumption  that  the  number  of  points  determines  the  number  of  units.  Upon  reflection,  many 

o  135 



students  revised  their  discrete  counting  plan  ard  redefined  their  representation  of  units  as  line  segments. 
However,  the  discrete  counting  schema  predominated  initially. 

The  representation  of  units  as  points  or  line  segments  appears  to  be  influenced  by  perceptual 
and  conceptual  factors.  The  points  are  perceptually  salient  to  the  ruler  and  partitioning  tasks. 
Attention  necessarily  is  centred  on  points  with  the  partitioning  task  and  often  centred  on  points  during 
the  ruler  task.  They  are  the  component  of  the  representation  acted  on  synchronously  with  the  verbal 
count,  exactly  the  same  actions  as  counting  discrete  units.  However,  it  is  indirectly  through  the  points 
that  line  segments  are  defined  as  linear  units  and  a  conceptual  understanding  of  this  is  necessary  in  the 
reflective  process.  One  has  to  attend  to  the  points,  think  about  line  segments,  and  keep  track  of  the 
relationship  between  the  count  of  points  and  the  number  of  line  segments.  Even  for  students  who  in 
other  situation  focussed  invariably  on  appropriate  relationships  between  the  count  of  points  and  the 
resultant  number  of  line  segments,  the  ruler  situation  generated  additional  attention  to  points.  The 
common  use  of  the  ruler  reinforces  the  notion  that  there  is  a  direct  relationship  between  the  count  of 
points  and  the  number  of  units.  Once  a  ruler  is  placed  correctly,  only  the  points  and  numerals  have  to 
be  attended  to  "to  read"  the  length.  The  discrete  counting  schema  appears  more  likely  to  predominate 
regardless  of  a  student's  understanding  of  geometric  relationships  between  the  points  and  lines  because 
the  numerals  and  points  are  juxtaposed.  The  need  to  attend  to  other  factors  besides  the  points  when 
representing  or  interpreting  units  is  not  recognized  universally  by  the  students. 

It  is  insufficient  to  conclude,  as  Hirstein,  Lamb,  &  Osborne.  (1978)  do,  that  a  child  who 
assumed  that  the  count  of  points  determines  the  number  of  linear  units,  "had  no  sense  of  a  linear  unit" 
(p.16).  Students  who  attend  only  to  points  in  one  measurement  situation  did  not  necessarily  do  so  in 
others.  Eventually  children  must  construct  a  conception  of  counting  which  admits  to  variable 
relationships  between  what  is  counted,  how  it  is  counted  and  the  implication  of  such  on  the  number 
units,  a  conception  of  counting  that  differs  from  their  experiences  in  discrete  situations. 

It  has  been  reported  extensively  that  the  number  line  is  a  significantly  difficult  form  of 
mathematical  representation  for  students  to  interpret  and  use  in  a  variety  of  instructional  contexts 
(Behr.  Lesh.  Post.  &  Silver,  1983;  Dufour-Janvier,  Bednarlz  &  Belanger,  1987:  En.'st,  1985;  Hart,  1981;  Novillis- 

I  triplications 

1  -  112 

Larson,  1980, 1987;  Payne,  197S;  Vergnaud,  1983).  The  tendency  for  the  discrete  counting  schema  to 
dominate  when  numerals  are  juxtaposed  to  points  along  a  line  may  be  one  of  the  factors  contributing  to 
students  alternative  interpretations  of  mathematical  relationships  represented  with  a  number  line. 
This  study  was  limited  to  students  In  the  middle  grades,  but  the  author  has  observed  the  same 
application  of  a  discrete  counting  schema  when  linear  representation  were  constructed  by  pre-service 
teachers.  It  would  appear  that  a  fully  flexible  counting  schema  appropriate  to  the  representation  of 
linear  units  is  long  in  developing. 


Babcock,  G.  R.  (1978).  The  relationship  between  basal  measurement  ability  and  rational  number  learning  at  three 
grade  levels.  Unpublished  doctoral  dissertation,  University  Of  Alberta,  Edmonton. 

Bailey,  T.  G.  (1974).  Linear  measurement  in  the  elementary  school.  Arithmetic  Teacher.  21, 520-525. 

Behr,  M.  J.,  Lesh,  R,  Post,  T.,  &  Silver.  E.A.  (1983).  Rational  number  concepts.  In  R  Lesh,  &  M.  Landau  (Eds.), 
Acquisition  ot  Mathematics  Concepts  and  Processes  (pp.  91-126).  New  York:  Academic  Press. 

Cannon,  P.  L.  (1991).  An  exploratory  study  of  students'  representations  ot  units  and  unit  relationships  In  tour 
mathematical  mnw«  Unpublished  doctoral  dissertation,  University  of  British  Columbia,  Vancouver.. 

Dutour-Janvier,  B,  Bednarz,  N.,  &  Belanger,  M  (1987).  Pedagogical  considerations  concerning  the  problem  of 
representation.  In  C.  Janvier  (Ed.),  Problems  of  Representation  in  the  Teaching  and  Learning  ot 
Mathematics  (pp.  109-122).  Hillsdale,  New  Jersey:  Lawrence  Erlbaum  Associates. 

Ernest,  P.,  (1985).  The  number  line  as  a  teaching  aid.  Frincatmnat  studies  m  Mathematics.  jfi,  41 1-421 . 

Hart,  K.  M.  (Ed.)  (1981 ).  Children's  Understanding  of  Mathematics:  11-16.  London:  John  Murray. 

Hkstein,  J.  J.,  Lamb.  C.E.,  &  Osborne.  A.  (1978).  Student  misconceptions  about  area  measure.  Arithmetic 
Teacher.  25.  10-16. 

NovUlis-Larson,  C.  (1980).  Locating  proper  fractions  on  number  lines:  effect  of  length  and  equivalence.  School 
Science  and  Mathematics  8JJ,  423-428. 

NovilHs-Larson,  C.  (1987).  Regions,  numberiines  and  rulers  as  models  for  fractions.  In  J.  C.  Bergeron,  N. 

Herscovlcs,  &  C  Kieren  (Eds.),  Proceedings  ot  the  Eleventh  International  Conference  tor  Psychology  ot 
u^h.mMi^  Prii^aHAn  Montreal.  Jury  19-25. 1 398-404. 

Payne.  J.  N.  (1975).  Review  of  research  on  fractions.  In  R.  A.  Lesh  (Ed.),  Numbar  and  Measurement:  Papers  from 
a  Research  Workshop  (pp.  145-187).  Athens,  Georgia:  The  Georgia  Center  tor  the  Study  ot  Learning  and 
Teaching  Mathematics. 

Vergnaud,  G.  (1 983).  Why  is  an  eplstemotogical  perspective  necessary  for  research  in  mathematics  education? 

Proceedings  of  the  Fifth  Annual  Meeting  ot  the  North  American  Chapter  of  the  International  flmup  tor  the 
Psychology  of  Mathematics  Frinratlnn  Montreal.  September  29  to  October  1.  1983. 1,  2-20. 


1  - 113 


Enrique  Castro  Martinez 
Luis  Rico  Romero 
Encarnaci6n  Castro  Martinez 

Departamento  de  Didactica  de  la  Matematica 
Universidad  de  Granada.  Espafia. 


In  litis  paper  we  present  the  results  of  a  study  about  the  processes  used  by  300  Primary  School 
Children,  in  the  5th  and  6th  levels  (II  and  12 years  old),  when  they  carry  out  compare  pro- 
blems. Tlte  analysis  of  written  protocols  has  been  made  taking  into  account  two  qualitative  soh-er 
variables:  the  choice  of  structure  (additive  or  multiplicative)  and  the  interpretation  of  the  relation 
(direct  o  reverse)  which  pupils  consider  when  they  solve  this  kind  of  problems.  Vie  obtained 
results  show  that  pupils' erroneous  processes  are  related  either  to  an  inadequate  choice  of 
structure  or  to  a  reverse  interpretation  of  Oie  relation. 

Multiplicative  word  problems  create  many  difficulties  to  children.  There  arc  several  reasons  for  this 
but  we  arc  convinced  that  one  of  the  most  important  is  that  pupils  arc  normally  confronted  with  a  very  scarce 
variety  of  standard  multiplicative  situations  in  their  daily  school  work. 

Multiplication  and  division  concepts  are  usually  taught  having  into  account  models  which  are  deeply 
rooted  in  settled  cultural  elements.  Multiplication  is  initially  presented  as  a  "repeated  addition",  even  though 
this  idea  was  temporally  abandoned  by  the  Cartesian  product  model.  There  is,  however,  no  unanimity  in  the 
way  to  introduce  the  division  concept. 

Hcndrickson  (1986,  p.26)  stated  that  division  is  usually  taught  as  a  repeated  subtraction,  but  this 
could  be  in  American  curriculum,  because  in  Spain  the  main  model  is  partitive  division  (cf.  Castellanos,  1980, 

Although  these  concepts  arc  taught  with  the  same  basic  model,  pupils  zrc  expected  to  solve  any  kind  of  one  step 
word  problems  of  multiplicative  structure,  with  the  meaning  Vcrgnaud  has  given  to  this  term.  Very  often  pupils 
cannot  transfer  what  they  have  learnt  from  a  specific  standard  word  problem  toanothcr  of  the  same  conceptual  field 
but  different  wording,  andthcy  will  probably  not  be  able  tosolvc  these  problems  by  themselves,  if  there  is  not  expli- 
cit teaching.  It  is  useful  therefore  to  establish  the  different  models  within  the  multiplicative  conceptual  field  so  that 
pupils  could  have  specific  instruction  to  overcome  this  gap.  This  is  in  line  with  the  exhaustive  investigation  program 
Vcrgnaud  proposed  tostudy  conceptual  fields.  Vcrgnaud(1990,pp.2.1-24)cstablishcssixdiffcrcnt  points  to  carry 
out  this  empirical  and  theoretical  work  systematically.  The  two  first  arc: 

-  Analyse  and  clarify  the  variety  of  situations  in  each  ctmcci>tual  field; 
■  Describe  precisely  the  variety  of  btftavimr,  procedures,  and  reasoning  that  students  exhibit  in 
dealing  with  each  class  of  situations. 

Vcrgnaud  (199(1;  p.24)  says  that: 

We  have  only  hits  and  pieces  of  insinuation  on  these  complimentary  lines  of  inquiry. 

1  - 114 

Both  points  have  been  widely  treated  in  relation  with  additive  structure  conceptual  field  (cf.  Carpen- 
ter, Moser,  Romberg,  1982;  De  Cortc  &  Verschaffel,  1987).  This  research  has  distinguished  the  very  well- 
known  semantic  categories  of  problems. 

Classification  and  analysis  on  multiplicative  structure  problems  has  been  done  from  different 
points  of  view  (Hart,  1981;  Vergnaud,  1983, 1988;  Schwartz,  1988;  Nesher,  1988;  Bell  et  all,  1988).  Though 
in  most  of  these  studies  multiplicative  compare  problems  have  not  been  considered,  now  the  number  of 
researchers  considering  this  semantic  category  are  increasingly  growing  (Hart,  1981;  Nesher,  1988;  Greer,  in 

Semantic  Factors  in  Compart  Problems. 

In  relation  with  addition  and  subtraction  word  problems,  two  semantic  factors  have  been  identified  as 
the  ones  which  may  influence  children's  strategy  when  solving  word  problems:  the  static  or  dynamic  character 
of  the  situation  and  the  position  of  the  unknown. 

Multiplicative  compare  problems  are  static  entities  located  at  a  time  TQ.  They  can  be  mathematically 
described  as  a  scale  function  between  the  referent  set  R  and  the  compared  set  C. 


R  >  C 

x  >  f(x)=-ax 

The  scalar  a  may  be  used  in  a  direct  o  reverse  way,  and  so  we  have  two  possibilities: 

xa  /a 

R  >  C        R  >  C 

x  >  ax        x  >  x/a 

If  we  work  only  with  natural  numbers,  the  scalar  shows  an  increasing  or  decreasing  comparison,  respectively. 

In  Spanish  increasing  comparison  are  usually  referred  to  with  the  following  relational  expressions:  "x 
veces  mas  que"  (x  times  more  than)  and  "tantas  veecs  como"  (times  as  many  as). 

Decreasing  comparison,  in  turn,  are  referred  to  with  relational  expressions  such  as  "x  veecs  menos 
que"  (x  times  less  than)  and  "como  una  parte  de"  (as  a  part  of).  Similar  expressions  can  also  be  found  in  the 
literature  in  other  languages.  For  example,  Hare),  Post,  Bchr  (1988:  pp.  373-74)  use  the  following  relational 

"Ruth  has  72  marbles. 

Ruth  has  6  times  as  many  marbles  as  Dan  lias. 
How  many  marbles  does  Dan  haver 

Hcndrickson  (1986,  p.  29)  uses  this  other  one: 

"There  are  12  girls  and  16  boy  s  in  a  room. 

The  number  of  girls  it  wliat  part  of  the  number  of  hoys?" 

The  relationship  "n  more  than"  can  be  interpreted  in  two  different  ways: 

(1)  as  the  additive  relationship  A-n  +  B  (i.e.,  n"  A-B)  or 

(2)  as  the  multiplicative  relationship  A  -n'B  (i.e.,  n-A/B). 

In  the  latter  situation  it  is  generally  referred  to  as  "n  times  as  many".  (Lcsh,  Post  and  Bchr:  1988,  p.101). 

a  139 

1  -  115 

Vergnaud  also  considers  relations  such  >"■  "three  times  more"  and  "three  times  less"  with*  multiplicative 
meaning,  expressing  ratios  (Vergnaud:  1988,  p.  156).  We  have  also  found  these  relations  in  Spanish  and  French  in 
old  arithmetical  text  books: 

"Se  dice  que  dos  canlidades  variables  son  proportionates  cuando  haciindose  una  de  ellas 
2,3,4,...  veces  mayor dmenor,  la  olra se hace  at  mismo  tiempo  2,3,4,...  veces mayor omenor' 
(Sanchez  y  Casado.1890,  p.  85). 

"Deux  quantiliis  sont  inversement  proponionelles  lorsque  la premiirt  devenanl  2,3,4...  fois  plus 
grande  ou  plus  petite,  la  deuxiimt  devient  au  contraire  2,3,4...  fois  plus  petite  o  plus  grand*. 

On  a  eu  pour  100  francs  24  mitres  d'itoffe;  si  on  veut  une  itoffe  2  3,4...  fois  plus  chire  pour  la 
mime  somme  de  100 fr.,  on  aura  23,4...fois  mains  de  mitres"  (Leyssenne:  1904,  p.  240). 

Grccr  (in  press)  points  out  another  important  semantic  factor  on  the  compare  multiplicative  pro- 
blems: the  cultural  dimension.  Some  performance  differences  with  compare  problems  between  English  and 
Hcbrcw-spcaking  students  can  be  explained  because  the  simplicity  of  the  Hebrew  compare  expression:  P-3 
instead  of  "3  times  as  many  as". 

All  this  makes  us  think  that  the  relational  expression  used  to  build  the  comparison  verbally  is  highly 
responsible  for  children's  successful  or  unsuccessful  performance  when  solving  problems. 

MacGrcgor  (1991)  has  also  pointed  out  the  influence  of  this  cultural  and  linguistic  component  and 
has  analysed  the  misunderstanding  between  the  relational  expressions  "times"  and  "times  more". 

In  our  study  we  are  going  to  use  four  different  propositions  to  establish  comparison  in  multipli- 
cative compare  problems:  "veces  mis  que",  "veces  menos  que",  "tantas  veecs  como"  and  "como  una  parte  de". 
Every  one  of  these  four  expressions  can  be  used  in  three  different  one-step  multiplicative  compare  word 
problems.  These  three  types  differ  in  the  unknown  quantity  (referent,  scalar,  or  compared)  of  the  comparative 

The  relational  term  and  the  unknown  quantity  arc  both  task  variables,  in  the  sense  which  Kilpatrick 
(1978)  gives  to  them. 

Having  into  account  these  two  task  variables,  we  establish  twelve  different  multiplicative  comparison 
word  problems  (sec  table  1).  We  claim  the  following  hypothesis: 

Error  patterns  need  to  be  explained  having  into  account  not  only  the  relational  term  or  the 
unknown  quantity  but  paying  attention  both  to  the  two  variables  simultaneously  and  to 
their  mutual  influence. 


The  subjects  were  3(H)  pupils  from  4  groups  of  fifth-grade  (11-ycars-old)  and  4  groups  of 
sixth-grade  (12-ycars-old)  in  four  Spanish  schools  at  Granada.  The  project  was  done  at  the  end  of 
the  academic  year.  According  to  the  math  curriculum,  the  notions  of  multiplication  and  division  arc 
introduced  in  the  third  grade. 



1  - 116 

Tools  tod  Procedure 

In  this  research  we  have  worked  on  twelve  multiplicative  comparison  word  problems.  These 
twelve  problems  arise  from  considering  the  task  variable  R  (relational  comparative  proposition) 
with  four  values: 

Rl  -  "times  more  than" 
R2 = "times  less  than* 
R3= "times  as  many  as" 
R4  =  "as  one  part  of 

together  with  the  task  variable  O  (unknown  quantity  on  the  relation)  with  three  values: 
01  =  "compared  unknown" 
02 ^"scalar  unknown" 
03  » "referent  unknown" 

We  have  controlled  the  following  task  variables:  syntax,  class  of  numbers  and  class  of  quantities.  We 
used  natural  numbers  and  discrete  quantities.  To  control  learning  effects,  three  homogeneous  paper- 
and-pencil  tests  consisting  of  4  one-step  problems  were  prepared.  In  every  test  the  items  were 
problems  that  incorporated  a  different  term  of  comparison  in  their  statement.  In  all  the  problems 
we  have  used  the  static  verb  "to  have".  In  the  three  tests  the  number  size  and  the  contexts  used  were 
controlled  variables.  The  number  triples  used  in  the  problems  were  (12, 6,  72),  (18,  3,  54),  (IS,  5, 
75),  (16,  4, 64).  Every  pupil  solved  one  4-itcra  test  in  a  free-response  form.  All  pupils  completed  the 
test  in  class.  There  was  no  time-limit  to  answer  the  test. 

Table  1 

Six  difftrtnt  types  of  the  problem  used  in  the  study 

Increase  comparison 

Decrease  comparison 

Compared  unknown 

Daniel  has  12  mrbles. 

Morla  has  6  tines  as  many 

Mrbles  as  Daniel  has. 

How  many  Mrbles  does  Maria  have?. 

nana  has  72  Mrbles. 
Daniel  has  as  Mny  Mrbles  as 
one  of  the  6  parts  that  Maria  has. 
How  Mny  Mrblts  does  Daniel  have?. 

Scale  unknown 

Maria  has  72  Mrbles. 
Daniel  has  12  Mrbles. 
How  Mny  tines  as  Mny  as 
Daniel  does  Maria  have?. 

Daniel  has  12  Mrbles. 

Maria  has  72  Mrbles. 

What  part  are  Daniel's  Mrbles  in 

comparison  to  Maria's. 

Referent  unknown 

Maria  has  72  Mrbles. 

Maria  has  6  tlMS  as  Mny 

Mrbles  as  Dan  has. 

How  Mny  Mrbles  does  Oaniel  have?. 

Oaniel  has  12  Mrbles. 

Daniel  has  as  Mny  Mrbles  as  one  of 

the  4  parts  that  Maria  has. 

How  Mny  Mrbles  does  Daniel  have?. 

Note.  Originally  problem  were  in  Spanish. 


Wc  have  classified  pupils'  answers  to  the  twelve  multiplicative  compare  word  problems  in  three 
groups:  right,  wrong,  and  not  answered.  Answers  arc  right  when  the  pupil's  process  leads  to  the  right  solution, 
but  wc  have  not  paid  attention  to  small  mistakes  with  operations.  When  the  pupil  has  not  given  any  solution, 
wc  have  considered  it  a  'not  answered'  reply. 

er|c  141 

1  - 117 

Wc  have  received  1200  different  answers  to  our  problems;  694  were  right  answers  (58%),  453  wrong 
(38%)  and  53  "not  answered"  (4%).  The  right  answers  have  been  analysed  in  Castro  et  all  (1991),  and  therefore 
wc  will  only  present  here  the  analysis  of  wrong  answers. 

Wc  present  in  table  2  the  number  of  all  the  different  processes  which  led  to  wrong  answers  found  in 
everyone  of  twelve  problems  and  in  table  3  the  number  of  the  different  processes  leading  to  right  answers.  As 
wc  can  see  there  is  a  great  number  of  different  wrong  processes  for  every  problem  but  only  a  few  of  the  right 

Table  2 

Number  of  different  processes 
leading  to  vrong  answers 

Table  3 

Somber  of  different  processes 
leading  to  right  answers 






































If  we  add  up  both  the  wrong  and  right  processes  wc  can  appreciate  the  variety  of  all  the  different 
processes  which  pupils  have  used  to  solve  every  problem  (table  4). 
Table  * 

Number  of  all  the  different  processes 
for  every  problem 





















After  the  observation  and  analysis  of  pupils'  written  protocols  wc  think  that  the  most  frequent  mistakes 
can  be  explained  under  two  basic  error  patterns: 

\.  Change  of  structure:  the  pupil  understands  the  problem  as  if  it  had  an  additive  structure  (with  the  mea- 
ning Vcrgnaud  gave  this  term). 
Fur  example,  in  the  problem 

Maria  has  54  marbles. 
Daniel  has  18  marbles. 

How  many  times  as  many  as  Daniel  docs  Maria  have?, 
change  of  structure  means  that  the  pupil  sets  up  as  solution  54-18  or  54  + 18. 

2.  Reversal  of  relation:  The  pupil  solves  the  problem  using  the  reversal  relation  of  the  one  which  appears  in 
the  statement. 

For  example,  in  this  problem: 

Maria  has  54  marbles. 

Maria  has  3  limes  as  many  marbles  as  Daniel  has. 
How  many  marbles  docs  Daniel  have?. 



1  - 118 

The  reversal  error  means  that  the  pupil  proposes  as  solution  54x3"  162. 

In  some  problems  the  errors  were  mainly  caused  by  only  one  of  these  patterns,  but  in  others  the 
errors  were  based  on  both  patterns  indistinctly.  In  table  5  we  present  the  main  error  pattern  for  every  one  of 
the  twelve  problems;  the  wrong  answers  percentage  over  all  the  solutions  (right,  wrong,  and  not  answered); 
and  the  most  usual  error  pattern  percentage  over  all  the  wrong  answers. 

We  try  to  show  that  pupils'  errors  have  been  mainly  produced  by  the  same  pattern,  and  although 
there  are  more  different  patterns,  these  appear  with  a  very  tow  percentage. 

Table  S 

Most  frequent  error  patterns  and  percentages  over  all  the 
solutions  and  over  all  the  wrong  answers  in  every  problem. 






Change  of 

Change  of 

Ch.  of  str. 
5%  (50%) 

of  relation 

9%  (60%) 
(*)  (**) 

20%  (61%) 

Rev.  of  rel. 
3%  (30%) 

11%  (70%) 


Change  of 

Change  of 

Change  of 

Change  of 

59%  (92%) 

60%  (90%) 

32%  (80%) 

32%  (91%) 


Ch.  of  str. 
15%  (46%) 

Ch.  of  str. 
23%  (56%) 

of  relation 

of  relation 

Rev.  of  rel. 
20%  (56%) 

Rev.  of  rel. 
18%  (39%) 

49%  (84%) 

30%  (86%) 

(*)  percentage  over  all  the  solutions. 

(**)  percentage  over  all  the  wrong  answers. 

The  causes  behind  the  error  patterns  detected  in  tabic  5  can  be  summarised  in  the  following: 

a)  Errors  in  the  four  O2  problems  arc  basically  due  to  change  of  structure.  This  type  of  error  is  even  bigger  in 
Rj  and  R2  problems,  where  we  find  the  words  "more"  and  "less",  respectively. 

b)  In  the  four  O3  problems,  errors  arc  mainly  of  reversal  relation,  whereas  in  R  |  and  R2  variables  we  find 
both  types  of  error  patterns  with  a  very  similar  percentage. 

c)  Oj  problems  has  a  very  low  percentage  of  errors. 

d)  Rj  and  Rj  problems  arc  mainly  caused  by  a  change  of  structure  error  pattern,  but,  as  we  have  said  in  (b),  in 
Oj  problems  wc  find  reversal  error  pattern  with  a  similar  percentage. 


1  - 119 


After  the  analysis  of  pupils'  errors  we  can  state  the  following  conclusions: 

1)  The  pupils  in  this  study  use  two  basic  models  to  solve  multiplicative  comparison  verbal  problems. 
The  two  basic  models  have  been:  cither  they  have  used  an  addtitivc  structure  pattern  or  they  misplace  the 
unknown  in  the  compared  (O,  problems).  This  is  consistent  with  Fischbcin  theory  of  implicit  models 
(Fischbcin  et  all.,  1985)  and  with  the  guiding  frame  model  for  understanding  word  problems,  proposed  by 
Lewis  and  Mayer  (Lewis  &  Mayer,  1987). 

2)  Errors  in  Scalar  unknown  problems  arc  mainly  caused  by  a  change  of  structure.  Pupils  usually  iden- 
tify this  class  with  additive  comparison  problems,  and  so  they  give  the  a-b  solutions  instead  off  the  i/b  ones. 
This  error  pattern  has  also  been  detected  with  ratio  problems  by  Piagct,  Karplus  and  Hart  (Hart,  1981). 

3)  Errors  in 'referent  unknown"  problems  arc  based  on  the  reversal  pattern.  Problems  are  solved  as 
multiplicative  structured  but  as  if  they  were  simple  "compared  unknown"  model.  Lewis  and  Mayer  arrive  at  the 
same  conclusion  using  the  relation  "times  as  many  as"  on  consistent  (compared  unknown)  and  inconsistent 
(referent  unknown)  compare  problems. 

4)  The  two  previous  conclusions  should  be  assessed  considering  the  distractcr  effect  produced  by 
the  relations  "veecs  mas  que"  (times  more  than)  and  "veecs  mcnos  que"  (times  less  than).  Problem  statements 
with  these  two  terms  lead  to  errors  of  change  of  structure.  For  this  reason  when  we  find  these  terms  "scalar 
unknown"  problems  (which  cause  the  same  error  pattern)  the  effect  of  both  variables  is  reinforced,  and  this  is 
why  in  both  cases  RjQj  and  R202  we  have  the  greatest  percentage  of  error  due  to  change  of  structure  pat- 
terns. When  these  relational  terms  appear  in  "unknown  referent"  problems  we  find  both  types  of  error  pat- 
terns indistinctly. 

Our  conclusions  have  to  be  understood  in  the  controlled  variables  frame.  That  is,  as  Bell  ct  all.  have 
explained  (1984, 1989),  number  size,  class  of  numN:rs  and  the  role  of  the  numbers  involved  in  a  multiplicative 
relation  could  have  influenced  in  the  operations  choice  on  multiplicative  word  problems.  Our  results  could 
always  have  been  affected  by  the  change  of  these  variables. 

REFERENCES  ,  .    .  .      .      ui       -rt.     «  .  r 

Bell  A  W  Fischbcin,  E.  y  Orccr,  B.  (1984).  Choice  of  operation  in  verbal  arithmetic  problems:  The  cttccts  ot 
number  sire,  problem  structure  and  context.  Educational  Studies  in  Mathematics,  IS,  129-147. 

Bell  A  Orccr  B  Grimison,  L.  y  Mangan,  C.  (1989).  Children's  pcrfomancc  on  multiplicative  word  problems: 
"  Elcmc'nls'of  a  descriptive  theory.  Journal  for  Research  in  Mathematics  Education.  20, 434-449. 

Carpenter,  T.  P.,  Moscr,  J.  M.,  y  Romberg,  T.  A.  (Eds.)  (1982).  Addition  and  subtraction:  A  cognitive  perspective. 
Hillsdale,  NJ:  Lawrence  Erlbaum  Associates. 

Castcllanos,  (i.  (i.  (1980).  Mathematics  and  the  Spanish-Speaking  Student.  Arithmetic  Teacher,  28(2),  lft. 

Castro  E  Rico  L ,  Balancro,  C.  y  Castro.  E.  (1991).  Dificullad  cn  prolilemas  dc  comparacion  mulliplicativa. 
'  En  F.  Furinghctti  (Ed.)  Proceedings  Fifteenth  PME  Conference.  Vol.1  (pp.  192-198).  Assisi  (Italy). 

Dc  Cortc  E  yVcrschaffcl.L.(l987).  The  effect  of  semantic  structure  on  first  grader's  strategics  for  solving 
addition  and  subtraction  word  problems.  Journal  for  Research  in  Mathematics  Education,  IS,  .Vi3-381. 

e  144 


Fischbein,  E.,  Deri,  M.,  Nello,  M.  S.,  y  Marino,  M.  S.  (1985).  The  role  of  implicit  models  in  solving  verbal 
problems  in  multiplication  and  division.  Journal  for  Research  in  Mathematics  Education.  16, 3-17. 

Ginsburg,  H.  (Ed.)  (1983).  The  development  of  mathematical  thinking.  Orlando,  FL:  Academic  Press. 

Grccr,  B.  (en  prensa).  Multiplication  and  division  as  models  of  situations.  En  D.  Grows  (Ed.),  Handbook  of 
research  on  learning  and  teaching  mathematics.  NCTM/Macmillan. 

Hart,  K.  (Ed.)  (1981).  Children's  understanding  of  mathematics:  11-16.  Londres:  John  Murray. 

Hendrickson,  A.  D.  (1986).  Verbal  multiplication  and  division  problems:  Some  dificulties  and  some  solutions. 
Arithmetic  Teacher,  33(i),  26-33. 

Kilpatrick,  J.  (1978).  Variables  and  methodologies  in  research  on  problem  solving.  En  L.  L.  Hatfield  y  D.  A. 
Bradbard  (Eds.),  Mathematical  problem  solving:  papers  from  a  research  worshop.  Columbus,  Ohio- 

Lcsh,  R.,  Post,  T.  y  Behr,  M.  (1988).  Proportional  reasoning.  En  En  J.  Hicbcrt  y  M.  Behr  (Eds.),  Number 
concepts  and  operations  in  the  middle  grades,  (pp.  93-118).  Hillsdale,  NJ:  Erlbaum/Reston,  VA: 

Lewis,  A.  B.  y  Mayer,  R.E.  (1987).  Students'  miscomprehension  of  relational  statements  in  arithmetic  word 
problems.  Journal  of  Educational  Psychology,  79, 363-371. 

Lcysscnne,  P.  (1904).  Le  dewdime  ann(e  d'Anthmetujue.  Paris:  Armand  Colin. 

MacGrcgor,  M.  (1991).  Understanding  and  expressing  comparison  of  quantities  confusion  between  'times"  and 
"more".  Paper  prepared  for  the  Fifteen  Annual  Conference  of  the  International  Group  for  the  Psycho- 
logy of  Mathematics  Education.  Assist,  June  29-July  4. 

Ncshcr,  P.  (1988).  Multiplicative  school  word  problems:  Theoretical  approaches  and  empirical  findings.  En  J. 

Hicbcrt  y  M.  Behr  (Eds.),  Number  concepts  and  operations  in  the  middle  grades,  (pp.  19-40).  Hillsdale, 
NJ:  Erlbaum/Reston,  VA:  NCTM. 

Sanchez,  y  Casado,  F.  (1890).  Prontuark)  dc  Aritmctica  y  Algebra.  Madrid. 

Schwartz,  J.  L.  (1988).  Intensive  quantity  and  referent  transforming  arithmetic  operations.  En  J.  Hicbert  y  M. 
Behr  (Eds.),  Number  concepts  and  operations  in  the  middle  grades  (pp.  41-52),  Hillsdale,  NJ:  Lawrence 
Erlbaum;  Rest  on,  VA:  NCTM. 

Vcrgnaud,  G.  (1983).  Multiplicative  structures.  En  R.  Lcsh  y  M.  Landau  (Eds.),  Adquisitions  of  mathematics 
concepts  and  processes  (pp.  127-174).  London:  Academy  Press. 

Vcrgnaud,  G.  (1988).  Multiplicative  structures.  En  J.  Hicbcrt  y  M.  Behr  (Eds.),  Number  concepts  and  opera- 
tions in  the  middle  grades  (141-161).  Hillsdale,  NJ:  Erlbaum;  Rcslon,  VA:  National  Council  of  Tea- 
chers of  Mathematics. 

Vcrgnaud,  G.  (1990).  Epistcmology  and  psychology  of  mathematics  education.  En  P.  Ncshcr  y  J.  Kilpatrick. 
Mathematics  and  Cognition:  A  research  synthesis  by  the  International  Croup  for  the  Psychology  of  Mathe- 
matics Education  (pp.  14-30).  Cambridge:  Cambridge  University  Press. 




1  - 121 


Olive  Chapman 

The  University  of  Calgary 

This  paper  is  based  on  a  study  involving  three  female  first  year  junior  college  students, 
enrolled  in  a  business  mathematics  course,  to  identify  and  understand  the  meaning  of 
any  uncharacteristic  problem  solving  behaviour,  from  their  perspective.  The  study  is 
framed  within  a  social  perspective  of  mathematics.  The  paper  discusses  the  students ' 
use  of  a  unique  process  of  sharing  "stories"  of  personal  experiences  which  led  to  a 
"connection  of  knowledge"  which  was  used  to  obtain  a  final  solution  to  problems  that 
had  a  context  they  could  relate  to  their  personal  experiences.  Based  on  this  process,  a 
conception  of  mathematics  as  experience  is  proposed. 

From  my  experience  as  a  mathematics  teacher,  it  became  evident  that  many  students 
engaged  in  problem  solving  processes  or  displayed  learning  behaviours  that  could  not  be 
meaningfully  explained  within  a  traditional  conception  of  mathematics  or  the  teaching  of  it  Such 
uncharacteristic  behaviours  seem  to  require  educators/researchers  to  look  beyond  the  "purely 
cognitive"  (Cobb  1986)  and  pedagogical  processes  (Easley  1980)  to  understand  the  deeper  meaning 
of  their  existence.  This  paper  reports  on  a  study  in  which  I  investigated  the  problem  solving 
behaviour  of  three  female  first  year  junior  college  students,  enrolled  in  a  business  mathematics 
course,  to  identify  and  understand  the  meaning  of  those  aspects  of  their  behaviour  that  seemed  to 
fit  this  uncharacteristic  label. 

The  study  focused  on  word  problems,  specifically  business  math  problems  because  of 

1  - 122 

the  students'  circumstances.  This  fpcus  became  important  because  of  my  experience  with  students 
who  often  paid  more  attention  to  the  social  context  instead  of  the  mathematical  context  of  these 
problems  when  trying  to  solve  them.  The  literature  also  suggests  that  many  students,  particularly 
poor  problem  solvers,  display  the  former  behaviour.  Traditional  approaches  to  problem  solving  tend 
to  favour  the  focus  on  mathematicalcontextanddiscourageafocuson  social  context  Consequently, 
in  this  study,  the  latter  behaviour  was  investigated  to  understand  the  implications  of  mathematics 
learning  embodied  in  it 

The  body  of  literature  which  provided  a  frame  for  this  study  falls  in  the  category  of  a 
social  perspective  of  mathematics  where  belief,  personal  meaning,  culture,  ...  are  important 
considerations(Fasheh  1982,  Gordon  1978,  Cobb  1986,  Schoenfeld  1985,  D'Ambrosio  1986).  Within 
this  framework,  the  social  circumstances  of  the  students  play  a  significant  role  in  their  learning.  In 
the  case  of  females,  some  attempt  has  been  made  to  explain  the  social  implications  of  their 
treatment  of  problem  context  in  terms  of  a  preferred  way  of  knowing  rooted  in  relationships  and 
connectedness  (Belenky  et  al  1986,  Buerk  1986).  Thus,  for  example,  they  are  likely  to  respond  to 
problem  context  in  terms  of  its  humane  qualities  instead  of  its  abstract  ideas.  However,  by  itself,  this 
does  not  provide  an  understanding  of  the  nature  of  the  problem  solving  process  these  students 
engage  in,  from  a  social  perspective.  This  study  provides  a  way  of  beginning  to  fill  this  gap. 

In  this  study  data  was  collected  through  interviews,  classroom  observations,  group  and 
individual  problem  solvingsessions,  and  journals.  The  analysisinvolved  various  levelsofcomparisons 
between  the  students'  social  biography,  math  biography  and  problem  solving  behaviour  in  terms  of 
teacher-student  relationships,  peer  interaction  and  problem  solving  strategies.  Because  of  the 
limitation  on  the  length  of  this  paper,  only  the  unique  aspects  of  their  problem  solving  strategy  as 
a  group  and  the  conception  of  mathematics  embodied  in  it  will  be  presented. 


1  - 123 


Tbeuniqueproblemsohnngbehaviourofthese  students  ocwrredwhentheywere  able 

to  relate  the  context  of  the  problems  to  their  personal  experiences.  Their  solutions  of  such  problems 

presented  a  different  way  of  viewing  problem  solving  in  math.  The  approach  was  not  to  focus  on  the 

mathematical  context  of  the  problem  to  understand  it,  but  on  the  social  context  It  involved  an 

exploration  of  the  problem  in  a  social  or  experiential  mode  instead  of  a  cognitive  mode.  The  process 

consisted  of  three  stages:  a  sharing  of  "stories"  of  personal  experiences,  which  led  to  a  "connection 

of  knowledge",  which  led  to  a  final  solution  of  the  problem  and  a  reflection  on  the  consistency  of 

their  answer  in  the  context  of  the  connected  knowledge.  It  seemed  to  be  a  different  way  of 

interpreting  Polya's(  1957)  problem  solvingmodel.  The  followingproblem  andexcerpt  of  the  group's 

solution  of  it  will  be  used  to  discuss  these  aspects  of  the  approach. 

Problem:  A  bookstore  buys  used  books  and  sells  those  that  zxe  slightly  damaged  at  20% 
below  the  cost  of  a  new  brx>k.  If  a  customer  paid  $45  for  a  use«i  *  ook,  how  much  did  she 


J:  ^  But  it  says  you  get  20%  for  slightly  damaged  books  and  the  customer  bought  a  used 
book,  we  don't  know  if  it's  slightly  damaged. ...  What  if  it's  more  than  slightly  damaged? 

L:  That  could  be,  which  is  why  it's  not  working  out. ...  I  don't  usually  buy  used  books,  like 
school  books,  because  I  find  they  sell  them  for  too  much,  even  when  they  are  more  than 
slightly  damaged.... 

M:  Well  I  always  buy  my  books  used ....  but  it's  from  other  people  selling  theirs  and  I  don't 
take  the  price  they  give  even  if  it's  slightly  damaged.  ...  No,  but  it's  true.  One  time 
somebody  was  giving  me  a  price  and  I  just ....  I  asked  somebody  else  to  give  me  a  better 
price  and  I  did.  I  got  lots  of  sh--  from  the  first  person  but  I  still  took  it 
J :  But  some  people  prefer  to  get  the  price  they  want  than  to  take  what  they  can  get  which 
is  so  stupid  if  you  don't  need  the  book  anyways.  I  know  people  like  that...  Like  this  guy 
wanted  to  sell  me  a  book  for  $30,  he  paid  35  for  it.  I  said,  "20".  He  said,  "forget  it"  ... 

L:  I  know. ...  But  what  you  said  makes  sense  because  it  could  be  more  off  if  it  is  damaged 
a  lot.  But  they  don't  give  us  that  information.  So ... 


1  - 124 

M:  But  we  are  dealing  with  a  bookstore  here.  So  they  would  want  to  sell  it  to  make  a 
profit  too. 

(...)  (p.s.t.Nov.88] 

In  solving  this  problem,  the  students'  first  approach  was  not  to  look  for  a  formula, 
recall  a  method  illustrated  by  their  teacher  or  us*  one  from  the  text  book,  as  they  did  with  problems 
with  abstract  or  "irrelevant"  context  They  suited  from  scratch,  depending  more  on  their  real  life 
personal  experiences  to  arrive  at  a  solution,  instead  of  an  abstract  connection  to  what  they  had  learnt 
in  class. 

The  "story  ing"  stage  started  when  their  initial,  individual  attempts  at  a  solution  failed. 
To  resolve  the  situation,  they  resorted  to  a  special  type  of  sharing.  It  wasn't  a  sharing  of  isolated 
opinions  of  what  was  wrong  and  how  to  fix  it.  It  wasn't  a  cause-effect  analysis  of  something  that 
happened  on  their  page.  It  was  a  differentpersonal  encounter,  one  involving  a  sharing  of  personal 
experiencesdirectly  and  indirectly  related  to  the  problem  context.  "J"  noted  a  "social"  concern  about 
the  problem.  She  pointed  out  that  the  problem  stated  that  books  which  were  slightly  damaged  would 
be  sold  at  a  20%  discount,  but  it  did  not  say  if  the  book  that  a  customer  purchased  was  slightly 
damaged.  It  only  said  that  the  customer  bought  a  used  book.  This  generated  a  "discussion"  of  this 
"defect"  in  the  context. 

This  "discussion"  portrayed  a  differentvoice  in  relation  to  traditional  problem  solving 
processes;  a  voice  that  would  likely  be  silenced  in  a  traditional  classroom  because  of  its  obvious 
deviation  from  the  "norm".  The  discussion  was  not  centered  around  "why  didn't  the  math  make 
sense",  but  "why  didn't  the  context  make  sense".  It  also  did  not  deal  with  the  context  in  a  general  way 
or  as  a  hypothetical  situation.  Instead,  it  personalized  the  context;  integrating  it  into  each 
participant's  personal  experiences. 

The  format  of  the  discussion  looked  more  like  a  narrative  process  as  each  person  took 
turn  at  sharing  a  "story"  of  a  personal  experience  related  to  the  context.  "M"  and  "J"  shared  an 


1  - 125 

experience  of  buying  used  books  whUe  "L"  talked  about  why  she  didn't  buy  used  books.  But  these 
weren't  just  any  "stories",  they  were  biographical.  They  contained  personal  information  that  the 
others  might  not  have  known.  For  example,  "M"  talked  about  buying  only  used  books,  about  not 
buyip^t-Eatons",  about  not  allowing  customers  at  work  to  take  advantage  of  her,  about  notto  give 
deals  but  to  get  them,  about  notto  sell  for  a  loss....  These  were  all  reflections  of  her  personal  world 
as  revealed  in  her  "social  biography".  They  were  manifestations  of  her  bargaining  tendencies  that 
defined  the  way  she  was  and  made  sense  of  her  world;  a  behaviour  rooted  in  her  childhood 
experiences.Sosheandtheothersseemedtobecontributingsomethingverypersonalin  this  sharing 


The  outcome  of  this  "story  sharing"  stage  was  a  social  construction  of  a  reality  of  the 
problem  as  it  was  experienced  by  each  of  them.  In  particular,  the  "merging"  of  their  personal 
experiences  resulted  in  a  "unique"  type  of  knowledge  used  to  solve  the  problems;  a  "personal 

connected  knowledge". 

I  conceptualized  "Personal  connected  knowledge"  as  the  knowledge  drawn  out  of  the 
personal  experiences  of  the  individual  members  of  the  group  by  real  images  in  the  problem  context 
and  provides  a  concrete  connection  to  the  abstraction  embodied  in  that  context.  It  is  an  experiential 
reconstruction  of  the  context  of  the  problem.  It  is  a  reconstructed  version  of  the  original  problem 
based  on  the  "truths"  extracted  from  their  experience  instead  of  those  given  in  the  problem.  Thus 

it  reflects  what  they  care  about;  their  meaning;  their  reality. 


was  through  the  group's  concern  about  the  negative  implications  of  the  context  -  the 
unreasonablenessof  the  businesspracticesin  the  given  situations,  for  example,  the  highcost  of  used 
books.  Such  concerns  were  usually  "restoried"  to  create  a  context  consistent  with  their  perceived 


r  o 


1  -  126 

Once  the  "personal  connected  knowledge"  was  established,  the  students  were  able  to 
get  to  a  solution  that  was  meaningful  to  them,  but  "correct"  only  when  the  "personal  connected 
knowlege"  did  not  conflict  with  the  intended  context  of  the  problem.  However,  they  seemed  to  be 
able  to  resolve  such  conflicts  with  appropriate  teacher  intervention. 


The  problem  solving  approach  and  other  related  mathematics  learning  behaviour  of 
these  students  suggest  a  different  conception  of  mathematics,  a  conception  that  is  necessary  to 
reflect  its  qualities  when  viewed  from  within  a  context  of  human  experiences.  The  conception  that 


that  is  experienced  in  terms  of  human  intention,  fear,  triumph,  hope  One  does  not  abstract  the 

cognitive  meanings  from  the  human  context,  but  deals  with  them  hoiistically.  These  students- 
learning  ol  mathematics  was  not  an  impersonal  application  of  algorithms  or  problem  solving 
strategies  to  some  phenomena  (real  or  fictitious)  embodied  in  the  problem,  external  to  themselves. 
Instead,  it  was  a  sharing  and  connecting  of  personal  experiences,  a  sharing  and  connecting  of  "self 
stories".  This  suggests  that  for  them,  the  experience  that  is  shared  and  connected  is  mathematics.  In 
the  businessmath  problems  they  solved,  mathematics  became  shopping,  bargaining,  selling...,  not 
the  manipulation  of  the  numbers  abstracted  from  the  experience.  Consequently,  from  this 
perspective,  not  only  the  learning  of  mathematics,  but  mathematics  itself  is  experience;  an  event  in 
their  life  story. 

This  conception  of  mathematics  provides  a  different  way  of  viewing  problem  context 
and  the  way  these  students  treated  it  For  them,  it  was  not  custom  made  clothing  for  some  abstract 
concept,  it  was  an  event  already  "storied",  or  a  "restorying"  of  one,  in  their  life  experiences.  THus 
"context"  is  viewed  as  the  "storying"  or  "restorying"  of  a  personal  experience.  Consequently,  to  ulk 




about  mathematics  as 

1  -  127 

embodied  in  the  context  is  to  talk  about  "self;  to  talk  about  personal 

This  summary  of  this  conclusion  drawn  from  the  study,  does  not  do  justice  to  the 
underlying  conceptual  considerations  of  the  perspective  being  proposed.  THe  goal,  however,  is  to 
draw  attention  to  an  important  dimension  in  considering  what  is  mathematics,  to  understand  some 
of  the  seemingly  "bizarre"  problem  solving  behaviour  of  many  students  and  to  deal  with  them 
meaningfully  for  the  students. 


The  outcome  of  this  study  is  suggesting  recognition  of  a  dimension  of  mathematics 

mathematics  in  this  mode  the  only  opportunity  to  engage  in  a  meaningful  learning  process.  Given 
the  current  shifts  in  philosophy  in  mathematics  education,  this  seem  to  be  a  timely  outcome  to  be 
included  in  the  broadened  definition  of  mathematics  in  the  school  curriculum.  Although  the 
NCTM's  Curriculum  and  Evaluation  Standards  (1989)  have  considered  mathematics  as  problem 
solving.reasoning.connections,  communications...  they  do  not  seem  to  go  far  enough  to  include  or 
explicitly  recognize  mathematics  as  experience  in  the  context  that  emerged  in  this  study.  Similarly, 

in  terms  of  what  is  considered  as  personal  experience  in  written  and  oral  communication  in  the 
leamingof  maths.  More  attention  is  needed  to  the  social  autobiographicalperspective  in  a  nanative 
context  to  facilitate  the  way  of  knowing  implied  in  this  study. 


1  -  128 

S&B&S^y* ^^"^^"f^"^^— r— f-lf  -irr unimiml  n, ,. 

Buerk,  D.  1986.  The  voices  of  women  making  meaning  in  mathematics.  Journal  of  Efttatjoj  167, 

Fasheh,  M.  1982.  Mathematics,  culture  and  authority.  For  the  Leamin.  0f  MathcjMfo  ,  2. 
^M^Conflict  and  deration:  persona,  aspects  of  the  mathematics  experience. 
Polya,  G.  1957.  How  to  solve  jt.  Garden  City,  New  York.  Doubleday. 
Schoenfeld,  A.H.  1985.  Mathematical  frgbjem  Selyjag  Academic  Press  Inc.,  New  York. 

o  153 



1  - 129 


G.  Chiappini  -  E.  Lemut 
Istituto  per  la  Matematica  ApplicaudelCN.R  -  Via  L.B.  Albert,  4-16132  Genova  -  Italy 

In  this  paper  we  analyse  the  role  that  the  computer  may  play  in  the  development  of  the  geometry  skills  that 
control  representation  by  parallel  perspective  and  orthogonal  views.  The  study  involved  students  of  12-13 
years  of  age.  Software  especially  designed  by  us  was  used  to  test  problems  of  representation  by  orthogonal 
views  of  a  poly-cube  structure  shown  in  parallel  perspective,  or  vice  versa.  The  a  priori  analysis  of  the 
teaching  situations  concerns  the  conceptual  aspects  inherent  in  the  use  of  software  with  respect  to  problem- 
solving  strategies  that  may  be  applied  by  the  students.  The  discussion  of  the  observations  made  concerns 
the  role  of  computer  mediation  in  the  development  of  the  students' strategies  with  respect  to  the  involved 

1 .  Introduction 

Various  studies  carried  out  during  the  last  few  years  stressed  the  difficulties  that  arise  in  the  mastery  of  the 
projective  system  underlying  parallel  perspective  and  orthogonal  projection  drawings  [11  [2]  [3]  [6]  [8]. 
The  difficulties  that  have  been  found  concern  the  geometric  conceptualization  involved  in  the  development 
of  such  notions  as  projection,  change  of  point  of  view,  and  adoption  of  a  reference  system.  Such  notions 
are  fundamental  to  permit  an  active  control  over  the  subject's  perception  with  reference  to  the  meaning- 
significant  contents  of  the  graphemes  of  perspective  drawing  or  for  the  coordination  of  the  points  of  view  in 
the  drawing  of  orthogonal  projections.  It  is  difficult  to  overcome  these  difficulties  because  in  teaching 
practice,  especially  at  comprehensive  school  level,  there  is  a  dearth  of  effective  and  tested  paths  towards 
development  of  skills  of  reading  and  of  representations  of  the  real  physical  space  according  to  parallel 
perspective  or  orthogonal  views. 

The  research  about  which  we  are  discuassing  in  this  paper  concerns  to  what  extent  the  computer  may  help 
the  growth  of  this  type  of  skills,  encouraging  the  development  of  the  concepts  that  the  subject  has  about  the 
projective  system  underlying  these  types  of  graphic  representation  [5].  The  research  that  we  carried  out  is 
circumscribed  within  the  graphic  space  to  tasks  of  reading  and  construction  requiring  the  transition  from  a 
parallel  perspective  drawing  to  the  corresponding  orthogonal  view,  and  vice  versa. 
Regarding  the  software  specifications  and  the  development  of  an  a  priori  analysis  of  teaching  situations,  we 
availed  ourselves  of  the  collaboration  of  Claire  Margolinas;  the  software  implementation  was  developed  by 
M.G.  Martinelli  for  the  thesis  she  wrote  for  her  mathematics  degree. 

2 .  General  Characteristics  of  the  Software  Implemented  and  Used 

The  software,  written  for  the  Autocad  environment,  exploits  the  potential  of  the  Autolisp  language.  It 
permits  to  address  the  following  two  types  of  problems: 

i)  given  a  representation  of  a  poly-cube  structure  in  orthogonal  views,  construct  the  parallel  perspective 
representation  of  the  same  structure; 

ii)  given  a  particular  parallel  perspective  representation  of  a  poly-cube  structure,  construct  the  corresponding 
orthogonal  projections. 

1  - 130 

The  second  type  of  problem  may  be  addressed  in  two  different  ways: 

-  constructing  on  each  view  the  projections  corresponding  to  any  number  of  cubes  before  passing  to  the 
representation  in  another  view  (MOD  1); 

-  constructing  on  each  view  the  representation  corresponding  to  only  one  cube  at  a  time  (MOD  2); 
The  software  was  designed  to  structure  the  student's  solution  process  in  three  distinct  phases: 

(1)  the  students  make  their  observations  on  the  statement  of  the  problem  and  gather  the  information  that  they 
think  may  be  useful  for  the  implementation  of  their  problem-solving  strategy  (solution  anticipation  phase); 

(2)  the  students  implement  their  problem-solving  strategy  exploiting  the  operational  features  of  the  software 
environment  used  (problem-solving  strategy  construction  phase); 

(3)  the  students  have  the  possibility  to  carry  out  a  validation  of  the  problem-solving  strategy  that  they  have 
implemented,  checking  the  correctness  of  the  results  obtained  with  respect  to  the  proposed  problem  (stage 
of  validation  of  the  implemented  strategy). 

Each  type  of  problem  requires  that  the  students  work  in  both  environments  (orthogonal  views  and  parallel 
perspective),  using  them  respectively  as  starting  and  validation  environment,  and  as  working  environment, 
and  vice  versa, 

The  basic  geometrical  elements  manipulated  by  the  used  software  are  squares  with  unit  side  (working  on 
orthogonal  views),  and  cubes  with  unit  edges  (working  on  parallel  perspective). 

3.      Learning  Situations  Proposed 

Here  below  we  shall  give  the  statements  of  the  problems  tackled  by  the  students  during  the  experiment.  The 
three  texts  were  introduced  by  a  verbal  question  of  the  experiment  leader,  such  as:  "What  you  see  on  screen 
is  a  representation  of  the  structure  of  an  object.  You  must  find  what  is  the  structure  and  represent  it  in  this 
new  environment". 

Prob  A  Prob  B  (MOD  1)  Prob  C  (MOD  2) 

For  problems  B  and  C,  specific  commands  are  available  with  the  visualization  of  the  statement  of  the 
problem.  These  commands  give  to  the  student  the  possibility  of  exploring  the  poly-cube  structure  from 
three  different  points  of  view.  Here  below  is  an  example  relevant  to  problem  C. 

1  -  131 

CI  C2  C3 

The  drawings  illustrated  here  represent  the  object  according  to  a  non-transparent  perspective  view.  Actually, 
by  default  the  software  represents  the  objects  according  to  a  wireframe  (transparent)  view,  but  a  specific 
command  permits  to  select  also  the  non-transparent  view  of  an  object.  We  should  also  notice  that  in  test  B, 
the  non-transparent  representation  does  not  permit  to  detect  the  presence  of  a  fifth  cube,  that  may  be 
perceived  in  the  transparent  view,  and  is  made  more  explicit  by  the  reading  and  coordination  of  the  other 
three  available  points  of  view. 

Notice  that  in  the  views  displayed  on  screen  both  the  planes  of  the  trihedron  and  those  of  the  orthogonal 
views  are  grids  and  coloured  so  as  to  help  the  student  see  the  correspondence  of  the  parts. 
The  problem-solving  strategy  for  test  A  is  developed  by  inserting  the  cubes  in  the  work-space  defined  by  a 
tri-rectangular  trihedron  identical  to  the  one  represented  in  tests  B  and  C.  The  cubes  are  inserted  using  the 
mouse  to  implicitly  define  on  the  horizontal  plane  the  coordinates  of  a  privileged  vertex  of  the  cube  (the 
vertex  whose  projection  on  the  horizontal  plane  is  the  nearest  to  the  intersection  of  the  three  planes  of  the 
trihedron),  and  assigning  to  the  cube  an  integer  between  0  and  5  to  define  its  height  in  relation  to  the 
horizontal  plane,  expressed  in  the  units  of  measurements  implicitly  defined  by  the  grids  of  the  planes  of  the 

Underlying  this  way  of  working  and  representing  is  a  conceptin  of  space  that  identifies  the  cube  by  means 
of  the  pair  of  coordinates  of  its  projection  on  the  horizontal  plane,  (seen  as  intersection  of  the  row/column 
on  the  grid)  and  of  the  coordinate  according  to  the  axis  orthogonal  to  it.  It  is  also  necessary  to  notice  that  the 
construction  of  each  cube  is  possible  only  by  specifying  the  coordinates  of  its  projection  only  with  respect 
to  the  horizontal  plane;  therefore,  the  three  directions  of  the  trihedron  are  not  equivalent 
The  construction  of  the  solution  strategy  for  tests  B  and  C  employs  a  mouse  to  insert  the  desired  projections 
on  the  grids  in  the  view  planes.  The  use  of  software  encourages  a  concept  of  space  according  to  which  each 
cube  is  identified  by  the  pairs  of  coordinates  of  its  projections  on  the  three  planes  of  a  tri-rectangular 
Euclidean  reference  system,  seen  as  intersection  of  row/column  on  the  grids  of  the  planes. 
It  should  be  noticed  that  in  the  solution  strategy  used  for  problem  B  (MOD  1),  the  demands  made  on  the 
students  to  put  into  correspondence  such  pairs  of  coordinates  are  not  pressing,  as  the  construction  of  each 
view  may  be  done  independently  from  the  others,  even  chronologically;  on  the  other  hand,  in  the  solution 
of  problem  C  (MOD  2)  the  coordination  of  such  pairs  of  coordinates  is  necessary  and  should  be  performed 
with  respect  to  the  parallel  perspective  representation,  in  order  to  permit  the  unambiguous  identification  of 
each  cube  whose  orthogonal  view  is  desired. 


1  - 132 

By  selecting  a  specific  command,  the  students  nave  the  possibility  of  automatically  accessing  the  orthogonal 
view  or  the  parallel  perspective  representations  corresponding,  respectively,  to  the  parallel  perspective  or 
orthogonal  view  constructed  by  them.  This  permits  the  validation  of  the  employed  solution  strategies  17]. 
Finally,  we  should  notice  that  the  test  of  a  problem,  the  students'  solution  and  the  representation 
automatically  offered  by  the  computer  never  can  be  simultaneously  visualized  on  the  screen.  The  students 
can  pass  from  each  one  to  another,  selecting  specific  commands. 

4.  Experimentation  Context  aad  Methodology 

Up  till  naw  the  experimentation  was  carried  out  on  5  pairs  of  students  of  12-13  years  of  age,  of  the  2nd 
year  of  comprehensive  school  (Grade  7).  Four  pairs  of  students  are  in  the  same  class.  The  teachers  of  these 
students  belong  to  our  research  group;  according  to  their  opinions,  these  students  belong  to  the  upper  half 
of  their  classes. 

The  problems  were  given  to  the  students  in  the  same  sequence  in  which  they  are  presented  in  this  paper.  To 
become  familiar  with  the  characteristic  of  the  software,  before  attempting  each  of  the  three  problems 
proposed,  the  students  worked  on  three  simple  problems  concerning  the  representation  of  only  one  cube. 
Also  those  problems  constituted  study  situations. 

Before  starting  the  software  activity,  we  asked  each  student  to  draw  by  her/himself  on  a  blank  sheet  of 
paper  the  parallel  perspective  image  of  a  cube  and  the  three  corresponding  orthogonal  views  (top,  front,  and 

No  student  had  any  difficulty  in  performing  the  task. 

The  students  were  instructed  to  use,  during  the  tests,  the  blank  sheets  available  to  take  notes.  The  entire 
work  session  of  each  pair  of  students  was  recorded  by  means  of  a  tape  recorder. 
The  authors  of  this  paper,  working  respectively  as  experimenter  and  as  observer,  along  with  one  of  the 
students'  mathematics  teacher,  assisted  to  the  tests  of  each  pair  of  students. 

The  analysis  of  the  results  is  based  on  the  transcription  of  the  recordings  made  on  tape,  on  the  notes  made 
by  the  students  on  the  sheets  of  paper,  on  the  printouts  (made  on  a  plotter)  of  the  various  validations 
performed  by  the  students,  and  on  the  notes  taken  by  the  observer  during  the  activity. 

5 .  Our  Hypotheses  on  the  Role  of  the  Computer 

Our  work  endeavours  to  verify  whether,  and  to  what  extent,  the  computer 

-  may  carry  out  an  active  mediation  role  in  the  students'  learning  process; 

-  may  encourage  the  implementation  of  actions  oriented  towards  the  pursuit  of  a  goal  meant  as  anticipation 
of  the  future  outcome  of  an  action; 

-  may  affect  the  inner  mental  processes  of  the  subjects  and  the  nature  of  the  communication  between  them. 
Within  this  framework,  we  formulate  the  following  hypotheses: 

a)  the  interaction  with  the  computer  may  take  the  form  of  a  social  interaction  in  which  action  and 
communication  may  integrate  dialectically; 

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b)  the  computer  may  have  this  important  role  in  the  learning  process  of  the  student  only  if  the  learning 
situations  that  the  student  tackles  with  the  mediation  of  the  computer  permit  the  validation  of  the  problem- 
solving  strategies  implemented  by  the  pair  of  students; 

c)  a  context  permitting  an  interaction  between  "equals"  affects  the  role  that  the  computer  may  play  in  the 
learning  process  of  the  stunts  involved. 

Our  hypotheses  take  in  account  the  research  findings  according  to  which  the  use  of  computers  may  give  to 
the  concept  of  "proximal  development  zone",  introduced  by  Vigotskij.  a  new  perspective.  Le..  "the  child 
may  do.  with  the  aid  of  computer  technology;  things  that  he  could  not  do  alone  or  with  the  assistance  of  an 
adult"  p]. 

i .      A  Priori  Analysis  of  the  Learning  Situations 

The  a  priori  analysis  concerns  the  conceptual  aspects  of  the  use  of  the  software  in  relation  to  the  possible 
problem-solving  strategies  that  the  students  may  employ  to  solve  the  assigned  problem  situations,  and  also 
the  conceptual  aspects  of  the  changes  of  strategy  in  the  course  of  the  computer  activity. 
The  considered  objects  are  abstract  geometrical  configurations  (cubes  and/or  poly-cube  structures),  whose 
shape  and  location  within  the  environment  displayed  on  screen  are  not  subject  to  any  balance  limitation. 
Since  these  are  not  objects  characterized  by  a  specific  function,  the  shaping  of  a  mental  image  of  the 
represented  object  cannot  be  based  on  the  identification  of  a  known  shape:  the  students  have  to  construct  it 
every  time. 

6.1  -  Problem  A  requires  the  interpretation  of  a  system  of  views  and  the  production  of  the  corresponding 
parallel  perspective  representation. 

The  strategies  implemented  by  the  students  may  correspond  to  quite  different  levels  of  knowledge  and  of 
anticipation,  and  may  be  linked  with  the  information  that  they  establish  before  acting. 
A  priori,  we  identified  four  possible  first  approaches  to  the  solution: 

-  proceeding  without  taking  notes,  trying  to  construct  the  representation  trusting  one's  memory, 

-  reproducing  the  views  on  a  sheet  of  paper,  even  if  in  different  ways:  sketch  of  the  outlines  alone,  without 
reproducing  the  positions  on  the  views'  planes;  position  sketches,  by  means  of  coordinates;  complete 
sketch  of  outlines  and  grids; 

-  sketching  on  a  sheet  of  paper  a  parallel  perspective  representation  of  the  object,  with  or  without  indication 
of  a  spatial  location; 

-  drawing  only  one  view  (mainly  from  above)  with  a  number  placed  over  each  square  to  indicate  the  number 
of  cubes  "present"  at  that  position. 

Generally  speaking,  we  may  expect  that: 

a)  the  behaviour  of  the  students  in  implementing  their  solution  strategy  depends  not  only  on  the  notes  they 
take,  but  also  on  their  knowledge  of  the  system  of  the  views.  For  instance,  if  the  students  never  worked 
with  orthogonal  projections  and  do  not  know  the  geometrical  rules  that  govern  «hem.  they  may  decide  to 
draw  three  different  objects  corresponding  to  the  three  views  as  seen  from  t.bove  or  three  objects 
corresponding  to  the  views  as  seen  from  the  front. 

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b)  at  first  the  attention  of  the  students  is  drawn  mainly  on  the  reconstruction  of  the  object's  shape;  only  later 
do  they  tackle  the  problem  of  the  correct  location  of  the  object  in  the  measured  and  oriented  Euclidean 
space,  especially  if,  in  their  anticipatioo,  the  coordinates  of  the  object  had  not  been  stressed  in  any  way. 
*2 .  To  solve  situations  B  and  C,  the  students  must  understand  what  is  the  shape  of  the  object  and  in 
which  way  it  is  placed  within  the  trihedral  space.  They  must  gather  the  necessary  information  to  be  able  to 
reproduce  the  views,  considering  that  the  single  representation  given  in  the  statement  of  the  problem  is  not 

A  priori  we  identified  four  possible  fust  approaches  towards  a  solution: 

-  proceeding  without  taking  notes,  trying  to  construct  the  representation  trusting  one's  memory; 

-  drawing  the  orthogonal  projections  directly  after  analysing  the  four  patal'el  perspective  representations 
offered  by  the  software,  recording  or  not  the  location  of  the  object: 

-  reproducing  the  parallel  perspective  drawing  on  a  sheet  of  paper  to  remember  its  shape,  taking  for  granted 
the  ability  to  obtain  from  it  all  the  information  necessary  to  the  representation  o'.  the  views; 

-  drawing  the  view  from  above,  with  a  number  placed  over  each  square  to  indicate  the  number  of  cubes 
present  at  that  position. 

Generally  speaking,  we  may  expect  that: 

a)  the  initial  strategies  of  the  students  are  quite  not  different  with  regard  to  the  construction  modality  to  be 
used  (MOD  1,  MOD  2);  they  tackle  problems  B  and  C  with  the  same  spirit.  Only  later  they  may  feel  the 
need  to  conform  their  strategy  to  the  operational  characteristics  of  the  available  environment; 

b)  the  solution  method  affects  the  setting  up  of  an  optimum  strategy  for  the  solution  of  the  two  types  of 
task.  When  it  is  possible  to  insert  all  the  projections  on  a  view,  and  later  on  the  other  views  (prob.  B),  the 
optimum  strategy  is  based  on  the  second  approach,  since  there  does  not  appear  any  strong  necessity  to  link 
the  object  to  the  views  .  If  it  is  necessary  to  insert  the  projections  corresponding  to  one  cube  at  a  time 
(prob.  C),  the  optimum  strategy  requires,  beside  the  second  approach,  also  the  third  or  the  fourth  one;  in 
fact  it  is  necessary  to  remember  the  shape  of  the  object,  since  for  each  projection  it  is  necessary  to  recognize 
to  which  cube  the  projection  corresponds,  in  order  to  be  able  to  coordinate  them  coherently. 

(.3  •  In  all  the  types  of  problem  tackled,  the  possibilities  of  learning  offered  by  the  software  are  lcnked  to 
the  students'  development  of  strategies  and  knowledge  in  the  course  of  the  activity.  During  the  process  of 
construction  of  the  solution  strategy,  the  effect  of  every  action  performed  by  the  students  is  visualized  on 
the  screen,  yet  without  any  indication  whether  it  is  adequate  to  the  assigned  task.  The  validation  of  one's 
own  strategy  or  of  a  particular  action,  can  be  performed  later,  by  comparing  the  test  of  the  problem  with  the 
representation  automatically  produced  by  the  computer  in  relation  with  the  solution  proposed  by  the 
students.  The  differences  that  students  can  possibly  observe  can  induce  them  to  go  back  to  the  performed 
actions,  hence  starting  a  dynamic  process  between  anticipations  and  validations,  that  we  believe  meaningful 
for  understanding  the  rules  related  to  the  representation  by  means  of  views  and  in  parallel  perspective. 
This  software,  hence,  offers  a  possibility  of  validation,  but  the  decision  whether ,  when,  and  how  to  use  it 
belongs  only  to  the  students.  Since  we  did  not  put  any  limitation  on  the  number  of  validations  that  the 
students  can  perform  during  an  exercise,  we  expect  that  this  can  influence  the  way  they  use  it,  hence  that 
this  can  finally  contribute  to  characterize  the  role  of  the  computer  in  the  students'  learning  process.  We 


1  - 135 

intend  to  evaluate  a  posteriori  the  students'  btUviour  in  relation  to  the  visual  feedback  offered  by  the 

7.  DitcuMMM  of  Sam*  Results  of  tkU  Study 

The  role  of  the  computer  as  mediator  in  the  students'  learning  process  emerged  from  an  analysis  of  the 
processes  by  which  the  students  learned  to  overcome  p»a  erron  and  to  construct  and  modify  their  solution 

In  our  context  we  note  that  the  computer-mediated  activity  allows  an  immediate  actualization,  by  means  of 
images,  of  the  students'  actions  which  can  be  judged  based  on  the  possibility  of  the  available  validation. 
During  our  experimentation,  we  have  observed  that  the  visual  feedback  connected  with  the  possibility  of 
validation  has  been  used  in  different  ways  by  the  students. 

In  same  cases,  overcoming  past  errors  (as  well  as  constructing  a  correct  solution)  has  been  based  on  a  trial 
and  error  practice.  The  visual  feedback  connected  with  the  validation  allowed  a  correctness  test  of  the 
involved  actions.  In  these  cases  the  students'  actions  depended  on  the  way  they  perceived  the  visual 
feedback  related  to  the  past  action. 

For  example,  in  problem  A,  all  pairs  of  students  have  used  a  trial  and  error  practice  to  develop  a  3-D 
reference  system  conception  suitable  to  the  software  operation. 

We  observe  that  the  trial  and  error  practice  requires  many  validations  and  is  characterized  by  the  low  level 
of  the  cognitive  processes  that  students  put  in  action. 

Even  though  in  some  contexts,  such  as  those  linked  with  the  discovery  of  the  software  operation,  a  trial  and 
error  practice  very  often  can  be  the  only  useful  approach  for  the  students,  we  observe  that  in  the  task 
solution  such  approach  is  scarcely  productive  for  the  students'  learning  process. 
We  have  observed  that  only  a  pair  of  students  has  been  captive  of  the  action  -  validation  cycle  that  is  typical 
of  the  trial  and  error  practice;  in  the  various  tasks  this  pair  has  continued  to  perform  actions  based  on 
successive  approximations,  relying  every  action  upon  the  memory  of  the  last  visual  "eedback,  without 
never  earring  out  a  global  anticipation  of  strategy.  We  note  that  only  this  pair  was  unable  to  solve  problem 
C,  for  which  was  necessary  to  work  out  a  more  articulate  strategy  than  the  trial  and  error  one. 
In  all  other  cases  the  trial  and  error  practice  was  used  only  at  the  very  begining;  subsequently  we  have 
observed  that  the  actualizations  through  images  offered  by  computer  have  allowed  the  students  to  observe 
"regularities"  in  the  operation  of  the  projection  system  underlying  the  use  of  the  software,  and  to  work  out, 
in  relation  to  them,  behavioural  schemes  which  have  allowed  the  students  to  single  out  more  articulated 
objectives  for  the  problem  solution. 

For  example,  the  observed  "regularities"  concern  the  correspondence  between  the  position  of  a  cube  in  the 
trihedral  space  and  the  localization  of  its  projections  onto  the  view  planes,  or  the  direction  of  the  cube's 
edges  with  respect  to  the  knots  of  the  grid  of  the  trihedral  planes  in  the  different  possible  points  of  view. 
We  observed  that  the  elaboration  of  a  behavioural  scheme  connected  with  regularities  observed  by  the 
students  is  the  result  of  a  non  formalized  analysis  produced  by  the  dialectic  between  anticipation  -  action  - 
actualization  by  means  of  images  -  validation,  which  is  realized  through  the  dialogue  with  the  computer.  A 
peculiar  characteristic  of  the  non  formal  analysis  performed  with  the  help  of  the  computer  is  the  binding 


1  - 136 

that  is  established,  while  students  of  a  pair  communicate  with  each  other,  between  the  observed  "regularity" 
and  one  or  mote  "key  words"  taken  from  the  vocabulary  of  both  of  them.  Examining  dialogues  between 
students,  results  that,  through  the  mediation  of  the  computer,  they  have  been  able  to  give  to  these  words 
particular  meanings  connected  to  the  geometrical  properties  related  with  the  observed  regularities:  such 
words  tend  to  be  implicitly  transformed,  during  a  work  session,  into  "conceptual  words"  related  to  the 
geometrical  knowledge  which  is  requires  by  the  problem  solution.  At  the  same  time,  also  actions  and 
controls,  that  the  students  have  put  in  action  in  discovering  a  regularity,  acquire  a  unitary  meaning  related  to 
the  naming  process.  This  process  leads  to  the  construction  of  a  behavioural  scheme  related  to  the  observed 
regularities,  and  places  at  disposal  tools  that  allow  students  to  stop  and  reflect  upon  their  own  solution 
strategy,  and  possibly  elaborate  new  anticipative  hypotheses  for  the  proposed  problem. 
Hence,  we  observed  that,  in  these  cases,  the  computer  mediates  the  individual  and  pair  activity  of  the 
students,  influencing  at  the  same  time  both  the  elaboration  capabilities  of  each  student  and  the  quality  of 

The  observations  made  during  this  experimentation  raise  two  important  problems. 

The  first  problem  is  whether  low  level  strategies,  like  trial  and  error,  are  intrinsic  in  any  software  with 

graphic  feedback,  hence  in  some  cases  avoidable  only  with  teacher's  assistence,  or  can  be  overcome  by 

adding  bindings  to  the  dialogue  between  student  and  computer,  suitably  modifying  the  software 


The  second  problem  concerns  how  to  formalize  the  non  formal  analysis  conducted  by  the  students  with  the 
mediation  of  the  computer,  transforming  what  actually  is  an  in-progress  knowledge  into  a  conscious 
learning  of  the  geometric  knowledge  under  consideration.  We  believe  that  this  problem  is  closely  correlated 
to  the  elucidation  of  the  role  that  the  teacher  may  play  during  and  after  the  students'  activity  with  the 
computer,  that  has  not  yet  been  studied  in  our  work. 


[1J  Baldy,  Dolle,  Higele,  Lebahar,  Rabardel,  Verillon,  Vermersch,  Weill-Fassina,  1988,  'Activites 
cognittves  dans  rapprentusage  et  I'utilisation  du  design  techinique:  etat  des  traveaux',  in  Didactique  et 
acquisition  de  connaistnces  scientifujues.  Edition  la  Pensee  Sauvage,  Pag  149-165 
[2]  Bautier T.,  Boudarel  J.,  Colraez  P.,  Parzysz  B.,  1988,  'Geometric  et  space .  Representation  plane  de 
figures  de  I  espace ,  in  Didactique  et  acquisition  de  connaistnces  scientifiques.  Edition  la  Pensee  Sauvage 
131  Bessot  A.,  Eberhard  M.,  1987,  'Representations  graphiques  d'assemblages  de  cubes  et  finalittsdes 
situation',  in  Le  dtsm  technique,  Hermes,  Pag  61  -  71 

[4]  Margolinas  C,  1989,  'Le  point  de  vue  deb  validation:  essai  de  syntese  et  d'analyse  en  didactique  des 
truuMmatiques',  These  pour  obutiir  le  litre  de  Demur  de  IVniversUt  Joseph  Fourier  -Grenoble  I 
[5]  Osta  I„  1988,  'L'ordinateur  comme  outile  a  I'enseignement.  Une  sequence  didactique  pour 
lmseignernent  du  reperage  dans  l'espace  a  l'alde  de  logiciels  graphiques'.  These  pour  obtenir  le  litre  de 
Docteur  de  IVmversai  Joseph  Fourier  -Grenoble  I 

[6]  Polo  M.,  1989,  'Sistema  di  riferimeoto  e  geotnetria  dello  spado:  analisi  di  comportamenti  spontanei  di 
bambini  di  8-9  anru ,  L  insegnamento  delta  maiematka  e  delle  selente  integrate.  Vol.  1 1  n.  7/8 
[71  Jildiomirov  O.,  1991,  Tirformation  age  and  Lev  Vygotsky's  theory1.  Lecture  at  the  Fourth  International 
conference  Children  m  the  information  age",  Alberta-Bulgaria 

[8]  Weill-Fassina  A.,  Wermersch  P.,  Zouggari  G.,  1987,  'devolution  des  competences  dans  la  lecture  de 
formes  tlemetaires  en  vuej,  onogonales',  in  Le  dessin  technique,  Hermes,  Pag  101-108. 



1  - 137 


David  J.  Claike  and  Peter  A.  Sullivan 
Mathematics  Teaching  and  Learning  Centre 
Australian  Catholic  University  (Victoria)  -  Christ  Campus 


The  classification  of  the  types  of  responses  provided  by  primary  and  secondary  schoolchildren  to 
a  particular  form  of  open-ended  mathematics  task  has  enabled  the  i  investigation  of  the  effects  of 
factors  such  as  collaborative  work,  age,  instruction,  question  format,  and  culture  or  school 
system.  The  demonstrated  reluctance  of  pupils  to  give  multiple  or  general  responses  has  been 
investigated  through  the  distinction  between  the  inclination  to  give  such  answers  and  the  ability  i. 
do  so.  Comparison  has  been  made  with  pupil  responses  to  open-ended  tasks  in  disciplines  other 
than  mathematics.  Study  has  been  made  of  pupils'  accounts  of  their  thinking  while  attempting 
such  tasks  and  their  justifications  for  their  answers.  Preliminary  findings  are  reported  of  an  on- 
going study  into  the  learning  outcomes  of  a  teaching  program  based  solely  on  the  use  of  such 
open-ended  mathematics  tasks.  The  implications  of  this  research  are  discussed  with  respect  to  the 
use  of  open-ended  mathematics  tasks  for  the  purposes  of  instruction,  assessment  and  as  a 
research  tool. 


This  paper  constitutes  a  progress  report  on  an  extensive  and  continuing  program  of 
research  in'.o  the  use  of  a  particular  form  of  open-ended  mathematics  task  for  the  purposes  of 
instruction  and  assessment  The  task  type  has  been  given  the  title  "Good  Questions",  and  the 
characteristics  of  these  Good  Questions  have  been  discussed  elsewhere  (Sullivan  &  Clarke, 
1988;  Clarke  and  Sullivan,  1990,  Sullivan  and  Clarke,  1991a  and  b,  Sullivan,  Clarke  and 
Wallbridge,  1991).  Examples  of 'good'  questions  are  as  follows: 

The  questions  are  different  from  conventional  exercises  in  two  major  ways.  First,  these 
questions  engage  the  students  in  constructive  thinking  by  requiring  them  to  consider  the 
necessary  relationships  for  themselves,  and  to  devise  their  own  strategies  for  responding  to  the 
questions.  Second,  the  questions  have  more  than  one  possible  correct  answer.  Some  students 
might  give  just  one  correct  response,  others  might  produce  many  correct  answers,  and  there  may 

A  number  has  been  rounded  off  to  5.8.  What  might  the  number  be? 
Draw  some  triangles  with  an  area  of  6  sq  cm. 
Find  two  objects  with  the  same  mass  but  different  volutM. 
Describe  a  box  with  a  surface  area  of  94  sq.  cm. 

1  - 138 

be  some  who  will  make  general  statements.  The  openness  of  the  tasks  offers  significant  benefits 
to  classroom  teachers  because  of  the  potential  for  students  at  different  stages  of  development  to 
respond  at  their  own  level. 


The  following  is  a  report  of  a  five-stage  project  which  sought  to  identify  the  way 
schoolchildren  respond  to  Good  Questions.  The  discussion  of  results  is  structured  around  the 
specific  research  questions  addressed  at  each  stage  of  the  project. 

General  Method  -  Task  administration 

In  a  typical  administration,  a  set  if  four  questions  was  given  to  participant  classes  of 
schoolchildren.  The  criterion  for  selection  of  classes  was  the  willingness  of  their  teachers  to 
participate.  Even  though  no  teacher  declined  the  invitation  no  claims  are  made  about 
representativeness  of  the  results  for  other  schools. 

In  the  first  administration  of  tasks,  the  questions  asked,  the  procedure  for  administration 
and  the  response  coding  system  were  as  follows: 

Last  night  I  did  a  subtraction  task.  I  can  remember  some  of  the  numbers. 

What  might  the  missing  numbers  have  been? 

A  number  has  been  rounded  off  to  5,8  What  might  the  number  have  been? 

A  rectangle  has  a  perimeter  has  of  30  m.  What  might  be  the  area? 
Fraction  as  operator 

j  of  the  pupils  in  a  schooi  play  basketball.  How  many  pupils  might  there  be  in  the  school  and 
how  many  might  play  basketball? 

The  format  for  administering  the  questions  was  the  same  in  each  class: 

i)  The  question:    "       +  «  10  What  might  the  missing  number  be?"  was  posed,  and 

the  responses  suggested  by  the  pupils  were  written  on  the  chalkboard.  The  pupils  in  the 
class  were  asked  to  comment  on  what  was  different  about  this  task  from  common 
mathematics  questions.  The  response  sought  was  that  there  are  many  possible  answers. 

ii)  The  first  two  questions  were  distributed  (subtraction,  rounding). 

iii)  The  papers  were  collected  and  the  answers  reviewed.  Again  the  possibility  of  multiple 
answers  was  discussed. 

iv)  The  second  two  questions  were  distributed  (area,  fraction  as  operator) 


-  7 


1  - 139 

The  responses  of  the  pupils  to  the  tasks  were  coded.  The  coding  was  as  follows: 

0  meant  no  correct  answers 

1  meant  only  one  correct  answer 

2  meant  two  or  three  correct  answers 

3  meant  all  or  many  correct  answers 

4  meant  that  a  general  statement  was  given 

To  illustrate  the  way  that  this  code  was  applied  to  the  rounding  question,  the  following  is 
the  meaning  of  the  codes.  Individual  correct  answers  were  numbers  such  as  5.82  and  5.78.  A 
code  of  "3"  was  given  to  a  response  like  "5.75  5.76  5.77  5.78  5.79  5.80  5.81  5.82  5.83  5.84". 
Examples  of  responses  which  were  considered  to  represent  a  general  statement,  "4",  were  "5.75 
...  right  up  to  5.84999..."  or  "between  5.75  and  5.849". 


Stage  1. 

Stage  1  of  the  project  addressed  two  questions: 

What  types  of  responses  do  primary  and  secondary  schoolchildren  give  to  such 
open-ended  tasks? 

Do  the  responses  of  the  pupils  vary  depending  on  whether  they  work  together 
or  individually? 

The  purpose  of  this  stage  of  the  investigation  was  to  ascertain  the  proportion  of  the 
pupils  who  gave  each  of  the  types  of  responses  and  to  compare  the  responses  of  pupils  in  the 
different  groups;  individual,  combined,  and  pair/ind.  There  were  39  pupils  who  completed 
individual  responses,  49  students  worked  in  pairs  but  submitted  individual  papers,  and  there 
were  39  pairs  of  students  who  gave  combined  answers.  It  was  confirmed  with  the  respective 
teachers  that  all  of  the  concepts  which  are  pre-requisite  to  these  questions  had  been  taught 
during  the  year  prior  to  this  study. 

No  clear  differences  between  the  groups  emerged.  The  proportions  who  responded  at 
each  level  did  not  appear  to  be  influenced  by  whether  they  worked  individually  or  with  a  partner. 
It  had  been  anticipated  that  the  pupils  who  had  worked  together  would  be  more  likely  to  give 
multiple  or  general  answers.  It  was  presumed  that  two  minds  might  view  each  task  differently 
and  produce  at  least  two  responses,  as  well  as  alerting  the  pairs  to  the  possibility  of  more  than 
one  correct  answer.  Whatever  thinking  or  expectation  is  necessary  to  stimulate  multiple  or 
general  answers  appeared  to  be  no  more  available  to  pairs  than  to  individuals.  Or  conversely, 
whatever  preconceptions  limited  the  potential  to  give  multiple  answers  affected  both  pairs  and 
individuals  alike. 



1  -  140 

Stage  2 

Stage  2  of  the  project  addressed  the  questions: 

What  effect  does  age  or  school  experience  have  on  the  types  of  responses  given 
to  'good'  questions? 

Does  the  distribution  of  pupil  response  types  differ  according  to  culture  or 
school  system? 

The  same  four  questions  were  given  to  99  year  10  students  at  two  outer  suburban  high 
schools  in  Melbourne,  to  97  year  10  students  at  a  specialist  mathematics/science  school  in  Penang 
State,  Malaysia,  and  to  86  year  X0  students  in  two  high  schools  in  the  USA.  The  questions  were 
translated  into  Bahasa  Malaysia  for  the  Malaysian  students.  Subsequent  independent  re- 
translation  of  the  Malaysian  questions  into  English  verified  the  accuracy  of  the  translation.  The 
protocol  for  the  delivery  of  the  questions  was  the  same  as  in  stage  1,  and  was  followed  in  each 
case.  The  year  10  students  worked  individually. 

The  year  10  pupils  were  able  to  give  better  responses  to  each  of  the  questions  than  the 
year  6  pupils.  Fewer  year  10  pupils  made  errors,  and  there  were  more  who  gave  multiple  and 
general  responses.  While  noting  that  the  pre-requisite  content  is  prescribed  at  the  primary  level  in 
curriculum  documents,  it  was  pleasing  that  most  year  10  students  were  able  to  give  satisfactory 
answers  to  the  questions. 

The  profile  of  the  responses  of  the  year  10  students  from  Penang  was  marginally  different 
from  the  year  10  Australian  students.' In  each  question  a  higher  proportion  of  the  Penang  students 
were  technically  accurate  in  their  responses  to  the  four  questions  than  the  Australian  students,  but 
fewer  gave  general  responses.  The  responses  of  the  American  students  resembled  the  Malaysian 
sample  on  the  first  two  tasks,  the  Australian  sample  on  the  third,  and  was  quite  distinctive  with 
regard  to  the  basketball  question. 

Overall,  while  there  were  more  students  at  year  10  level  than  at  year  6  who  gave  multiple 
and  general  responses,  there  was  still  a  significant  number  of  year  10  students  who  gave  a  single 
response  even  though  it  seemed  to  the  researchers  that  a  request  for  multiple  answers  had  been 
implied  by  the  wording  of  the  question. 

Stage  3  addressed  the  following  research  questions: 

Does  instruction  increase  the  number  of  pupils  who  give  multiple  or  general 

Does  question  format  affect  the  number  of  pupils  who  give  multiple  or  general 

Stage  3 

1  -  141 

This  third  stage  was  an  attempt  to  investigate  the  cueing  inherent  in  the  questions.  On  one 
hand,  it  was  hypothesised  that  the  pupils  may  have  had  too  much  experience  at  the  single-answer 
type  of  question,  and  would  need  more  experience  than  that  provided  in  the  protocol  of  stage  1 . 
On  the  other  hand,  it  was  possible  that  the  word  "might"  in  the  question  may  not,  in  a 
mathematical  context,  imply  that  more  than  one  possible  answer  was  required. 

Although  the  two  research  questions  for  this  stage  are  to  some  extent  separate  they  were 
investigated  concurrently.  Three  of  the  grade  6  classes  who  had  participated  in  stage  one  were 
given  instruction  and  another  three  of  the  grade  6  classes  were  asked  the  questions  in  a  different 

The  lesson  taught  to  the  three  classes  aimed  to  broaden  the  pupils'  view  of  what  such 
questions  arc  seeking.  The  first  step  in  the  lesson  was  to  focus  on  the  word  "might".  Questions 
relating  to  both  everyday  and  mathematical  situations  were  used  to  discuss  both  multiple  and 
general  answers.  These  classes  are  called  the  "instruction"  group. 

The  other  three  classes  were  not  taught  a  lesson  but  were  given  the  same  mathematical 
tasks  in  a  slightly  different  format.  Instead  of  phrasing  the  question  like  "What  might  the  answer 
be?",  words  similar  to  "Give  as  many  possible  answers  as  you  can"  were  used.  This  was 
intended  to  alert  the  pupils  directly  to  both  the  possibility  of  multiple  answers  and  to  the 
requirement  to  give  as  many  answers  as  they  could.  These  classes  are  called  the  "question 
format"  group. 

There  was  a  clear  trend  that  the  classes  which  responded  to  the  questions  with  the  revised 
format  gave  more  multiple  responses  than  the  classes  which  had  had  instruction.  It  appears  likely 
that  there  are  pupils  who  are  able  to  give  multiple  responses  but  who  do  not  consider  that  the 
"might"  questions  invite  such  replies. 

There  was  some  indication  that  pupils'  responses  to  particular  questions  were 
disproportionately  influenced  by  the  specific  concept  or  concepts  invoked.  Because  of  the 
extended  nature  of  open-ended  tasks,  the  form  of  the  individual  question  assumes  greater 
significance  than  is  the  case  with  conventional  closed  tasks. 

Stage  4. 

Stage  4  involved  the  implementation  of  a  teaching  experiment  employing  Good  Questions  and 

addressed  the  question: 

What  are  the  learning  outcomes  of  a  teaching  program  based  solely  on  the  use  of  such 

open-ended  mathematics  tasks? 

Stage  4  was  conducted  in  a  suburban  Catholic  primary  school.  The  school  had  a  high 
proportion  of  students  from  non-English  speaking  backgrounds  and  served  a  predominantly  lower 
socio-economic  community. 



1  - 142 

The  experimental  group  was  taught  a  unit  on  length,  perimeter  and  area  over  seven  one- 
hour  lessons.  The  control  group  was  taught  the  same  topic.  The  teacher  of  the  control  class  was 
instructed  to  follow  the  program  presented  in  the  most  commonly  used  text.  This  was  to  simulate 
a  standard  approach  to  the  topic.  Data  collected  included  amtudinal  data,  achievement  data  and 
observation  data  regarding  classroom  practices.  » 

There  were  two  interesting  results.  First,  the  experimental  class  were  able  to  respond  to 
the  skill  items  as  welt  as  the  control  group,  even  though  there  had  been  no  teaching  or  practice  of 
skills  in  that  class.  Second,  even  though  there  more  students  in  the  experimental  group  who 
could  give  one  correct  response  to  the  Good  Questions  than  in  the  control,  no  students  in  either 
group  attempted  multiple  or  general  answers.  The  meaning  of  this  is  unclear.  During  the 
program,  many  students  in  the  experimental  group  were  willing  and  able  to  give  multiple  and 
general  responses  to  'good'  questions.  It  is  not  clear  why  they  did  not  give  such  responses  to  the 
test  items. 

Stage  5 

Stage  5  sought  to  address  the  questions: 

Is  the  reluctance  of  pupils  to  give  multiple  or  general  responses  to  open-ended 
mathematics  tasks  replicated  with  open-ended  tasks  in  other  disciplines? 
Does  the  use  of  open-ended  tasks  for  assessment  purposes  disadvantage  any 
identifiable  groups  of  students? 

For  stage  5  of  the  project,  students  at  years  7  and  10  from  three  schools  (one 
single-sex  boys,  one  single-sex  girls,  and  one  co-educational)  were  asked  to  respond  to 
open-ended  items  from  a  variety  of  academic  contexts. 
For  instance,  one  question  was: 

In  a  Victorian  country  town,  the  population  fell  by  50%  over  a  period  of  5  years. 

Why  might  this  have  happened? 

The  protocol  guiding  the  administration  took  two  basic  forms  derived  from  that  for 
stage  one.  The  administration  varied  task  order  and  whether  or  not  the  requirement  of 
multiple  answers  was  made  explicit  in  the  task  format 

Analysis  disregarded  the  correctness  of  student  response  in  applying  the  coding  of 
stage  one.  As  a  consequence,  the  discussion  which  follows  documents  student  intended 
response  levels. 

Order  of  task  administration  did  not  affect  student  response  types. 


1  -  143 

Inferences  which  might  be  drawn  from  these  results  included: 

•  that  the  inclination  to  give  singie  responses  (or  the  reluctance  to  give  multiple 
responses)  is  a  product  of  schooling,  and  not  peculiar  to  mathematics.  Both  year  7  and 
year  10  pupils  were  similarly  recluctant  to  given  multiple  answers  in  all  four  academic 

•  that  the  explicit  request  of  multiple  responses  produces  a  significant  increase  in  the 
response  level  in  other  academic  contexts,  but  not  necessarily  mathematics: 

•  that  the  ability  to  give  multiple  responses  increases  significantly  with  age,  except  in  the 
context  of  literature. 

Mathematical  power  has  been  identified  with  the  capacity  to  solve  non-routine  problems 
(NCTM,  1989),  and  open-ended  tasks  are  seen  as  an  appropriate  vehicle  for  instruction  and 
assessment  of  students'  learning  in  this  regard.  Further  justification  for  the  use  of  open-ended 
questions  for  instructional  purposes  can  be  found  in  the  work  of  Swelter  and  his  associates  (see, 
for  instance,  Swelter,  1989),  where  the  use  of  goal-free  tasks  was  associated  with  effective  schema 

Are  students'  responses  to  open-ended  tasks  constrained  unduly  by  their  preconceptions 
about  the  nature  of  an  acceptable  response?  It  would  appear  that  students  in  both  year  7  and  year 
10  possessed  a  comparable  reluctance  to  provide  multiple  responses.  However,  when  multiple 
responses  were  explicitly  requested,  there  was  a  significant  increase  in  the  proportion  of  multiple 
responses  offered  by  pupils  in  Literature,  Science  and  Social  Science,  but  only  to  some  items  in 
Mathematics.  This  finding  may  be  task-specific  and  warrants  further  investigation. 

Are  students'  responses  to  open-ended  tasks  necessarily  indicative  of  either  mathematical 
understanding  or  capability?  Certainly  it  appears  that  young  children  find  it  substantially  more 
difficult  than  older  children  to  provide  multiple  answers  to  mathematics  tasks.  This  suggests  that  it 
may  be  inappropriate  to  ask  primary  school  age  children  to  give  multiple  responses.  The  legitimacy 
of  relating  student  responses  to  non-routine  and  open-ended  tasks  to  c utricular  content  currently 
being  studied  continues  to  be  the  subject  of  research.  Given  current  curriculum  initiatives  which 
employ  open-ended  tasks  for  assessment  purposes  (for  example,  CAP,  1989;  VCAB,  1990),  the 
results  of  this  research  assume  some  significance.  Substantial  additional  research  is  required  if  we 
are  to  understand  the  significance  of  the  meanings  constructed  by  students  in  responding  to  open- 
ended  tasks.  Such  research  must  address  those  student  conceptions  of  legitimate  mathematical 
activity  on  which  their  response  inclinations  are  predicated  (Clarke,  Wallbridge  &  Fraser,  1992, 
and  this  research),  and  issues  of  cognitive  load  or  working  memory  capacity  and  related 
developmental  theories  of  learning  outcomes  which  determine  student  response  capability  (for 


1  - 144 

example,  Collis,  1991;  Sweller,  19S9).  These  associated  nutters  of  inclination  and  capability  must 
be  understood  if  we  are  to  employ  such  Usks  with  success  in  mathematics  classrooms  for  the 
purposes  of  either  instruction  or  assessment. 

•  The  invaluable  assistance  of  Ursula  Spandel  and  Margarita  Wallbridge  is 
gratefully  acknowledged. 

•  This  project  benefitted  from  the  support  of  an  Australian  Research  Council  grant 
for  research  infrastructure. 


Califomian  Assessment  Program  (CAP)  (1989).  A  question  of  thinking.  Sacramento,  CA: 

California  State  Department  of  Education. 
Clarke,  D.J.  and  Sullivan,  P.  ( 1990)  Is  a  question  the  best  answer?  The  Australian  Mathematics 

Teacher  46(3),  30  -  33. 

Clarke,  D.J.,  Wallbridge,  M.  and  Eraser,  S.  (1992).  The  other  consequences  of  a  problem-based 

curriculum.  Research  Report  No.  3.  Oakleigh,  Vic:  Australian  Catholic  University, 

Mathematics  Teaching  and  Learning  Centre. 
Collis,  K.F.  (1991).  Assessment  of  the  Learned  Structure  in  Elementary  Mathematics  and 
Science.  Paper  presented  to  the  Conference  on  Assessment  in  the  Mathematical  Sciences, 
held  at  the  Institute  of  Educational  Administration,  Geelong  East,  November  20  -  24, 1991. 
National  Council  of  Teachers  of  Mathematics  (NCTM)  (1989)  Curriculum  and  Evaluation 

Standards  for  School  Mathematics.  Reston.VA:  NCTM. 
Sullivan,  P.  and  Clarke,  D.J.  (1988)  Asking  Better  Questions.  Journal  of  Science  and 

Mathematics  Education  in  South  East  Asia,  June,14  - 19. 
Sullivan,  P.  and  Clarke,  D.J.  (1991a)  Catering  to  all  abilities  through  "Good"  questions. 

Arithmetic  Teacher  39(2),  14  -  21. 
Sullivan,  P.  and  Clarke,  D.J.  (1991b)  Communication  in  the  Classroom:  The  Importance  of 

Good  Questioning.  Geelong:  Deakin  University  Press. 
Sullivan,  P.,  Clarke,  DJ.,  &  Wallbridge,  M.  (1991)  Problem  solving  with  conventional 

mathematics  content:  Responses  of  pupils  to  open  mathematical  tasks.  Research  Report  I . 

Oakleigh:  Mathematics  Teaching  and  Learning  Centre  (MTLC),  Australian  Catholic 

University  (Victoria). 

Sweller,  J.  (1989).  Cognitive  technology:  Some  procedures  for  facilitating  learning  and  problem 
solving  in  mathematics  and  science.  Journal  of  Educational  Psychology  81(A),  437-466. 

Victorian  Curriculum  and  Assessment  Board  (VCAB)  (1990)  Mathematics:  Study  Design. 
Melbourne:  VCAB. 

Wertsch,  J.V.  ( 1991).  Voices  of  the  mind:  A  sociocultural  approach  to  mediated  action. 
CambridgcMA:  Harvard  University  Press. 


1  -  145 


M.A.  fKtn)  flfm™^.  Nerida  F.  Ellerton 

Faculty  of  Education  Faculty  of  Education 

Deatan  University  (Geclong)  Dcakin  University  (Geelong) 

The  Newman  procedure  for  analysing  errors  on  written  mathematical  tasks  is  summarised 
and  data  from  studies  carried  out  in  Australia,  India.  Malaysia,  PapmNew  GuMufa  and 
Thailand  are  reported.  These  data  show  that,  in  each  country; the  SXZtot 
a  large  percentage  (more  than  50%  in  four  of  the  five  countrie7)ofthTe^rsSZ 

ft  JH^  <*Pl«d  Paces'  skills  such  as  the  four  operations.  AdditiZalZtash^im t  hit 
Indian  primary  school  students  perform  significantly  better  than  Australian  studem  of  a 
TrHrJgZ,°n  '"".SWorward  computational  exercises,  but  significZlywor^e  onlrUhmltic 
TV*  q^S,hn  ,here  « an  over-emlhasis  onprocZ Swfh 

insufficient  attention  being  given  to  the  role  of  language  factors  in  mathematics  learning. 

The  Newman  Hierarchy  of  Error  Causes  for  Written  Mathematical  Tasks 

Since  1977,  when  Newman  (1977a,b)  first  published  data  based  on  a  system  she  had 
developed  for  analysing  errors  made  on  written  tasks,  there  has  been  a  steady  stream  of  research 
papers  reporting  studies,  carried  out  in  many  countries,  in  which  her  data  collection  and  data 
analysis  methods  have  been  used  (see,  for  example,  Casey,  1978;  Clarkson,  1980,  1983,  1991; 
Clements,  1980, 1982;  Marinas  &  Clements,  1990;  Watson,  1980). 

The  findings  of  these  studies  have  been  sufficiently  different  from  those  produced  by  other 
error  analysis  procedures  (for  example,  Hollander,  1978;  Lankford,  1974;  Radatz,  1979),  to  attract 
considerable  attention  from  both  the  international  body  of  mathematics  education  researchers  (see, 
for  example,  Dickson,  Brown  and  Gibson,  1984;  Mellin-OIsen,  1987;  Zcpp,  1989)  and  teachers  of 
mathematics.  In  particular,  analyses  of  data  based  on  the  Newman  procedure  have  drawn  special 
attention  to  (a)  the  influence  of  language  factors  on  mathematics  learning;  and  (b)  the 
inappropriateness  of  many  "remedial"  mathematics  programs  in  schools  in  which  there  is  an  over- 
emphasis on  the  revision  of  standard  algorithms  (Clarke,  1989). 

The  Newman  Procedure 

According  to  Newman  (1977a,b;  1983),  a  person  wishing  to  obtain  a  correct  solution  to  an 
arithmetic  word  problem  such  as  "The  marked  price  of  a  book  was  $20.  However,  at  a  sale,  20% 
discount  was  given.  How  much  discount  was  this?",  must  ultimately  proceed  according  io  the 
following  hierarchy: 

1.  Read  the  problem; 

2.  Comprehend  what  is  read; 

3.  Carry  out  a  mental  transformation  from  the  words  of  the  question  to  the  selection  of  an 
appropriate  mathematical  strategy; 

4.  Apply  the  process  skills  demanded  by  the  selected  strategy;  and 

5.  Encode  the  answer  in  an  acceptable  written  form. 


1  -146 

Newman  used  the  word  "hierarchy"  because  she  reasoned  that  failure  at  any  level  of ^  above 
sequence  prevents  problem  solvers  from  obtaining  satisfactory  solutions  (unless  by  chance  they 
arrive  at  correct  solutions  by  faulty  reasoning). 

Of  course,  as  Casey  (1978)  pointed  out,  problem  solvers  often  return  to  lower  stages  of  the 
hierarchy  when  attempting  to  solve  problems,  especially  those  of  a  multi-step  variety.  (For 
example,  in  the  middle  of  a  complicated  calculation  someone  might  decide  to  reread  the  question  o 
check  whether  all  relevant  information  has  been  taken  into  account.)  However,  even  .f  some  of  the 
steps  are  revisited  during  the  problem-solving  process,  the  Newman  hierarchy  provides  a 
fundamental  framework  for  the  sequencing  of  essential  steps.   

■—      ~~~7~    i  ~  Interaction  Between  the  Question 

5SSE    J   and  the  Person  Attempting  it 

',  |  ENCOONG   


.  L_      ^  >s  



Figure  I.  The  Newman  hierarchy  of  error  causes  (from  Clements.  1980.  p.  4). 
Clements  (1980)  illustrated  the  Newman  technique  with  the  diagram  shown  in  Figure  1 
According  to  Clements  (1980.  p.  4).  errors  due  to  the  form  of  the  question  are  essentially  Afferent 
Zn  those  in  the  other  categories  shown  in  Figure  1  because  the  source  of  difficulty  resides 
fundamentally  in  the  question  itself  rather  than  in  the  interaction  between  the  problem  solver  r^th 

p  aced  beside  the  five-stage  hierarchy.  Two  other  categories.  "Carelessness"  and  "Motivation, 
ave  also  been  shown  as  separate  from  the  hierarchy  although,  as  indicated,  the*  types  of  errors 
can  occur  at  any  stage  of  the  problem-solving  process.  A  careless  error,  for  example,  cou  d  be  a 
read"  error,  a  com  rehension  enor.  and  so  on.  Similarly,  someone  who  had  ^™P« 
^worked  out  an  appropriate  strategy  for  solving  a  problem  might  decline  to  proceed  furtherin  *e 
hlrchytcause  of  a  lack  of  motivation.  (For  example,  a  problem-solver  might  exclaim:  What  a 
trivial  problem.  It's  not  worth  going  any  further") 

Newman  (1983.  p.  1 1)  recommended  that  the  following  "questions  or  requests  be  used  m 
interviews  that  are  carried  out  in  order  to  classify  students'  errors  on  written  mathematical  tasks: 

1  Please  read  the  question  to  me.  (Reading) 

2  Tell  me  what  the  question  is  asking  you  to  do.  (Comprehension) 

3.  Tell  me  a  method  you  can  use  to  find  and  answer  to  the  question.  (Transformation) 



1  - 147 

4.  Show  me  how  you  worked  out  the  answer  to  the  question.  Explain  to  me  what  you  are 
doing  as  you  do  it.  (Process  Skills) 

5,  Now  write  down  your  answer  to  the  question.  (Encoding) 

If  pupils  who  originally  gave  an  incorrect  answer  to  a  word  problem  gave  a  correct  answer 
when  asked  by  an  interviewer  to  do  it  once  again,  the  interviewer  should  still  make  the  five  requests 
in  order  to  investigate  whether  the  original  error  was  due  to  carelessness  or  motivational  factors. 

Example  of  a  Newman  Interview 

Mellin-Olsen  (1987,  p.  150)  suggested  that  although  the  Newman  hierarchy  was  helpful  for  the 
teacher,  it  could  conflict  with  an  educator's  aspiration  "that  the  learner  ought  to  experience  her  own 
capability  by  developing  her  own  methods  and  ways."  We  would  maintain  that  there  is  no  conflict 
as  the  Newman  hierarchy  is  not  a  learning  hierarchy  in  the  strict  Gagne  (1967)  sense  of  that 
expression.  Newman's  framework  for  the  analysis  of  errors  was  not  put  forward  as  a  rigid 
information  processing  model  of  problem  solving.  The  framework  was  meant  to  complement  rather 
than  to  challenge  descriptions  of  problem-solving  processes  such  as  those  offered  by  Polya  (1973). 
With  the  Newman  approach  the  researcher  is  attempting  to  stand  back  and  observe  an  individual's 
problem-solving  efforts  from  a  coordinated  perspective;  Polya  (1973)  on  the  other  hand,  was  most 
interested  in  elaborating  the  richness  of  what  Newman  termed  Comprehension  and  Transformation. 

The  versatility  of  the  N-swman  procedure  can  be  seen  in  the  following  interview  reported  by 
Ferrer  (1991).  The  student  interviewed  was  an  1 1-year-old  Malaysian  primary  school  girl  who  had 
given  the  response  "AH"  to  the  question  "My  brother  and  I  ate  a  pizza  today.  I  ate  only  one  quarter 
of  the  pizza,  but  my  brother  ate  two-thirds.  How  much  of  the  pizza  did  we  eat?"  After  the  student 
had  read  the  question  correctly  to  the  interviewer,  the  following  dialogue  took  place.  (In  the 
transcript,  "I"  stands  for  Interviewer,  and  "S"  for  Student.) 

I:  What  is  the  question  asking  you  to  do? 

S:  Uhmm  . . .  It's  asking  you  how  many  . . .  how  much  of  the  pizza  we  ate  in  total? 
I:  Alright.  How  did  you  work  that  out? 

S:  By  drawing  a  pizza  out ...  and  by  drawing  a  quarter  of  it  and  then  make  a  two-thirds. 
I:  What  sort  of  sum  is  it? 
S.  A  problem  sum! 

I:  Is  it  adding  or  subtracting  or  multiplying  or  dividing? 
S:  Adding. 

I:  Could  you  show  me  how  you  worked  it  out?  You  said  you  did  a  diagram.  Could  you 

show  me  how  you  did  it  and  what  the  diagram  was? 
S:  (Draws  the  diagram  in  Figure  1A.)  I  ate  one-quarter  of  the  pizza  (draws  a  quarter*). 

1  - 148 



Figure  I.  Diagrammatic  representations  of  the  pizza  problem. 

I:  Which  is  the  quarter? 

S:  This  one.  (Points  to  the  appropriate  region  and  labels  it  1/4.) 
I:  How  do  you  know  that's  a  quarter? 

S:  Because  it's  one-fourth  of  the  pizza.  Then  I  drew  up  two-thirds,  which  my  brother  ate. 

(Draws  line  x  -  see  Figure  IB  -  and  labels  each  part  1/3) 
I:  And  that's  1/3  and  that's  1/3.  How  do  you  know  it's  1/3. 
S:  Because  it's  a  third  of  a  pizza. 

The  interview  continued  beyond  this  point,  but  it  was  clear  from  what  had  been  said  that  the  original 
error  should  be  classified  as  a  Transformation  error  -  the  student  comprehended  the  question,  but 
did  not  succeed  in  developing  an  appropriate  strategy.  Although  the  interview  was  conducted 
according  to  the  Newman  procedure,  the  interviewer  was  able  to  identify  some  of  the  student's 
difficulties  without  forcing  her  along  a  solution  path  she  had  not  chosen. 

In  her  initial  study,  Newman  (1977a)  found  that  Reading,  Comprehension,  and  Transformation 
errors  made  by  124  low-achieving  Grade  6  pupils  accounted  for  13%,  22%  and  12%  respectively  of 
all  errors  made.  Thus,  almost  half  the  errors  made  occurred  before  the  application  of  process  skills. 
Studies  carried  out  with  primary  and  junior  secondary  school  children  in  Melbourne,  Australia,  by 
Casey  (1978),  Clements  (1980),  Watson  (1980),  and  Clarkson  (1980)  obtained  similar  results,  with 
about  50%  of  errors  first  occurring  at  the  Reading,  Comprehension  or  Transformation  stages. 
Casey's  study  involved  116  Grade  7  students,  Clements's  sample  included  over  700  children  in 
Grades  5  to  7,  Watson's  study  was  confined  to  a  preparatory  grade,  and  Ciarkson's  sample 
contained  13  low-achieving  Grade  7  students.  In  each  study  all  students  were  individually 
interviewed  and  with  the  exception  of  Casey,  who  helped  interviewees  over  early  break-down 
points  to  see  if  they  were  then  able  to  proceed  towards  satisfactory  solutions,  error  classification 
was  based  on  the  first  break-down  point  on  the  Newman  hierarchy. 

The  consistency  of  the  findings  of  these  Melbourne  studies  involving  primary  and  junior 
secondary  students  contrasted  with  another  finding,  also  from  Melbourne  data,  by  Clarkson  (1980) 

(From  Ferrer,  1991,  p.  2) 

Summary  of  Findings  of  Early  Australian  Newman  Studies 

1  - 149 

that  only  about  15%  of  initial  errors  made  by  10th  and  1 1th  Grade  students  occurred  at  any  one  of 
the  Reading,  Comprehension  or  Transformation  stages.  This  contrast  raised  the  question  of  whether 
the  application  of  the  Newman  procedure  at  different  grade  levels,  and  in  different  cultural  contexts, 
would  produce  different  error  profiles. 

Some  Recent  Asian  and  Papua  New  Guinea  Newman  Data 

Since  the  early  1980s  the  Newman  approach  to  error  analysis  has  increasingly  been  used  outside 
Ausiralia.  Clements  (1982)  and  Clarkson  (1983)  applied  Newman  techniques  in  error  analysis 
rest  arch  carried  out  in  Papua  New  Guinea,  and  more  recently  the  methods  have  been  applied  to 
mathematics  and  science  education  research  studies  in  Brunei  (Mohidin,  1991),  India  (Kaushil, 
Sajjin  Singh  &  Clements,  1985),  Indonesia  (Ora,  1992),  Malaysia  (Kim,  1991;  Kownan,  1992; 
Marinas  &  Clements,  1990),  Papua  New  Guinea  (Clarkson,  1991),  the  Philippines  (Jimenez, 
1992),  and  Thailand  (Singhatat,  1991;  Sobhachit,  1991). 

Rather  than  attempt  to  summarise  the  data  from  all  of  these  Asian  studies,  the  results  of  four 
studies  which  focused  on  errors  made  by  children  on  written  mathematical  tasks  will  be  given 
special  attention  here.  The  four  studies,  which  have  been  selected  as  typical  of  Newman  studies 
conducted  outside  Australia,  are  those  by  Clarkson  (1983),  Kaushil  et  al.  (1985),  Marinas  and 
Clements  (1990),  and  Singhatat  (1991).  Pertinent  features  of  these  studies,  conducted  in  Papua 
New  Guinea  (PNG),  India,  Malaysia,  and  Thailand,  respectively,  have  been  summarised  in  Table 

Table  1  „  .. 

Background  Details  of  the  Asian  and  PNG  Studies 


Country    Grade  Sample  Number  Language  of  test 
level     size     of  errors    &  of  Newman 



















Was  the  interview  in 
student's  language 
of  instruction? 

Clarkson  (1983)  PNG 
Kaushil  et  al.  (1985)  India 

Marinas  & 
Clements  (1990) 

Singhatat  (1991) 




•  Note  that  the  38  errors  attributed  by  Singhatat  to  "lack  of  motivation"  have  not  been  taken  into  account  for  the 
purposes  of  this  Table. 

The  percentage  of  errors  classified  in  each  of  the  major  Newman  categories  in  these  four  studies 
is  shown  in  Table  2.  The  last  column  of  this  Table  shows  the  percentage  of  errors  in  the  categories 
when  the  data  from  the  four  studies  are  combined. 


1  - 150 

From  Table  2  it  can  be  seen  that,  in  each  of  the  studies,  over  50%  of  the  initial  errors  made  were 
in  one  of  the  Reading,  Comprehension,  and  Transformation  categories.  The  right-hand  column  of 
Table  2  shows  that  60%  of  students'  initial  breakdown  points  in  the  four  studies  were  in  one  of  the 
Reading,  Comprehension,  and  Transformation  categories.  This  means  that,  for  most  errors, 
students  had  either  not  been  able  to  understand  the  word  problems  or,  even  when  understanding 
was  present,  they  had  not  worked  out  appropriate  strategies  for  solving  the  given  problems. 

Table  2 

Percentage  of  Initial  Errors  in  Different  Newman  Categories  in  the  Four  Studies 



(n  =  1851  errors) 

(n  =  329  errors) 

Marinas  & 
Clements  (1990) 
(n  =  382  errors) 

(n  =  220  errors) 





















Process  Skills 



















The  high  proportion  of  Comprehension  and  Transformation  errors  in  Table  2  suggests  that 
many  Asian  and  Papua  New  Guinea  children  have  considerable  difficulty  in  understanding  and 
developing  appropriate  representations  of  word  problems.  This  raises  the  question  of  whether  too 
much  emphasis  is  placed  in  their  schools  on  basic  arithmetic  skills,  and  not  enough  on  the 
peculiarities  of  the  language  of  mathematics. 

Further  evidence  for  a  possible  over-emphasis  on  algorithmic  skills  was  obtained  in  the  Indian 
study  (Kaushil  et  al.,  1985)  when  the  performances  of  the  Delhi  Grade  5  sample  on  a  range  of 
mathematical  problems  where  compared  with  those  of  Australian  fifth-grade  children  on  the  same 
problems.  It  was  found  that  the  Indian  children  consistently  and  significantly  outperformed  a  large 
sample  of  Australian  children  on  tasks  requiring  straightforward  applications  of  algorithms  for  the 
four  arithmetic  operations  (for  example,  940  -  586  =  □).  However,  on-  word  problems,  the 
Australian  children  invariably  performed  significantly  better  (see  Table  3).  Clements  and  Lean 
(1981)  reported  similar  patterns  when  the  performances  of  Papua  New  Guinea  and  Australian 
primary  school  students  were  compared  on  tasks  similar  to  those  shown  in  Table  3. 

Interestingly,  Faulkner  (1992),  who  used  Newman  techniques  in  research  investigating  the 
errors  made  by  nurses  undergoing  a  calculation  audit,  also  found  that  the  majority  of  errors  the 
nurses  made  were  of  the  Comprehension  or  Transfon.iation  type. 

o  175 


1  - 151 

Table  3 

Percentage  of  Indian  and  Australian  Grade  5  Children  Correct  on  Selected  Problems 
(from  Kaushil  et  al.,  1985). 

to  uiuion 

sample  correct 

OL.  Aiictration 

sample  correct 

940  -  586  =  □ 



273  +  7  =  □ 



A  shop  is  open  from  1  pm  :o  4  pm.  For  how 
many  hours  is  it  open? 



It  is  now  5  o'clock.  What  time  was  it  3  hours  ago? 



Suniti  has  3  less  shells  than  Aarthi.  If  Suniti  has  5  shells, 
how  many  shells  does  Aarthi  have? 



The  high  percentage  of  Comprehension  and  Transformation  errors  found  in  studies  using  the 
Newman  procedure  in  the  widely  differing  contexts  in  which  the  above  studies  took  place  has 
provided  strong  evidence  for  the  importance  of  language  factors  in  the  development  of  mathematical 
concepts.  However,  the  research  raises  the  difficult  issue  of  what  educators  can  do  to  improve  a 
learner's  comprehension  of  mathematical  text  or  ability  to  transform,  that  is  to  say,  to  identify  an 
appropriate  way  to  assist  learners  to  construct  sequences  of  operations  that  will  solve  a  given  word 
problem.  At  present,  little  progress  has  been  made  on  this  issue,  and  it  should  be  an  important 
focus  of  the  mathematics  education  research  agenda  during  the  1990s. 


Casey,  D.  P.  (1978).  Failing  students:  A  strategy  of  error  analysis.  In  P.  Costello  (Ed.),  Aspects 
of  motivation  (pp.  295-306).  Melbourne:  Mathematical  Association  of  Victoria. 

Clarke,  D.  J.  (1989).  Assessment  alternatives  in  mathematics.  Canberra:  Curriculum  Development 

Clarkson,  P.  C.  (1980).  The  Newman  error  analysis  -  Some  extensions.  In  B.A.  Foster  (Ed.), 

Research  in  mathematics  education  in  Australia  1980  (Vol.  1,  pp.  1 1-22).  Hobart:  Mathematics 

Education  Research  Group  of  Australia. 
Clarkson,  P.  C.  (1983).  Types  of  errors  made  by  Papua  New  Guinean  students.  Report  No.  26. 

Lac:  Papua  New  Guinea  University  of  Technology  Mathematics  Education  Centre. 
Clarkson,  P.  C.  (1991).  Language  comprehension  errors:  A  further  investigation.  Mathematics 

Education  Research  Journal,  3  (2),  24-33. 
Clements,  M.  A.  (1980).  Analysing  children's  errors  on  written  mathematical  tasks.  Educational 

Studies  in  Mathematics,  //(l),  1-21. 
Clements,  M.  A.  (1982).  Careless  errors  made  by  sixth-grade  children  on  written  mathematical 

tasks.  Journal  for  Research  in  Mathematics  Education,  13(2),  136-144. 
Clements,  M.  A.,  &  Lean,  G.A.  (1981).  Influences  on  mathematics  learning  in  Papua  New 

Guinea.  Report  No.  13.  Lae:  Papua  New  Guinea  University  of  Technology  Mathematics 

Education  Centre. 

Dickson,  L.,  Brown,  M.,  &  Gibson,  O.  (1984).  Children  learning  mathematics:  A  teacher's  guide 
to  recent  research.  Oxford:  Schools  Council. 



1  - 152 

Faulkner,  R.  (1992).  Research  on  the  number  and  type  of  calculation  errors  made  by  registered 
nurses  in  a  major  Melbourne  teaching  hospital.  Unpublished  M.Ed,  research  paper,  Deakin 

Ferrer,  L.  M.  (1991).  Diagnosing  children's  conceptions  of  fractions  and  decimals.  Understanding 

and  Doing:  What  Research  Says.  1(1),  1  -7.  (RECSAM  publication.) 
Gagne\  R.  M.  (1967).  The  conditions  of  learning.  New  York:  Holt,  Reinhart  &  Winston. 
Hollander,  S.  K.  (1978).  A  literature  review:  Thought  processes  employed  in  the  solution  of  verbal 

arithmetic  problems.  School  Science  and  Mathematics.  78, 327-335. 
Jimenez,  E.  C.  (1992).  A  cross-lingual  study  of  Grade  3  and  Grade  5  Filipino  children's 

processing  of  mathematical  word  problems.  Unpublished  manuscript,  SEAMEO-RECSAM, 

Penang.  „ 
Kaushil,  L.  D.,  Sajjin  Singh,  &  Clements,  M.  A.  (1985).  Language  factors  influencing  the  learning 

of  mathematics  in  an  English-medium  school  in  Delhi.  Delhi:  State  Institute  of  Education  (Roop 

Nagar).  , 
Kim,  Teoh  Sooi  (1991).  An  investigation  into  three  aspects  of  numeracy  among  pupils  studying  in 

Year  three  and  Year  six  in  two  primary  schools  in  Malaysia.  Penang:  SEAMEO-RECSAM. 
Kownan,  M.  B.  (1992).  An  investigation  of  Malaysian  Form  2  students'  misconceptions  of  force 

and  energy.  Unpublished  manuscript,  SEAMEO-RECSAM,  Penang. 
Lankford,  F.  G.  (1974).  What  can  a  teacher  learn  about  a  pupil's  thinking  through  oral  interviews? 

Arithmetic  Teacher,  21,  26-32. 
Marinas,  B.,  &  Clements,  M.  A.  (1990).  Understanding  the  problem:  A  prerequisite  to  problem 

solving  in  mathematics.  Journal  for  Research  in  Science  and  Mathematics  Education  in 

Southeast  Asia,  13  (1),  14-20. 
Mellin-Olsen,  S.  (1987).  The  politics  of  mathematics  education.  Dordrecht:  Reidel. 
Mohidin,  Hajjah  Radiah  Haji  (1991).  An  investigation  into  the  difficulties  faced  by  the  students  of 

Form  4  SMJA  secondary  school  in  transforming  short  mathematics  problems  into  algebraic 

form.  Penang:  SEAMEO-RECSAM. 
Newman,  M.  A.  (1977a).  An  analysis  of  sixth-grade  pupils'  errors  on  written  mathematical  tasks. 

In  M.  A.  Clements  &  J.  Foyster  (Eds.),  Research  in  mathematics  education  in  Australia,  1977 

(Vol.  2,  pp.  269-287).  Melbourne:  Swinburne  College  Press. 
Newman,  M.  A.  (1977b).  An  analysis  of  sixth-grade  pupils'  errors  on  written  mathematical  tasks. 

Victorian  Institute  for  Educational  Research  Bulletin,  39, 3 1  -43. 
Newman,  M.  A.  (1983).  Strategies  for  diagnosis  and  remediation.  Sydney:  Harcourt,  Brace 


Ora,  M.  (1992).  An  investigation  into  whether  senior  secondary  physical  science  students  in 

Indonesia  relate  their  practical  work  to  their  theoretical  studies.  Unpublished  manuscript, 

Polya,  G.  (1973).  How  to  solve  it:  A  new  aspect  of  mathematical  method.  Princeton,  NJ: 

Princeton  University  Press. 
Radatz,  H.  (1979).  Error  analysis  in  mathematics  education.  Journal  for  Research  in  Mathematics 

Education,  10,  163-172.  . 
Singhatat,  N.  (1991).  Analysis  of  mathematics  errors  of  lower  secondary  pupils  in  solving  word 

problems.  Penang:  SEAMEO-RECSAM.  . 
Sobhachit,  S.  (1991).  An  investigation  into  students'  understanding  of  the  electrochemical  cell  and 

the  electrolytic  cell.  Penang:  SEAMEO-RECSAM.  ,,«..*. 
Watson,  I.  (1980).  Investigating  errors  of  beginning  mathematicians.  Educational  Studies  in 

Mathematics.  11(3),  319-329. 
Zepp,  R.  (1989).  Language  and  mathematics  education.  Hong  Kong:  API  Press. 


1  -  153 


In  this  paper  we  argue  that  while  the  function  concept  should  play  a  central  role  in  the  secondary 
curriculum,  current  curriculum  based  on  formal  definitions  of  function  restricts  both  the  conceptual 
and  operational  understandings  students  need  to  develop.  We  argue  that  building  an  understanding 
of  functions  through  multiple  representations  and  contextual  problems  provides  an  alternative 
epistemological  approach  to  functions  which  suggests  that  experience  working  in  functional 
situations,  in  doing  functions,  is  more  important  than  learning  static  definitions  which  mask  its 
basis  in  human  activity. 

Introduction.  The  importance  of  the  function  concept  in  the  secondary  curriculum  is  virtually 
undisputed.  Calls  for  the  reform  of  mathematics  place  it  at  the  center  of  the  curriculum  as  an  integrating 
concept  and  locate  its  importance  in  its  modeling  capacity.  Moreover,  with  the  increased  power  of 
computers  and  graphing  calculators  to  display  multiple  and  dynamic  representations,  one  might  expect  the 
treatment  of  the  function  concept  in  the  curriculum  to  be  modified:  however,  there  is  little  evidence  that  a 
theoretical  framework  for  these  changes  has  been  carefully  specified.  In  this  paper,  we  oulline  the 
conventional  educational  view  of  functions  and  then  suggest  a  three-part  framework  based  on  the  use  of : 
1)  dynamic  multi-representational  software,  2)  contextual  problems  and  3)  student  interviews. 
The  Conventional  View.  In  the  conventional  treatment  of  functions,  the  definition  typically  given  is 
"a  function  is  a  relation  such  that  for  each  element  of  the  domain,  there  is  exactly  one  element  of  the 
range."  Such  a  definition  does  not  necessarily  preclude  a  multiplicity  of  approaches  to  functions; 
however,  in  textbooks  and  on  assessment  measures,  one  sees  that  a  restricted  view  of  functions  emerges 
in  which  the  algebraic  presentation  dominates  the  underlying  assumptions  about  functions.  These 
restrictions,  which  are  overlapping,  include  an  undue  emphasis  on : 

1)  the  algebraic  presentation  with  graphs  as  secondary  and  with  tables  a  distant  third. 

2)  a  correspondence  model  which  requires  one  to  treat  a  function  as  a  relationship  between  x  and  a 
corresponding  y,  rather  than,  for  example,  a  covariation  model  where  one  can  describe  how  the  y 
values  change  in  relation  to  each  other,  for  given  x  changes  (or  vice-versa); 

3)  a  functional  format  of  "y="  so  the  equation  is  solved  for  y,  and 

4)  a  directionality  of  the  relation,  so  that  one  can  predict  a  y  value  for  a  given  x. 


Jere  Confrev  and  Erick  Smith,  Cornell  University,  Ithaca,  New  York 

1  - 154 

Although  there  is  nothing  formally  erroneous  about  such  a  concept  of  function,  we  wish  to 
demonstrate  that  it  is  insufficient  for  approaches  to  functions  which  emphasize  the  use  of  multiple 
representations  and  contextual  problems.  We  also  wish  to  suggest  that  it  makes  it  difficult  to  discuss, 
describe  and  evaluate  the  complexity  and  richness  of  a  student's  goal-directed  investigation  of  functional 

I.  An  Epistemotocy  of  Multiple  Representations.  When  one  works  in  an  environment  that 
allows  students  to  explore  multiple  representations  of  functions,  one  must  learn  to  legitimize  the  use  of  a 
variety  of  forms  of  representation  to  describe  functions.  In  Function  Probe®  (Confrey,  1991),  a  multi- 
representational  software  using  graphs,  tables,  calculator  buttons  and  algebraic  approaches  to  functions, 
the  student  must  coordinate  a  variety  of  representational  forms.  Meanings  of  the  concept  of  function  vary 
across  these  forms.  To  illustrate,  we  will  work  with  the  exponential  function  and  demonstrate 
characteristics  of  the  function  which  are  more  easily  visible  in  different  representations. 
The  Graph.  A  problem  we  have  used  with  students  involves  setting  up  a  time-line  for  a  list  of  events 
which  occurred  over  the  earth's  geologic  history.  We  prepared  a  list  of  'events'  with  dates.  These  events 
turned  out  to  have  a  fairly  uniform  distribution  when  plotted  on  a  log  scale.  One  student  tried  plotting  the 
number  of  years  ago  on  the  y-axis  and  the  log  of  that  number  on  the  x-axis.  The  points  lie  on  the  graph  of 
y=IOV  An  idealized  simulation  of  her  graph(with  d;ffering  y-scales)  is  shown  below: 

'       -1%**  1  r'» 

loqCyrs  »flo) 
Fig.  1 

Uf(vrs  «9«) 

While  scaling  the  y-axis,  she  noticed  two  things:  First,  no  matter  what  value  she  used  as  the  high  y- 
value  the  graphs  had  the  same  shape.  Geometrically,  they  were  congruent.  Second,  the  points  were 
always  bunched  together  near  the  origin,  spread  out  near  the  top,  and  nicely  spaced  around  the  'curve'. 


1  -  155 

Thus  she  suggested  that  with  appropriate  scaling,  she  could  have  'nicely-spaced'  points  for  whatever 
period  she  wanted.  Seeing  multiplicative  self-similarity  as  characteristic  of  the  exponential  graph  can  help 
students  form  a  number  of  insights,  including  the  equivalence  of  vertical  stretches  and  horizontal 
translations,  or  the  equivalency  of  half-life  or  doubling  time.  This  quality  of  similarity  is  also  easily 
recognizable  in  other  visualizations  of  the  exponential  function  and  is  a  powerful  way  to  recognize  when 
an  exponential  function  will  prove  an  appropriate  modeling  device  (sunflowers,  nautilus  shells,  rams 
horns  etc.). 

The  Table.  When  students  encounter  a  table  for  the  exponent  *1  for  the  first 
time,  they  will  often  apply  the  strategies  they  have  learned  about  polynomial 
functions,  for  example  looking  at  the  differences  in  the  y's  as  the  x  values 
change  at  a  constant  rate.  For  polynomials  looking  at  differences  and 
differences  of  differences,  etc.  eventually  leads  to  a  column  of  constant 
differences  and  this  is  often  what  students  expect  (figure  2).  When  they  do 

this  with  the  exponential  function,  they  may  also  see  a  repeating  pattern  emerging.  However,  if  they  are 
seeking  to  find  a  difference  column  that  becomes  constant  over  repeated  application,  they  quickly  learn  of 
its  impossibility,  for  all  difference  columns  maintain  the  original  constant  ratio  between  terms  (figure  3a). 

In  Function  Probe,  we  added  a  new  resource,  the 









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>  2 



>  3 

>  2 



>  5 

Fig.  2 












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>  1 

>  1 


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>  2 

>  4 

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>  8 




Fig.  3a 

•  3c 

ability  to  take  ratios  as  well  as  differences  and 
created  a  notation  for  doing  so.  Students  learn  to 
see  a  constant  ratio  as  indicative  of  en  exponential 
relationship  (figure  3b).  A  verbal  description,  that 

for  a  constant  change  in  x,  the  y  values  change  by  a  constant  ratio,  is  another  signal  for  recognizing 
exponential  situations.  A  second  resource  of  Function  Probe  is  the  accumulation  command.  When  one 
accumulates  the  exponential  function,  an  exponential  is  produced(figure  3c).  Between  the  accumulate  and 
difference  command,  we  can  anticipate  why  the  derivative  and  the  integral  of  the  exponential  produce  the 



1  - 156 

n».  rairt.l.fnr  Logarithms  often  provide  difficulty  tor  students.  We  would  attribute  this  both  to  their 
formal  treatment  in  the  curriculum  and  to  the  limitations  of  tables  and  most  calculators  which  restrict 
students  to  either  base  10  or  base  e.  Thus  they  must  invoke  the  somewhat  mysterious  change-of-base 
formula  whenever  they  build  an  exponential  or  logarithmic  model  where  the  base  (i.e.  constant  ratio)  is  not 
10  or  e.  For  example,  in  a  situation  where  $100.00  is  invested  at  9%  interest,  students  will  work  out  the 
annual  multiplier  (constant  ratio)  of  1.09  and  eventually  model  the  situation  as  an  exponential  function,  P 
«  100(1.09)'.  If  they  have  made  a  table  of  graph  for  the  problem,  they  leam  quickly  that  theycan 
'inverse'  the  function  by  changing  the  order  of  the  table  columns  or  reflecting  the  graph.  However  when 
trying  to  inverse  the  equation,  they  can  easily  write  it  as  P/100  =  (1.09)',  but  then  are  faced  with  the  'log 
problem'.  Taking  the  log  of  both  sides  is  a  procedure  that  somehow  works  but  has  no  apparent  connection 
to  their  original  way  of  making  the  equation.  Because  of  this  particular  concern,  we  built  two  features  into 
the  calculator  of  Function  Probe,  a  way  to  save  procedures  as  user-defined  buttons,  and  a  button  that  takes 
any  base  as  the  input  for  either  an  exponential  or  a  log  function.  Figure  4a  represents  a  string  of  calculator 
keystrokes  which  calculates  the  amount  accumulated  after  five  years.  By  placing  the  variable'  over  the 

five  in  figure  4b,  the  user  has  designated  this  as  a  button,  j  1 ,  that  can  now  take  any  value  for  time  as  an 
input.  If  the  student  now  wants  to  calculate  what  input  creates  the  output  of  say  325,  she  can  imagine 
undoing  the  button  from  tight  to  left  Thus  she  might  enter  the  set  of  keystrokes  shown  in  figure  5a: 

After  dividing  325  by  100,  they  have  to  decide  how  to  undo  1 .09'.  The  "log,x"  key  becomes  a  key  that 
undoes  the  action  of  an  exponential  This  ability  to  build  and  unbuild  procedures  and  to  'undo'  an 
exponential  allows  the  log  to  play  a  stronger  and  more  intuitive  role  in  student's  problem-solving.  In 
addition  the  process  of  building  algebraic  equations  from  the  linear  procedures  represented  by  calculator 
keystroke  records  and  also  building  keystroke  records  from  algebraic  equations  can  provide  strong 
assistance  in  helping  student  come  to  better  understand  the  operational  basis  of  algebra. 

5©  1 .09  -  *  100-  1 153.86 
Fig.  4a 

J1:  Q©  1.09 -*  100-  |QD 
Fig.  4b 

1  - 157 

Integration  of  the  Rcpresentatinm  In  developing  this  example,  we  are  not  arguing  that  the  traits  of  the 
exponential  that  are  displayed  cannot  be  seen  across  representations.  In  fact,  searching  for  how  to  see 
them  and  learning  to  recognize  them  independent  of  the  other  representations  is  a  valuable  learning 
experience  on  the  software. 

II.  The  Impact  of  Placing  Functions  in  Contextual  Situations.  The  treatment  of  functions 
within  multiple  representations  such  as  those  described  above  ignores  the  question  of  how  the  function  is 
generated,  and/or  where  its  application  is  witnessed  or  warranted.  Others  have  argued  for  the  value  of 
contextual  problems  on  the  grounds  that  they  are  more  socially  relevant,  realistic,  open-ended,  data  driven, 
and  inviting  to  students  (Monk,  1989;  Treffers,  1987;  deLange,  1987)  In  addition  to  these  important 
qualities,  we  see  the  value  of  placing  functions  into  contextual  situations  as  a  challenge  to  the  belief  that 
abstraction  requires  one  to  decontextualize  the  concept  from  its  experiential  roots.  Instead  we  see 
abstraction  as  the  integration,  reconciliation,  juxtaposition  of  multiple  schemes  of  action  for  a  given 

In  our  work,  we  have  chosen  the  contextual  problems  to  highlight  aspects  of  the  function  that  are 
grounded  in  human  anions.  Piagetian  research  stressed  the  importance  of  the  evolution  of  human  schemes 
through  the  actions  and  operations  one  carries  out  on  those  actions.  Reflective  abstraction  is  the  process 
by  which  the  practical  usefulness  of  those  actions  is  acknowledged  and  the  actions  and  operations  become 
part  of  our  mental  repertoire  in  the  form  of  schemes. 

Accordingly,  we  stress  the  development  of  an  operational  schemes  for  understanding  functions.  For 
instance,  to  recognize  contexts  in  which  the  exponential  scheme  is  useful,  we  have  postulated  an 
underlying  scheme  called  "splitting".  Splitting,  we  suspect,  has  its  roots  in  early  childhood  in  sharing  and 
congruence,  primarily  the  binary  split,  and  forms  a  basis  for  division  (and  multiplication)  that  is  not  well 
described  by  repeated  subtraction  (or  addition).  Doubling  and  halving  are  the  simplest  instantiations  of  it, 
and  contrary  to  repeated  addition  views  of  multiplication,  the  split  is  a  primitive  multiplicative  action  which 
is  often  embedded  in  a  repeated  division  and  multiplication  structure.  The  invariance  in  the  operation  is  the 
constant  ratio  of  1:2  or  2:1.  We  witness  students  solving  problems  such  as  42671 8  by  going  18 , 36  (2 
tallies).  72  (4  tallies),  144  (8  tallies),  288  (16  tallies),  476  (32  tallies)  and  then  adjusting  10  get  the  exact 

1  - 158 

result  (Confrey.  1992a).  The  children  understand  that  they  can  ''retmitize"  (Confrey,  1992b)  from  the  unit 
18, 36,. . .  to  the  unit  288,  to  reach  their  goal  more  quickly.  We  have  also  argued  for  the  coordination  of 
the  splitting  structure  with  similarity  as  an  underlying  basis  for  the  exponential  function. 

Such  an  approach  to  the  concept  of  function  locates  a  function  within  a  family  of  functions  and 
examines  how  that  prototypic  function  is  fit  to  the  existing  data  or  situation  (Confrcy  and  Smith,  1991). 
Thus,  in  a  compound  interest  function,  the  principle  is  multiplied  times  the  constant  rate  of  growth  factor 
multiplied  by  itself  n  tiroes.  In  such  an  approach,  a  prototypic  function  might  operate  inetaphorically.  If 
the  initial  situation  were  of  bunnies  reproducing,  then  the  principle  functions  as  the  initial  number  of 
bunnies  and  the  reproductive  rate  is  cast  as  the  interest  rate.  It  is  not  so  much  the  specifics  of  the  situations 
that  remain  invariant  as  the  characters  required  in  various  contextual  roles  and  the  actions  they  each  carry 
out  in  relation  to  each  other. 

The  impact  of  such  a  view  on  the  multiple  representations  is  that  one  is  encouraged  to  seek  out  how  the 
actions,  operations  and  roles  are  carried  out  and  made  vWWe  (more  or  less)  in  the  different 

III.  Functioning  as  a  Human  Activity,  Many  mathematics  educators  prefer  to  speak  of 
"mathematizing"  to  emphasize  the  role  of  the  students)  and/or  teachers  in  doing  mathematics.  The  reasons 
for  this  switch  in  language  is  that  the  process  of  doing  mathematics  is  emphasized  rather  than  the 
acquisition  and  display  of  traditionally  accepted  responses.  We  too  find  this  shift  to  aid  us  in  the 
understanding  of  mathematical  ideas,  for  when  we  try  to  answer  the  question,  'what  is  a  function?',  our 
answers  vary  dramatically  from  when  we  seek  to  explain,  'what  is  the  experience  of  understanding 
functions  like?' 

An  Illustration.  To  illustrate  this,  consider  one  student's  path1  through  the  following  problem:  The 
tuition  of  Cornell  University  is  $  11,700.  For  the  last  five  years,  the  average  tuition  hike  has  been  11.3%. 
What  can  you  expect  to  pay  when  your  children  wish  to  attend  the  University?  When  will  the  tuition 
exceed  1  million  dollars? 

1  This  is  a  m*Hi<actured  example,  but  it  representative  of  the  kinds  of  ipprachej  we  have  witnessed  repeatedly  by  students 
working  on  this  problem. 

1  - 159 

The  student,  Ann,  inputs  1 1,700  into  the  table.  She  calculates  .113  s  1 1.700  and  adds  this  to  1 1,700. 
Finding  this  tedious,  she  builds  a  button  to  cany  out  the  actions.  Her  keystrokes  for  the  button  look  like 
this:  Corrwll  tuition:  ©•.113  +  ©*  |QD.  To  figure  out  how  many  times  to  hit  it,  she  figures, 
shell  have  children  in  ten  years,  and  then  in  eighteen  years  they  will  go  to  the  University.  So,  she  hits  the 
button  28  times.  To  her  astonishment,  she  sees  the  value  over  $234,000.  To  answer  the  second  question, 
she  wants  a  table,  so  she  opens  that  window.  She  types  in  1 1,700  in  a  column  she  then  names,  "cost" 
informally  and  c  in  the  formal  labeL  She  uses  the  fill  command  and  types  in  fill  from  11,700  to  1,000,000 
and  chooses  multiplication  by  .1 13.  The  computer  gives  her  a  warning  that  she  has  filled  30  entries  and 
asks  if  she  wants  to  continue.  Her  values  have  gone  down  and  only  the  first  entry  is  what  she  wanted  it  to 
be.  After  answering  no,  she  goes  over  to  the  calculator  and  types  1 1,700  hits  the  button,  Jl,  and  sees 
13022. 10.  She  think*  she  has  figured  out  the  problem  and  then  goes  to  her  table  and  types  in  n  (for  new) 
*  c  +  1 1,700.  She  realizes  this  produces  only  the  correct  first  value,  and  feels  frustrated  at  still  not  getting 
the  other  values,  but  persists  long  enough  to  create  a  column  which  has  the  values  she  gets  from  the 
calculator  button  listed  next  to  the  column  labeled  n. 

Amivoi  The  description  shown  above  describes  the  richness  and  complexity  of  the  evolving  function 

concept  Some  characteristics  of  the  functioning  experience  for  Ann  are:  1)  it  is  embedded  in  a  goal- 

directed  activity  of  predicting  cost  as  a  function  of  time  for  t=«28;  2)  a  covariation  approach  is  used 

describing  now  cost  changes  as  time  increments  by  years;  3)  an  entry  through  numeric  calculations  is 

easily  accomplished  and  she  uses  the  repetition  of  the  operation  to  create  a  button;  4)  the  results  of  the  first 

question  surprise  her  and  give  her  firsthand  experience  with  the  rapid  growth  of  the  exponential;  5)  she 

seeks  out  the  table  to  create  a  record  of  her  interim  values  asid  to  be  able  to  seek  out  the  $1  million  figure; 

6)  she  thinks  her  method  of  filling  by  multiplication  of  .1 13  is  the  same  as  her  calculator  actiotuK  7)  she 

recalculates  the  first  value  to  set  herself  a  specific  goal;  8)  she  diagnoses  her  problem  as  needing  to  add  the 

value  $  1 1 ,700;  9)  she  achieves  her  local  goal  but  not  her  longer  term  goal;  and  10)  she  cannot  find  a  path 

immediately,  and  sets  an  interim  goal  of  writing  down  her  desired  values. 

The  concept  of  function  which  emerges  allows  the  specification  and  inclusion  of: 
1.  epistemological  obstacles  such  her  experiences  of  failing  to  curtail  U,700*.U3+  11,700  into  a  single 
expression  of  (11,700)*1.1 13  to  allow  the  use  of  the  fill  command  and  see  the  repeated  addition. 




1  - 160 

2.  affect  into  one's  epistemological  investigations,  such  as  the  surprise  at  the  rapid  growth  rate,  or  her 
sense  of  ownership  of  the  problem; 

3.  goal-directed  behaviors,  goals  and  subgoals,  such  as  matching  previous  values  or  creating  records; 

4.  inquiry  skills  where  strategies  for  finding  are  expressed  along  with  basic  assumptions;  and 

5.  contrasting,  conflicting  and  supportive  uses  of  multiple  representations  demonstrating  the  sequence  and 
purpose  of  each  representation. 

The  result  of  a  revision  of  the  function  concept  to  incorporate  such  data  would  be  to  include  in  the 
function  concept  the  idea  of  it  representing  a  set  of  coherent  stories  to  capture  the  evolutionary  paths  of 
student  investigations. 

Conclusions.  In  this  paper,  we  suggest  that  the  formal  definitional  approaches  to  the  descriptions  of  the 
function  concept  fail  to  present  a  rich  aud  complex  enough  framework  for  guiding  the  development  of 
instructional  methods.  We  explain  how  that  framework  must  be  revised  to  include  the  use  of  multi- 
representational  approaches,  to  allow  for  the  action-based  schemes  and  conceptual  rales  that  can  result 
from  placement  in  contexts  and  to  describe  the  "functioning"  experience  as  a  personal  cr  social  experience, 
We  suggest  that  such  an  approach  is  akin  to  the  idea  of  a  "concept  image"  expressed  by  Vinner  (1983), 
and  that  a  richer  description  of  mathematical  concepts  is  necessary  to  create  the  knowledge  base  for  more 
effective  forms  of  assessment 

Confrey,  Jere  (1992a).  Learning  to  see  children's  mathematics:  Crucial  challenges  in  constructivist  reform. 

Ic  Ken  Tobin  (ed),  Constructivist  Perspectives  in  Science  and  Mathematics.  Washington,  DC 
Confrey,  Jere  (1992b).  Splitting,  similarity,  and  rate  of  change:  New  approaches  to  multipneatioa  and 

exponential  function!,  In  G.Harel  and  J.  Confrey  (eds.).  The  Development  cf  Multiplicative 

Reasoning  in  the  Learning  of Mathematics.  Albany,  NY 
Confrey,  Jere  (1991).  Function  Probe©  [Co:«KiKr  Program].  Santa  Barbara,  CA:  Intellimarion  Library 

for  the  Macintosh. 

Confrey,  Jere  and  Smith,  Brick  (1991).  A  fnnx  -'-.A.  for  functions:  Prototypes,  multiple  representations, 
and  transformations.  In  Robert  Underbill  ana  Catherine  Brown  (eds.).  Proceedings  of  the  1 3th  Annual 
Meeting  of  P  MEN  A.  Blacksburg,  VA.Ost  1M9,  .591.  (pp.  57-63) 

de  Lange,  Jan  J.  (1987).  Mathematics:  Insight  and  t.feuning.  Utrecht,  Netherlands:  Rijksumvenjteit 

Monk,  Steven  (1989)  Student  understanding  of  function  as  a  foundation  for  calculus  curriculum 
development.  A  paper  presented  at  the  Annual  Meeting  of  the  American  Educational  Research 
Association,  San  Francisco,  March  27-31, 1989. 

Trrffrr«  Adrian  MQaTt  Thw»  nimtminnt  DordrCCht  NMhwhnHf  Reidel 

Vinner,  S.  (1983).  Concept  definition,  concept  image,  and  the  notion  of  function.  Journal  for  Research  in 
Mathematics  Education,  14  (3),  pp.  293-305 


1  -  161 


Kathryn  Crawford 

The  University  of  Sydney 

Earlier  research  has  supported  the  relationship  between  the  setting  in  which  learning  occurs  and  the 
cognitive  processes  used  as  students  approach  tasks,  and  the  quality  of  the  resulting  learning.  Early 
school  experience  'often  traditional)  forms  the  basis  of  student-teachers'  ideas  about  teaching  and 
Mathematics  and  has  a  powerful  influence  on  their  initial  classroom  behaviour.  The  results  of  an 
educational  intervention  aimed  at  providing  experiences  as  a  basis  for  an  alternative  rationale  are 
reported.  Initial  results  suggest  that  many  pre-service  student  teachers  are  able  to  develop  a  rationale 
for  teaching  practice  based  on  their  knowledge  of  how  learning  occurs  and  apply  their  developing 
rationale  in  practice. 


Earlier  research  on  children's  learning  (see  Crawford  1983,1984,1986,)  supported  the  ideas  of  Luria 
(1973,  1982),  Vygotsky  (1978)  and  Leont'ev  (1981)  of  the  relationship  between  the  social  context  in 
which  learning  occurs  and  the  qualities  of  the  cognitive  processes  used  as  students  approach  tasks  and 
the  resulting  learning  outcomes.  In  particular,  the  results  indicated  that  the  major  cognitive  demands  of 
traditional  teacher-centred  instructional  settings  were  for  cognitive  processes  associated  with 
memorization  of  declarative  knowledge  and  imitation  of  teacher  demonstrations.  In  contrast, 
mathematical  problem  solving  and  enquiry  made  demands  on  students'  metacognitive  processes  and 
higher  order  intellectual  processes  (simultaneous  processing  in  Luria's  model)  associated  with  the 
formation  of  abstract  concepts.  All  undergraduate  student-teachers  have  many  years  of  experience  of 
instruction  in  school.  In  many  schools  in  Australia,  educational  practice  in  mathematics  is  "little 
different  from  what  it  was  20  years  ago"  (Speedy,  1989:16). 

Experience  as  a  student  in  school  forms  the  basis  of  student-teachers'  ideas  about  being  a  teacher, 
about  how  learning  occurs  and  even  about  Mathematics.  Research  (Crawford,1982;Ball,1987) 
suggests  that  these  early  experiences  powerfully  influence  the  classroom  behaviour  of  beginning 
teachers.  In  particular,  there  is  evidence  (Crawford  1982,  Speedy  1989)  to  suggest  that  traditional 


forms  of  teacher  education  have  been  largely  ineffective  in  changing  teaching  practice  or  even  student 
teachers'  beliefs  about  teaching  and  learning. 

Pressures  for  changes  in  teaching  practice  have  never  been  so  great.  Cobb  (1988)  has  described  the 
present  tensions  between  theories  of  learning  and  modes  of  instruction.  In  Mathematics,  the  advent  of 
information  technologies  has  significantly  changed  the  role  of  Mathematics  in  societies  and  the  roles  of 
humans  as  mathematicians.  The  mechanical  routines  that  have  played  such  a  large  pan  in  traditional 
mathematics  curricula  arc  now  largely  the  function  of  electronic  machines.  As  Speedy  suggests: 

To  be  skilled  in  mechanics  is  no  longer  sufficient.  To  skilled  in  applying  mathematical 
knowledge  across  the  whole  of  real  life  situations  is  imperative.(Ibid) 

The  report  below  describes  an  attempt  to  apply  theories  about  learning  to  the  education  of  teachers. 
A  Theoretical  Description  of  the  Problem. 

According  to  Leont'ev's  (1981)  activity  theory,  cognitive  development  occurs  as  the  result  of 
conscious  intellectual  activity  in  a  social  context.  The  actual  thinking  that  occurs  during  activity 
depends  on  the  perceived  needs  and  goals  of  an  individual  or  group  and  the  resulting  ways  in  which 
they  approach  the  tasks  at  hand.  Leont'ev  distinguishes  betwee..  activities  and  operations.  An  activity 
involves  conscious  reasoning  that  is  subordinated  to  a  goal,  operations  are  largely  automated, 
unavailable  for  review  and  usually  used  unconsciously  as  a  means  to  an  activity..  The  quality  of  the 
learning  outcomes  reflects  the  quality  of  activity  involved.  The  ways  in  which  these  factors  effect 
learners  is  also  described  by  Lave  (1988:25)  when  she  writes  f '  out  a  "setting"  as  a  dynamic  relation 
between  the  person  acting  and  the  arena  in  which  they  act.  Engestrom  (1989)  extends  the  idea  of 
Activity  as  proposed  by  Leont'ev  with  his  notion  of  an  "activity  system".  That  is,  a  group  of  related 
people  working  together  each  bring  with  them  their  own  needs  and  goals.  In  addition,  any 
institutionalised  system,  such  as  a  school,  has  an  established  set  of  cultural  expectations  about  the 
relationships  between  the  people  who  act  within  it.  Like  Lave  (1988),  Engestrom  focusses  in  his 
system  on  the  dynamic  relationships  between  the  actors  and  the  context  in  which  they  act  Thus  a 

1  -  163 

school,  or  more  specifically  a  Mathematics  classroom,  may  be  thought  of  as  a  very  complex  activity 
system  or  "setting"  in  which  a  large  number  of  people  interact  according  to  largely  subjective 
perceptions  of  expectations,  needs  and  goals. 

Student-teachers  have  long  experience  of  teaching  and  learning  in  school.  The  Activity  in  the 
classrooms  that  they  have  experienced  was  largely  directed  by  the  teacher.  The  teacher  did  most  of  the 
higher  order  intellectual  activity.. .the  posing  of  questions,  the  planning,  the  interpretation  and  the 
evaluation.  As  students  they  have  learned  ABOUT  Mathematics  from  the  teacher  and  "how  to"  carry 
out  selected  techniques.  What  they  know  about  being  a  teacher  is  the  result  of  their  experiences. 
Many  of  them  have  chosen  to  be  teachers  on  the  basis  of  such  knowledge.  They  are  attached  to  it  and,, 
understandably,  many  resist  reviewing  and  modifying  their  beliefs  in  the  course  of  their  pre-service 

At  universities,  a  similar  "setting"  often  persists  in  relation  to  their  teacher  education.  Lecturers  tell 
them  about  educational  theories..they  select  the  theories,  set  the  assignments  and  evaluate  them. 
Successful  students  Ieam  to  talk  (and  write)  ABOUT  teaching  and  learning  in  the  appropriate  ways.  In 
Mathematics  they  generally  learn  the  mechanics  of  using  existing  techniques  to  solve  problems. 
Because  teaching  practice  in  Mathematics  has  changed  little,  student-teachers'  practical  experience  in 
schools  tends  to  support  their  initial  perceptions  about  the  role  of  the  teacher. 

In  order  to  break  the  cycle,  it  seemed  essential  to  provide  student-teachers  with  a  wider  range  of 
experiences  both  as  learners  and  teachers.  It  also  seemed  important  to  construct  a  "setting"  for  their 
learning  which  facilitated  a  shift  of  attention  away  from  preconceived  notions  of  teaching  towards  an 
examination  of  the  learning  of  the  children  in  their  care.  The  "setting"  should  also  encourage 
intellectual  activity  directed  towards  the  development  of  a  practical  rationale  for  teaching  practice  that  is 
centred  on  a  working  theory  of  how  learning  occurs.  To  this  end  a  teaching  experiment  was 
conducted  with  final  year  students  at  the  university. 


1  -  164 

The  teaching  experiment  and  outcomes 

All  final  year  students  (n=45)  of  a  Primary  Bachelor  of  Education  course  were  provided  with 
information  in  the  form  of  readings  and  tutorials  about  recent  theories  of  learning.  In  the  beginning, 
tutorial  sessions  focussed  student  discussions  on  the  possible  implications  of  research  on  learning  for 
classroom  practice  in  Mathema.ics.  Thus  a  beginning  was  made  in  mapping  knowledge  about  theory 
onto  a  known  practical  situation.  Then  students  were  then  required  to  work  in  groups  of  three  or  four 
to  plan,  implement  and  evaluate  a  mathematical  learning  environment  in  which  autonomous  learning 
behaviour  was  encouraged  and  pupils  were  involved  in  investigation  and  enquiry  for  a  large  part  of 
each  session.  Students  worked  in  a  local  inner-city  school  one  morning  a  week  for  seven  weeks.  The 
morning  routine  involved  a  half  hour  planning  session  involving  all  students,  an  hour  in  charge  of  a 
class  (working  in  groups  of  three),  and  a  review  session  of  approximately  45  minutes.  Tertiary  staff 
were  available  for  consultation  and  advice  and  observed  student  work  in  the  classroom.  Student- 
teachers  met  in  working  groups  for  an  hour  between  sessions  to  plan  and  discuss.  School  staff  agreed 
to  negotiate  with  each  group  of  students  about  the  content  and  scope  of  activity  in  each  class  and  to 
thereafter  take  a  low  profile  and  allow  the  students  to  take  responsibility  for  the  implementation  of  their 
planned  projects..  Students  were  advised  to  take  turns  at  being  "teacher",  facilitator  and  observer. 
They  were  later  required  to  conduct  a  similar  project  alone  as  part  of  their  practicum  experience  in 
another  school. 

Initially  all  student-teachers  were  enthusiastic  about  the  prospect  of  allowing  children  a  more  active 
role  in  Mathematics  learning.  Many  had  negative  memories  of  their  own  mathematics  education  and 
expressed  a  commitment  to  ensuring  that  pupils  in  their  care  did  not  have  the  same  kind  of  experience. 
All  had  written  at  length  about  recent  research  based  learning  theories  in  other  parts  of  their  course. 
All  except  one  had  studied  Mathematics  at  matriculation  level  with  some  success.  Despite  this  positive 
beginning,  the  process  of  applying  theory  in  practice  was  fraught  with  tensions  and  inconsistencies. 
Some  of  these  are  listed  in  point  form  below: 

1  -  165 

1.  Without  exception  student-teachers  were  confronted  by  deeply  held  beliefs  about  the  need  to 

always  tell  pupils  about  Mathematics  before  allowing  them  to  begin  an  investigation.  This 
belief  dominated  their  behaviour  for  several  sessions  in  spite  of  their  awareness  of  the 
inconsistency  of  their  behaviour  in  terms  of  their  stated  aims. 

2.  Despite  a  specific  focus  in  tutorials  on  the  use  of  open  ended  questions  and  instructions  as 
stimulus  material  and  for  evaluation  purposes,  all  began  with  an  expressed  need  for  all 
children  to  complete  the  set  tasks  in  the  "correct"  way. 

3 .  Most  initially  failed  to  distinguish  between  their  needs,  expectations  and  goals  and  those  of 
the  children. 

4.  All  initially  found  it  difficult  to  shift  their  attention  away  from  the  intentions  of  the  teacher 
to  the  responses  of  the  children. 

5.  All  found  working  collaboratively  in  a  group  for  a  common  goal  difficult.  They 
empathised  with  the  similar  difficulties  experienced  by  children  doing  group  projects  at  the 

By  the  end  of  the  first  three  weeks  most  were  very  frustrated.  They  were  still  taking  charge  of  activity 
in  the  classroom  and  the  children  were  colluding.  School  staff  also  believed  that  the  children  needed  to 
be  "told  what  to  do".  Many  children  appeared  to  lack  the  social  skills  to  work  effectively  in  groups. 
The  discontinuity  between  their  behaviour  in  the  classroom  and  the  facilitative  role  that  was  implied  by 
learning  theory  was  troubling  most  student-teachers. 

Allowing  the  children  ownership  of  the  activity  became  a  major  focus  of  review  sessions.  Student- 
teachers  experimented  with  gains  and  role-play  as  a  means  to  help  children  take  more  assertive  and 
socially  collaborative  roles.  Gradually,  for  all  groups  of  student-teachers  there  was  a  change  in  the 
dynamics  of  the  classroom.  One  spoke  of  the  initial  experience  as  follows:  "It  felt  like  coming  through 
a  gate  into  a  place  that  I  didn't  know  existed."  As  the  children  became  confident  that  their  ideas  were 

1  - 166 

respected  and  a  more  active  role  in  the  Mathematics  projects  was  appropriate,  most  responded 
enthusiastically.  Both  student-teachers  and  school  staff  "were  amazed"  at  the  knowledge  of  the 
children.  The  student-teachers.paradoxically,  also  became  much  more  confident  about  choosing  to 
take  a  directive  role  when  it  was  perceived  to  be  advantageous.  Many  also  began  to  recognise  the 
scope  and  limitations  of  modelling  materials  as  aids  to  learning. 

A  survey  was  carried  out  at  the  end  of  the  course.  All  students  responded  Some  of  the  results  are 
summarized  below: 

97%  indicated  that  they  intended  to  use  group  learning. 
95%  indicated  that  they  would  use  modelling  materials. 
94%  said  they  were  likely  to  use  games. 
92%  said  they  would  encourage  self  directed  learning. 

These  are  real  options  for  the  student-teachers  after  experience  in  applying  the  strategies  in  two 
different  school  contexts.  They  also  indicate  very  different  expectations  of  learning  in  mathematics 
from  those  they  remembered  from  their  own  schooling. 

89%indicated  that  they  would  develop  programs  of  work  in  mathematics. 
87%  indicated  that  they  would  use  direct  instruction  in  some  circumstances. 
86%  said  they  would  use  enquiry  based  learning  techniques. 

It  was  clear  from  the  responses  that  the  student-teachers  had  not  merely  adopted  a  new  "method".  All, 
with  varying  degrees  of  confidence,  felt  that  they  were  able  to  use  a  range  of  teaching  techniques  as 
different  situations  and  different  learning  needs  required.  One  student  commented: 

"1  not  only  understand  what  to  do  in  different  situations,  I  also  know  how  to  do  it  and  can 
explain  why  to  anyone  who  asks". 

Many  had  experimented  with  a  wide  range  of  activities  in  their  efforts  to  facilitate  an  active  role  for  all 

65%  felt  they  would  hold  excursions  in  mathematics 
70%  felt  they  would  set  writing  tasks  in  mathematics. 

1  -  167 

73%  felt  it  was  likely  that  they  would  become  involved  in  school  policy  making  and  curriculum 
development  in  Mathematics. 

Despite  their  apparent  confidence  and  demonstrated  ability  to  facilitate  active  involvement  in 
Mathematics  classes,  many  were  less  positive  about  their  own  learning  experience. 

40%  found  working  in  a  group  difficult. 

42%  indicated  that  the  tertiary  support  for  their  practicum  assignment  was  unsatisfactory.  Many  more 
commented  on  the  lack  of  support  for  investigative  and  enquiry  based  learning  in  schools  (We  now 
hold  workshops  for  .practicum  supervisors  and  are  taking  steps  to  ensure  their  active  involvement  in 
supporting  practicum  in  mathematics. 

The  discomfort  and  confusion  that  occurs  during  a  major  review  of  attitudes  and  beliefs  is  well 
recognized.  (Mandler  (1980),  Gibbons  &  Phillips  (1978))  These  students  were  jusi  beginning  to  giin 
confidence  as  teachers  when  their  basic  beliefs  about  what  a  teacher  is  and  how  learning  occurs  were 
called  in  question.  They  had  wrestled  with  a  very  difficult  educational  task.  It  was  not  all  enjoyable. 

In  contrast,  87%  rated  the  practicum  assignments  positively.They  recognized  the  value  of  the  practical 
assignment  which  gave  them  a  chance  to  explore  the  implications  of  what  they  had  learned  in  a  second 
school  setting. 

Interestingly,  there  was  a  significant  (p>.025)  positive  correlation  between  formal  achievement  in 
mathematics  and  positive  attitudes  to  group  work  in  the  classroom.  Students  found  that  they  needed  to 
be  very  clear  about  the  mathematics  involved  in  a  theme  to  consult  effectively  with  a  number  of  small 

The  experience  of  teaching  this  course  suggests  that  providing  opportunities  for  students  to  revise  their 
beliefs  about  Mathematics  Education  is  an  effective  way  of  enhancing  the  range  and  adaptability  of 
their  teaching  practice.  An  opportunity  to  confront  and  review  their  strongly  held  beliefs  about 
learning  and  teaching  Mathematics  is  at  least  as  important  for  student-teachers  as  a  range  of  learning 
experiences  in  Mathematics.  Perhaps  the  most  important  change  for  our  students  was  the  recognition 


1  - 168 

of  the  practical  implications  of  the  fact  that  children  come  to  the  classroom  with  knowledge  of  their 
own  and  that  they  can  use  this  as  a  basis  for  further  learning.  This  understanding,  seems  a  necessary 
prerequisite  for  a  child-centred  teaching  style  which  responds  to  the  needs,  knowledge,  purposes  and 
priorities  of  the  learners.  In  the  process  of  schooling  the  teacher  is  a  powerful  element  in  the  "activity 
system"  of  a  classroom.  Thus  teacher  perception  of  their  role,  their  expectations  of  students  and  their 
needs  and  goals  are  major  influences  on  student  approaches  to  learning  —  influences  on  the  quality  of 
learning  outcomes.  In  addition  to  knowledge  of  mathematics,  it  seems  highly  desirable  that  teachers 
leave  pre-service  education  with  a  range  of  teaching  strategies.  Most  importantly,  as  educators  they 
nted  to  have  a  strong  professional  rationale  as  a  basis  for  deciding  which  teaching  strategies  are 
appropriate  for  different  students  and  an  understanding  of  the  learning  outcomes  that  are  to  be  expected 
when  particular  "settings"  are  facilitated. 


Ball  D.  (1987)  'Unlearning  to  teach  mathematics'  For  the  Learning  of  Mathematics.  8(1)  40-47. 

Cobb,  P.  (1988),  The  tension  between  theories  of  learning  and  instruction  in  Mathematics, 
Educational  Psychologist,  23  (2),  87-104.  . 

Crawford  K  P  (1982),  Critical  variables  in  Mathematics  education  and  assessment  in  early  schoo 
y°ar's*  in  S.Plummer  (Ed.)  Acquisition  of  Basic  Skills  in  Early  School  Years.ACT,  National 
Council  of  Independent  Schools.  .  .         ,_ . . 

Crawford  K.P.(1983)  Past  Experience  and  Mathematical  Problem  Solving,  in  R.  Herkowitz  (Ed.) 
Proceedings  of  the  Seventh  Annual  Conference  of  the  International  Group  for  the  Psychology  of 
Mathematics  Education,  Israel.  _  ,    ,       _,  „.„„«•,,,. 

Crawford,  K.P.  (1984)  Some  Cognitive  Abilities  and  Problem  Solving  Behaviour:  The  Role  of  the 
Generalised  Images  and/or  Simultaneous  Processing.  Proceedings  of  the  Eighth  Conference  of 
the  International  Group  for  the  Psychology  of  Mathematics  Education,  Sydney,  August, 

Crawford  K.P.  (1986)  'Cognitive  and  social  factors  in  problem  solving  behaviour  in.  The 
proceedings  of  the  Tenth  Conference  of  the  International  Group  for  the  Psychology  of 
Mathematics,  July,  415-421. 

Eneestrom  Y.  (1989)  'Developing  thinking  at  the  changing  work-place:  Towards  a  redefinition  ot 
expertise.'  Technical  Report  130-The  Centre  for  Human  Information  Processing,  University  of 
California,  San  Diego,  USA.  .  ,       _  , 

Gibbons  M  &  Phillips,  P.(1978),  'Helping  Students  Through  the  Self  evaluation  Crisis.  Phi  Delta 
Kappa  60(4)  296-300.  tl  .      .  _ 

Lave,  J.  (1988)  Cognition  in  Practice,  NY,  Cambridge  University  Press 

Leont'ev,  A.N.(1981)  'The  Problem  of  activity  in  psychology  in  J  Wertsch  (ed.)  The  Concept  of 

Activity  in  Soviet  Psychology  U.S.,  M.E.  Sharpe. 
Luria.A.R.  (1973)  The  Working  Brain, UK,  Penguin 
I  uria  A  R  (\9%2)Languaxe  and  Cognition,  rrans.J.V.  Wertsch,  U.i.,  Wiley, 
Mandler  G  (1980)  The  generation  of  emotion.'  In  Pluchik  &  Kellerman  (Eds)  Emotion:  Theory, 

Research  and  Experience.  Vol.1.  Theories  of  emotion,  NY  Academic  Press. 
Speedv,  G.  (1989)(Chairman)  Discipline  Review  of  Teacher  Education  in  Mathematics  and  Science, 

vol  1,  Canberra,  Australian  Government  Printing  Service. 
Vygotsky,  L.S..(  1978)  Mind  in  Society,  M.  Cole  et  Al.  (Eds),  U.S.  Harvard  University  Press. 

Er|c  193 

1  - 169 


Linda  Davenport  and  Ron  Narode 
Portland  State  University 

The  transition  from  a  traditional  classroom  torn  inquiry  classroom  is 
exceedingly  problematic  for  most  teachers.  This  study  builds  on  an  earlier  study 
designed  to  examine  the  kinds  of  questions  asked  by  three  mathematics  teachers 
attempting  to  adopt  an  inquiry  approach  to  mathematics  instruction.  The  focus  of 
this  paper  is  on  qualitative  changes  in  the  questioning  practices  of  these  three 
teachers  over  that  same  year. 

Ushering  In  a  new  paradigm  Is  never  an  easy  task.  Recent  attempts  to  Institute  the 
instructional  shift  In  mathematics  education  advocated  by  the  NCTM  Curriculum  and  Evaluation 
Standards  fnr  School  Mathematics  (1989)  abound,  but  the  transition  from  the  traditional 
classroom  which  presumes  a  transmission  view  of  knowledge  to  a  classroom  where  students 
construct  knowledge  from  genuine  mathematical  Inquiry  and  discourse  Is  exceedingly 

It  Is  our  observation  that  inquiry- based  curriculum  and  methods  of  instruction  do  not 
necessarily  result  in  inquiry  math  discourse.  In  spite  of  the  efforts  to  encourage  teachers  to 
foster  such  discourse,  Instruction  may  still  bear  many  of  the  characteristics  of  school  math.  In 
an  earlier  paper  (Davenport  &  Narode,  1991 )  we  described  the  questioning  practices  of  three 
teachers  as  they  attempted  to  engage  students  In  mathematical  inquiry.  We  found  that  although 
the  teachers  in  our  stuc,  religiously  eschewed  the  didactic  approach  to  instruction  in  favor  of 
inquiry,  an  analysis  of  the  frequency  and  types  of  questions  asked  indicated  that  the  ensuing 
discourse,  at  least  during  the  first  half  of  the  year ,  was  largely  "school  math".  This  paper 
attempts  to  look  more  carefully  at  patterns  among  the  types  of  questions  asked  by  these  teachers 
over  the  entire  course  of  the  same  year. 

Rmvch  Fr«MWork 

The  constructive  view  of  mathematics  learning  (von  Glasersfeld,  1983;  Steffe,  Cobb, 
«,  von  Glasersfeld,  1988;  Richards,  1991 )  asserts  that  discourse  is  a  universal  and  critical 






feature  of  concept  development  In  mathematics.  The  construction  of  knowledge  Is  idiosyncratic 
in  that  it  Is  Individual;  it  is  consensual  In  that  knowledge  cannot  stand  alone,  it  Is  mitigated 
through  social  interaction  and  mediated  through  language.  For  discussion  to  occur,  there  must 
first  develop  a  discourse  community  whereby  the  discussants  Implicitly  acknowledge  shared 
assumptions  which  gives  the  appearance  that  the  discussants  are  acting  In  accord.  Individuals 
exist  in  communities  where,  according  to  Richards  ( 1991 ),  membership  Is  developed  through  a 
"gradual  process  of  mutually  orienting  linguistic  behavior".  The  shared  community  of  the 
mathematics  classroom  presupposes  that  students  and  teachers  accept  Implicit  assumptions  as  to 
their  roles  and  responsibilities.  These  form  the  basis  of  their  linguistic  behavior. 

Richards  ( 1991 )  distinguishes  four  mathematical  communities  where  qualitatively 
different  mathematiu.;  discourse  occurs.  The  four  different  discourses  are  raaaarch  bmkpj,  or 
the  spoken  mathematics  of  professional  mathematicians  and  scientists,-  inquiry  Math,  or  the 
mathematics  of  "mathematically  literate  adults";  Journal  math,  or  the  language  of 
mathematical  publications  which  feature  "reconstructed  logic"  which  is  very  different  from  a 
logic  of  discovery;  and  school  math,  or  discourse  consisting  mostly  of  "inltiatlon-reply- 
evaluation"  sequences  (Mehan,  1979)  and  "number  talk"  which  is  useful  for  solving  "habitual, 
unreflective,  arithmetic  problems."  Bauersfeld  ( 1988)  also  draws  similar  distinctions 
between  what  might  be  characterized  as  school  math  and  inquiry  atath  and  identifies  a 
funneling  pattarn  of  interaction  that  often  comes  into  play  when,  In  school  math,  teachers 
attempt  to  lead  students  to  correct  solutions.  The  distinction  between  inquiry  math  and  school 
math  is  fundamental  in  appraising  the  success  of  present  reforms  in  mathematics  education. 

Raaaarch  Methodology 

The  subjects  in  this  study  are  three  teachers  who  are  part  of  an  on-going  project 
Involving  an  effort  to  Implement  many  of  the  recommendations  contained  In  the  NCTM  Standards 
( 1989)  through  the  use  of  a  curriculum  developed  with  support  from  the  National  Science 






Foundation.  The  teachers  participated  In  90  hours  of  staff  development  during  the  Spring  and 
Summer  In  which  they  explored  many  of  the  activities  Included  In  the  curriculum.  Since  this 
Initial  staff  development,  teachers  have  continued  to  meet  with  project  staff  on  a  regular  and 
frequent  basis  to  discuss  Issues  relevant  to  implementation,  with  much  of  the  focus  on 
explorations  of  student  thinking. 

Two  sources  of  data  are  examined  In  this  study:  ( 1 )  classroom  transcripts  and  (2) 
teacher  journals.  Transcript  analysis  focused  on  the  first  two  days  of  a  three-day  period  of 
videotaping  In  the  Fall  and  Spring.  All  6th-grade  teachers  were  teaching  approximately  the 
same  lessons.  Sequences  of  questions  were  examined  for  patterns  which  were  suggestive  of 
school  math  or  Inquiry  math  for  all  three  teachers  over  time. 

journal  analysis  focused  on  passages  pertaining  to  questioning  and  classroom  dlscouroo. 
The  journals  Included  reflections  by  teachers  throughout  the  year  as  well  as  responses  to  more 
structured  questions  posed  during  staff  development,  including  questions  designed  to  be 
addressed  as  teachers  reflected  on  the  videotaped  lessons  of  their  classroom  practice. 

In  the  Fall  of  1991 ,  sequences  of  questions  asked  by  all  teachers  were  highly 

reminiscent  of  "school  math"  as  described  by  Richards  ( 1991 )  and  Bauersfeld  ( 1988). 

Representative  sequences  Include  the  following: 

T*  1 :  (After  placing  the  first  three  pattern  block  train  on  the  overhead)  What 
will  the  fifth  one  look  like,  hard? 

SA:  It  will  have  2  trapezoids  and  a  hexagon  then  2  trapezoids  . . . 

T*  1  -.  It wiU  have  how  many  pairs  of  trapezoids?  . . .  It  will  have  6  pairs  of 

trapezoids  and  how  many  hexagons? 

SB:  (A  student  responds  quietly.) 

T*  1 :  OK,  so  the  5th  one  will  look  like  3  sets  of  trapezoids  and  2  hexagons.  Can 
we  say  anything  else  about  that?  Maybe  so  we  know  how  to  arrange  thern? 
SC:  It  would  be  a  trapezoid  then  hexagon  then  . . . 

T*  1 :  What  word  did  we  use  yesterday  to  describe  things  that  go  back  and  forth? 
SO:  Alternating. 

T*1-  Alternating.  Can  we  use  that  word  to  describe  Ws?(Wr1t1ngon  the 
overhead  as  he  speaks)  3  sets  of  trapezoids  will  alternate  with  two  hexsgons.  . 

Results  and  Discussion 


1  - 172 


T*2:  Do  you  guys  all  agree  that  ( the  pattern  block  train)  would  start  with  a 
triangle?  What  would  it  end  with? 
SA:  mumble 

T*2:  OK,  here's  what  you  guys  can  do. . .  Here's  the  first  train.  The  second  train 

looks  kind  of  like  this.  Can  somebody  describe  it  for  me?  Who  wants  to  describe 

the  second  one?  Trisha? 

SB:  Another  redone. 

T*2:  How  many  sets  of  squares? 

SB:  Three 

T*2:  OK,  the  first  train  is  2  sets  of  squares  and  a  trapezoid. 

T*3:  (Asking  student  to  build  the  fifth  pattern  block  train  made  up  of 

trapezoids):  Yeah.  What  do  you  have  there? 

SA:  Two  and  a  half. 

T*3:  Do  you  see  two  and  a  half? 

SA:  Urn. 

T*3:  It's  five  trapezoids,  isn't  it? 
SA:  Uh-huh. 

T*3:  How  many  trapezoids  make  a  hexagon? 
SA:  Two. 

T*3:  OK,  so  for  every  two  you're  going  to  make  a  hexagon.  Is  that  right? 
SA:  1  don't  know. 

T*3:  Look  at  it.  For  every  two  trapezoids,  do  I  have  a  hexagon? 
SA:  Yeah. 

T*3:  (walking  up  to  the  overhead  projector)  We  have  how  many? 
SA:  Two. 
T»3:  Two? 
SA:  Two  and  a  half. 
T*3:  How  many  trapezoids? 
SA:  Two? 
.   T*3:  Trapezoids!  Count. 
SA:  Five. 

T*3:  Count  by  two's.  Pull  them  aside,  go  ahead. 
SA:  Two,  four,  six. 

T*3:  Count  by  two's,  pull  two  aside.  So  how  many  two's  did  you  take  out  of  five? 
SA:  Two. 

T*3:  How  many  two's  can  you  take  out  of  7?  Oo  ahead,  count. 
SA:  2,4,6. 

T*3:  How  many  whole  ones? 
SA:  Three  and  a  half. 

This  sequence  is  reminiscent  of  the  traditional  discourse  In  which  teachers  Initiate,  students 
respond,  and  then  teachers  evaluate  for  closure.  Although  there  are  some  open-enced  questions 
asked  that  might  be  suggestive  of  Inquiry,  on  the  whole,  the  discourse  was  tightly  controlled  and 
displayed,  to  some  extent  a  funnellng  pattern  of  Interaction.  This  Is  especially  the  case  of  In  the 

1  - 173 


dialogue  with  teacher  *3  who  worked  very  hard  to  lead  a  student  to  a  correct  solution. 

This  observed  pattern  of  questioning  is  contrary  to  the  ways  teachers  write  about  their 

practice  in  their  journals.  In  the  Fall ,  one  teacher  described  the  ideal  classroom  as  follows: 

T*l  (Sept):  Activities  are  structured  so  that  students  interact  with  the 
materials  and  have  opportunities  to  explore  and  make  connections.  Process  is 
emphasized  over  content  and  the  teacher  facilitates  the  learning  with  questioning 
and  discussion  rather  than  dispensing  procedural  knowledge.  . .  Lesson  "plans" 
are  by  necessity  more  fluid  and  open-ended. . . 

Journal  entries  for  this  time  of  year  from  other  teachers  contain  similar  remarks. 

While  teachers  can  describe  their  "ideal  classroom"  using  language  that  is  suggestive  of 

mathematical  inquiry,  they  acknowledge  that  such  questioning  practices  are  problematic  for 

them.  Teachers  felt  that  students  were  not  well-prepared  for  responding  to  more  open-ended 

questions  which  probed  their  thinking: 

T*2-  (Oct)  These  kids  are  not  used  to  dealing  with  open-ended  questions.  It 
makes  it  tough  for  classroom  management  when  you  move  to  a  setting  that  allows 
for  a  more  open-ended  approach. . .   I  think  I  am  discouraged  from  asking  these 
kinds  of  questions  from  the  poor  quality  of  response  I  often  get  on  them. 

T*3-  (Oct)  I  saw  that  too  many  kids  were  not  having  success. . .  I  can't  buy 
the  idea  that  kids  don't  feel  bad  starting  off  with  what  they  perceive  to  be  failure. 

The  kids  need  to  succeed  very  badly. . .  Once  the  kids  have  success,  they  will 
try  harder  and  it  won't  need  to  be  structured  the  same  way. 

Arguing  the  importance  of  providing  students  with  success,  the  teachers  justified  a  need  to 
provide  more  structure  for  activities  and  explorations  included  In  the  curriculum.  Structure 
was  often  interpreted  by  teachers  to  mean  the  use  of  questions  which  were  leading  and  "set  up" 
so  that  students  were  likely  to  respond  correctly. 

By  Spring  of  1 99 1 ,  the  sequence  of  questioning  had  changed  in  some  important  ways  for 
two  of  the  three  teachers,  showing  some  movement  from  school  math  to  inquiry  math: 

T*1-  (Two  students  at  overhead  sharing  their  solution  to  a  problem  they  had  all 
worked  on  in  small  groups.)  Can  you  explain  a  little  bit  about  what  you  did 

SA&SB:  (Mumbling about  550's  and  3) 

T*1-  What  were  you  trying  to  do  there?  What  was  the  purpose  for  doing  that? 


1  - 174 


SA:  Add  them  all  up. 

T*  1 :  Why  did  you  multiply  550  times  3? 
SA&SB:  (Mumble) 

T*1 :  OK,  and  what  if  you  had  gotten  a  number  like  2150.  What  would  you  have 

SA&SB:  (Mumble) 

T*  1 :  OK,  any  questions  for  what  their  method  is  here?  (Students  ask  questions 
and  the  teacher  goes  on  to  ask  for  other  solutions  which  are  then  discussed. ) 

T*2:  Would  you  please  look  at  Casey's  that  she  has  in  front  of  her?  It's  4  cubes 
tall  and  3  cubes  long  and  2  cubes  wide.  You  should  all  have  that  in  front  of  you, . , 
Once  you've  got  that,  count  up  the  surface  area. 
SA:  I  know  what  It  Is. 

T*2:  OK,  can  you  give  me  an  explanation  of  how  you  got  it?  Also,  find  the 
volume.  How  many  cubes  are  in  that  thing? 
SA:  I  got  it.  I  know  the  volume. 
T*2:  What  is  it? 
SA:  24. 

T*2:  There  are  24  cubes  in  it,  aren't  there?  So  its  volume  is  24.  Now,  1  heard 
somebody  say  46  on  the  surface  irea.  That's  close,  but  you'll  want  to  check  again. 

( Students  have  been  working  on  problems  in  small  groups.  The  teacher 
announces  that  she  would  like  them  to  discuss  the  second  problem.  A  student 
volunteers  to  come  to  the  overhead  and  explain  his  solution.) 
SA.  (At  the  overhead)  First  I  draw  a  circle  and  divide  it  into  5  parts.  So  there  is 
5  parts  and  I  put  20*  in  each  of  them,  OK  ...  1  put  $  1000  at  the  top  and  I  put 
$200  under  each  of  them  ( drawing  as  he  talks)  and  that  equals  $  1 000. 
T*3:  Robby,  did  you  immediately  think  to  draw  a  circle  when  you  read  the 
problem?  What  did  you  think? 
SA:  A  circle. 

T*3:  Does  anybody  have  any  Questions?  Go  ahead  and  call  on  people. 
SB:  I  don't  understand  (mumble). 

SA:  Cause  there's  20*.  He  made  20*  of  each  painting  and  you  wanted  to  know 

how  much  the  painting  cost.  6et  it?  . . .  You  get  it  now? 

SB:  No,  I  still  don't  understand  why  you  put  20*  in  each  space. 

SA:  You  are  making  it  very  hard  for  me. 

t*3:  Maybe  he  doesn't  understand  your  question. 

SB:  OK ,  you  know  how  it  says  (mumble)?  The  part  I  don't  understand  is 


T*3:  OK  Robby,  how  did  you  know  to  put  5  parts  to  your  circle?  Why  didn't  you 
put  3  parts  and  put  20%  in  each  part?  or  10  parts? 
SA:  "Cause,  it's  like,  it  equals  a  thousand. 

(The  teacher  continues  to  ask  for  questions,  then  invites  other  students  to  share 
their  solutions  which  are  then  also  discussed.) 

Here,  In  the  discourse  of  Teachers  *1  and  3,  we  begin  to  see  features  of  Inquiry  mathematics. 

The  discourse  seems  more  genuine.  Teachers  are  asking  students  to  explain  their  thinking,  to 




1  -  175 


talk  about  their  Ideas,  to  explain  their  reasoning.  Other  students  are  encouraged  to  participate 
In  the  discourse  and  seem  willing,  even  eager,  to  ask  questions  or  share  their  observations.  On 
the  other  hand,  the  discourse  of  Teacher  *2  seems  to  have  grown  even  more  like  school  math 
over  the  year. 

An  examination  of  teacher  Journals  over  the  year  suggests  a  growing  awareness  of  the 
limitations  of  their  questioning: 

T*  1 :  ( Mar)  I  think  1  am  trying  to  do  more  probing  but  that  sometimes  I  miss 
opportunities  to  do  so.  I  have  tried  to  Increase  the  questions  phrased  "What 
caused  you  to  try  that?",  "what  was  in  the  problem  that  prompted  you  to  do 
that?" .  etc.  . .  I  need  to  be  more  alert  to  students  who  don't  respond  a  lot  In  class 
and  take  advantage  of  their  responses  and  questions  to  probe  their  thinking. 

T*2:  (Feb)  1  was  asking  lots  of  questions.  But  as  I  wrote  down  the  questions  It 
seemed  that  almost  none  of  them  were  probing  student  thinking.  Rather .  on  many 
of  them  I  had  a  specific  answer  in  mind. 

T*3;  (Nov)  I  think  I  probably  become  more  directive  If  kids  are  off  task  or  not 
responding  the  way  I  want  them  too.  I  am  willing  to  risk  empowering  the  kids 
with  their  own  learning  but  I  take  back  tha  power,  not  really  consciously, 
whenever  things  don't  go  my  way.  That  is  not  truly  empowering  the  kids  and 
believing  that  they  can  succeed,  that  they  have  Ideas. . .  If  I  really  believe  that 
the  kids  are  capable.  I  will  stop  reverting  to  teacher -directed  every  time  I  feel 
Insecure. . .  I  need  to  trust  my  kids  more.  They  will  learn.  (Mar)  This  year  I 
have  really  struggled  with  the  questioning.  I  tell  myself  I  am  going  to  ask  better 
questions,  be  less  directive.  At  first  I  wasn't  that  aware  of  my  questions  but  now 
I  cringe  sometimes  at  the  questions  1  ask.  I'm  asking  more  genuine  questions 
now,  but  not  as  many  as  I  need  to  do. 

Teachers  *  1  and  3  wrote  extensively  In  their  Journals  over  the  course  of  the  year .  looking 

critically  at  their  own  practice  and  talking  with  colleagues  and  project  staff  about  the  nature  of 

their  classroom  discourse.  Teacher*2.  on  the  other  hand,  wrote  little  and  seemed  to  be  less 

engaged  In  looking  critically  at  his  own  practice.  In  addition.  Teachers  *1  and  3  often  spent 

class  time  helping  students  learn  the  behaviors  that  one  might  associate  with  Inquiry  math: 

sharing  Ideas,  explaining  one's  thinking,  asking  questions,  and  looking  for  multiple  solutions. 

Teacher  »2  engaged  students  In  these  kind  of  discussions  to  a  much  lesser  extent,  and,  one  might 

note,  did  not  actually  support  these  behaviors  In  his  classroom. 



1  -  176 



The  shift  from  "school  math"  to  "Inquiry  math"  Is  a  challenge  to  Instruction.  Even  fn 
cases  where  teachers  are  working  with  currlcular  materials  designed  to  reflect 
recommendations  contained  In  such  documents  as  the  NCTM  Standards  ( 1 989),  genuine 
mathematical  inquiry  and  discourse  are  problematic.  Questions  are  not  in  themselves  evidence 
of  inquiry.  One  happy  observation  Is  that  the  teachers  are  becoming  more  aware  of  the 
problems  surrounding  their  practice  and  some  are  making  significant  changes  In  their 
questioning.  A  better  understanding  of  how  that  change  occurs  Is  central  to  the  reform 
movement  In  mathematics  education.  Further  discussions  with  teachers  who  are  making  those 
changes,  as  well  as  with  those  who  are  not,  cannot  help  but  Inform  our  efforts  to  help  teachers 
create  an  environment  which  supports  genuine  mathematical  Inquiry  and  discourse. 

Bauersfeld,  H.  (1988)  Interaction,  construction,  and  knowledge?  alternative 

perspectives  for  mathematics  education.  In  D.A.  Grouws,  T.J.  Cooney,  &  D.Jones 

( eds. )  Effective  Mathematics  Teaching.  Reston ,  YA:  Lawrence  E r  1  baum 

Associates  and  National  Council  of  Teachers  of  Mathematics. 
Brousseau,  G.  ( 1 984).  The  crucial  role  of  the  didactical  contract  In  the  analysis  and 

construction  of  situations  in  teaching  and  learning  mathematics.  In  Steiner  et  al.  (eds. ) 

Theory  of  Mathematics  Education,  occasional  paper  54.  Bielefeld.  I  DM,  pp  110-119. 
Cobb,  P.;  Wood,  T.;  &  Yackel,  E.  (1991).  A  construct! vist  approach  to  second  grade 

mathematics.  In  E.  von  Glasersfeld  (ed)  Radical  Constructivism  In  MathBmattcs 

Education.  Dordrecht,  The  Netherlands:  Kluwer  Academic. 
Davenport,  L.  &  Narode,  R.  Open  to  question:  an  examination  of  teacher  questioning.  Psychology 

of  Mathematics  Education  Proceedings.  Oct  1991. 
Mehan.H.  (1979).  The  competent  student.  Soclollngulstlc  working  paper  61.  Austin,  TX: 

Southwest  Educational  Development  Laboratory. 
Richards,  J.  (1991).  Mathematical  discussions.  In  E.  von  Glasersfeld  (ed)  Radical 

Constructivism  in  Mathematics  Education.  Dordrecht,  The  Netherlands:  Kluwer 


Steffe,  L. ;  Cobb ,  P. ;  &  von  Glasersfeld,  E.  (1988).  Construction  of  Arithmetic  Meanings  and 

Strategies.  New  York,  NY:  Sprlnger-Yerlag. 
von  Glasersfeld,  E.  ( 1 983).  Learning  as  a  constructive  activity.  In  «J.  Bergeron  &  N. 

Herscovlcs  (eds)  Proceedings  of  the  Fifth  Annual  Meeting  of  PME-NA.  Montreal, 

pp  41-69. 





Gary  DAVIS:  The  Institute  of  Mathematics  Education,  La  Trobe  University.  Melbourne 

/  detail  a  case  study  of  problem  solving  in  an  advanced  mathematical  setting.  The  study  shows  clearly 
the  false  starts  and  detours  that  occurred  prior  to  a  solution.  It  also  shows  the  interactive  catalytic 
effect  of  a  group  in  problem  solving.  The  study  is  presented  in  part  as  a  counter-example  to  the  notion 
that  good  problem  solving  abilities  can  be  equated  with  the  automation  of  domain  specific  rules.  Such 
automation  is  important  and,  in  most  cases,  necessary;  it  is  however  far  from  sufficient. 

INTRODUCTION  In  this  article  I  detail  the  background  to  the  solution  of  an  elementary  but 
important  result  in  dynamical  systems.  The  technical  solution  of  this  problem  will  appear  in  the 
American  Mathematical  Monthly  (Banks  et  al.  to  appear).  In  what  follows  I  have  referred  to  the  five 
authors  of  the  Banks  et  al  article  as  "A.  B.C.  D.  and  E"  in  a  random  order.  I  will  consider  the  attributes 
of  this  group  and  its  individual  members  that  seemed  to  contribute  to  a  successful  solution  to  the 
problem.  I  have  presented  an  account  that  attempts  to  reconstruct  a  successful  group  attempt  at 
mathematical  problem  solving.  The  problem  posed  was  novel  for  any  person  to  whom  it  was  posed. 
This  is  because  it  was  at  the  time  not  only  an  unsolved  problem  but.  as  far  as  we  are  aware,  one  that  had 
not  even  been  previously  posed  because  the  connections  it  established  were  not  suspected  (a  possible 
exception  is  evidenced  in  the  article  by  Peters  and  Pennings,  1991.  in  which  they  speculate  on  the 
interdependence  of  the  three  conditions  for  chaos  that  we  outline  below). 

The  problem  stems  from  a  mathematical  definition  of  chaos  given  by  Devaney  (1989).  In  order  to 
discuss  the  problem,  and  the  steps  to  its  solution,  a  modicum  of  notation  and  basic  concepts  from  dy- 
namical systems  is  necessary.  In  Devaney's  book  a  dynamical  system  is  determined  by  a  continuous 
function  on  a  suitable  topological  space.  In  fact,  in  order  to  state  the  most  important  condition  for  chaos, 
Devaney  assumes  that  he  has  a  continuous  function  defined  on  a  metric  space.  This  is  a  topological 
space  in  which  measurement  is  possible  in  very  general  terms,  subject  only  to  a  few  axioms,  the  most 
important  of  which  is  the  triangle  inequality.  These  axioms  say  that  when  we  have  a  method  for  assign- 
ing to  all  pairs  of  points  x,  y  from  our  space,  a  "distance"  d(x,y),  then  the  function  d  satisfies  the  fol- 
lowing laws: 

•  d(x.y)  >  0.  with  equality  exactly  when  x=y 

•  d(x,y)  =d(y.x) 

•  d(x.y)  S  d(x.z)  +  d(z,y) 

Probably  the  best  known  example  is  the  metric  d  defined  on  the  Euclidean  plane  by  : 
d(x.y)  =  Vte-c^+fb-d)2,  where  x  =  (a.b)  and  y  =  (c.d). 

A  discrete  dynamical  system  is  then  determined  by  a  continuous  function  f  on  a  metric  space  X.  A 
good  example  to  bear  in  mind  for  all  that  follows  is  the  case  when  X  is  the  Euclidean  plane  consisting  of 
all  pairs  (a.b)  where  a,  b  are  real  numbers,  d  is  the  metric  described  above,  and  f  is  the  function  defined 
by  f«a.b))  =  (|a|-b+l.a  ).  This  very  simple  function  has  quite  complicated  dynamics  in  the  plane, 
Devaney  (1983,  1988).  These  dynamics  can  be  investigated  empirically  with  a  small  computer  and  a 
simple  programming  language  (as  simple  as  Basic,  for  example). 

CONDITIONS  FOR  CHAOS  There  are  three  conditions  that  Devaney  (1989)  requires  for  chaos.  I 
will  adumbrate  these  conditions  in  relation  to  the  specific  function  f  defined  above. 

The  first  condition  is  that  f  is  transitive.  In  its  simplest  from  this  means  that  there  is  a  point  (a,b)  for 

which  the  orbit  of  (a.b)  -  that  is  the  set  of  points  (a.b),  f(a.b).  f(f(a,b)))  =  f2(a.b).  f(f2(a.b))  =  f3(a.b)  

and  so  on  -  passes  arbitrarily  close  to  any  prescribed  point  of  the  plane.  The  example  f  that  I  described 
above  is  not  transitive  in  this  sense.  It  is  however  transitive  on  the  set  X  shown  in  black  below. 


1  - 178 

This  set  X.  indicated  in  black,  is  invariant  under  f  -  that  is.  if  (a.b)  is  a  point  in  X  then  f(a.b)  is  also  a 
point  in  the  region  X  -  and  f  is  transitive  on  this  invariant  set.  It  is  not  immediately  obvious  that  f  is 
transitive  on  the  set  X  shown,  but  in  fact  any  point  (a.b)  in  that  set  where  a  and  b  are  irrational  numbers 
will  have  a  dense  orbit  -  that  is.  an  orbit  that  passes  arbitrarily  close  to  any  point  of  the  invariant  set. 
Devaney  (1984). 

The  second  condition  on  the  function  f  to  define  a  chaotic  dynamical  system  is  that  arbitrarily  close 
to  any  point  there  is  a  periodic  point.  A  periodic  point  is  one  that  eventually  returns  to  itself,  under  the 
action  of  the  function,  after  a  finite  number  of  steps.  For  example  the  point  (1.1)  is  a  fixed  point  of  the 
function  f  above  (since  f(l.l )  =  (1,1)),  and  the  point  (0.0)  is  periodic  with  period  6.  Devaney  (1989.  p. 
50)  refers  to  the  above  condition  -that  is.  density  of  periodic  points  -  as  "an  element  of  regularity." 

The  final  condition  is  widely  thought  of  as  the  essential  ingredient  of  chaos:  the  function  f  should 
have  "sensitive  dependence  on  initial  conditions".  This  means.  roughly.<that  there  is  some  constant  K  so 
that  if  we  take  two  distinct  points  x  and  y  and  iterate  them  under  the  function  f  sufficiently  many  times, 
we  will  get  points  at  least  a  distance  K  apart  It  is  this  condition  that  says  that  in  a  chaotic  dynamical 
system  small  experimental  errors  are  eventually  magnified  to  large  errors.Technically.  the  condition  is 
as  follows:  there  is  a  constant  K  >  0  so  that  if  x  is  any  point  and  n  is  a  positive  integer  then  there  is  a 

point  y  whose  distance  from  x  is  -  or  less,  and  an  integer  p  so  that  the  distance  from  fl>(x)  to  fP(y)  is  K  or 

This  third  condition,  of  sensitivity  to  initial  conditions,  is  different  from  the  other  two  conditions  in 
that  it  depends  on  the  metric,  and  is  not  entirely  a  topological  property.  In  the  development  of  dynamical 
systems  this  creates  a  difficulty.  The  difficulty  is  that  the  best  notion  of  equivalence  of  dynamical 
systems  seems  to  be  topological  equivalence,  and  not  the  stronger  notion  of  metric  equivalence.  The 
reason  is  simply  that  attracting  behaviours  such  as  those  shown  below  are  commonly  thought  to  de- 
scribe part  of  the  same  dynamical  behaviour:  however  such  dynamical  systems  can.  in  general,  only  be 
topological^  equivalent,  and  not  metrically  equivalent. 

SPECULATION  ON  A  BASIC  QUESTION  A  question  arose  in  the  mind  oi  B  whether  chaos,  as  the 
conjunction  of  Devaney"s  three  conditions,  is  a  topological  property.  If  it  were  not  this  would  be  most 
unfortunate,  because  it  would  say  that  chaos  was  a  metric  but  not  a  topological  property,  so  that  a  dyn- 
amical system  could  be  chaotic  whilst  another  dynamical  system  essentially  the  same  as  it.  from  the 
topological  point  of  view,  might  not  be  chaotic.  This  is  a  basic  consideration  in  all  structural  math- 
ematics: to  determine  what  sort  of  mappings  preserve  a  given  property.  B  gave  a  nice  simple  argument 
to  show  that  chaos  is  a  topological  property  in  the  case  that  the  underlying  metric  space  X  is  compact 
(that  is.  when  every  sequence  in  X  has  a  convergent  sub-sequence).  This  includes  such  important  metric 
spaces  as  closed  intervals  and  closed  discs:  in  general  it  includes  all  closed  and  bounded  subsets  of 
Euclidean  space,  of  any  finite  dimension.  However  in  many  examples  of  chaotic  dynamical  systems  the 
underlying  metric  space  is  not  unnpact.  so  the  more  general  question  remained  open. 




1  - 179 

FIRST  STEPS  TO  A  THEOREM  As  a  result  of  B's  problem.  A  speculated  that  the  metric  property  of 
sensitivity  to  initial  conditions  might  be  a  logical  consequence  of  the  other  two  properties  for  chaos. 
This  conjecture  seemed  surprising  and  somewhat  naive  to  some  other  members  of  the  chaos  study  group 
when  it  was  presented.  A  proved,  via  a  remarkably  short  and  transparent  argument,  that  his  conjecture 
was  correct  in  the  special  case  when  the  space  X  is  unbounded  -  that  is  when  the  set  of  distances  d(x.y), 
with  x  and  y  points  of  X,  is  an  unbounded  collection  of  real  numbers.  This  is  a  sort  of  opposite  case  to 
that  when  the  space  X  is  compact. 

CONVICTION  A's  argument,  presented  at  a  seminar,  stimulated  D  to  give  a  general  proof  by  reduc- 
ing the  bounded  case  to  the  unbounded  case.  This  is  an  opposite  procedure  to  what  is  a  common  trick  in 
analysts,  so  the  idea  came  from  a  resonance  with  bounded-unbounded  and  seemed  highly  plausible,  acc- 
ompanied by  a  strong  feeling  of  "I've  seen  this  before."  Consequently.  D  presented  his  proof  to  B.  only 
to  realise  that  the  proof  worked  in  detail  only  for  those  bounded  spaces  in  which  the  diameter  of  the 
space  is  not  achieved:  spaces  such  as  the  open  disc  below,  but  not  the  closed  disc,  nor  the  half  closed 

An  open,  closed,  and  half -open  disc,  respectively.  The  points  shown  in  the  closed  and  half-open  discs  are 
as  far  apart  ax  the  diameter  of  the  disc.  There  are  no  such  point*  in  the  open  disc. 

However  it  now  seemed  highly  likely  to  B  and  D  that  A's  conjecture  was  indeed  true,  and  that  the 
condition  of  not  achieving  the  diameter  of  the  space  was  a  technical  hitch  that  could  be  patched  up. 

A  BREAKTHROUGH  IDEA  A  day  or  so  later  C .  in  contemplating  A 's  argument,  presented  the  out- 
line of  an  argument  to  show  that  the  conjecture  was  true,  at  least  in  a  fairly  general  and  natural  setting.  1 
present  below  the  first  sentence  of  C's  statement  because  it  shows  the  intuitive  feel  for  being  on  the 
right  track  that  characterizes  creative  problem  solving  in  mathematics.  It  also  shows  too  how  one  makes 
a  leap  of  faith: 

"In  the  following  f:X— X  is  continuous  and  X  is  some  topological  space  with  enough 
properties  to  make  everything  work  (a  suivre  ...  V 

What  were  the  sufficient  properties  "to  make  everything  work"  and  which  were  to  (eventually) 
follow!?  C's  idea,  stemming  form  his  work  in  differential  geometry,  was  to  show,  by  way  of  contradic- 
tion, that  if  the  first  two  conditions  for  chaos  held  in  conjunction  with  the  negation  of  the  third 
condition,  then  the  period  of  any  given  periodic  point  would  be  forced  to  be  arbitrarily  long  -  a 
contradictory  situation.  His  idea  was  to  base  an  argument  on  volume  estimates,  assuming  that  volume 
could  be  measured  in  X  in  some  way  (for  example,  so  that  for  each  6  >  0  the  collection  of  balls  B5  (x)  = 
{y  I  d(x.y)  <  6  1  had  a  measure  that  was  bounded  over  x.  the  least  upper  bound  for  which  tended  to  0  as 
5  approached  0.) 

CRITICAL  SIMPLIFICATION  Unfortunately  it  was  not  clear  to  which  classes  of  metric  spaces 
with  well-defined  notions  of  volume  this  argument  would  apply:  in  other  words,  it  was  not  clear  how 
general  the  argument  would  be.  However  it  seemed  then,  and  still  does  seem,  a  very  potent  idea  that 
shed  considerable  light  on  the  question.  Then  E.  in  trying  to  understand  C's  written  demonstration, 
concluded  by  a  most  pertinent  but  elementary  argument  that  we  could  dispense  with  any  idea  of  volume 
simply  by  interchanging  the  order  of  two  operations,  and  we  could  simplify  a  technical  part  of  the 
argument  by  an  elementary  but  subtle  use  of  the  triangle  inequality. 

THE  FINAL  ARGUMENT  The  one  catch  was  that  Devaney  (1989)  actually  had  a  somewhat  more 
general  notion  of  transitivity,  of  which  the  dense  orbit  notion  is  an  important  specialisation.  We  had 



1  -180 

therefore,  in  writing  down  a  version  of  the  proof  of  the  conjecture  for  publication,  to  make  an  ass- 
umption that  the  metric  space  X  had  a  special,  but  important,  property  (technically,  it  had  to  be  a 
separable  Baire  space).  After  submission  of  the  article  for  publication  we  woiUed  individually,  and  as  a 
group,  for  some  time  to  rid  ourselves  of  this  restriction.  A  seminar  visitor  pointed  out  to  us  that  over  any 
metric  space  X  there  lived  a  separable  Baire  space  to  which  a  function  on  X  could  be  lifted.  This  seemed 
to  offer  the  hope  we  were  after  when  the  grim  news  arrived  that  the  editor  of  the  American 
Mathematical  Monthly  had  rejected  our  article!  The  referee's  remarks  appear  below.  They  are  of 
interest  here  for  two  reasons.  First  because  these  remarks,  in  part,  stimulated  A  to  find  an  even  better 
proof.  Second,  the  referee's  remarks  indicate  a  genuine  gap  between  a  certain  sort  of  applied 
mathematics,  where  one  may  use  words  in  a  somewhat  loose  way,  and  what  is  commonly  thought  of  as 
"pure  mathematics''  where  precise  definitions  are  de  rigeur. 

"Toe  paper  is  a  reasonable  remark,  which  1  believe  is  correct.  1  have  read  through  the  paper  but  no!  with  a  magnifying 
glass,  and  I  can  more-or-less  imagine  a  direct  proof.  The  writing  is  fluent  (both  in  the  sense  of  fluency  of  language  - 
which  shouldn't  surprise  us  -  and  of  fluency  of  exposition).  1  am  not  aware  of  any  published  proof  of  the  theorem.  I 
even  suspect  that  it  will  be  of  interest  to  a  number  of  Monthly  readers. 

So  why  am  1  unenthusiastic?  1  think  it's  the  first  two  pages,  which  seem  to  put  the  wrong  sL-ess  on  (ji'c).  The  popular- 
ity of  Gleick's  book  (and  I  hope  soon  Stewart's  book)  and  the  wonderfully  evocative  buziword  "chaos"  has  inspired  a 
lot  of  armchair  scientists,  and  in  in  particular  it  seems  to  be  the  "in"  thing  to  try  to  argue  about  the  definition  of  the 
buziword.  chaos.  (Imagine  trying  it  with  "art"  or  -democracy"  or  "truth"  ■  you  get  chaos.)  In  order  to  make  their 
comment  weighty,  the  authors  spend  two  pages  discussing  the  "definition"  of  "chaos".  1  tend  to  yawn  at  such  discus- 
sions, but  1  also  wonder  at  their  reliance  on  Devaney's  text  for  the  authoritative  statement  of  such  a  definition.  What 
do  Collet  Eckntann  or  Mane  say?  If  you're  going  to  discuss  the  "usual"  definition,  point  to  more  than  one  source  for 

I  think,  if  it  were  pulled  together  a  bit.  the  paper  could  be  a  perfectly  reasonable  note  for  the  Monthly.  The  discussion 
motivating  the  theorem  should  be  shortened  considerably,  and  1  suspect  the  proof  can  be  done  witLjut  invoking  con- 

I  think  you  could  reasonably  either  refuse  the  paper  or  ask  for  a  rewrite.  1  don't  favor  publication  as  is." 

Our  paper  had  been  deliberately  written  with  the  provocative  title  "What  is  Chaos?"  We  did  this  to 
highlight  what  seemed  to  us  to  be  a  fact:  namely  that  no  one  yet  had  apparently  come  up  with  a  satisfac- 
tory mathematical  definition  of  chaos.  This  title,  we  concluded,  had  upset  the  referee,  so  we  answered 
the  remarks  by  changing  the  title  and  attending  to  a  few  other  minor  matters.  Some  of  us  were  puzzled 
by  what  the  referee  referred  to  in  the  statement  "the  proof  can  be  done  without  invoking  contradiction", 
since  logically,  if  not  psychologically,  a  proof  by  contradiction  is  as  direct  as  any  other  proof  (simply 
change  the  statement  of  the  result).  The  referee  was  also  operating  in  a  different  theatre  to  us:  he  was 
apparently  taking  "chaos"  as  an  intuitive  undefined  term  in  mathematics.  This  would  be  a  revolutionary 
idea  indeed,  so  we  preferred  to  stick  with  the  usual  mathematical  practice  of  making  precise  mathemati- 
cal definitions.  The  definition  of  Devaney  (1989,  p.  50)  was.  as  far  as  we  know,  the  only  general 
mathematical  definition  of  chaos  in  print  at  that  time,  and  we  believed  we  had  established  an  elementary 
but  important  result  that  showed  an  appropriate  definition  of  chaos  was  still  not  yet  clear.  That  is,  our 
theorem  was  a  mathematical  criticism  of  Devaney's  definition. 

However  A  was  moved,  in  part  by  the  referee's  remarks,  to  re-consider  the  entire  proof  and.  using  the 
same  circle  of  ideas,  came  up  with  a  shorter,  more  direct,  and  compelling  argument  in  which  we  could 
use  Devanev's  more  general  condition  of  transitivity.  The  final  argument  had  the  compelling  features  of 
technical  simplicity  and  complete  generality.  We  sent  the  revised  paper  to  the  (new)  editor,  and  were 
relieved  to  hear  that  it  was  accepted  for  publication. 


Group  work.  It  would  appear,  even  to  a  casual  observer,  that  we  understood  the  benefits  of  group 
work.  We  also  seem  to  understand  how  to  implement  group  work  in  practice.  Indeed  surges  of 
excitement  came  in  waves  as  we  got  deeper  into  the  problem  and  the  excitement  of  one  member  of  the 


1  - 181 

group  spurred  others  on  to  better  things.  How  did  we  cooperate  -  by  writing,  by  talking,  or  both  ?  The 
answer  is.  of  course,  both.  Our  habitual  way  of  working  is  to  let  one  member-of  the  group  talk  until  we 
have  a  serious  criticism,  a  misunderstanding  that  needs  clarification,  or  until  the  speaker  dries  up.  This 
speaking  is  not  ordinary  conversation:  it  is  more  like  thinking  aloud  and  is  usually  done  at  a  blackboard 
This  talking  is  almost  always  done  in  a  room  other  than  the  speakers'  room.  The  reason  is  that  the 
speaker  has  an  idea  and  an  urge  to  talk  about  it.  He  goes  looking  for  an  audience,  and  so  the  talking 
begins.  The  listening  is  never  passive,  and  sometimes  it  can  be  difficult  to  talk  easily,  especially  if  the 
ideas  are  only  half-formed:  at  that  stage  the  talker  wants  a  critical  but  sympathetic  audience. 

When  talking  is  temporarily  done  it  is  time  for  writing  thoughts  down  carefully  in  a  mathematical 
format,  and  time  for  reflection  -  on  ideas  just  conveyed  or  on  new  ideas  forming.  In  practice  we  seem  to 
form  most  ideas  alone,  after  much  cogitation,  or  calculation,  or  both.  This  time  is  essentially  time  spent 
in  finding  quality  data  pertinent  to  the  problem  and  the  arguments  we  have  used,  or  intend  to  present. 
But  precious  ideas  need  to  be  subjected  to  a  searchlight  of  criticism,  and  that  is  where  talking  is  essen- 
tial, to  us  at  least. 

False  leads  There  were  three  obvious  false  leads  that  were  important  in  the  problem  solving  process. 
The  first  was  the  result  that  said  the  theorem  is  true  for  bounded  metric  spaces  in  which  the  diameter  of 
the  space  is  not  achieved.  Although  this  did  not  appear  anywhere  in  the  final  theorem,  nor  did  the  ideas 
used  there  assume  any  importance  later  in  the  argument,  this  result  on  the  way  was  a  catalyst  that 
stimulated  us  to  look  for  a  proof  of  the  main  result,  which  a  number  of  us  now  believed  to  be  true.  In 
other  words  this  subsidiary  result,  which  we  abandoned,  gave  us  the  feeling  that  we  had  to  find  a  general 
argument  for  a  palpably  true  result.  This  is  a  situation  that  mathematician's  delight  in,  because  feelings 
run  strongly  positively  that  success  will  soon  follow. 

The  second  false  lead  was  the  excursion  into  volume  arguments.  This  involved  a  beautiful  circle  of 
ideas  that  gave  us  great  exultation  at  the  time,  but  they  rapidly  became  superseded  by  a  very  elementary 
argument,  based  simply  on  on  the  triangle  inequality.  As  irrelevant  as  the  volume  idea  was  to  the  final 
proof  it  buoyed  us  up  enormously,  because  we  now  felt  that  we  had  a  deeper  understanding  of  a  reason 
why  our  hoped-for  theorem  was  true,  and  we  had  a  water-tight  proof  for  some  important  special  cases. 

The  third  false  lead  was  the  simplification  of  the  transitivity  condition  to  that  of  the  important  sub- 
case of  a  dense  orbit,  and  the  necessary  assumption  that  we  were  working  in  a  separable  Baire  space. 
For  a  long  time  we  could  not  see  how  to  weaken  this  condition,  and  it  was.  in  part,  the  referee's 
comments  which  stimulated  us  to  reflect  sufficiently  to  give  a  proof  in  which  these  restrictive  conditions 
were  completely  removed. 

Whilst  these  three  paths  were  eventually  abandoned,  they  were  each  important  in  leading  us  to  a 
completely  general,  simple,  proof. 

Critical  reflection  Much  of  our  time  after  the  volume  argument  was  spent  critically  examining  our 
assumptions,  and  the  restrictions  we  had  imposed  in  order  to  get  a  moderately  general  result  In  this  per- 
iod many  original,  and  some  fantastic,  ideas  were  dreamed  up  to  try  to  remove  all  restrictions  in  the 
statement  of  the  theorem.  All  but  two  were  abandoned  as  being  without  sufficient  import.  Only  the  sug- 
gestion of  our  seminar  visitor,  alluded  to  above,  and  the  final  argument  of  A  resolved  the  matter,  the 
latter  most  decisively. 

Some  individual  reflections  on  critical  steps  in  the  problem  solving  process 

I  present  below  the  re-collections  of  A,  C,  end  D  about  the  problem-solving  process.  The  other 
members  of  the  group  either  could  not  recall  how  they  came  to  the  arguments  they  did.  or  were  not 
available  for  interview. 

A:  "My  argument  came  out  of  B's  question  of  whether  topological  equivalence  was  the  appropriate 
equivalence  for  chaotic  dynamical  systems,  or  whether  the  conjunction  of  Devaney's  three  conditions 
was  the  appropriate  notion.  This  was  an  obvious  problem  to  answer  •  the  whole  notion  of  chaos  in  the 
sense  of  Devaney  depended  on  the  answer.  After  B  gave  his  proof  for  the  compact  case  I  was  looking  at 
unbounded  spaces  as  a  sort  of  opposite  to  compact  one.  It  was  the  resuil  for  the  unbounded  case  that 
made  me  conjecture  the  result  was  true  in  general.  1  think  I  was  just  basically  trying  to  produce  some 

1  -182 

simple-minded  proof  that  the  three  conditions  taken  together  were  preserved  by  conjugacy.  It  just 
popped  out  of  that. 

All  the  way  through  I'd  been  unhappy  about  the  proof  by  contradiction.  I  didn't  think  it  gave  you 
much  intuitive  insight  into  what  was  going  on.  I  tried  to  go  through  the  proof  by  contradiction  and  con- 
vert it  into  a  more  direct  proof.  Arthur  (a  colleague)  had  said  if  you've  got  a  fixed  point  then  the  map 
stays  near  there  for  a  while  (thinking  of  flows).  I  thought :  yes.  but  that  doesn't  give  you  a  number.  On 
the" other  hand  if  you  had  nvo  fixed  points  you'd  be  O.K!  So  I  wrote  the  argument  for  the  special  case  of 
two  fixed  points  first..  I'd  been  investigating  another  matter  related  to  periodic  orbits.  It  occurred  to  me 
then  that  if  you  set  things  up  the  right  way  you  could  get  the  orbits  separating  properly." 

C:  "I'm  not  sure  what  started  me  off  on  volume  - 1  was  trying  to  capture  what  was  inherent  in  A's 
argument.  I  remember  it  was  an  interesting  problem.  The  book  had  been  around  for  a  long  time  and  it 
seemed  that  the  other  guys  (A,  B  and  0)  might  be  right  My  initial  attitude  was  to  find  a  counter- 
example. There  was  something  similar  in  my  past,  but  I  wasn't  conscious  of  it  at  the  time.  In  my  PhD  ! 
was  looking  at  the  problem  of  whether  paracompact  spaces  are  regular:  the  obvious  argument  didn't 
work,  but  a  more  delicate  analysis  -  refining  the  ideas  -  did.  I  was  just  trying  to  make  A's  argument 
more  subtle." 

D  "When  A  gave  his  argument  in  a  seminar  he  made  what  seemed  to  me  to  be  a  very  strange  assu- 
mption: that  the  metric  space  was  unbounded.  This  was  the  opposite  sort  of  assumption  to  that  normally 
made  in  analysis.  As  he  talked  I  immediately  had  the  realisation  that  a  standard  trick  of  passing  from  un- 
bounded to  bounded  metrics  could  be  used  in  reverse.  All  I  had  to  do  was  to  check  that  the  three  condi- 
tions for  chaos  passed  from  one  case  to  the  other.  This  I  did  very  easily  that  evening.  Unfortunately.  B 
pointed  out  to  me  next  morning,  when  I  presented  my  argument  to  him.  that !  got  an  unbounded  metric 
from  a  bounded  one  only  when  the  diameter  of  the  space  was  not  achieved.  Still,  I  had  substantially 
broadened  the  spaces  to  which  A's  conjecture  applied,  and  I  now  believed  it  to  be  completely  true." 

Automation  of  domain-specific  rules 

Sweller  (Sweller,  Mawer  and  Ward,  1983:  Owen  and  Sweller.  1989:  and  Sweller.  1990:  see  also 
Law  son.  1990)  has  argued  that,  good  mathematical  problem  solvers  are  good  principally  because  they 
have  access  to  relevant  schemas  and  they  have  automated  domain  specific  rules,  so  reducing  cognitive 
load.  My  own  view  is  that  Sweller,  like  many  psychologists  who  venture  into  a  mathematical  domain, 
may  not  be  talking  about  problem  solving  in  the  way  in  which  mathematicians  and  the  mathematics 
education  community  in  general  understand  problem  solving.  In  one  sense  problem  solving  skills  and 
strategies  are  what  apply  when  automation  of  domain  specific  skills  no  longer  helps.  However  let  us 
look  at  what  schema  and  domain  specific  skills  may  have  helped  in  solving  the  problem  reported  here. 
En  route  certain  specific  techniques  were  important  First  there  was  the  idea  that  the  bounded  case  could 
be  related  to  the  unbounded  case  via  a  specific  trick  in  analysis.  Then  there  was  the  idea  of  using 
volume  estimates,  with  which  one  of  us  was  quite  familiar,  to  get  a  reasonably  general  argument.  Then 
again  there  was  a  standard  analytic  technique  of  bounding  a  finite  set  of  points  away  from  another  finite 
set.  It  is  eminently  reasonable  therefore  to  argue  that  knowledge  of  specific  analytic  techniques  proved 
very  useful  en  route  to  a  solution. 

The  results  of  this  study  are  entirely  in  accord  with  Kilpatrick  (1985).  who  said: 

"Studies  of  expert  problem  solvers  and  computer  simulation  models  have  shown  that  the  solution  of  a  complex 
problem  requires  ( I )  a  rich  store  of  organized  knowledge  about  the  content  domain.  (2)  a  set  of  procedures  for 
representing  and  transforming  the  problem,  and  (3)  a  control  system  to  guide  the  selection  of  knowledge  and 
procedures  It  is  easy  to  underestimate  the  deep  knowledge  of  mathematics  and  extensive  expenence  in  solving 
problems  that  underlie  proficiency  in  mathematical  problem  solving.  On  the  other  hand  it  is  easy  to  underestimate  the 
control  processes  used  by  experts  to  monitor  and  direct  their  problem  -solving  activity  "  (pp  7-8) 
Our  experience  also  supports  the  remarks  of  Thompson  (1985)  when  he  says: 

•  "Several  studies  in  cognitive  psychology  and  mathematics  education  haw  also  shown  the  importance  of  structure  in 
one  s  thinking  in  mathematical  problem  solving  "  (p.  195) 

This  is  evidenced  by  the  emphasis  on  such  structural  features  as  the  distinction  and  connections  be- 
tween bounded  and  unbounded  metric  spaces,  the  role  played  by  compact  metric  spaces,  the  role  of 

1  -  183 

volume  in  providing  estimates  on  size,  and  the  role  of  fixed  points  of  continuous  functions.  The  remark 
of  C.  quoted  above,  also  emphasized  heavily  a  structural  approach  to  the  problem. 

So  there  is  a  sense  in  which  Sweller's  argument  cannot  be  easily  dismissed  by  this  example  of 
problem-solving  at  an  advanced  level.  Indeed,  in  many  respects  it  supports  Sweller's  thesis:  knowledge 
of  a  subject  and  ready  recall  of  pertinent  skills  can  be  of  great  assistance  in  solving  mathematical  prob- 
lems. In  practice  however  the  converse  is  most  often  encountered:  without  the  ready  recall  of  pertinent 
skills  the  solution  of  genuine  mathematical  problems  will  usually  be  impossible.  A  terrible  catch  is:  how 
do  we  know  beforehand  what  is  pertinent,  or  useful? 

The  problem  considered  in  this  study  had  a  particularly  simple  conceptual  scheme:  A  &  B  —  C. 
However  this  logical  formulation  of  the  problem  was  of  no  assistance  in  telling  us  why  we  should  expect 
the  condition  of  .sensitivity  to  initial  conditions  to  be  a  logical  consequence  of  the  conditions  of 
transitivity  and  density  of  periodic  points.  What  we  needed  were  useful  ideas . 

When  we.  as  teachers,  set  difficult  or  challenging  mathematical  problems  for  our  students  we  think 
we  know  what  is  pertinent  Davis'  (1984)  long  term  study  suggest  strongly  that  whilst  for  most  students 
we  are  right,  for  highly  capable  students  we  are  wrong.  Pertinence  is  especially  difficult  to  judge  when 
the  problem  is  unsolved:  that  is,  when,  as  far  as  we  are  aware,  no  one  knows  a  solution.  Browder  and 
MacLane  ( 1978.  p.)  comment  on  pertinence,  or  usefulness: 

■The  potential  usefulness  of  a  mathematical  concept  or  technique  in  helping  to  advance  scientific  understanding  has 
very  little  to  do  with  what  one  can  foresee  before  that  concept  or  technique  has  appeared. ...  Concepts  or  techniques 
are  useful  if  they  can  be  eventually  put  in  a  form  which  is  simple  and  relatively  easy  to  use  in  a  variety  of  contexts. 
We  don't  know  what  will  be  useful  (or  even  essential)  untH  we  have  used  it.  We  can't  rely  upon  the  concepts  and 
techniques  which  have  been  applied  in  the  past,  unless  we  want  to  rule  out  the  possibility  of  significant 
innovation. "(My  italics) 

However  this  question  of  pertinence,  or  usefulness,  is  of  critical  importance.  It  is  a  variable  that 
needs  to  be  considered  deeply  because  it  is  at  the  heart  of  the  process  of  creative  problem-solving.  Once 
the  usual  ideas  and  domain-specific  rules  seem  to  be  exhausted,  how  is  it  that  successful  prob >  em- 
solvers  proceed^  I  believe  they  create.  They  create  new  ideas  and  concepts  which  they  hope  will  be 
useful  in  solving  the  problem.  The  processes  of  concept  creation,  and  its  dual  of  concept  annihilation 
due  to  the  constraints  of  the  problem  and  the  critical  comments  of  colleagues,  is  I  believe,  an  example  of 
evolution  in  microcosm.  This  it  seems  to  me.  is  where  mathematics  is  born,  ever  new.  and  this.  I 
believe,  is  where  we  should  concentrate  our  efforts  on  understanding  the  problem-solving  process  in 

Finally,  the  problem  we  worked  on  was  a  universal  problem:  it  was  a  problem  for  every  per*011 10 
whom  it  was  posed.  The  mathematics  education  literature  has  often  had  difficulty  with  the  relative  na- 
ture of  "problems"  -  for  whom  is  a  problem  a  problem?  -  and  many  of  the  examples  elucidated  in  Silver 
(1985).  for  example,  are  problems  only  for  relative  novices.  I  suggest  that  as  a  research  community  we 
will  learn  more  about  the  important  creative  processes  involved  in  problem  solving  when  we  concen- 
trate on  student/group  interaction  with  universal  problems:  those  that  are  known  not  to  have  been  solved 
at  a  particular  time.  An  example  is  the  following: 

-  A  small  boat  has  travelled  2  kilometres  out  to  sea  from  a  straight  shore  line.  Fog  descends,  and  visibility  is  almost 
nil.  There  is  no  wind  and  no  current.  The  people  on  the  boat  do  not  know  in  which  direction  the  shore  lies.  They  decide 
to  wvel  at  constant  speed  to  conserve  fuel.  What  are  the  shortest  path  or  paths  they  could  take  so  »« to  be  certain  that 
they  will  reach  the  shore?  " 

This  was  an  unsolved  problem  at  the  time  of  writing  (Croft  et  al .  1991,pp.  40-41).  Such  problems, 
capable  of  being  stated  in  elementary  terms,  are  useful  in  that  they  largely  dispense  with  the  notion  o, 
utility  or  pertinence  of  an  idea  to  a  solution,  since  no  one  knows  what  will  be  pertinent..  The  problem 
poser  -  usually  a  mathematics  teacher  -  cannot  then  occupy  a  position  of  knower  in  respect  of  a  solution 
to  the  problem.  The  advantage  of  such  a  situation  is  that  it  forces  a  teacher  to  judge  proposed  solutions 
for  appropriateness  to  the  problem  at  hand  and  for  inventiveness,  rather  than  scrutinise  them  as 
approximations  to  a  "correct"  solution.  Since  we  don't  know  what  will  work  we  are  obliged  to  take 


1  - 184 

students  ideas  seriously  and  consider  them  carefully.  I  think  by  focusing  on  such  problems  we  will  learn 
much  about  mathematical  concept  creation  in  individual  brains,  and  much  about  teachers  critical 


Banks.  J..  Brooks,  J..  Cairns.  C  Davis.  G.  and  Stacey.  P.  (to  appear)  On  Devaney's  definition  of  chaos. 
The  American  Mathematical  Monthly. 

Browder,  F.  and  Mac  Lane.  S.  ( 1978)  The  relevance  of  mathematics.  In  L.  Steen  (Ed)  Mathematics 
Today.  Twelve  Informal  Essays,  p.  348.  New  Yotk:  Springer  Verlag 

Croft.  H.T.,  Falconer.  KJ.  and  Guy.  RK.  (1991)  Unsolved  Problems  in  Geometry.  Problem  Books  in 
Mathematics.  Unsolved  Problems  in  Intuitive  Mathematics,  volume  2.  New  York:  Springer  Verlag. 

Davis.  R.B.  (1984)  Learning  Mathematics.  The  Cognitive  Science  Approach  to  Mathematics.  Education. 
London:  London:  Groom  Helm 

Devanev.  R.  L.  (1984)  A  piecewise  linear  model  for  the  zones  of  instability  of  an  area-preserving  map. 
Phys'ica  10D,  387  -  393. 

Devaney.  R.L.  (1988)  Fractal  patterns  arising  in  chaotic  dynamical  systems.  In  H.-O.  Peitgen  and  D. 
Saupe  (Editors)  77if  Science  of  Fractal  Images  ..pp.  1 37- 1 68.  New  York:  Springer  Verlag. 

Devaney,  R.  L.  (1989)  An  Introduction  to  Dynamical  Systems.  Second  Edition.  Redwood  City, 
California:  Addison-Wesley. 

Kilpatrick,  J.(  198S)  A  retrospective  account  of  the  past  25  years  of  research  on  teaching  mathematical 
problem  solving.  In  E  Silver  (ed.)  Teaching  and  Learning  Mathematical  Problem  Solving:  Multiple 
Research  Perspectives,  pp.  1-16.  New  Jersey:  Lawrence  Erlbaum. 

Lawson.  M.  (1990)  The  case  for  instruction  in  the  use  of  general  problem-solving  strategies  in 
mathematics  teaching:  A  comment  on  Owen  and  Sweller.  journal  for  Research  in  Mathematics 
Education,  21, 401-410. 

Owen.  E.  and  Sweller.  J.  (1989)  Should  problem  -  solving  be  used  as  a  teaming  device  in  mathematics? 
Journal  for  Research  in  Mathematics  Education,  20,  322-328. 

Peters,  J.  and  Pennings,  T.  (1991)  Chaotic  extensions  of  dynamical  systems  by  function  algebras. 
Journal  of  Mathematical  Analysis  and  Applications .  159, 345-360. 

Silver,  E  A.  (Ed.)  (1985)  Teaching  and  Learning  Mathematical  Problem  Solving:  Mutiple  Research 
Perspectives  .Hillsdale.  N.J.:  Lawrence  Erlbaum. 

Sweller.  J.  (1990)  On  the  limited  evidence  for  the  effectiveness  of  teaching  general  problem  solving 
strategies.  Journal  for  Research  in  Mathematics  Education,  21,  411-415. 

Sweller.  J..  Mawer.  R.F.  and  Ward.  M.R.  (1983)  Development  of  expertise  in  mathematical  problem 
solving.  Journal  of  Experimental  Psychology:  General,  112, 639-661 . 

Thompson.  P.W.  (1985)  Experience,  problem  solving,  and  learning  mathematics:  Considerations  in 
developing  mathematics  curricula.  In  E  Silver  (ed.)  Teaching  and  Learning  Mathematical  Problem 
Solving:  Multiple  Research  Perspectives,  pp.  189-236.  New  Jersey:  Lawrence  Erlbaum. 

1  - 185 


Linda  J.  DeGuire 
California  State  University,  Long  Beach  (U.S.A.) 

Teachers'  perceptions  of  the  development  of  their  own  problem-solving  abilities  during  a 
course  on  problem  solving  seem  to  be  reflected  in  their  perceptions  of  their  students' 
development  of  problem-solving  abilities.  The  data  were  collected  during  a  15-week 
course  on  the  teaching  of  problem  solving  and  consisted  primarily  of  Journal  entries  of 
reflections  during  the  course.  The  subjects  were  all  6  students  in  the  course  (out  of  18) 
who  were  also  teaching  full  time.  The  results  are  presented  in  groups  of  3  subjects  each  in 
which  the  subjects  in  each  group  were  similar  at  the  beginning  of  the  course  and  during 
the  course  in  their  conceptions  of  problem  solving  and  levels  of  confidence,  and  reported 
similar  developments  within  their  students  in  their  own  classrooms. 

Within  the  literature  on  mathematical  problem-solving,  few  studies  have  studied 
the  role  of  the  classroom  teacher  in  developing  students'  problem-solving  abilities.  Clark 
and  Peterson  (1986),  after  extensive  review  of  studies  in  the  broader  educational  literature 
on  teacher  thinking  and  decision  making,  concluded  that  teachers'  theories  and  beliefs 
provide  a  frame  of  reference  for  planning  and  interactive  decisions  which  affect  their 
actions  and  effects  in  the  classroom.  Thompson  (1985,  1988)  reported  a  study  in  which 
teachers'  beliefs  about  problem  solving  were  changed  and  the  changes  in  some  subjects 
positively  affected  their  abilities  to  teach  problem  solving.  The  purpose  of  the  present 
paper  is  to  explore  the  possibility  that  teachers'  perceptions  of  the  development  of  their 
own  problem-solving  abilities  during  a  course  on  problem  solving  will  be  reflected  in  their 
perceptions  of  their  students'  development  of  problem-solving  abilities.  The  data  were 
drawn  from  a  larger  data  set  gathered  to  study  the  development  of  metacognitlon  during 
mathematical  problem  solving  (DeGuire,  1987, 1991a,  1991b). 

The  Course  and  Data  Sources 

The  data  were  gathered  throughout  a  semester-long  course  (one  3-hour  session 
per  week  for  15  weeks)  on  problem  solving  In  mathematics.  The  course  began  with  an 
introductory  phase,  that  is,  3  sessions  devoted  to  an  introduction  to  several  problem-solv- 
ing strategies.  The  course  then  progressed  from  fairly  easy  problem-solving  experiences 
to  quite  complex  and  rich  problem-solving  experiences,  gradually  introducing  discussions 



1  - 186 

of  and  experiences  with  the  teaching  of  and  through  problem  solving  and  the  integration  of 
problem  solving  into  one's  approach  to  teaching.  Throughout  the  course,  subjects  dis- 
cussed and  engaged  in  reflection  and  metacognition. 

A  variety  of  data  sources  were  used— journal  entries,  written  problem  solutions 
with  explicit  "metacognitive  reveries,"  optional  videotapes  of  talking  aloud  while  solving 
problems,  and  general  observation  of  the  subjects.  Subjects  also  wrote  a  journal  entry 
each  week.  The  topics  of  the  journal  entries  were  chosen  to  encourage  reflection  upon 
their  own  problem  solving  processes  and  their  own  development  of  confidence,  strategies, 
and  metacognition  during  problem  solving.  Each  subject  chose  a  code  name  to  use  for 
their  journal  entries.  The  code  names  of  the  6  subjects  for  this  paper  were  Apple,  Euclid, 
Galileo,  Hobie.  Simplicius,  and  Thales.  The  data  for  the  present  paper  were  taken  primari- 
ly from  the  journal  entries,  though  their  other  sources  of  data  were  used  secondarily. 

The  subjects  in  the  entire  data  set  were  18  students,  all  inservice  and  preservice 
teachers  of  mathematics,  mostly  on  the  middle-school  level  (grades  6-8,  ages  11-14),  but 
with  some  teachers  on  the  intermediate  level  (grades  4-6)  and  some  on  the  secondary 
level  (grades  9-12).  The  subjects  had  chosen  to  take  the  course  as  part  of  degree  pro- 
grams in  which  they  were  involved. 

The  subjects  for  the  present  paper  were  all  6  students  in  the  course  who  were  also 
teaching  full  time.  Of  this  subset.  4  (Apple.  Hobie.  Simplicius.  and  Thales)  were  teaching 
on  the  middle-school  level  and  2  (Euclid  and  Galileo)  were  teaching  on  the  secondary 
level;  all  had  substantial  teaching  experience,  with  8  years  being  the  minimum.  Regarding 
their  mathematics  backgrounds.  2  (Galileo  and  Simplicius)  had  completed  Masters  in 
mathematics,  1  (Euclid)  had  completed  an  undergraduate  major  in  mathematics,  and  3 
(Apple.  Hobie.  and  Thales)  had  completed  enough  mathematics  to  be  certified  to  teach 
mathematics  in  the  middle  grades  (that  is.  about  7  courses  on  the  college  level,  including 
at  least  1  course  in  calculus  and  perhaps  one  course  beyond  calculus).  All  came  to  the 
course  with  some  exposure  to  problem  solving  through  inservice  workshops  varying  in 
length  from  2  to  10  contact  hours,  sessions  at  professional  meetings,  or  professional 
reading;  none  had  taken  a  problem  solving  course  before.  (Throughout  this  paper,  direct 
quotes  are  from  the  subjects'  journal  entries.) 



1  - 187 


Apple,  Euclid,  and  Thales 

The  stories  of  Apple,  Euclid,  and  Thales  begin  similarly  in  that  each  felt  some 
apprehension  about  problem  solving  and  a  lack  of  confidence  in  their  own  problem-solv- 
ing abilities.  However,  each  also  exhibited  some  growth  in  confidence  quite  early.  Apple 

expressed  her  apprehensions  and  budding  confidence  as  follows: 

When  I  first  entered  this  class  on  problem  solving,  I  was  very  apprehensive.  .  . . 
Now  that  class  has  been  in  session  for  two  weeks,  some  of  my  fears  have  been 

alleviated  I  feel  a  certain  excitement  when  I  leave  class,  and  my  first  inclination 

is  to  hide  somewhere  and  work  on  the  problems  With  every  new  technique  and 

problem,  my  enthusiasm  has  increased. 
Thales  expressed  feelings  similar  to  Apple.  "I  consider  my  problem  solving  abilities  to  be 
minimal  but  increasing.  In  the  past,  when  confronted  with  a  problem  solving  task.  .  .[I 
would]  panic.  ...  My  frustration  levels  are  decreasing  somewhat."  Euclid  summarized 
similar  feelings  in  an  interesting  way.  He  said  that  "I  felt  somewhat  that  problem  solving 
had  to  be  caught  rather  than  taught. .  .and  I  seemed  to  not  catch  it  frequently!" 

All  3  subjects  also  began  the  course  with  very  limited  conceptions  of  problems 
and  problem  solving,  conceptions  that  were  rapidly  expanded.  In  his  very  first  journal 

entry  (the  second  week  of  class),  Euclid  explained  his  expanded  conceptions  as  follows: 
I'm  not  sure  that  I  had  the  appropriate  definition  and  understanding  of  the  nature  of 
problem-solving  at  the  beginning  of  class.  ...  My  horizons  have  already  been 
broadened  and  enriched  from  the  distinction  made  between  exercises  and  problem 
solving  and  the  practice  that  we  have  had  in  problem  solving  My  prior  percep- 
tions of  problem  solving  centered  on  the  word  problem  experience. 

Thales  expressed  a  similar  conception  of  problems  before  beginning  the  course. 

My  experiences  with  problem  solving  have  been  very  similar  to  those  discussed  in 
this  class  as  a  misconception.  As  a  student,  I  can  remember  many  occasions  when 
we  were  asked  to  solve  a  series  of  "problems"  where  the  operations  and  proce- 
dures were  evident  I  have  had  little  experience  solving  actual  "problems*. 

Thales  soon  realized  in  the  course  that,  if  she  saw  an  immediate  solution  to  the  task,  then  it 

was  not  really  a  problem.  Thus,  her  conception  of  problems  had  expanded.  Apple  initially 

expressed  a  similar  misconception  of  "problem"  by  describing  problems  as  "textbook  word 

problems  used  to  teach  one  basic  particular  skill;  formulas  with  different  arrangements  of 

addition,  subtraction,  multiplication,  and  division;  geometry  and  algebra  word  problems." 

As  the  course  progressed,  each  of  the  three  grew  in  confidence  and  enthusiasm  in 

their  own  problem-solving  abilities.  Apple  chronicle  her  growth  as  follows: 


1  - 188 

[  About  a  third  of  the  way  through  the  course:]  I  feel  much  more  familiar  with  some  of 
the  problem  solving  techniques.  Also  I  don't  feat  "not  finding  the  answer"  as  much 
as  I  have  in  the  past.  I  concentrate  more  on  my  attack  to  the  problem.  [About  half- 
way through  the  course:]  I  am  definitely  more  aware  of  the  process  going  on  in  my 

head.  [Towards  the  end  of  the  course:]  I  understand  a  lot  more  than  I  did  before  

I  feel  much  more  qualified  to  solve  a  problem  now  than  I  did.  [After  the  final  exam:] 
The  answer  that  I'm  about  to  write  for  that  question  [How  confident  are  you  now?] 
surprises  me.  Even  one  month  earlier  or  possibly  one  week  earlier,  my  answer 
would  have  been  different.  After  having  worked  with  the  final  exam,  I  feel  a  lot  more 
confident.  For  some  reason,  ideas  that  I  thought  I  had  learned  did  not  really 
become  whole  until  that  exam. 

Apple's  growth  chronicled  above  was  mirrored  in  the  changes  in  her  problem  solutions; 

they  became  progressively  richer  in  appropriate  strategies  and  metacognitions  and  correct 

solutions,  as  well  as  in  alternative  and  generalized  solutions.  Thafes'  development  of 

confidence  in  her  own  problem-solving  abilities  is  similar  to  Apple's  but  not  as  thoroughly 

chronicled  in  her  journal  entries.  Her  change  in  emotional  response  moved  from  "panic"  at 

.  the  beginning  of  the  course  to  "enjoyment".  Her  confidence  also  grew. 

[About  halfway  through  the  course:]  I  think  that  during  the  last  few  weeks,  my  prob- 
lem solving  skills  have  improved. . . .  This  course  is. .  .making  me  a  more  confident 
problem  s.lver.  [Towards  the  end  of  the  course:]  I'm  sure  that  my  problem  solving 
skills  have  increased  over  the  last  few  months.  However  I'm  still  not  an  overly  confi- 
dent problem  solver.  There  have  been  problems  on  each  of  the  problem  sets  which 
I  found  to  be  particularly  frustrating.  However  at  least  now  I  don't  panic  when  I  read 
the  problem  and  a  method  for  solving  it  isn't  immediately  obvious.  [After  the  final 
exam:]  My  first  reaction  upon  seeing  the  exam  was  panic. . . .  However,  as  I  began 
to  more  carefully  study  the  problems,  I  became  more  confident. ...  I  enjoyed  work- 
ing on  the  problems  which  I  selected. 

Euclid  admits  to  some  confusion  on  certain  aspects  of  the  course  (especially  met^cogni- 

tion),  a  confusion  that  was  not  completely  cleared  up  even  at  the  end  of  the  course. 
[About  halfway  through  the  course:]  I  think  I  have  become  more  aware  of  cognitive 
processes  since  the  beginning  of  the  course.  However,  I  am  not  sure  that  I  really 
understand  yet. .  .what  I  am  trying  to  be  really  aware  of. . . .  Some  of  the  early  prob- 
lems seem  quite  simple  now.  [After  the  final  exam:]  I  feel  as  though  I  am  definitely 
a  better  problem  solver. . . .  [Yet]  I  feel  quite  a  bit  of  frustration. ...  I  feel  fairly  com- 
fortable with  my  success  on  the  exam. . . .  [Yet]  I  am  still  not  as  clear  as  I  should  be 
about  the  distinctions  between  teaching  problem  solving,  teaching  a  problem,  and 
teaching  through  problem  solving. 

His  confusion  was  evident  in  his  exam  responses,  both  in  his  solution  of  problems  (even 

though  he  had  a  mathematics  major  in  college)  and  in  his  essay  responses  to  items  such 

as  distinguishing  teaching  problem  solving,  teaching  a  problem,  and  teaching  through 

problem  solving. 

1  -  189 

By  about  a  third  of  the  way  through  the  course,  each  of  Apple,  Euclid,  and  Thales 

began  to  report  attempts  to  introduce  problem  solving  into  their  own  classrooms.  The 

parallel  of  their  perceptions  of  their  students'  success  in  and  reactions  to  problem-solving 

experiences  and  their  reports  of  their  own  development  (as  above)  are  striking  and  are 

even  explicitly  referred  to  by  both  Apple  and  Thales.  Apple  reported  the  following  on  her 

attempts  to  introduce  problem  solving  into  her  classroom: 

[About  a  third  of  the  way  through  the  course:]  One  real  effect  that  this  course  is  hav- 
ing is  that  my  reaction  to  my  students  has  changed.  I'm  far  more  concerned  with 

their  attack  on  the  problems  than  with  their  answers  The  students  in  my  classes 

are  experiencing  a  change  in  their  success  rate  in  solving  problems.  .  .  .  [About 
two-thirds  of  the  way  through  the  course:]  Lately,  my  students  expect  to  be  solving 

problems  as  a  regular  part  of  the  routine  Some  very  positive  results  seem  to  be 

happening.  .  .  .  I  have  to  use  different  problems  in  different  classes  because  the 
students  get  excited  and  tell  each  other  all  about  the  problems.  [At  the  end  of  the 
course:]  More  than  ever,  I  feel  that  I  see  a  real  difference  in  "understanding"  as  it  is 
applied  to  me  personally  and  in  "understanding"  as  the  work  is  taught. ...  I  can  see 
my  strategies  and  my  attitudes  (enthusiasm!)  picked  up  by  the  students  My  stu- 
dents have  become  very  enthusiastic  about  problem  solving.  And  successful  too! 

Thales  admitted  to  mixed  success  in  initial  attempts  to  introduce  her  students  to  "real" 

problem  solving.  Several  of  her  comments  (underlined  below)  consciously  mirror  her  own 


[About  a  third  of  the  way  through  the  course:]  t  have  used  some  of  the  problems  we 
have  discussed  in  class  with  my  seventh  graders,  with  mixed  results.  ...  I'm  sure 
that  they  have  mixed  emotions  about  their  abilities  now,  just  as  I  often  have  about 
mine.  ...  It  is  not  uncommon  for  many  of  my  students  to  read  the  problem  and 
immediately  say,  "I  don't  understand!"  I  know  that  what  they  are  really  saying  is  that 
they  don't  see  an  obvious  solution  and  they're  not  sure  where  they  should  begin,  a 
panin  that  I  certainly  understand  well.  [About  halfway  through  the  course:]  At  the 
beginning  of  this  course,  |  viewed  problem  solving  with  some  of  tha  same  wariness  I 
now  see  in  my  students. 

Thales  did  not  make  further  comments  in  her  journal  about  the  success  or  enthusiasm  of 

her  students  in  problem-solving.  Euclid  did  not  comment  in  his  journal  on  attempts  to 

introduce  problem  solving  into  his  classes  until  about  a  third  of  the  way  through  the  course. 
For  years,  the  extent  of  my  problem  solving  activities  in  the  classroom  involved  word 
problems,  t  have  taught  a  plan  to  try  to  solve  these  problems.  Most  students  seem 
to  have  a  great  deal  of  trouble  and  difficulty  with  word  problems. ...  I  have  consis- 
tently thrown  in  to  the  classes  some  problem  solving  activities.  I  have  found  that 
students  enjoy  them  and  gives  [sic]  a  different  pace  to  the  classroom.  However, 
success  rates  are  mixed. 

1  - 190 

Euclid  only  very  briefly  included  further  journal  references  to  his  students'  problem-solving 
experiences.  They  seemed  to  indicate  mixed  success  and  many  reservations  about  the 
possibility  of  success  for  all  students.  Euclid's  perception  of  his  students'  mixed  success 
mirrored  his  own  mixed  success  in  the  course. 
Galileo,  Hobie,  and  Simplicius 

Note:  The  stories  of  Galileo  and  Simplicius  have  been  chronicled  and  their  jour- 
nal entries  quoted  extensively  in  at  least  two  other  places  (DeGuire,  1991a,  1991b).  Thus, 
due  to  space  limitations  here,  conclusions  about  these  2  subjects  will  be  cited  here  but 
supporting  quotes  will  be  limited.  Hobie's  story  will  be  chronicled  more  thoroughly. 

The  stories  of  Galileo,  Hobie,  and  Simplicius  are  also  somewhat  similar  but  quite 
different  from  those  of  Apple,  Euclid,  and  Thales.  Unlike  the  earlier  3  subjects,  Galileo, 
Hobie,  and  Simplicius  all  began  the  course  feeling  quite  confident  about  their  own 
problem-solving  abilities  and  very  enthusiastic  about  problem  solving.  Galileo  and  Simpli- 
cius had  extensive  mathematics  backgrounds  and  some  previous  problem-solving  experi- 
ence; they  were  both  immediately  very  successful  with  the  problems  in  the  course.  Their 
confidence,  enthusiasm,  and  richness  of  solutions  and  metacognitions  grew  throughout 
the  course.  Hobie  had  the  least  mathematics  background  of  all  6  subjects  in  this  paper  but 
had  had  some  previous  experiences  in  problem  solving  in  a  mathematics  methods  course 
the  previous  semester.  She  began  the  course  feeling  quite  confident  in  problem  solving 
but  soon  realized  the  limitations  of  her  knowledge.  She  also  consistently  recorded  enjoy- 
ment of  the  problem-solving  experiences.  She  reported: 

[At  the  beginning  of  the  course:]  I  felt  okay  about  problem  solving  before  this  class 
or  at  least  I  thought  I  did.  It  is  amazing  what  one  can  learn  from  just  one  class.  I  can 
already  tell  that  my  problem  solving  strategies  were  somewhat  weak.  . . .  [About  a 
third  of  the  way  through  the  course:]  The  more  we  get  into  the  class,  the  more  I  re- 
alize how  little  I  really  did  know  Before  the  class,  I  had  pretty  much  confidence 

in  myself  as  a  problem  solver.  After  the  first  night,  t  had  lost  some  of  that.  However, 
as  each  class  ends  and  as  I  solve  more  problems  and  read  more  articles  my  confi- 
dence moves  up  a  step  again. ...  I  am  thoroughly  enjoying  these  activities.  [About 
halfway  through  the  course:]  I  find  that  writing  the  metacognitive  reveries  in  solving 
the  problems  has  really  helped  me. .  .to  become  a  better  problem  solver.  [Towards 
the  end  of  the  course:]  I  know  I  have  become  better  at  problem  solving,  mostly  be- 
cause I  can  take  a  problem  apart,  and  concentrate  on  the  process.  [After  the  final 
exam:]  I  felt  pretty  good  about  the  exam.  It  is  amazing  to  me  how  sitting  down  and 
working  on  something  can  be  so  rewarding. ...  I  feel  like  my  problem  solving  skills 
have  really  improved. 


1  -  191 

Though  Hobie's  problem  solutions  were  never  as  mathematically  rich  as  Galileo's  and 
Simplicius',  they  did  exhibit  mathematical  richness  in  line  with  Hobie's  more  limited 
mathematics  background.  All  three  subjects  also  began  the  course  with  conceptions  of 
"problem"  and  "problem  solving"  that  were  essentially  congruent  with  the  widely-accepted 
meanings  of  the  words. 

Just  as  all  3  of  these  subjects— Galileo,  Hobie,  and  Simplicius— were  consistently 
confident  and  successful  in  their  own  problem  solving,  so  their  perceptions  of  their  stu- 
dents' confidence  and  success  in  problem-solving  experiences  was  consistently  positive. 
Hobie  had  already  begun  to  implement  some  of  problem-solving  experiences  into  her 
classroom  as  a  result  of  her  experiences  in  the  methods  course  the  previous  semester. 
Even  at  the  beginning  of  the  course,  she  reported  her  students'  excitement  about  problem 
solving.  Several  of  her  journal  entries  make  explicit  statements  (underlined)  that  con- 
sciously reflect  her  own  development: 

[At  the  beginning  of  the  course:]  We  have  already  gone  over  the  problem  solving 
strategies  and  have  used  several  of  them.  So  far,  my  students  as  wall  as  me  are 
vary  excited  about  it.  [About  a  third  of  the  way  through  the  course:]  I  hope  to 
become  so  confident  when  the  class  is  over  that  some  of  it  will  spill  over  to  mv 
students  So,  far,  I  feel  my  students  are  really  enjoying  doing  the  problem  solv- 
ing, just  as  I  am.  [About  halfway  through  the  course:]  Beginning  this  quarter,  I  am 
going  to  begin  having  my  students  write  down  their  metacognitive  reveries.  ...  I 
think  that  this  will  really  help  them,  juSLasJLhaaJIlft.  [At  the  end  of  the  course:]  Not 
only  has  this  course  helped  me,  but  it  is  doing  wonders  for  my  classroom. 

From  the  perceptions  that  Hobie  reports  in  her  journal,  it  would  appear  that  her  students 
have  become  successful  and  enthusiastic  problem  solvers.  Both  Galileo  and  Simplicius 
make  explicit  references  to  the  influence  of  the  course  on  their  teaching,  with  Galileo 
providing  evidence  of  his  students'  development  reflecting  his  own.  In  commenting  on 
implementing  a  problem-solving  approach  to  teaching,  Galileo  observes,  "This  almost 
becomes  contagious  to  the  student.  I  have  noticed  students  beginning  to  imitate  the  very 
same  processes  which  I  utilize  in  confronting  problems."  As  Simplicius  expressed  in  her 
journal,  "I  feel  that. .  .my  ability  as  a  teacher  has  blossomed.  I  have  definitely  made  more 
effort  to  incorporate  problem  solving  into  the  curriculum.  ...  I  feel  that  this  course  has 
fundamentally  changed  my  attitude  toward  teaching  and  what  the  focus  of  my  teaching 
should  be." 

1  - 192 


The  parallels  between  the  subject's  development  of  problem-solving  abilities, 
confidence,  and  enthusiasm  and  their  perceptions  of  their  students'  development  of 
problem-solving  abilities,  confidence,  and  enthusiasm  is  quite  interesting.  The  present 
conclusions  have  been  uo-od  on  self-report  data.  As  with  all  self-report  data,  one  must 
assume  that,  to  a  certain  extent,  the  subjects  reported  what  they  feel  the  researcher  wants 
to  hear  or  read.  Such  data  has  many  deficiencies  and  problems  and  is  not  here  triangu- 
lated with  other  data  sources.  The  problems  and  issues  with  self-report  data  have  been 
discussed  well  in  Brown  (1987).  It  is  unfortunate  that  it  was  not  possible  to  follow  these 
teachers  into  their  classrooms  to  obtain  independent,  observation  data  on  what  problem 
solving  they  incorporated  into  their  classrooms  and  how  they  did  so.  However,  the  data 
seem  to  present  an  interesting  hypothesis  for  further  exploration,  that  is,  that  students' 
development  of  problem-solving  abilities,  confidence,  and  enthusiasm  will  mirror  their 
teachers'  development  of  these  qualities. 


Brown,  A.  L.  (1987).  Metacognition,  executive  control,  self-regulation  and  other  more 
mysterious  mechanisms.  In  F.  E.  Weinert  &  R.  H.  Kluwe  (Eds.),  Metacognition.  moti- 
vation, and  understanding  (dp.  65-1 16).  Hillsdale,  NJ:  Erlbaum. 

Clark,  C.  M.,  &  Peterson,  P.  (1986).  Teachers  thought  processes.  In  M.  Witt  rock  (Ed.), 
Handbook  of  research  on  teaching  (pp.  255-296.)  New  York:  Macmillan. 

DeGuire,  L.  J.  (1987).  Awareness  of  metacognitive  processes  during  mathematical  prob- 
lem solving.  In  J.  C.  Bergeron,  N.  Herscovics,  &  C.  Kieran  (Eds.),  Proceedings  of  the 
Eleventh  Annual  Meeting  of  the  International  Group  for  the  Psychology  of  Mathematics 
Education  (pp.  215-221).  Montreal:  University  of  Montreal. 

DeGuire,  L.  J.  (1991a,  April).  Metacognition  during  problem  solving:  Case  studies  in  its 
development.  Paper  presented  at  a  research  session  of  the  annual  meeting  of  the  Na- 
tional Council  of  Teachers  of  Mathematics,  New  Orleans. 

DeGuire,  L  J.  (1991b).  Metacognition  during  problem  solving:  Advanced  stages  of  its 
development.  In  R.  G.  Underbill  (Ed.),  Proceedings  of  the  Thirteenth  Annual  Meeting  of 
the  North  American  Chapter  of  the  International  Group  for  the  Psychology  of  Mathema- 
tics Education  (Volume  2,  pp.  147-153).  Blacksburg,  VA:  Virginia  Tech. 

Thompson,  A.  G.  (1985).  Teachers'  conceptions  of  mathematics  and  the  teaching  of  prob- 
lem solving.  In  E.  Silver  (Ed.),  Teaching  and  learning  mathematical  problem  solving: 
Multiple  research  perspectives  (pp.  281-2941  Hillsdale,  NJ:  Lawrence  Erlbaum. 

Thompson,  A.  G.  (1988).  Learning  to  teach  mathematical  problem  solving:  Changes  in 
teachers'  conceptions  and  beliefs.  In  R.  I.  Charles  &  E.  A.  Silver  (Eds.).  The  teaching 
and  assessing  of  mathematical  problem  solving  (Volume  3  of  the  Research  Agenda  for 
Mathematics  Education,  pp.  232-243).  Reston,  VA:  National  Council  of  Teachers  of 
Mathematics,  and  Hillsdale,  NJ:  Erlbaum. 


1  -  193 


M.  Ann  Dirkes 
Indiana  University-Purdue  University  at  Fort  Wayne 

A  study  of  productive  thinking  in  elementary  school  and  college  students 
suggests  (a)  longitudinal  effects  of  current  traditional  teaching  methods 
on  thinking  habits  and  (b)  the  effects  of  self-directed  strategies  on 
thinking.  A  program  was  designed  to  diversify  thinking  by  helping 
students  produce  ideas  in  a  search  for  understanding;  multiple  formats, 
and  connection-making.  Research  supporting  the  program  includes 
studi  is  on  problem  solving  as  a  constructive  enterprise,  learning  as  a 
generative  process,  thinking  perspectives,  and  metacognition. 


For  Students  to  develop  insight  and  transfer  knowledge  to  new  contexts,  they 
need  to  manage  thinking  consciously  and  stretch  their  own  development.  In  this 
sense  they  share  the  direction  of  thinking  with  teachers. 

For  me  to  construct  ideas,  I  must  be  in  charge  of  my  own  thinking. 

For  me  to  use  my  uniqueness  to  do  your  mathematics,  I  need 
to  monitor  the  learning  strategies  I  use. 

We  know  little  about  what  students  are  able  to  do  in  this  domain,  especially 

through  a  deliberate  cultivation  of  self-directed  thinking.  The  work  has  begun, 

however.  Notable  examples  of  groundwork  include  the  research  of  Feuerstein  (1980) 

on  student  generation  of  new  knowledge,  Schoenfeld  (1985)  on  student  beliefs  and 

Whimbey  and  Lochhead  (1982)  on  collaborative  problem  solving  that  helps  students 

use  what  they  know.  From  a  different  perspective,  Vosniadou  and  Ortony  (1991 )  bring 

together  diverse  studies  that  reexamine  the  roles  of  analogical  reasoning  in  learning 

for  children  and  adults.  This  work  includes  self-direction  insofar  as  it  treats  individual 

plans  and  goals,  and  thinking  that  students  can  initiate,  e.g.,  the  identification  of 

surface  features  as  cues  to  underlying  structures. 

Before  children  are  introduced  to  academic  learning,  they  use  a  global 


1  -  194 

approach  to  learning  language  and  certain  quantitative  relationships.  With  a  sense  of 
thinking  autonomy  during  the  early  years  (Kamii  &  DeClark,  1985),  students  engage  in 
spontaneous  qualitative  activity  to  gain  enough  understanding  to  connect  symbols  to 
the  world  (Piaget,  1973).  Although  we  do  not  expect  young  children  to  analyze 
thinking  strategies,  we  do  expect  them  to  think  in  the  ways  that  they  are  able.  Children 
in  the  second  grade,  for  example,  can  learn  to  strengthen  their  beliefs  about  the 
importance  to  conforming  to  the  solution  methods  of  others  (Cobb  et  al,  1991). 
Teachers  giving  informal  reports  describe  children  who  create  ideas  freely  and  decide 
consciously  to  use  self-help  aids  instead  of  asking  for  help  unnecessarily.  These 
teachers  say  that  they  emphasize  self-help  because  they  cannot  know  the  precise 
dimensions  of  thinking  possible  for  every  student  at  a  given  time. 

Projecting  self-direction  to  later  stages  of  development,  we  might  expect  that 
children  and  adults  would  select  different  learning  strategies  for  themselves.  Children 
would  choose  to  manipulate  objects  and  interact  with  peers  as  aids  to  understanding 
mathematics  and  adults  would  use  an  even  larger  array  of  thinking  strategies, 
including  the  manipulation  of  objects,  drawing,  and  analysis  represented  by  spatial 
patterns  and  symbolic  equations.  Yet  we  know  that  in  school  students  rely  on  very  few 
strategies,  largely  memory  and  speed  that  suppresses  informal  thinking  (Resnick, 


This  study  emanates  from  three  integrated  research  directions:  (a)  problem 
solving  as  a  constructive  enterprise  (Steffe,  1990;  Confrey,  1985),  (b)  thinking 
perspectives  (Greeno,  1989)  and  (c)  self-direction  or  metacognition  (Schoenfeld, 
1987;  Lester,  1985). 

As  problem  solvers,  individuals  interpret  mathematical  content,  context, 
structure,  and  heuristics  (Hatfield,  1984)  and  manage  a  repertoire  of  strategies  to  meet 
challenges.  Performance  weighs  heavily  on  accessing  learned  content  (Silver,  1982) 



1  -  195 

and  on  searching  and  elaborating  extensively  (Mayer,  1985).  This  generative  process 
is  characteristic  of  learning  (Wittrock,  1977;  Dirkes,  1978)  and  of  the  thinking  that 
students  do  to  understand  and  solve  problems.  There  is  an  expectation  that  students 
will  construct  possible  models  of  reality  without  allowing  perceptions  of  predetermined 
absolutes  to  restrict  their  thinking.  (Glasersfeld,  1984). 

According  to  Robert  Davis  (1984),  the  true  nature  of  mathematics  involves 
processes  that'  demand  thought  and  creativity.  Doing  mathematics  means  confronting 
vague  situations  and  refining  them  to  a  sharper  conceptualization;  building  complex 
knowledge  representation-structures  in  your  own  mind;  criticizing  these  structures, 
revising  them  and  extending  them;  analyzing  problem,  employing  heuristics,  setting 
subgoals  and  conducting  searches  in  unlikely  corners  of  your  memory.  If  this  is  so, 
students  must  assume  an  active  role,  one  that  they  initiate  and  monitor. 

Treating  mathematics  as  an  ill-structured  discipline  is  a  step  toward  both  the 
dispositional  and  cognitive  changes  required  for  the  construction  of  meaning  (Resnick, 
1984).  For  students  working  in  familiar  situations,  algorithms  and  heuristics  fit  neatly 
into  a  structure.  For  unfamiliar  and  complex  situations,  however,  students  must  not 
only  create  a  plan  to  help  them  organize  data  and  select  mathematical  strategies 
(Kulm,  1984),  they  also  accept  ambiguity,  set  aside  time  for  problem  solving,  and  find 
connections  among  possibilities  that  they  produce. 

Choosing  to  think  and  claiming  the  authority  to  produce  ideas  are  commitments 
that  rely  on  the  development  of  metacognition,  the  awareness  of  mental  functions  and 
executive  decisions  about  when  to  use  them  (Flavell,  1979;  Sternberg,  1984). 
Metacognitive  functions  help  students  regulate  (a)  cognitive  operations,  e.g.,  recall, 
infer,  and  compare;  (b)  strategies,  e.g.,  draw  and  list  possibilities;  and  (c) 
metacognitive  action  to  plan  strategy,  monitor  it,  and  allocate  time  for  thinking.  These 
functions  supplement  what  teachers  do,  beginning  with  the  regulation  of  thinking 
unique  to  individuals. 


1  - 196 


SELF-DIRECTION  IN  MATHEMATICS  (SDM)  is  a  program  designed  to  engage 
students  productively  in  active  problem  solving  from  elementary  school  through 
college.  Ten  groups  of  college  students  enrolled  in  methods  courses  for  teaching 
mathematics  experienced  the  entire  program  for  a  semester  and  ten  groups  of 
students  in  grades  two  through  seven  participated  in  two  or  three  sessions  on  idea 
listing.  Numerous  one-on-one  interactions  with  students  in  public  schools  were  also 
recorded  for  study. 

Six  components  integrate  the  program. 

1 .  MATHEMATICS  Students  construct  meaning  and  problem  solutions  in 
response  to  a  wide  range  of  challenges,  and  teachers  use  oral  techniques 
to  prompt  student  connections.  Current  local  and  national  recommendations 
direct  the  choice  of  mathematical  topics  and  instructional  strategies. 

2.  ROUTINE  AND  SELF-DIRECTED  THINKING  Students  monitor  and  regulate 
thinking  strategies  and  beliefs.  They  allocate  time  for  thinking;  produce 
alternate  interpretations;  and  make  connections  among  ideas,  drawings  and 

3.  IDEA  LISTING  Students  produce  ideas  freely  to  tackle  novelty  and 
complexity,  to  clarify  concepts,  and  create  problem  solutions.  Resources 
include  recall,  observation,  imagination  and  peer  interaction.  A  checklist 
guides  their  thinking  into  mathematical  concerns  and  informal  prompts 
develop  a  climate  for  problem  solving. 

4.  PROCEDURE  For  challenge  problems  students  (a)  list  many  ideas  about 
given  facts;  (b)  restate  questions  to  insure  meaning;  (c)  list  many  ideas  in 
drawings,  words  and  symbols  that  might  lead  to  solutions;  and  (d)  select 
their  best  ideas.  They  solve  given  word  problem  and  those  in  which  they 
add  facts  and  a  question.  A  modified  version  of  this  plan  helps  them 



respond  to  social  situations  and,  when  needed,  divert  their  efforts  to  skill 

5.  MATERIALS  Diverse  materials  and  technology  stimulate  thinking  and 
multiple  representations. 

6.  ASSESSMENT  Productive  thinking  described  in  Components  3  and  4 
reach  beyond  most  paper-and-pencil  instruments  to  support  self-directed 
problem  solving.  Portfolios  and  two-stage  tests  show  the  development  of 
thinking  strategies,  dispositions,  and  mathematical  knowledge. 

An  examination  of  many  idea  lists  shows  that  students  in  elementary  and 
middle  school  can  learn  to  access  what  they  know  and  use  ideas  in  new  ways.  With 
appropriate  strategies,  college  students  begin  to  use  self-u'irection  for  thinking  in 
mathematical  situations  and  for  managing  learning.  At  first,  their  lists  generally  do  not 
demonstrate  more  ideas  or  more  quality  ideas  than  younger  students.  Checklists  that 
cue  mathematical  concerns  and  other  strategies,  however,  improve  the  quality  of  their 
thinking  and  enlarge  their  perceptions  of  the  nature  of  mathematics. 

The  SDM  program  uses  self-direction  because  it  is  a  term  more  familiar  than 
metacognition  and  also  suggests  specifically  that  students  be  the  ones  to  examine  and 
regulate  cognitive  operations,  strategies  and  metacognitive  action.  Monitoring  their 
own  thinking,  students  decide  when  to  probe  long-term  memory  and  when  to  combine 
ideas  into  new  inventions. 

To  optimize  thinking  that  encompasses  physical  and  social  contexts  as  well  as 
personal  beliefs  and  understanding  about  cognition  (Greeno,  1989),  students 
construct  concepts  and  solutions  by  connecting  ideas  within  mathematics,  other 
disciplines  and  life  outside  the  classroom.  An  active  assimilation  of  ideas  prepares 
them  to  elaborate  on  what  they  know  and  develop  representations  that  communicate 
their  knowledge  and  problem  solutions  to  others.  Implications  extend  to  what  students 


1  - 198 

believe  about  mathematics  and  themselves. 

SDM  activities  center  on  an  autonomous  production  of  ideas  in  a  search  for 
new  con  9&ions  that  develop  concepts  and  solve  word  problems  The  word,  idea, 
suggests  that  students  expect  to  produce  possibilities  for  answers  or  for  the  direction  of 
complex  problem  solutions.  Whereas  answers  are  to  be  correct,  the  immersion  of 
ideas  into  subject  matter  (Prawat,  1991)  introduces  problem  solving  that  encourages 
growth  and  revision.  What  is  your  idea,  Susan?  What  else  might  be  important?  List 
many  possible  ideas.  What  do  you  want  to  revise?  The  discourse  created  builds 
understanding  (Lampert,  1989).  Where  expectations  for  a  uniform  development  of 
meaning  do  not  interfere,  students  take  intellectual  risks  that  reach  beyond  minimal 
prescriptions  and  perceive  that  an  extensive  generation  of  ideas  is  as  much  a  part  of 
school  performance  as  the  reproduction  of  definitions  and  algorithms. 

Mathematical  power  comes  with  the  direction  of  strategies.  Students  search  for 
understanding  by  producing  alternate  interpretations  stating  questions  and 
interpretations  in  their  own  words,  and  producing  ideas  as  they  reread  to  construct 
meaning.  They  demonstrate  a  willingness  to  think  by  speaking  extensively,  drawing 
and  writing,  and  organizing  ideas  for  future  reference.  Producing  multiple  ideas  that 
might  be  connected  is  a  process  that  complements  long-term  recall  and  making  sense 
of  mathematics.  This  is  the  kind  of  thinking  students  do  outside  of  school. 

When  understanding  is  not  forthcoming,  a  flexible  production  of  ideas  under 
deferred  judgment,  alone  and  in  groups,  breaks  down  barriers  and  suggests 
connections.  Producing  prior  knowledge  and  new  inventions,  students  unify  their 
consciousness  of  facts,  questions,  and  solution.  Self-direction  and  understanding 
reinforce  positive  attitudes  toward  thinking  so  that  they  face  novelty  and  complexity 
with  a  sense  that  /  can  think  and  /  always  know  how  to  begin. 

Cobb,  P.,  Wood,  T.,  Yackel,  E.,  Nicholls,  J.,  Wheatley,  G.  Trigatti,  B.,  &  Periwitz,  M. 



1  - 199 

(1 991 ).  Journal  tor  Research  in  Mathematics  Education,  22(1 ),  3-29. 

Confrey,  J.  (1985).  A  constructivist  view  of  mathematics  instruction.  Paper  presented 
at  the  annual  meeting  of  American  Educational  Research  Association,  Chicago. 

Davis,  R.B.  (1984).  Learning  mathematics:  The  cognitive  science  approach  to 
mathematics  education.  New  Jersey:  Ablex. 

Dirkes,  M.A.  (1978).  The  role  of  divergent  production  in  the  learning  process. 
American  Psychologist,  33(9),  815-820. 

Flavell,  J.H.  (1979).  Metacognition  and  cognitive  monitoring.  American  Psychologist, 
34(10).  906-911. 

Glasersfeld,  van  E.  (1984).  An  introduction  to  radical  constructivism.  In  P.  Watzlawicz 
(Ed.),  The  invented  reality  (pp.  17-40).  NY:  W.W.  Norton. 

Greeno,  J.G.  (1989).  A  perspective  on  thinking.  American  Psychologist,  44(2), 

Hatfield,  L.L.  (1984).  They  study  of  problem-solving  processes  in  mathematics 
education.  In  G.  Goldin  &  McClintock  (Eds.),  Task  variables  in  mathematical 
problem  solving.  Hillsdale,  NJ:  Erlbaum. 

Kamii,  C.  &  DeClark,  G.  (1985).  Young  children  re-invent  arithmetic.  NY:  Teachers 
College  Press. 

Kulm  G  (1984).  The  classification  of  problem  solving  research  variables.  InG. 
Goldin  &  E.  McClintock  (Eds.),  Task  variables  in  mathematical  problem  solving. 
Hillsdale,  NJ:  Lawrence  Erlbaum. 

Lampert,  M.  (1989).  Choosing  and  using  mathematical  tools  in  classroom  discourse. 
In  J.  Brophy  (Ed.),  Advances  in  research  on  teaching  (Vol.1 ,  pp.  223-264). 
Greenwich,  CT:  JAI  Press. 

Lester,  F.  (1985).  Methodological  considerations  in  research  on  mathematical 
problem-solving  instruction.  In  E.A.  Silver  (Ed.),  Teaching  and  learning 
mathematical  problem  solving.  Hillsdale,  NJ:  Lawrence  Erlbaum. 

Mayer,  R.  (1985).  Implications  of  cognitive  psychology  for  instruction  in  mathematical 
problem  solving.  In  E.A.  Silver,  (Ed.),  Teaching  and  learning  mathematical 
problem  solving.  Hillsdale,  NJ:  Lawrence  Erlbaum. 

Piaget,  J.  (1973).  To  understand  is  to  invent.  NY:  Grossman. 

Prawat,  R.S.  (1991).  The  value  of  ideas:  The  immersion  approach  to  the  development 


1  -200 

of  thinking.  Educational  Researcher,  20(2),  3-10. 

Resnick,  L.B.  (1989).  Treating  mathematics  as  an  ill-structured  discipline.  In  R. 
Charles  &  E.A.  Silver  (Eds.),  The  teaching  and  assessing  of  mathematical  problem 
solving  (Vol.3,  pp.32-60).  Ertbaum,  National  Council  of  Teachers  of  Mathematics. 

SchoenfeW,  AM.  (1985).  Metacognitive  and  epistemological  issues  in  mathematical 
understanding.  In  E.A.  Silver  (Ed.),  Teaching  and  learning  mathematical  problem 
solving:  Multiple  research  perspectives  (pp.361 -380).  Hillsdale,  NJ:  Eribaum. 

Silver,  E.A.  (1982).  Knowledge  organization  and  mathematical  problem  solving. 
In  F.K.  Lester  &  J.  Garofalo  (Eds.),  Mathematical  problem  solving:  Issues  in 
research  (pp.1 5-25).  Hillsdale,  NJ:  Lawrence  Eribaum. 

Steffe,  LP.  (1990).  Mathematics  curriculum  design:  Constructivist's  perspective.  In  LP. 
Steffe  (Ed.),  Transforming  children's  mathematics  education  (pp.389-398). 
Hillsdale,  NJ:  Eribaum. 

Sternberg,  R.J.  (1984).  What  should  intelligence  tests  test?  Educational  Researcher, 

Vosniadou,  S.  &  Ortony,  A.  (Eds.).  (1989).  Similarity  and  analogical  reasoning. 
Cambridge,  England:  Cambridge  University. 

Whimbey,  A.  &  Lochhead,  J.  (1982).  Problem  solving  and  comprehension.  Hillsdale, 
NJ:  Lawrence  Eribaum. 

Wittrock,  M.  (1977).  Learning  as  a  generative  process.  In  Wittrock,  M.  (Ed.), 
Learning  and  instruction.  Berkeley,  CA:  McCutchon. 


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ftyhan  I  Dougherty 

University  of  Hawaii  -  Manoa 

This  project  investigates  teacher  change  in  intermediate  and  secondary  classrooms.  Using  methodology 
consisting  of  interviews  and  observations,  movement  to  a  process  teaching  model  is  documented.  Data 
lutve  revealed  that  teachers  can  make  behavioral  changes  bat  the  richness  oftliose  changes  is  related  to  tlie 
match  between  teacher  philosophical  structures  and  the  teaclung  approach.  Additionally,  materials 
supporting  both  the  philosophy  and  specific  pedagogical  actions  is  an  important  contributing  factor  in  the 
change  process. 

Project  DELTA  (Determining  the  Evolution  of  the  Learning  and  the  Teaching  of  Algebra)  is  a  research 
program  investigating  teacher  change.  The  project  focuses  on  teachers  at  the  intermediate  and  high  school  level  as 
they  implement  curricular  materials  and  an  associated  teaching  style.  These  materials  were  developed  by  the  Hawai 
Algebra  Learning  Project  (HALP)  (NSF  grant  MDR-8470273)  and  incorporate  a  process  approach  to  teaching. 

Background  and  Premises 

With  recommendations  that  teaching  ■  ove  from  a  traditional  or  lecture  approach  to  one  with  more  student 
involvement,  many  descriptors  such  as  process  teaching,  inquiry-based  approach  or  problem-solving  instruction 
ha«e  been  tossed  about,  each  with  specific  characteristics.  Since  mathematics  educators,  practitioners  and 
researchers  alike,  do  not  agree  on  what  secondary  classrooms  would  specifically  look  like  using  nomraditional 
instructional  methods,  DELTA  first  sought  to  identify  characteristics  of  process  teaching  to  ease  communications 
and  to  establish  specific  areas  that  are  different  from  traditional  instruction.  These  areas  would  then  be  associated 
with  those  most  related  to  teacher  change.  This  preliminary  work  was  conducted  in  classrooms  using  the  HALP 
curriculum  {Algebra  I:  A  Process  Approach,  Rachlin,  Wada,  and  Matsumoto,  1992)  as  a  means  of  assuring 
consonance  between  the  process  teaching  method  and  materials  supporting  that  method. 

The  HALP  materials  are  intended  to  be  a  complete  Algebra  I  curriculum  for  intermediate  or  high  school 
grades.  They  were  developed  through  a  seven-year  intensive  classroom-based  research  program  conducted  with 
ninth-grade  students.  Curriculum  developers  served  as  classroom  teachers  and  piloted  draft  materials  in  their 
classes.  After  each  class,  individual  students  were  interviewed  on  a  regular  basis  to  "think-aloud"  as  they  solved 
problems  that  would  appear  in  the  next  lesson  "s  problem  set.  These  problem  sets  were  constructed  to  model 
Krutetskiian  problem-solving  processes  (Krutetskii.  1976).  Combining  these  problem  sets  with  a  Vygotskian 
perspective  on  learning  ( Vygotsky,  1 978),  the  HALP  created  an  algebra  curriculum  that  included  materials  and  an 
intertwined  leaching  approach  (process  teaching) 



1  -202 

In  put  curriculum  projects,  teachers  have  been  afforded  little  or  no  implementation  support.  Because  this 
curriculum  presents  algebra  content  in  a  different  way  and  is  based  on  a  non traditional  teaching  style,  a  45- hour 
workshop  was  designed  to  help  teachers  use  the  curriculum.  While  it  is  required  for  those  planning  to  use  the 
curriculum,  other  teachers  may  also  enroll. 

During  the  workshop,  participants  read  current  articles  about  algebra,  problem  solving,  and  teaching. 
Homework  assignments  from  the  text  are  given  to  involve  participants  in  thinking  about  algebra  through  a 
problem-solving  context  rather  than  an  algorithmic  one.  Participants  experience  the  problem-solving  processes  of 
generalization,  flexibility,  and  reversibility  by  solving  problems  that  exemplify  each  process.  Most  importantly, 
videos  of  individual  high  school  students  with  varying  abilities  and  of  secondary  classrooms  are  shown  to 
stimulate  participants  to  question  their  beliefs  about  algebra  and  its  instruction.  Even  though  instructors  model 
process  leaching  through  the  workshop  pedagogy,  no  explicit  teaching  methods  are  given  to  participants. 

The  methodology  was  designed  in  three  phases.  Phase  one  focussed  on  ascertaining  characteristics  of 
process  teaching  and  instrumentation.  Phase  two's  purpose  was  pilot  testing  and  phase  three  is  currently 
concerned  with  study  redesign. 

Phase  One:    Process  Teaching  Characterization  and  Instrumentation 

Methodology  of  phase  one.  A  member  of  the  HALP  team  was  chosen  for  pilot  classroom 
observations.  Her  ninth  grade,  heterogeneous  class  was  observed  bi-weekly  in  consecutive  three-day  periods  for 
two  months.  Scripted  field  notes  and  audiotapes  were  used  to  record  class  proceedings.  The  relatively  set  pattern 
of  instruction  and  lesson  format  in  traditional  classes  did  not  hold  for  process  classes;  they  were  much  more 
complex.  Even  with  audiotaping  it  was  difficult  to  script  everything  that  was  occurring.  An  observation  coding 
instrument  (OC1)  was  constructed  to  ease  data  collection. 

Its  construction  first  required  characterizing  process  teaching  based  on  the  pilot  observations.  The  feature 
of  process  leaching  could  be  divided  into  quantitative  and  qualitative  aspects.  The  quantitative  features  included 
time  and  frequency.  The  amount  of  time  spent  on  the  lesson  segments  of  content  development,  seatwork.  and 
management  was  particularly  relevant.  These  three  lesson  segments  appeared  to  be  dramatically  different  from 
traditional  lessons  in  that  a  much  larger  portion  of  the  class  period  (42  minutes  of  43  minute  periods)  is  spent  on 
content  development  and  negligible  time  on  seatwork. 

Methodology  and  Results 



Frequency  collection  documented  the  number  of  factual  (ie..  what  is3x  +2x  ?).  process  (ie..  is  the 
answer  unique  and  how  do  you  know  that?),  and  managerial  (did  everyone  mm  in  a  paper?)  questions  asked  and 
answered  by  teacher  and  students.  In  traditional  classes  questions  tend  to  be  factual  but  in  the  process  classes, 
process  questions  are  more  frequent.  The  person  responding  to  questions  was  also  different  In  process  classes 
students  tend  to  respond  more  often  than  the  teacher  due  to  the  active  student  participation. 

It  is  not,  however,  just  measurable  aspects:  the  quality  of  responses  and  of  content  discourse  is  even  more 
important.  The  OCI  was  constructed  to  allow  for  this  documentation.  Actual  dialogue  could  be  captured  during 
observation  periods  or  reconstructed  with  audiotapes  of  those  sessions.  Particular  attention  is  given  to  the 
dialogues  of  each  problem  discussed  during  the  lesson  because  the  mathematical  content  that  evolves  in  the 
dialogue  comes  from  students  and  is.  therefore,  reliant  upon  the  teaching  method  that  allows  for  and  encourages 
student  input.  This  mathematical  content  could  not  be  neglected  since  it  affects,  and  is  affected  by.  the  teaching 
approach.  The  descriptions  of  developing  algebraic  ideas  enhanced  snapshots  of  the  classrooms  in  the  way  in 
which  students  discussed  particular  ideas.  For  example,  in  one  lesson,  generalizations  now  carried  the  student's 
name  that  "discovered"  it.  These  development  ideas  suggested  richer  views  of  the  classroom  culture  and  of  the 
construction  of  mathematical  knowledge. 

While  teacher  and  classroom  behaviors  are  certainly  one  way  to  document  changes,  all  teachers 
confronting  change  may  not  demonstrate  it  through  their  teaching  behaviors.  Based  on  previous  work  (Grouws. 
Good,  &  Dougherty,  1990),  a  semi-structured  interview  was  considered  to  ascertain  attitudes  and  beliefs  about 
mathematics,  algebra,  their  instruction,  and  individual  and  school  demographics.  The  protocol  questions  clustered 
about  four  main  research  areas:  ( 1 )  teacher  views  of  algebra  and  mathematics.  (2)  teaching  strategies  and/or  style. 
(3)  student  aspects  including  teacher  expectations,  and  (4)  enhancements  to  the  change  process. 

Results  of  phase  one.  Preliminary  data  were  analyzed  with  particular  attention  to  characterizing 
process  teaching  and  validating  the  appropriateness  of  the  instruments.  Important  features  of  process  teaching 
included:  (I)  class  periods  devoted  to  active  discussion  of  mathematics.  (2)  "teacher  talk"  kept  to  a  minimum.  (J) 
questions  from  teacher  and  students  wore  more  related  to  "why"  than  to  "how",  (4)  students  assumed  leadership 
roles  in  the  learning  process,  and  (5)  mathematical  content  developed  from  a  concept  level  to  skill. 

The  instruments  were  modified  slightly  after  initial  data  analysis.  The  methodology  was  reviewed  by  an 
external  consultant  and  was  determined  to  be  appropriate  for  this  stage  of  the  study. 



Phase  Two:   Pilot  Tcsllnc 

Methodology  of  phase  two.  Phase  two  began  with  an  HALP  workshop  offered  in  Honolulu. 
Hawaii.  Teachers  from  across  the  United  States  were  enrolled  and  those  that  indicated  they  were  using  the  HALP 
materials  in  the  next  school  year  were  asked  to  participate  in  the  study.  Six  teachers  volunteered  and  represented 
intermediate  and  high  sc.'iool  grade  levels,  varying  class  types  (ie..  high  ability,  heterogeneous,  and  acceleratedl. 
and  a  range  of  teaching  experience  (3  to  18  years). 

At  the  beginning  of  the  school  year  (October)  each  teacher  was  interviewed  with  the  protocol.  Their 
responses  were  audiouped  and  transcribed  for  later  data  analyses.  At  the  same  time  their  Algebra  I  classes  were 
observed  using  the  OCT.  Classes  were  observed  again  in  December  and  March.  The  last  observation  period  also 
included  administering  the  interview  protocol  again. 

Results  of  phase  two.  Data  analyses  using  Hyperqual  showed  interesting  patterns  within  and  across 
teachers.  When  both  data  sets  were  analyzed,  teacher  beliefs  and  teaching  actions  did  not  necessarily  match.  Then 
were  three  cases:  (1)  beliefs  were  more  traditional  and  teaching  was  process  oriented.  (2)  beliefs  were  process 
oriented  and  teaching  was  more  traditional,  and  (3)  beliefs  and  teaching  were  process  oriented.  There  were  also 
noticeable  differences  in  the  ease  in  which  teachers  adapted  to  process  teaching  and  the  depth  to  which  they  were 
able  to  implement  it  in  the  classroom.  These  findings  motivated  a  look  beyond  teacher  beliefs  and  actions  after  the 

Phase  Three:    Study  Redesign 

Methodology  of  phase  three.  Data  collection  design  was  restructured  to  allow  for  capturing 
information  about  teachers  prior  to  any  workshop  intervention.  Two  sites  where  workshops  would  be  held  in  the 
summer  were  selected.  Teachers  pre-enrolled  in  the  workshop  in  those  locations  were  contacted.  Thirteen  teacher 
from  a  Midwest  city  and  five  from  an  Eastern  site  agreed  to  be  involved  in  the  study. 

A  graduate  student  from  the  University  of  Missouri  worked  with  this  phase  in  the  Midwest  as  part  of  her 
dissertation.  At  the  beginning  this  phase,  she  came  to  University  of  Hawaii  where  we  viewed  classrooms  and 
videotapes  of  classrooms  to  establish  observer  reliability  (.92).  In  May  of  the  spring  semester  prior  Vi  the 
workshop,  sample  teachers  were  visited.  Their  Algebra  classes  (or  another  class  if  they  were  not  teac'iing  Algebra 
were  observed  over  three  consecutive  days.  These  observations  were  anticipated  to  involve  classes  taught  in  a 
traditional  manner  and  required  the  OCI  be  altered  slightly  to  accommodate  both  traditional  and  process  instruction. 

1  -205 

Also,  an  interview  protocol  that  had  been  adapted  from  phase  two  was  administered  during  these  pre-workshop 
observations.  The  pre-workshop  interview  protocol  was  to  be  used  prior  to  any  workshop  intervention  so  that 
questions  related  to  how  teachers  were  implementing  workshop  ideas  were  not  asked. 

Phase  three  also  involved  the  workshop  itself.  Since  explicit  implementation  strategies  are  not  given  in  the 
workshop,  we  wanted  to  determine  if  workshop  instructors  were  teaching  in  a  way  similar  to  what  was  expected  o 
teachers  after  the  workshop.  The  graduate  student  observed  and  audiotaped  the  Midwest  workshop,  including 
interviewing  the  instructors  with  the  pre-workshop  interview  protocol. 

The  school  year  following  the  workshop,  project  teachers  were  observed  in  November  and  March  for 
consecutive  three-day  periods  in  Algebra  and  other  mathematics  classes  as  available.  The  postworkshop  interview 
protocol  was  also  administered  in  March.  This  protocol  was  identical  to  the  pre-workshop  form  with  the  inclusion 
of  questions  related  to  the  workshop  and  to  the  implementation  of  the  workshop  ideas  or  materials. 

Results  of  phase  three.  Analyses  of  data  sets  from  both  sites  are  currently  being  conducted  but 
preliminary  analysis  on  data  from  the  Eastern  site  is  available.  Pre-workshop  observations  documented  that  all  but 
one  of  the  sample  teachers  taught  in  a  traditional  manner.  At  the  Eastern  site  the  amount  of  time  spent  on  content 
development  averaged  3.5  minutes  in  a  45-minute  period  and  the  average  time  spent  on  seatwork  (starting  the  next 
day's  homework)  was  24  minutes.  Student  talk  time  to  the  whole  class  and  student-student  interaction  was 
negligible.  Questions  involved  students  telling  the  teacher  what  "to  do  next."  That  is,  students  would  give  a  step 
in  the  problem  or  answer  a  factual  question  such  as  "what  is  three  times  21?"  The  mathematical  content  was 
introduced  at  the  skill  level  and  received  developmental  attention  for  one  class  period  and  again  on  the  next  day 
when  homework  answers  were  given. 

Pre-workshop  interview  responses  also  indicated  traditional  views  about  mathematics,  algebra,  and  their 
instruction.  Teacher  responses  emphasized  skill  aspects  with  reference  to  rules,  procedures,  and  application  of 
those  in  appropriate  ways.  Expectations  for  the  students  except  for  one  teacher  focussed  on  the  retention  of  skills. 
Problem  solving  was  a  separate  topic  included  in  the  lesson  as  determined  by  the  textbook's  presentation  of  word 

One  teacher,  even  though  she  discussed  mathematics  and  algebra  in  a  procedural  manner,  taught  in  a 
slightly  different  way  than  the  others.  She  engaged  the  students  in  active  discussion  of  topics  and  made  an  effort  tc 
connect  the  new  material  with  something  that  students  had  previously  done  in  class.  Questions  emphasized  studeni 


reasoning  and  alternative  ways  of  approaching  a  problem.  The  classroom  atmosphere  was  student-centered  in  that 
student  ideas  were  regarded  with  respect  by  the  teacher  and  other  students. 

Pbstworkshop  observation  data  indicate  varying  levels  of  implementation  of  workshop  aspects.  Two  of 
the  project  teachers  are  using  the  HALP  materials.  In  one  Algebra  classroom,  there  was  use  of  certain  questions 
such  as  "did  anyone  do  it  a  different  way"  or  "is  the  answer  unique."  There  was.  however,  a  lack  of  depth  in 
natural  questions  that  arose  from  mathematical  discourse.  Students  and  teacher  were  discussing  content,  led 
predominantly  by  the  teacher.  While  these  instructional  strategies  do  not  precisely  fit  the  process  teaching  model, 
there  was  a  definite  movement  from  the  traditional  style  used  prior  to  the  workshop,  albeit  superficial.  Her 
responses  to  interview  questions  related  to  belief  structures  were  less  rigid  than  comparable  pre- workshop 

In  the  other  teacher's  classroom,  her  teaching  style  was  more  open  than  before  the  workshop.  Previously 
she  had  encouraged  student  interaction  under  her  direction  but  now  students  assumed  more  leadership  in  initiating 
questions  and  responses.  She  allowed  the  mathematical  content  to  develop  over  longer  time  through  the 
exploration  of  techniques  and  strategies  in  the  developmental  stage  of  new  concept  formation.  Using  prior 
knowledge  to  solve  problems  was  encouraged.  Her  interview  responses  indicated  movement  to  the  creative 
aspects  of  mathematics  and  algebra. 

Postworkshop  observations  of  teachers  in  classes  not  using  the  HALP  materials  such  as  general 
mathematics  or  geometry  are  not  consistent  with  the  HALP  classes.  There  is  a  tendency  for  all  teachers  to  teach 
more  traditionally,  especially  those  that  are  not  concurrently  teaching  any  HALP  classes.  Teacher  rationales  vary: 
for  some,  they  feel  their  students  in  those  classes  are  different  from  those  in  the  HALP  classes.  This  is  especially 
true  for  those  teaching  where  students  are  tracked.  Others  commented  that  it  is  too  time  consuming  and  too 
difficult  to  change  existing  materials  to  fit  a  problem-solving  approach  to  instruction.  Restructuring  homework 
assignments  so  that  development  over  time  can  occur  and  students  are  exposed  to  new  ideas  through  problem 
solving  requires  an  expertise  teachers  felt  they  did  not  have. 

Discussion  of  Results 

All  data  have  not  been  analyzed,  but  the  results  from  the  second  and  third  phases  of  the  project  suggest 
some  interesting  ideas  about  teacher  change.  The  superficiality  of  changes  in  teaching  strategies,  while  not  an  ideal 
application  of  the  process  teaching  model,  appears  to  be  an  important  link  in  substantial  changes.  Using  Hall. 


Loucks.  Rutherford,  and  Newlove's  levels  of  use  (1975).  one  cm  find  evidence  of  teachers  at level  HI  (uses  new 
strategies  while  struggling  with  problem  of  classroom  management  related  to  implementation).  However,  there  is 
more  to  change  than  the  physical  implementation  of  strategies.  Teachers  must  also  cope  with  the  philosophy  that 
underlies  the  specific  teaching  strategies.  The  crux  of  process  teaching  lies  with  students  as  they  construct 
meaningful  ideas  about  mathematics  and  teachers  incorporating  those  ideas  into  the  lesson.  The  use  of  student 
ideas  that  may  be  different  from  what  has  been  accepted  as  conventional  or  traditional  algebra  content  appears  to  be 
the  most  difficult  aspect  of  implementing  process  teaching.  The  unpredictability  for  teachers  not  knowing  what 
direction  the  lesson  is  heading  challenges  them  to  be  flexible  enough  to  recognize  mathematical  ideas  used  in 
creative  ways.  And,  the  worth  of  student  ideas  or  their  mathematical  validity  is  almost  inconceivable  for  those 
teachers  who  have  constructed  their  own  mathematical  knowledge  in  a  rigid  fashion,  especially  as  the  amount  of 
time  increases  since  their  workshop  exposure. 

Teachers  are  capable  of  using  questions  that  are  consistent  from  lesson  to  lesson  such  as  "did  anyone  do 
the  problem  a  different  way?"  but  struggle  with  creating  questions  when  mathematical  opportunities  present 
themselves  in  the  lesson.  HA  LP  teacher  materials  have  attempted  to  suggest  appropriate  questions  but  again, 
student  experiences  vary  and  often  novel  ideas  appear  in  discussions.  Some  teachers  have  commented  that  their 
inability  to  perceive  patterns  quickly  or  to  note  subtle  references  to  other  mathematical  ideas  may  account  for 
difficulties  in  asking  higher-level  questions  so  they  resort  to  factual  ones. 

The  comfortableness  teachers  feel  with  the  mathematics  they  are  teaching  is  also  another  consideration. 
For  example,  in  one  classroom,  students  suggested  that  it  may  be  possible  to  have  three  axes  when  graphing 
instead  of  two.  The  teacher  ignored  that  suggestion  because,  as  she  indicated  later,  she  was  unfamiliar  with  three- 
dimensional  graphing  and  could  not  think  of  how  to  pursue  their  ideas  since  she  could  not  discuss  it.  Additionally 
if  the  teacher's  mathematical  knowledge  is  limited,  it  is  difficult  to  assess  the  validity  of  student  arguments.  Rather 
than  cope  with  that,  some  teachers  opt  to  force  the  discussion  in  the  way  they  feel  most  comfortable  to  handle. 
This  may  create  an  ambivalent  classroom  setting;  at  one  time  it  is  appropriate  for  students  to  guide  the  discussion 
and  other  times  it  is  very  directed  by  the  teacher. 

This  ambivalence  also  occurs  when  teachers  only  implement  the  strategies  and  do  not  change  other 
classroom  aspects  that  support  those  strategies.  The  most  common  occurrence  is  to  have  student  evaluation  based 
wholly  on  tests  and  quizzes  while  ignoring  other  means.  On  the  one  hand,  student  discussion  is  encouraged,  but  i' 


1  -208 

ignored  in  ihe  evaluation  process.  This  slows  student  adaptation  to  a  different  classroom  environment  and 
frustrates  the  teacher  when  students  do  not  respond  as  they  had  expected. 

Our  study  supports  the  obvious:  if  teacher  beliefs  are  similar  to  the  philosophy  of  a  new  curriculum,  it  is 
easier  to  implement  change.  But  what  about  the  teachers  whose  philosophy  is  diametrically  opposed  with  that  of  a 
new  curriculum?  The  day-to-day  coping  with  the  classroom  forces  a  mechanistic  application  of  strategies  while 
teachers  begin  to  bridge  the  chasm  between  the  curriculum  and  their  own  beliefs.  To  a  casual  observer  it  would 
nppear  that  implementation  was  well  underway  but  closer  inspection  indicates  a  superficiality  that  may  precede  a 
return  to  previous  teaching  methods  or  movement  toward  a  closer  match  with  the  process  leaching  model. 

Three  factors  seem  to  influence  the  perseverance  to  move  to  a  richer  application  of  process  teaching.  Most 
important  is  the  use  of  materials  that  support  the  instructional  approach.  More  than  ever,  we  are  aware  that 
pedagogy  and  content  must  be  tightly  intertwined.  Secondly,  a  philosophical  shift  to  match  the  teaching  approach 
must  occur.  Finally,  an  integration  of  classroom  practices  into  a  global  entity  rather  than  isolated  segments  such  as 
instruction  and  evaluation  provides  a  cohesive  environment  that  allows  students  to  change  and  adapt  to  the 
classroom  environment  as  teachers  change. 

Workshop  data  have  not  been  fully  analyzed  at  this  point.  However,  aspects  of  the  workshop  will  be  tied 
to  teaching  actions  and  philosophical  issues  documented  in  postworkshop  data  collection.  It  is  hoped  that  through 
this  data  analyses,  workshop  features  can  be  modified  to  encourage  greater  success  in  the  implementation  stage. 


Grouws.  D.  A.,  Good.  T.  A.,  &  Dougherty,  B.  J.  (1990).  Teacher  conceptions  about  problem  solving  and 

problem-solving  instruction.  In  G.  Booker,  P.  Cobb.  &  T.  N.  de  Mendicuti  (Eds.),  Proceedings  of  the 
Fourteenth  PME  Conference  (pp.  135-142).  Mexico  City,  Mexico. 

Hall.  G.  E.,  Loucks.  S.  F..  Rutherford.  W.  L..  and  Newlove.  B.  W.  ( 197S  Levels  of  use  of  the  innovation:  A 
framework  for  analyzing  innovation  adoption.  The  Research  and  Development  Center  for  Teacher 

Krutctskii.  V.  A.  (1976).  The  psychology  of  mathematical  abilities  in  school  children  (J.  Kilpatrick  &  I. 
Wirszup.  Eds.).  Chicago:  University  of  Chicago. 

Vygotsky.  L.  S.  (1978).  Mind  in  society:  The  development  of  higher  psychological  processes.  Cambridge: 


1  -209 

Laurie  D.  Edwards 
University  of  California  at  Santa  Cruz 

Ten  first  year  high  school  students  were  asked  to  judge  simple  statements 
about  combining  odd  and  even  numbers  as  true  or  false.  They  were  also  asked  to 
give  justifications  or  explanations  for  their  decisions.  All  of  the  students  initially 
reasoned  purely  inductively,  appealing  to  specific  cases  and  justifying  their 
answers  with  additional  examples  when  presses.  However,  three  students  went 
beyond  this  empirical  reasoning  and  created  idiosyncratic,  personal  arguments  for 
their  decisions.  None  of  the  students  used  algebraic  notation  in  any  of  their 
reasoning.  Two  of  the  students  used  a  visual  representation  of  odds  and  evens  in 
making  their  arguments. 


Generalization,  and  testing  the  limits  of  generalization  through  proof,  may  be  said  to  be  at  the  heart 
of  mathematics.  An  acknowledgement  of  the  importance  of  this  kind  of  thinking  in  the  mathematics 
curriculum  can  be  found  in  the  Curriculum  j  :A  Evaluation  Standards  for  School  Mathematics,  published 
recently  in  the  United  States  by  the  National  Council  of  Teachers  of  Mathematics  ( 1989).  In  the 
Standards,  mathematical  reasoning  is  set  forth  as  a  goal  for  all  students  of  mathematics,  at  all  ages  and 
levels.  This  term,  "mathematical  reasoning"  is  defined  to  include  a  range  of  capabilities.  According  to  the 
Standards,  students  should  be  able  to: 

"  •  recognize  and  apply  deductive  and  inductive  reasoning; 

•  understand  and  apply  reasoning  processes, with  special  attention  to  spatial  reasoning... 

•  make  and  evaluate  mathematical  conjectures  and  arguments; 

•  formulate  counterexamples; 

•  formulate  logical  arguments; 

•  judge  the  validity  of  arguments ..." 

(NCTM.  1989.  p.  81  and  143) 

1  -210 

These  skills  have  often  been  addressed  only  in  geometry  classes,  in  the  context  of  carrying  out  formal, 
two-column  proof  on  triangles. 'circles  and  other  figures.  Yet  it  has  long  been  acknowledged  that  the 
teaching  of  proof  in  such  classes  is  often  unsuccessful,  and  may  lead  to  shallow,  syntactic  knowledge 
rather  than  deep  understandings  of  the  mathematics  involved  (Schoenfeld.  1988;  Hanna,  1983). 

The  study  described  here  was  concerned  with  mathematical  reasoning  and  explanation  outside  of, 
and  prior  to,  formal  instruction  in  a  geometry  class.  Instead,  the  focus  was  on  the  reasoning  skills  of  ten 
first-year  high  school  students,  who  were  volunteers  in  a  project  on  the  use  of  computer-based 
microworlds  for  mathemadcs.  In  order  to  understand  some  of  the  difficulties  involved  in  teaming  and 
teaching  proof,  it  may  be  useful  to  look  at  the  cognitive  precursors  to  formal  proof;  that  is,  the  kind  of 
informal  explanations  that  students  offer  when  confronted  with  mathematical  patterns  or  regularities.  Such 
an  approach,  which  takes  a  constructivist  or  genetic  stance  toward  the  development  of  students'  reasoning 
abilities,  may  clarify  difficulties  and  suggest  instructional  strategies  for  assisting  students  in  learning  this 
specialized  kind  of  thinking. 

Objectives  of  the  Research 

The  results  reported  here  were  gathered  as  part  of  a  study  of  high  school  students'  interactions  with 
a  computer  microworld  for  transformation  geometry  (Edwards.  1990;  1991).  The  objectives  of  the 
research  project  as  a  whole  were  to  investigate  the  land  of  reasoning  which  high  school  students  applied  to 
situations  involving  composition  of  reflections,  a  task  which  had  been  previously  investigated  with 
middle-school  students  (Edwards,  1988).  This  task  was  determined  to  be  useful  in  eliciting  students' 
strategies  for  discovering  and  testing  hypotheses,  using  the  computer  microworld.  and  for  engaging  in 
mathematical  generalization  (for  a  report  of  research  addressing  similar  questions,  using  a  different 
computer  environment,  the  Geometric  Supposcr.  see  Chazan,  1990). 

One  of  the  research  questions  was  whether  the  opportunity  to  use  a  computer  microworld  to 
generate  and  test  hypotheses  and  conjectures  would  improve  the  students'  abilities  to  reason 
mathematically.  In  order  to  test  this,  a  simple  task-based  interview  was  carried  out  before  and  after  the 
students'  experience  with  the  microworld.  The  objective  of  the  interview  was  to  discover  the  kind  of 


reasoning  the  students  already  employed,  in  a  domain  unrelated  to  transformation  geometry.  If  there  was  a 
change  in  this  reasoning  at  the  conclusion  of  the  study,  then  it  could  be  argued  that  the  microworid  was 
effective  in  helping  the  students  to  learn  how  to  reason  mathematically. 


The  students  who  participated  in  the  study  were  10  first  year  algebra  students, ages  14-15,. 
including  four  girls  and  six  boys.  The  students  worked  in  a  small  research  lab  at  the  university  for  a 
period  of  5  weeks.  During  the  first  and  last  session,  the  students  were  interviewed  individually  using  the 
task  described  below;  for  the  remaining  sessions,  they  worked  in  pairs  with  the  microworid  (written  in 

The  task  used  to  assess  the  students'  reasoning  consisted  of  a  set  of  statements  printed  on  cards  of 

the  form: 

"Odd  plus  odd  makes  even" 
The  students  were  asked  to  decide  whether  the  statement  was  true  or  false,  and  then  to  tell  the  investigator 
why  they  made  their  decision.  A  final  card  was  presented,  showing  the  following  pattern: 



For  this  card,  the  students  were  asked  to  add  two  more  lines  which  showed  the  same  pattern,  and  to 
explain  the  pattern. 

The  sessions  were  video-  and  audio-taped,  and  transcribed.  A  full  analysis  of  the  protocols  is  still 
underway,  but  the  initial  analysis,  which  showed  some  surprising  and  intriguing  results,  will  be  presented 


1  -212 


The  pretest  consisted  of  the  following  statements,  two  of  which  are  true  and  one  false: 
"Even  x  odd  makes  even" 
"Odd  +  odd  makes  odd" 
"Even  +  even  makes  even" 
The  post-test  consisted  of  the  following  statements: 
"Odd  +  even  makes  odd" 
"Even  x  odd  makes  odd" 
"Odd  x  odd  makes  odd" 

One  unexpected  outcome  was  that  a  few  students  (3  or  4)  had  some  difficulty  in  establishing  the  truth  or 
falsity  of  the  first  statement.  This  seemed  to  be  attributable  to  two  factors:  first,  many  students  answered 
very  quickly,  apparently  without  much  thought  When  they  were  asked,  "Are  you  sure?"  they  quickly 
self-corrected.  The  other  source  of  error  on  the  first  item,  "even  x  odd  makes  even"  was  to  interpret  'even 
times  odd  makes  odd"  as  a  misapplication  of  the  "rule"  for  positive  and  negative  numbers:  "positive  times 
negative  makes  negative."  For  example,  one  student,  when  asked  for  a  justification,  stated: 

NR:  A  positive  and  a  positive  makes  a  positive  and  a  negative  and  a  negative  makes 
a  positive,  uh,  something  like  that,  I  don't  know... 

It  turned  out  that  the  students  had  recently  been  studying  positives  and  negatives  in  class,  and  that 
this  "rule"  was  salient  in  their  memories.  This  evidently  interfered  with  thrir  interpretation  of  the  "odd  and 
even"  questions. 

This  result  in  itself  was  interesting,  in  that  it  indicated  the  syntactic  nature  of  these  students' 
learning  in  mathematics  -  while  they  might  have  remembered  the  form  of  a  rule,  they  did  not  pay  attention 
to  its  meaning.  Nor  did  'hey  attend  to  the  meaning  of  the  items  presented  in  the  pretest.  Instead,  they 
seemed  to  make  a  cognitive  mapping,  associating  "even"  with  "positive"  and  "odd"  with  "negative,"  and 
then  applying  a  rule  they  had  recently  memorized. 


The  experimenter  modified  the  introduction  to  the  pretest  after  this  error  appeared  in  the  first  two 
subjects.  The  interview  was  started  with  the  statement,  "These  questions  are  about  odd  and  even 
numbers.  What  are  some  odd  numbers?  What  are  some  even  numbers?"  This  prompt  was  effective  in 
orienting  the  students  to  the  question  at  hand,  and  the  "positive/negative"  error  was  thereafter  not  repeated. 

A  more  significant  pattern  of  responses  was  found  in  the  students'  explanations,  provided  after 
they  had  correctly  decided  whether  a  particular  statement  was  true  or  was  false.  It  was  expected  that  at 
least  some  of  the  students,  after  a  year  of  algebra,  would  use  their  algebraic  knowledge  in  simple  proofs 
for  the  statements  which  they  stated  were  true.  For  example,  when  asked  why  "Odd  plus  odd  makes 
even,"  it  was  anticipated  that  some  students  would  present  a  proof  such  as  the  following: 

"Odd  numbers  can  be  written  as  2n+ 1 

(2n+l)  +  (2n+l)  =  4n+2  =  2(n+l) 

2(n+ 1 )  is  divisible  by  2  and  therefore  evea" 

Noneofthe  10  students  offered  an  algebraic  proof  of  this  kind.  In  fact,  all  of  the  students  initially 
offered  a  purely  inductive  or  empirical  rationale  for  their  decisions.  When  asked  why  a  statement  was 
false,  they  would  offer  a  counterexample.  When  asked  why  a  statement  was  true,  they  would  reply  with 
statements  to  the  effect  of.  "I  tried  it,  and  it  works." 

When  pressed  to  justify  their  answers,  most  of  the  students  simply  tried  more  cases.  For  example, 
the  following  dialogue  took  place  after  the  first  item  had  been  answered  correctly: 

JG:,  so  even  times  odd  makes  even. 

LE:  Is  there  anything  else  you  want  to  say  or  add  about  that, 

or  any  way  you  could  explain  or  prove  to  somebody  that  it  was  true? 

JG:  The  only  thing  that  I  could  do  is  just  try  a  few... 

In  total.  7  out  of  the  10  students  reasoned  in  a  way  which  could  be  described  as  purely  inductive  or 




Beyond  empiricism,  before  formal  proof 

The  three  students  who  offered  explanations  which  went  beyond  simple  induction  did  not  use 
algebraic  notation  or  appear  to  be  using  specific  knowledge  gained  in  their  algebra  class.  Instead,  each 
offered  an  idiosyncratic  argument,  which  in  two  cases  was  based  on  a  change  in  representation  of  the 

In  one  case,  the  student,  CM,  answered  all  of  the  questions  quickly  and  accurately,  working  out 
examples  mentally  and  only  writing  down  the  specific  numbers  he  tried  when  asked  to  by  the  investigator. 
When  pressed  to  give  a  reason  or  explanation  for  the  fact  that  "Odd  ♦  odd  makes  odd"  is  false,  CM  offered 
an  explanation  based  on  sketches  of  tick  marks  corresponding  to  odd  and  even  numbers,  as  indicated  in 
Figure  1. 

Figure  1:   CM's  sketch/explanation 

CM  explained  that  odd  numbers  always  had  one  "left  over",  and  showed  with  his  sketch  that  when 
two  odd  numbers  were  combined,  the  "left  overs"  made  up  pairs,  so  that  the  sum  would  be  even  (a  set  of 

This  visual  and  verbal  explanation  indicated  that  CM  was  willing  to  go  beyond  empirical 
justification,  and  actually  look  at  the  structure  of  even  and  odd  numbers  in  order  to  generate  a  valid 
argument  for  his  decision.  He  used  a  similar  argument  for  a  number  of  the  other  items  in  the  test. 



1  -215 

In  the  second  case,  the  student  also  used  a  visual  representation  to  support  her  reasoning.  She 
created  a  number  line,  and  used  a  similar  argument  as  that  made  by  CM.  involving  "jumps"  with  gaps  of 

two,  or  gaps  of  two  and  one  more. 

The  final  case  involved  a  somewhat  more  complicated  verbal  argument,  presented  to  justify  the 
sutement  "Even  ♦  even  makes  even."  In  this  argument,  the  student  noted  that  all  two^igit  even  numbers 
end  in  0. 2. 4. 6.  or  8.  and  since  the  sum  of  any  pair  of  these  single  digit  numbers  is  even,  then  the  sum  of 
any  pair  of  even  numbers  must  be  even.  This  student  did  not  present  anything  like  this  argument  on  any 
of  the  other  items,  instead  appealing  only  to  examples. 


Hanna  has  pointed  out  the  importance  of  differentiating  between  "proofs  that  prove"  and  "proofs 
that  explain"  (Hanna.  1989).  Before  students  are  taught  to  prove,  they  can  be  provided  with  the 
opportunity  to  engage  in  less  formal  mathematical  reasoning,  by  being  asked  toexplain  Simple 
mathematical  regularities.  A  well-noted  difficulty  encountered  in  this  area  is  for  students  to  see  the  need  to 
go  beyond  empirical  or  inductive  reasoning  at  all  (Chaan.  1990). 

The  study  described  above  suggests  that  some  students  at  the  beginning  of  high  school,  even 
without  instruction  in  formal  proof,  will  go  beyond  empirical  reasoning  and  offer  informal  proofs  or 
explanations  of  their  findings.  The  results  reported  here  are  extremely  limited  in  scope,  and  in  fact,  plans 
for  the  next  phase  of  research  are  to  extend  the  study  both  in  duration  (a  school  year)  and  population  (two 
first  year  high  school  classes)  in  order  to  more  fully  investigate  reasoning  among  students  of  this  age. 
However,  me  results  are  consistent  with  previous  findings  for  British  students,  working  witha  written  test 
(Bell.  1976).  Itisintemtingthatfortwoofmesmdenumthisstudy.wmmgupwift 
involved  a  change  in  representation  of  the  problem.  Each  student  "translated"  the  problem  into  a  visual 
form  in  order  to  build  his  or  her  argument  This  may  have  helped  them  to  see  better  the  structure  of  the 
.  mathematics  underlying  the  simple  regularities  involved  in  combining  odd  and  even  numbers.  In  this 
sense,  these  were  "proofs  that  explain."  or  at  least,  held  explanatory  power  for  the  students  concerned.  In 
future  research,  the  cognitive  territory  which  comes  before  formal  proof  will  continue  to  be  explored,  in 




order  to  provide  a  better  understanding  of  how  more  sophisticated  and  powerful  kinds  of  mathematical 
reasoning  might  be  learned  by  students  in  secondary  school. 


bfuriSSffi  Ma  dy  °f  PUpilS'  pr0Of-exPbnationJ  ta  mathematical  situations.  Educational  Stmtb. 
Oaam.D.  (1990).  Quasi-empirical  views  of  mathematics  and  mathematics  teaching.  Interchange  ?' 

Edwards,  L  (1988).  Children's  learning  in  a  transformation  geometry  microwortd.  Proceedings  of  the. 
n^Wm^S^SSj  „fl^tg"atk>nal  rtmm  fnr     Psvcholngv  "f  ^th^TMS^Pr^ 

p£022ir  L  ( r9??  c  The  TOle  of  comPuter  microworlds  in  the  construction  of  conceptual  entities. 

^^SZ^^^rA^^1"  ^ transformation  to* 

Hanna.  G.  (  1983).  Rigorous  pmnf  in  mathjnurir»  atogfioj  t~~,.».  OISE  Press 

r^.n" ( T^r0OfS.that,PI?Ve an.d Prooft "P1™-  Proceeding nf  the Thirt^nth  Ann..a| 

Conference  "f  the  International  Oroup  for  the  Psvcholngv  atSSSSg;  VouffrMS  it) 

iSStesSTA^CTM ^*"natiCS  (1989)'  Q"iM'im  «d  conation  standard,  yhml 

Schoenfeld.  A.  ( 1988).  When  good  teaching  leads  to  bad  results:  The  disasters  of  well  taueht 
mathematics  courses.  Educational  Psychologist.  23  Ml  US-lftt  6 



1  -217 


Pier  Luigi  Ferrari 
Dipartimento  di  Matematica  -  Universita  di  Genova 


The  analysis  of  the  role  of  figure  may  explain  some  differences  in  problem-solving  between  the 
arithmetical  and  the  geometrical  setting.  The  aim  of  the  study  I  am  reporting  is  to  begin  an  analysis 
of  the  interactions  between  figure  and  strategy  in  the  resolution  of  problems  in  geometrical  setting, 
with  particular  regard  to  problems  related  to  the  notions  of  area  and  perimeter  of  plane  surfaces. 
The  analysis  of  the  protocols  suggeststhat  the  perception  of  the  figure  as  an  object,  autonomous 
from  the  graphic  constructions  performed,  is  achieved  after  a  difficult  and  contradictory  process.  It 
suggests  also  that  the  ability  at  mentally  trasforming  figures  may  help  pupils  in  planning  and 
describing  complex  strategies  in  geometrical  setting,  since  a  figure  may  embody  part  of  a  complex 
procedure  and  thus  contract  its  temporal  dimension.  It  is  also  pointed  out  that  a  procedure  may  be 
grounded  in  a  particular  time  without  necessarily  losing  its  generality. 


1.1.  Object  of  the  research 

In  Bondesan  and  Ferrari  (1991)  some  data  axe  given  that  seem  to  stress  the  role  of  the  figure  in  the 
resolution  of  problems  in  geometrical  setting.  In  fact,  it  is  reported  that  in  geometrical  problems  children 
are  more  willing  to  search  for  alternative  strategies  and  a  larger  amount  of  pupils  who  do  not  master 
verbal  language  in  order  to  organize  their  reasoning  can  build  effective  strategies;  moreover,  the 
comparison  of  strategies,  carried  out  in  the  classroom,  gives  rise  to  the  diffusion  of  the  ability  at  planning 
(or,  at  least,  performing)  complex  strategies  and  the  increase  of  the  number  of  strategies  produced  for 
each  problem.  It  is  argued  that  the  figure  is  crucial  on  account  of  its  heuristic  role  in  the  search  for  a 
strategy,  as  pupils  may  'manipulate'  it  (cutting,  superposing,  measuring,  ...)  by  means  of  suitable 
representations.  Moreover,  it  allows  pupils  to  effectively  represent  the  problem-situation  (as  far  as  it 
allows  them  to  simultaneously  perceive  multiple  relationships)  as  well  as  the  resolution  procedures  (as  far 
as  it  may  embody  the  sketch  or  the  record  of  a  procedure).  This  seem  to  fit  very  well  with  learning 
processes  based  on  verbal  interactions  among  pupils. 

The  goal  of  the  study  I  am  reporting  is  to  proceed  deeper  in  the  explanation  of  these  phenomena,  with 
particular  regard  to  the  interactions  between  figure  and  the  construction  of  a  strategy.  In  particular  I  was 
interested  at  testing  the  likelihood  of  my  hypotheses,  stating  more  precise  ones  and  focusing  some  aspects 
of  the  subject.  More  systematic  research  is  needed  to  validate  the  results  presented  here.  The  whole 
subject  obviously  concerns  problem-solving  in  geometrical  setting,  but  it  may  have  implications  for 
problem -solving  in  other  settings,  such  as  arithmetic,  where  representations  seem  to  strongly  affect  the 
performances  of  pupils  (in  particular,  low-level  pupils). 


1  -218 

1.2.  Theoretical  frame 

In  the  last  years  the  role  of  visualization  in  mathematics  and  mathematics  learning  has  been  widely 
analyzed  (see  for  example  Dreyfus  (1991)  for  a  review).  The  status  of  visual  reasoning  is  not  yet  clearly 
explained,  but  a  lot  of  studies  has  stressed  the  c;  icial  role  of  figures  in  geometry.  Figures  are  regarded  as 

thus  distinct  from  both  pure  concepts  and  drawings.  Recently,  research  has  pointed  out  the  complexity  of 
the  interactions  between  different  symbolic  systems  (such  as  verbal  language  and  spatial  representations) 
which  have  been  regarded  as  a  characteristic  feature  of  teaming  processes  in  geometrical  setting  (see  for 
example  Arsac  (1989),  Caron-Pargue  (1981),  Laborde  (1988),  Parzysz  (1988)).  Computer  models  have 
been  regarded  as  intermediate  objects,  different  from  both  figures  and  drawings  (see  for  example. 
StrSsser  and  Capponi  (1991)). 

Related  to  the  study  I  am  reporting  the  results  of  Mesquita  (1989,  1990, 1991)  are  quite  interesting,  in 
particular  as  far  as  they  concern: 

=  the  analysis  of  status  of  a  figure  (figures  that  are  'objects'-or  models-  in  the  sense  that  the  geometrical 
properties  used  in  their  construction  may  be  evinced,  and  figures  that  are  only  'illustrations'  if  it  is  not 
the  case) 

» the  analysis  of  the  role  of  a  figure  (figures  may  only  describe  a  problem-situation,  as  far  as  they  supply 
a  simultaneous  insight  of  the  properties  involved,  or  may  also  promote  the  construction  of  a  resolution 

=  the  stress  on  pupils'  representations  of  algorithms  in  geometry;  three  fundamental  kinds  of 
representation  (flgural,  functional  and  structural)  are  recognized  that  do  not  depend  upon  age. 

1.3.  The  role  of  figure:  some  hypotheses 

Related  to  the  issues  mentioned  in  1.1. 1  have  stated  the  following  hypotheses  about  the  aspects  of  the 
status  of  figures  that  may  affect  performances  in  geometrical  setting: 

=  a  figure  is  an  autonomous  object  on  which  pupils  can  operate  and  reflect;  it  can  simultaneously 
represent  complex  systems  of  spatial  relationships; 

=  a  figure  can  represent  complex  resolution  procedures;  the  temporal  dimension  of  the  procedures 
represented  is  contracted;  this  means  that  pupils  who  master  mental  manipulation  of  figures  are 
expected  to  manage  complex  procedures  better  and  more  generally; 

=  pupils  may  perceive  a  figure  and  operate  on  it  at  different  levels  (material  manipulation,  measurement, 
symbolic  manipulation,  'game  of  hypotheses',...;  see  also  Mesquita  (1991));  these  levels  may  be 
simultaneously  present  (a  pupil  may  use,  at  the  same  time,  measurement  arguments  or  more  abstract 
relationships  in  order  to  discover  or  verify  a  property). 


The  research  that  is  reported  is  not  a  large-scale  systematic  one;  I  have  gathered  a  large  amount  of 
protocols  from  2  classes  of  grade  5  (about  40  pupils).  These  classes  have  experienced  the  Genova 

complex  units,  with  both  conceptual  and  spatial  properties  (such  as  Fischbein's  'flgural  concepts')  and 

1  -219 

Group's  Project  since  first  grade.  The  materials  I  have  analyzed  are  normal  working  materials  (pupils' 
copybooks,  papers  and  so  on)  or  assessment  tests  usually  administered  during  the  school  year  and 
concern  the  following  tasks: 

-  find  the  area  and  the  perimeter  of  a  polygon  (not  necessarily  regular  nor  convex)  drawn  on  the  paper, 
=  find  the  area  of  a  region  on  a  scale  map  (the  pupils  were  given  the  map  on  a  blank  sheet); 
=  explain  to  some  friend  of  yours  how  to  fulfil  the  previous  task. 

For  a  general  information  on  the  Genova  Group's  Project  see  for  example  Boero  (1989),  Boero  ( 1991)  or 
Ferrari  (1991).  The  concepts  of  area  and  perimeter  have  been  introduced  during  grade  5  according  to  the 
following  steps: 

=  discussion  in  the  classroom  of  the  meaning  of  words  such  as  area,  surface,  extension  in  everyday-life; 
=  cutting  (with  scissors)  or  drawing  on  a  squared  sheet  different  shapes  with  the  same  extension, 

comparison  of  extensions  by  superposition  and  so  on; 
=  doubling  or  halving  the  extensions  of  triangles  and  rectangles; 

=  measurement  of  the  extension  of  rectangles  by  counting  of  the  squares  and  using  different  units; 
=  construction  of  an  area  unit  of  one  square  meter, 
=  formula  for  the  area  of  a  rectangle; 

=  different  ways  to  compare  the  extensions  of  plane  surfaces:  counting  of  squares,  superposition, 

transformation,  formulas; 
*  boundary  of  a  plane  surface;  perimeter  as  the  measure  of  the  boundary  of  a  plane  surface; 
=  comparison  of  the  boundaries  of  a  plane  surface; 
=  change  of  units  of  area  and  length; 
=  comparison  of  strategies  in  problems  of  area  and  perimeter, 

=  formula  for  the  area  of  a  triangle  (by  means  of  material  and  graphic  transformations);  heights  of  a 

»  measures  with  decimal  fractions;  change  of  decimal  units; 

=  area  of  polygons  (not  necessarily  regular  nor  convex)  by  (exact)  covering  with  triangles; 

=  approximate  area  of  geographic  regions  by  approximate  covering  of  a  scale-map  and  balancing; 

=  formulas  for  the  area  of  regular  polygons  (by  means  of  graphic  transformations)  and  of  the  circle. 

The  problem  of  the  reliability  of  written  reports  related  to  the  Genova  Group's  Project  has  been  discussed 

in  Ferrari  (1991).  For  a  general  discussion  of  this  issue  see  Ericsson  and  Simon  (1980). 

Throughout  the  paper  by  'good  problem-solvers'  I  mean  pupils  who  are  able  to  give  acceptable  solutions 

to  most  of  the  problems  (either  contextualized  or  not)  they  are  administered  during  the  year,  not  regarding 

too  much  the  quality  of  the  resolution  processes  or  the  reports.  By  'poor  problem  solvers'  I  mean  pupils 


1  -220 

Nevertheless,  these  limitations  do  not  seem  to  damage  the  skills  at  transformating  the  figure  even 
mentally  and  planning  complex  strategies.  There  is  also  a  number  of  pupils  (about  20%)  who  manage  to 
transform  the  figure  (for  example  by  decomposing  it,  or  including  it  in  other  figures),  but  cannot  use  their 
constructions  in  order  to  solve  a  problem. 

3.2.  Figure  and  strategy 

From  a  general  analysis  of  the  protocols  concerning  the  approximate  covering  of  a  scale-map  of  a  region 
with  triangles  or  rectangles  in  order  to  estimate  the  area  of  the  region,  we  have  noticed  three  different 
kinds  of  constructions: 

S 1.  the  strategy  is  built  according  to  some  previous 
mental  schema,  without  taking  into  account  the 
specificity  of  the  figure,  in  spite  of  contrary 
statements  (for  example,  pupils  who  use  only 
rectangles  to  cover  a  scale-map  of  Great  Britain, 
or  only  triangles  in  order  to  cover  a  scale-map  of 
Portugal  or  Sardegna); 

S2.  the  strategy  is  built  according  to  some  previous 
mental  schema  which  can  be  adapted  to  the 
specific  needs  (for  example,  pupils  who  change 
their  strategy  according  to  the  map  they  want  to 
cover,  or  who  use  both  rectangles  and  triangles 
with  the  explicit  purpose  of  reducing  the 
calculations  or  the  errors); 

S3,  the  resolution  is  built  by  means  of  graphic 
operations  without  any  strategy  or  schema 
previously  thought  (for  example,  pupils  who 
cover  the  map  with  a  large  number  of  small 
triangles  drawn  at  random,  or  who  do  not  take 
into  account  the  need  for  reducing  the  errors). 




who  almost  never  «re  able  to  design  some  strategy  to  solve  complex  problems  and  often  meet  with 
difficulties  even  when  solving  simple  problems. 

3.1  Figure  as  an  object 

Pupils  succeed  in  perceiving  a  figure  as  an  autonomous  object  only  after  a  difficult  process.  At  first  they 
perceive  the  figure  as  the  record  of  sequence  of  the  graphic  operations  they  have  performed  te  I  jild  it. 
They  write,  for  example:  "/  change  this  triangle  by  putting  another  one  at  the  side;  so  it  is  now  a  rectangle 
. . ."  Only  few  pupils  (less  than  20%),  in  the  first  problems  on  triangles,  seem  to  identify  the  figure  as  a 
product  of  their  constructions,  equipped  with  relations,  which  does  not  entirely  depend  upon  the  graphic 
operations  performed.  The  elements  of  the  drawing  preserve  the  functions  they  have  had  in  the  graphic 
construction  or  in  the  manipulation,  and  are  not  included  in  a  system  of  relationships.  The  height  of  a 
triangle  is  perceived  (by  about  90%)  as  "the  thing  that  allows  me  to  divide  the  drawing..."  and  the 
operation  of  drawing  it  is  regarded  as  a  transformation  of  the  figure  (as  it  is  a  transformation  of  the 
drawing).  To  the  question  "why  the  area  of  a  triangle  is  b  x  h/2  and  not  b  x  I?"  they  (about  80%)  give 
answers  based  on  calculations  or  counting  of  the  square  ( "because  b  x I  gives  a  wrong  number. ..").  In 
the  successive  problems,  even  if  more  complex,  the  figure  as  an  object,  with  some  relationships  among  it 
elements  seems  to  appear. 

When  searching  for  the  area  of  a  trapezium  most  pupils  work 
without  any  difficulty  on  the  figure  transformed  by  adding  a 
small  triangle  on  the  left;  this  triangle  (which  allows  pupils  to 
regard  the  trapezium  as  a  part  of  a  rectangle)  loses  its  procedural 
function  of  graphic  construction  and  becomes  a  stable  element 
of  the  new  figure. 

Some  pupils  (about  50%)  begin  to  recognize  some  relationship  among  the  components,  such  as  the 
congruence  of  the  small  triangles,  even  if  only  few  (less  than  25%)  explicitly  recognize  that  the  sides  are 
pairwise  equal. 

Analogously,  in  the  figure  on  the  left,  the  operation  of  adding 
the  triangle  on  the  left-upper  part  is  regarded  by  more  than  60% 
of  the  sample  as  an  operation  on  the  trapezium,  not  on  the  whole 

Moreover,  many  pupils  state  (about  70%)  that  the  quadrilateral 
they  have  built  at  ihe  bottom  is  a  rectangle,  but  very  few  use  the 
fact  that  the  short  sides  are  equal  when  calculating  the  area  of  the 
external  small  triangles.  Many  of  them  (more  than  50%) 
measure  either  side  and  someone  even  finds  different  values. 




Pupils  belonging  to  groups  S I  or  S2  seem  to  perform  the  graphic  operations  on  the  drawing  according  to 
some  figural  schema  previously  thought;  in  the  whole  sequence  they  have  given  significantly  better  results 
in  tasks  requiring  mental  transformations  of  figures. 

Asked  to  deal  with  the  error  in  their  approximation,  the  following  behaviours  have  been  noticed  in  the 
first  problems: 

El .  pupils  who  do  not  realize  the  need  for  estimating  the  error, 

E2.  pupils  who  deal  with  the  problem  from  a  geometrical  point  of  view,  and  try  to  improve  their 

approximation  by  means  of  progressive  refinements  of  the  covering; 
E3.  pupils  who  deal  with  the  problem  from  an  arithmetical  point  of  view  and  modify  their  covering 

according  to  calculations  previously  made  (explicitly  or  not). 
In  the  following  problems  of  the  sequence  almost  all  pupils  adopt  this  last  procedure:  they  mainly  use 
rectangles  to  approximate  the  area,  and  only  few  keep  on  us;"g  triangles;  in  the  last  problems  no  pupil 
use  more  than  three  polygons  to  cover  the  region. 

The  task  "write  down  to  a  friend  of  yours  how  to  find  out  the  area  of  a  geographic  region  by  the 
approximate  covering  of  a  scale  map,  and  to  estimate  the  error"  has  provided  some  interesting  data.  All 
pupils  have  obviously  given  descriptions  steadily  grounded  on  time  and  organized  as  sequences  of 
suggestions  (or  prescriptions)  temporally  structured  by  connectives  such  as  'before',  'afterwards'  'next' 
and  so  on;  beyond  this  common  feature,  three  different  kinds  of  text  may  be  recognized: 
Tl .  pupils  who  reconstruct  the  procedure  in  a  particular  situation  and  time  (about  20%);  they  use  verbs 
and  connectives  that  put  the  stress  on  the  reconstruction  of  their  own  experience  ("suppose  we  must 
find  the  area  of  Argentina;  now  I  draw  two  triangles  here.  I  call  this  point  A  and  this  B;  now  I  draw  a 
line  here...");  all  these  pupils  belong  to  the  group  S3  described  above  and  their  descriptions  are 
always  incomplete;  no  pupil  in  this  group  is  a  good  problem  -solver, 
T2.  pupils  who  reconstruct  the  procedure  in  a  particular  time  ("  we  draw, ...and  now  we  have 
almost  get  it...)  but  do  not  r  fer  to  any  particular  situation  (about  15%);  they  use  verbs  and 
connectives  like  Tl  but  the  procedure  is  described  in  general;  in  this  group  have  been  found  both 
good  and  poor  problem-solvers; 
T3.  pupils  who  reconstruct  the  procedure  as  a  process  with  the  temporal  dimension  but  place  it  in  an 
abstract  time  (about  65%);  they  use  connectives  such  as  'before',  'after'  but  never  'now'  or  'at 
present';  in  this  group  there  are  also  the  pupils  who  in  the  description  of  the  procedure  put  some 
stress  on  aspects  different  from  the  temporal  structure  of  the  steps,  taking  into  account  the  'logical' 
organization  or  some  constraints  involved  in  the  problem-situation;  also  in  this  group  there  are  both 
good  or  poor  problem-solvers. 

1  -223 


=»  Pupils'  re  presentations  of  a  figure  are  undoubtedly  relevant  related  to  the  planning  of  a  procedure;  the 
one  described  in  Tl  (which  seems  roughly  correspond  to  the  attitude  called  'figural'  by  Mesquita 
(1991))  seems  correlated  to  a  low  ability  at  mentally  transformating  the  figure.  Nevertheless,  the 
transition  from  a  kind  of  representation  to  another  is  not  clear-cut;  it  seems  likely  that  a  lot  of  pupils 
remain  quite  long  in  a  level  similar  to  the  one  called  'functional'  by  Mesquita  and  connected  with  the 
notion  of  'schema';  the  emergence  of  the  'structural*  (or  'algorithmic')  perception  of  the  figure  is 
anyway  difficult  and  contradictory; 
» the  ability  at  mentally  transforming  figures  and  regarding  them  as  objects  seem  to  affect  the  ability  at 
representing  and  managing  complex  procedures,  which  become  much  simpler  as  far  as  figures  may 
embody  sequences  of  operations  and  substantially  reduce  their  temporal  complexity;  good  problem- 
solvers  sometimes  manage  to  go  on  even  without  it,  but  it  seem  crucial  for  weaker  pupils,  related  to 
their  problem-solving  performances  in  geometrical  as  well  as  in  arithmetic  settings; 
=  pupils  with  a  rigid  ('geographic')  perception  of  the  figure  meet  with  difficulties  when  asked  to 
reconstruct  some  procedure  in  a  general  position;  nevertheless  the  opposition  between  the  figure  (with 
its  supposed  specificity)  and  the  algorithm  (with  its  supposed  generality,  see  Mesquita  (1991))  is  not 
entirely  satisfactory;  there  are  pupils  who  perceive  algorithms  in  a  close  connection  with  time,  and 
when  describing  them  they  seem  to  run  over  the  steps  again  in  a  sort  of  identification;  among  these 
subjects  there  are  also  some  very  good  problem-solvers;  this  way  of  perceiving  algorithms  does  not 
seem  an  intermediate  level  between  the  understanding  'by  examples'  and  'structural'  understanding, 
but  a  characteristic  feature  of  a  particular  learning  (and  thinking)  style,  which  does  not  seem  to  prevent 
the  achievement  of  high  level  of  abstraction. 
=  the  trend  of  almost  all  pupils  is  to  give  simpler  and  simpler  answers  from  the  computational  point  of 
view  (using  mainly  rectangles,  and  often  only  one)  is  most  likely  a  consequence  of  the  too  rapid 
transition  from  area  as  a  magnitude  to  area  as  a  real  number  (see  Douady  and  Perrin-Glorian  (1989)); 
pupils'  behaviour  becomes  more  and  more  similar  to  their  behaviour  in  arithmetical  problems  (very  few 
alternative  strategies,  one  strategy  which  spreads  over  the  classroom,  ...see  Bondesan  and  Ferrari 
(1991));  even  the  lack  of  distinction  between  area  and  perimeter  (which  is  more  frequent  among  pupils 
who  cannot  transform  figures  mentally)  may  be  explained  by  similar  arguments. 

Arsac,  G.:  1989,  'La  construction  du  concept  de  figure  chez  les  eleves  de  12  ans'.  Proceedings  PME  13, 
vol.1,  85-92. 

Boero,  P.:  (1989),  'Mathematical  literacy  for  all:  experiences  and  problems'.  Proceedings  PME  13, 
vol.1,  62-76. 

Boero,  P.:  (1991),  'The  crucial  role  of  semantic  fields  in  the  development  of  problem-solving  skills'. 
Proceedings  of  the  NATO  Seminar  on  Problem-Solving  and  Information  Technology,  Springer- 
Verlag  (in  press). 


1  -224 

Bondesan,  M.G.  and  Ferrari,  P.L.:  1991,  'The  active  comparison  of  strategies  in  problem-solving:  an 

exploratory  study'.  Proceedings  PME  15,  vol.1, 168-175. 
Caron-Pargue,  J.:  198 1 ,  'Quelques  aspects  de  la  manipulation  -  manipulation  materielle  et  manipulation 

symbolique',  Recherches  en  Didactique  des  Mathimatiques,  2/3.  S-3S. 
Dorfler,  W:  1991,  'Meaning:  image  schemata  and  protocols'.  Proceedings  PME  15,  vol.1, 17-32. 
Douady,  R.:  1986,  'Jeux  de  cadres  et  dialectique  outil-objet',  Recherches  en  Didactique  des 

Mathimatiques,  7/2, 5-31. 
Douady,  R.  and  Perrin-Glorian,  M.J.:  1989,  'Un  processus  d'apprcntissage  du  concept  d'aire  de  surface 

plane',  Educational  Studies  in  Mathematics,  20, 387-424. 
Dreyfus,  T.:  1991,  'On  the  status  of  visual  reasoning  in  mathematics  and  mathematics  education'. 

Proceedings  PME  15,  vol.1,  33-48. 
Ericsson.  K.A.  and  Simon,  H.A.:  (1980),  'Verbal  reports  as  data'.  Psychological  Review,  vol.87, 215- 


Ferrari,  P.L.:1991,  'Aspects  of  hypothetical  reasoning  in  problem-solving".  Proceedings  of  the  NATO 

Seminar  on  Problem-Solving  and  Information  Technology,  Springer- Verlag  (in  press). 
Fischbein,  E.:  1987,  Intuition  in  Science  and  Mathematics,  Dordrecht,  Reidel. 

Johnson-Laird,  P.N.:  (1975),  'Models  of  reasoning",  in  Reasoning:  representation  and  process  in 

children  and  adults,  R.J.Falmagne  and  N.J.Hillsdale  eds.,  Lawrence  Erlbaum  ass.. 
Johsua,  M.A.  and  Johsua,  S.:  1987,  'Les  fonctions  didactiques  de  l'expenmental  dans  l'einsegnement 

scientifique',  Recherches  en  Didactique  des  Mathimatiques,  8/3.,  231-266 
Laborde,  C,  1988,  'L'einsegnement  de  la  geome"trie  en  tant  que  terrain  d"exploration  de  phe"nomenes 

didactiques',  Recherches  en  Didactique  des  Mathimatiques,  9.3,  337-364. 
Mariotti,  M.A.:1991,  'Age  variant  and  invariant  elements  in  the  solution  of  unfolding  problems', 

Proceedings  PME  15,  vol.2.  389-396. 
Matos,  J.M.:1991,  'Cognitive  Models  in  Geometry  Learning',  Proceedings  of  the  NATO  Seminar  on 

Problem-Solving  and  Information  Technology,  Springer- Verlag  (in  press). 
Mesquita,  A. L:  1989,  'Sur  une  situation  d'eveil  a  la  deduction  en  geometric'.  Educational  Studies  in 

Mathematics,  20,  55-77. 

Mesquita,  A.L.:1990,  'L'influence  des  aspects  figuratifs  dans  le  raisonnement  des  eleves  en  geomitrie'. 

Proceedings  PME  14,  vol.2, 291-296. 
Mesquita,  A.L:  1991,  'La  construction  algorithmique:  niveaux  ou  stades?'.  Proceedings  PME  15,  vol.3, 


Pareysz,  B.:  1988,  '"Knowing"  vs  "seeing".  Problems  of  the  plane  representation  of  space  geometry 

figures'.  Educational  Studies  in  Mathematics,  19, 79-92. 
Presmeg,  N.C.:1986.  *  Visualization  in  High  School  mathematics'.  For  the  Learning  of  Mathematics,  6.3. 


Rogalski,  J.:  (1982),  'Acquisition  de  notions  relatives  a  la  dimensionality  des  mesures  spatiales 
(longueur,  surface)',  Recherches  en  Didactique  des  Mathimatiques,  3/3, 343-396. 

Strisser,  R.  and  Capponi,  B:  1991.  'Drawing  -  Computer  Model  -  Figure.  Case  studies  in  student's  use 
of  geometry-software'.  Proceedings  of  PME  XV,  vol.3, 302-309. 


R.  Gamti.  I  M  A  -C.N.R.  Genova:  P.  Boero,  Dipartimento  Malemalica  Universita,  Genova 

The  report  concerns  an  exploratory  study  performed  about  a  sequence  of6  proportionality problems 
proposed  in  two  classes  by  the  same  teacher  over  a  period  of about  ten  months.  The  problems 
concern  different  settings  (geometrical  setting  and,  aAer,  arithmetical  setting)  and  different  contexts . 
The  purpose  of  the  study  was  to  explore  the  transition  to  a  multiplicative  model ,  the  conditions 
which  may  enhance  it  and  the  difficulties  connected  with  the  transfer  of  a  model  costructed  in  the 
geometrical  setting  loan  arithmetical  one. 

I  .Introduction 

The  studies  and  surveys  of  the  past  decade  concerning  problem-solving  have  posed  the  question  of  the  relationship 
between  "laboratory"  research  on  problem-solving  and  the  study  of  the  possible  implications  for  teaching  (in 
general,  see  Lester  &  Charles,  l99l:asregarus,inr^icular,prcix)rtic^ityprc*lems,seeKarplus  &C..I983; 
Toumiaire  and  Pulos,  1985;  Grugnetti,  199 1 ).  We  believe  that  this  is  a  relevant  question,  as  the  research  findings 
on  proportionality  problem  solving  do  not  seem  in  the  least  to  have  affected  the  most  widespread  teaching 
methods  (consisting,  in  Italy  and  other  countries,  of  training  students  to  mechanically  apply  the  A:B-C:X  scheme). 
This  is  an  exploratory  study  of  7  teaching  situations  presented  in  two  classes  by  the  same  teacher  over  a  period  of 
approximately  10  months.  These  teaching  situations  concern  "paper  and  pencil  explanation  missing  values 
proportion  problems"  (see  Toumiaire  and  Pulos,  1985).  The  study  involves  the  complete  knowledge  by  the 
teacher-researcher-observer  of  the  teaching  activities  carried  out  during  the  whole  period  considered.  For  this 
reason,  we  believe  that  it  may  provide  reliable  elements  on  which  to  base  further  studies  concerning  the 
"engineering  of  teaching"  relevant  to  proportional  reasoning  and  on  the  learning  processes  involved,  even 
considering  the  limits  ensuing  from  the  small  number  of  students  and  from  the  singularity  of  the  experience. 
This  study  is  characterized  by  the  following  aspects: 

The  first  five  situations  concern  geometrical  proportionality  problems  referring  to  physical  situations 
(sunshadbws)  evoked  or  directly  experienced  in  real  life  (at  first  through  problems  without  explicit  numerical 
data).  These  problems.requiring  a  physical  knowledge  in  addition  to  their  proportional  content,  permit  us  to  view 
separately,  to  a  certain  extent,  students'  difficulties  and  behaviours  due  to  numerical  values  from  their  master)'  of 
the  relationships  between  the  physical  variables  (see  Harel  &  C ,  1 99 1 ).  This  choice  appears  to  be  significant  in 
relation  to  the  hypothesis  that  working  with  numerical  values  and  the  meanings  of  division  may  constitute  in  itself 
an  element  of  difficulty.  The  problems  posed  permit,  in  particular,  an  exploration  of  the  transition  from  the 
qualitative  concept  of  dependence  between  proportional  quantities  ("if  one  grows  then  the  other  grows,  too")  to  the 
quantitative  concept  ("if  one  goes  into  the  other  a  certain  number  of  times,  then  the  other,  too...") 
The  other  two  situations  involve  a  change  of  context,  the  first  one  (body  proportions:  see  Hoyles&C.I989,199l  ) 
still  in  the  physical-geometrical  setting  (Douady,  1985);  the  second  in  the  arithmetical  setting.  These  problems  were 
proposed  to  explore  the  difficulties  encountered  by  students  in  transferring,  to  more  or  less  similar  contexts,  the 
models  established  in  the  first  context.  It  should  be  noted  that,  in  this  sequence,  the  work-in  the  geometrical  setting 
precedes  that  in  the  arithmetical  setting,  and  that  numerical  data  never  suggest  easy,  exact  proportionality 
relationships  (from  this  point  of  view,  these  problems  may  be  classified  as  "difficult"  .according  to  Hart,  1981) 

In  this  study  our  focus  ha  been,  above  all,  to  the  short-  and  long-term  effect  of  particular  teaching  choices 
on  the  emergence  and  evolution  of  problem-solving  strategies,  and  to  the  nature  of  such  evolution.  In  particular,  w 
have  studied:  I )  the  effects  of  the  presence  of  a  real  physical-geometrical  situation,  initially,  and  over  a  long  period. 


experienced  directly,  referred  to  as  an  "experience  fiel(T(  Boero,  1989)  that  provides  meaning  and  consistency  to 
the  problem  posed  (determination  of  a  height  that  cannot  be  directly  measured);  2)  effects  of  initially  proposing 
problem  situations  without  explicit  numerical  data:  3)  the  role  of  classroom  discussion  and  of  the  active  comparison 
of  strategies  (Ferrari.  1 99 1 )  in  overcoming  the  additive  model  and  realising  a  conscious  transition  to  the 
multiplicative  model  in  the  geometrical  setting:4)  the  steps  involved  in  this  transition:  and  5)  the  problems  inherent 
in  the  subsequent  transfer  of  the  multiplicative  model  to  other  contexts  and  settings  (especially  to  the  arithmetical 

In  our  findings  we  have  observed  (see  par.5)  that  real  physical-geometrical  situations  directly  experienced  are  not. 
in  themselves,  able  to  lead  students  (at  the  age  of  1 1 )  to  constructing  proportionality  relationships  between  the 
geometrical-physical  variables  involved,  but  that  (if  appropriately  handled  by  the  teacher)  they  may  have  an 
important  role  for  many  students  in  constructing  such  relationships(cfr.  Karplus&  C,  1983).  However,  a 
complete  mastery  of  the  multiplicative  model  -  transferable  to  other  geometrical  contexts  and  well  established  over 
time  -  seems  to  require  also  the  mastery  of  the  link  between  geometrical  proportionality  relationships  and 
arithmetical  operations  on  the  numerical  values  that  represent  the  measurements. 

An  issue  that  we  deem  important  and  that  remains  an  open  question  is  the  role  of  additive-type  reasoning  in  the 
transition  to  multiplicative  strategies.  This  problem  appears  to  be  more  complex  and.  in  part,  different  from  what 
has  been  highlighted  so  far  by  the  research  on  proportional  reasoning.  Another  important  question  concerns  the 
interpretation  of  the  difficulties  that  students  have  in  transferring  strategies  outside  the  geometrical  setting. 

The  study  examined  37  students.  ofGrades  6and  7,  most  of  whom  (30  out  of  37)  were  between  1 1  and  12ycars 
old  at  the  beginning  of  the  study.  They  were  enrolled  in  two  classes,  of  average  level,  of  a  school  in  Carpi  (North 
Italy).  The  study  was  conducted  from  March.  1991  (Grade  6)  to  January,  1992  (Grade  7).  It  has  also  been  possible 
to  compare  some  of  the  data  resulting  from  the  observation  of  these  two  classes  with  data  obtained  from  other 
classes  of  the  same  grade.  All  me  classes  we  are  considering  are  involved  in  the  project  of  the  Genoa  Group  for 
an  integrated  teaching  of  mathematics  with  the  experimental  sciences  in  the  comprehensive  school  .The  following 
characteristics  of  the  project  are  relevant  to  this  study :  systematic  work  in  "experience  fields"  (Bocro,  1 989)  in  the 
construction  of  mathematical  concepts  and  skills  as  "knowledge  tools":  systematic  recouisc  to  verbalization  in 
problem-solving,  and  in  comparing  problem-solving  strategies :  extended  work  on  the  (open)  applied  mathematical 
problems,  including  some  problems  in  which  numerical  values  are  not  made  expliciCaltemalion  between  periods  of 
individual  work  (e.g.  during  the  resolution  of  mathematical  problems)  and  of  class  discussions  (e.g.  during  the 
comparison  and  evaluation  of  problem-solving  strategies  proposed  in  the  classroom);  systematic  exclusion  of  the 
"automation"  of  the  solution  to  proportionality  problems  through  the  adoption  of  such  models  as  A:B"C:X. 
The  observations  concern:  individual  solution  of  open  questions,  some  asked  as  "story  problems"  (as  in  Situations 
5  ,6.7):  recorded  discussions  (in  particular  in  Situations  1  and  2):  reports  by  individual  students  (sec  Sit.2  and  4). 

3.  The  sequence  of  teaching  situations 

These  were  the  sole  situations  in  which  the  two  classes  tackled  the  problem  of  the  height  of  an  object  that  cannot  be 
measured  directly,  and  of  an  additional  situation  of  arithmetical  type.  During  the  period  of  the  study  (from  March 
1991  to  January  1992)  no  other  proportionality  problems  were  posed. 



Situation  I:  the  problem  of  height  of  the  street-lamp  ( during  an  outing:  Grade  VI,  March) 
The  students  go  on  a  one-hour  outing  to  observe  sun  shadows.  During  the  outing  the  teacher  poses  the  problem  of 
determining  the  height  of  a  street-lamp  (almost  4  meters),  whose  shadow  is  seen  on  the  ground.  Near  the  street- 
lamp  the  students  observe  various  shadows  cast  by  objects  of  accessible  height,  in  particular  by  fence-posts,  just 
over  one  metre  high;  the  teacher  brings  these  shadows  to  the  students'  attention. This  is  a  verbal  arithmetical 
problem,  without  explicit  numerical  data  and  with  the  presence  of  a  physical  •  geometrical  reference  that  permits 
the  students  to  tackle  the  problem  without  worrying  about  the  actual  calculation  of  the  numerical  result. 

This  problem  situation  falls  within  the  teaching  unit  devoted  to  the  phenomenon  of  sun  shadows,  which  constitutes 
one  of  the  most  important  pans,  both  in  terms  of  content  and  of  the  time  invested,  of  the  Mathematics  and  Science 
activities  of  the  project  for  Grade  6.  In  particular,  the  problem  is  posed  after  some  observation  and  discussion  of 
the  "fan"  of  shadows  during  the  day.  During  these  activities,  the  students  realize,  among  other  things,  that  "when 
the  sun  is  high  in  the  sky.  shadows  are  short:  when  it  is  low.  shadows  are  long",  and  that  "longer  objects  cast,  at 
the  same  point  in  time,  longer  shadows". 

In  the  process  of  "rationalization"  of  the  shadows  phenomenon,  this  problem  situation  represents  the  introduction 
to  its  quantitative  analysis.  If.  with  qualitative  observation,  a  crisis  was  triggered  with  the  model  "strong  sun  -  long 
shadow"  that  most  of  the  students  hold,  with  this  situation  we  move  to  the  quantitative  aspect  of  the  relationship 
"longer  object  -  longer  shadow". 

Situation  2:  the  problem  of  the  height  of  the  street-lamp,  in  the  classroom  (the  day  after  the  outing): 
"On  4  March,  between  I 1  AM  and  I PM  we  went  out  to  determine  the  height  of  a  street-lamp.  Recount  what 
happened  and  find  a  way  to  determine  the  height  of the  street-lamp,  "(individual  work) . 

Later,  the  teacher  moves  to  the  analysis  and  "active  comparison"  (  Ferrari.  1991)  of  the  solutions  produccd:shc 

selects  two  of  the  solutions  produced  by  the  students,  one  of  multiplicative  type  (correct)  and  the  other  of  additive 

.  type  (incorrect),  and  asks  the  students  first  to  determine  which  of  these  solutions  their  own  strategy  followed,  then 

to  follow  the  other  strategy,  and  finally  to  evaluate  both  of  them.  Only  after  these  activities  are  completed,  are  (he 

measurements  of  the  shadows  and  of  the  fence-post  used  to  verify  the  different  results  produced  by  the  two 

strategies  and  discuss  in  depth  their  correctness.  Situation  2  required  over  three  hours  of  work. 

The  work  on  shadows  continues,  with  activities  concerning  parallelism  and  the  movement  of  shadows  on  the 

ground  (angles,  and  so  on). 

Situation  3:  the  problem  of  the  two  nails  (as  an  evaluation  test .  a  few  days  later) 

"The  drawing  represents,  fromabove,  the  shadows  cast  at  IIAMandat  I '2  noon  in  Genoa  by an  8-cm  nail placed 
at  position  A.  At  position  B  there  is  another  nail.  6  cm  in  length.  Do  you  think  you  can  draw  precisely  the  shadows 
cast  by  the  nail  at  position  B.  determining  their  lengths  and  positions?  Explain  your  reasoning. " 

The  problem  was  posed  to  explore  the  difficulties  the  students  experienced  because  of  the  presence  of  numerical 
data,  and  the  text  evoked  the  situation  which  they  had  previously  experienced.  The  "a  priori"  analysis  of  the 
problem  identified  as  additional  difficulties  those  ensuing  from  the  presence  of  a  decimal  ratio  and.  above  all.  from 
the  fact  that  the  unknown  length  was  less  than  (he  known  length. 

FIG.  I  (here  reduced  in  scale) 

1  -228 

The  work  on  shadows  continues,  with  activities  concerning  the  (angular)  height  of  the  sun  in  the  sky.  the 

movement  of  the  sun  in  the  sky,  and  so  on  . 

Situation  4:  individual  report  about  the  work  performed  during  the  year 

During  the  summer  holidays  the  students  were  asked  to  find  and  to  reconstruct  the  main  stages  of  the  teaching  unit 
devoted  to  the  phenomenon  of  sun  shadows  (February  -  June),  making  explicit  the  knowledge  gained  and  the 
difficulties  encountered,  so  as  to  evaluate  -in  particular-  whether  the  students  are  able  to  correctly  "reconstruct"  the 
experience  of  the  street-lamp  and  the  strategies  that  emerged  from  the  discussion. 

Situation  5:  the  problem  of  the  height  of  the  clock  tower  of  the  Pio  castle  (October  1 99 1 .  grade  VII) 
"Yesterday  I  was  in  the  square  at  Carpi  and  met  one  of  the  masons  that  are  restoring  Pio  Castle,  While  we  were 
talking  he  told  me  that  the  documents  concerning  the  building  of  the  castle  tower  ( the  clock  tower)  had  been  lost. 
Then,  a  little  worried,  he  told  me:  "I  have  to  call  a  crane,  but  it  would  be  better  to  know  to  what  height  it  must 
reach,  to  enable  us  to  work  on  the  clock  tower. "  "If  you  want  to  know  the  approximate  height  of  the  tower, 
without  measuring  it  directly,  you  can  measure  its  shadow:  that  is  much  easier  to  measure!  But  you  must  also  know 
the  length  of something  else  and  of the  shadow  it  casts,  at  the  same  moment"  I  told  him.  "Really?,  "he  asked  me. 
astonished,  "let's  try  it,  then!"  We  started  to  take  measurements  and  chose,  for  comparison,  my  height.These  are 
the  measurements  found:lcacher's  height:  1.60  m;  length  of  the  teacher's  shadow:  2.08  m:  length  of the  sliadow 
cast  by  the  tower:  32. 5  m. 

Can  you  determine  the  height  of the  clock  towefi  Explain  and  give  the  reasons  for  the  method  used. " 
The  problem  was  proposed  to  verify  the  medium-term  persistence  of  the  mastery  of  the  multiplicative  model  in  a 
problem  situation  given  in  the  text,  very  similar  to  the  "street-lamp  problem",  but  with  numerical  data  made  explicit, 
this  time. 

Situation  6:  the  problem  of  the  height  of  the  statue  (December  1991 .  grade  VII) 

"A  recent  archaeological  excavation  in  Calabria  found  the  remains  of  a  Greek  statue,  probably  of  a  warrior,  that  had 
stood  in  front  of  a  temple.  The  only  intact  part  of  the  statue  is  3  foot,  approximately  76  cm  in  length.  We  would  like 
to  know  approximately  how  tall  the  statue  was.  We  know  the  dimensions  ofMichelangelo's  David  which  are:  loot 
length  54  cm.hcighl  of  the  statue 432 cm. 

Try  to  find  how  tall  the  statue  was.  Explain  your  hypotheses  and  your  method". 

The  problem  was  proposed  in  order  to  verify  the  possibility  of  transferring  strategies  of  the  multiplicative  type  to 
geometrical  situations  that  arc  partially  different  from  previous  ones,  due  to  the  different  context.  The  elements  of 
diversity  essentially  consist  in  the  fact  that  they  are  proportional  parts  of  the  same  "object",  and  not  length 
relationships  between  different  "objects"  (object  that  casts  a  shadow,  and  its  shadow),  as  in  the  previous  cases. 
Moreover,  the  context  of  the  problem  may  bring  to  mind  a  "natural"  idea  of  proportionality  to  which  the  students 
may  refer  (see  also  Hoyles.  Noss.  Sutherland.  1989  and  1991). 

Situation  7:  the  problem  of  theJaroJ  January  1992.  during  the  four-monthly  evaluation*  ■  Grade  VI 1 ) 
"Last  year  Mrs.  Pina  made  plum  jam.  She  had  13  kg  of  plums,  from  which  she  obtained  5.5  kg  of  jam. 
This  year  she  wants  to  obtain  8  kg  of  jam.  What  quantity  of  plums  does  she  need? 
Explain  and  give  the  reasons  for  your  procedure". 

The  problem  was  proposed  to  verify  the  tra  lsferability  of  multiplicative  strategics  to  arithmetical  problems  without 
any  immediate  physical-geometric  reference.  Both  the  context  of  reference  of  the  problem  and  the  nature  of  the 
variables  concerned  are  thus  modified  with  respect  to  the  previous  problems. 

Situations  3. 4.  5. 6  were  proposed  without  any  subsequent  comment  and  explicit  evaluation  of  the  work  done  by 
the  students.  After  Situation  7.  approximately  6  hours  of  work  were  carried  out  (alternating  between  individual 
work  situations  and  discussions),  that  lead  the  students  to  think  (under  their  teacher's  guidance )  about  the  nature  of 
the  problems  proposed,  so  as  to  rccogni/c  common  aspects  and  possible  common  problem-solving  strategics. 

Er|c  253 

1  -229 

4.  Analysis  of  the  students'  behaviour  aai  evolution  of  their  strategies 
This  table  summarizes  the  results  of  the  analysis  performed  on  students' strategies: 

Sit.  1 

Sit.  2 

"Sit.  3 

Sit.  4 

Sit.  5 

Sit.  6 

Sit.  7 





■  Ma 




















A  D 


































































Mb  " 

R  V 



















































































































A  " 

























R  V 









■  A 















A  — 


A  D 





A  -  additive  ■  M"  multiplicative  complete:  Ma=  multiplicative  interwoven  with  additive  considerations:  Mb- 
building  up  (Hart.  1981):  Mu  -  reduction  to  unity;  11=  blockage:  M-B»  begias  with  multiplicative  considerations, 
then  blockage:  ;  R  V=  remembers  and  verbalizes  cxaustivcly:  R=  only  remembers:  D-  clearly  distinguishes  the 
two  strategics  :  *  -abscnl 




1  -230 

4a.  Further  information  about  itudenu'  stratef  ie» 

(i)  In  Sit.2  ,  only  6  students  proposed  a  correct  strategy:  purely  multiplicative  (M).or  partially  additive  (Mat- 
probably  influenced  by  the  strategies  proposed  verbally  by  their  classmate  in  Sit.  I  .In  Sit.  3 .notwithstanding  a  more 
difficult  problem  than  the  previous  one.  1 9  students  seemed  to  have  a  clear  idea  of  the  proportionality  relationship 
between  the  quantities.In  other  classes,  in  which  no  active  comparison  and  evaluation  of  the  strategies  for*  the 
"problem  of  the  street-lamp''  took  place,  less  than  20%  of  students  produced  proportional  reasoning  in  the 
"problem  of  the  two  nails". 

(ii)  In  Sit. 3.  the  analysis  of  the  multiplicative  strategies,  complete  or  not.  clarifies  the  nature  of  the  difficulties 
foreseen  in  the  "a  priori"  analysts:  7  students.  (M)or(Ma),  solved  the  problem  correctly  and  completely;  7  students 
(Mb)  clashed  with  the  decimal  value  of  the  shadow/nail  ratio  (19:8*2.3).  They  calculated  it,  made  explicit  that  it 
was  "the  times  that  the  nail  goes  into  its  shadow  at  1 1  AM",  but  did  not  identify  the  arithmetical  procedure  to  be 
used.  To  solve  the  problem,  they  give  "a  bit  more  than  twice"  the  length  of  nail  B(for  instance  6+6+1.5.  or 
6+6+2).  This  type  of  strategy  is  similar  to  that  described  by  Hart  ( 198 1)  and  Lin  (1989)  called  the  "building  up" 
method.Five  students  (M-B)  followed  again  the  strategy  of  the  "street-lamp"  problem  and  calculated  the  ratio 
between  the  two  nails  (8:6- 1 ,3).  made  it  explicit  that  this  was  "how  many  times  nail  Bgoes  into  nail  A",  after 
which  they  did  not  manage  to  correctly  use  this  ratio.  They  would  need  to  use  the  inverse  scalar  operator 
(Vergnaud.  1 98 1  ).but  the  students  did  not  succeed  to  give  a  meaning  to  "divide  for  a  certain  number  of  times". 

(iii)  In  Situation  7  only  10  students  solved  the  problem  correctly:5  by  using  a  strategy  of  "reduction  to  the  unit" 
(Mu).  calculating  the  weight  of  the  plums  needed  to  make  I  kg  of  jam  ( no  strategy  of  this  kind  was  performed 
before) ;  and  5  by  "building  up"  strategies  (Mb) 

(iv )  An  analysis  following  the  evolution  in  time  of  the  students'  strategies  in  the  geometrical  setting  is  particularly 
interesting:for  six  students  the  multiplicative  model  is  present  from  the  start  (in  Situation  2)  and  remains  well 
established  over  time;  for  8  students  there  is  a  progress,  without  lapses,  from  the  additive  model  used  at  first  to  the 
multiplicative  model.  All  these  students  (6+8)  were  able  to  recognize  the  model  adopted  for  the  solution  of  the  first 
problem,  and  made  explicit  in  this  occasion,  or  later,  the  reasons  why  the  other  model  was  not  valid.For  ten 
students  the  progress  from  the  additive  model  to  the  multiplicative  model  does  not  appear  to  be  steady .  Among 
these  students.  5  had  not  been  able  to  recognize  with  clarity,  during  the  comparison  of  the  strategics.the  one  they 
had  used,  neither  had  they  been  able  to  explain  why  the  additive  modefdoes  not  work".  For  seven  students  that 
had  initially  adopted  an  additive  progress  is  foundahey  were  not  able  to  recognize  their  strategy  as 
analogous  to  the  strategy  selected  by  the  teachenand  so  much  the  less*  to  acknowledge  that  it  was  not  correct. 

(v)  It  may  also  be  noticed  that  all  the  problems  posed  would  permit  to  proceed  both  with  strategies  of  the  "Between 
or  scalar  ratio"  type  and  with  strategies  of  the  "Within  or  function  ratio"  type  (according  to  Vergnaud.  1 98 1  -  sec 
also  Karplus  &  C. .  1 983).  The  choice  between  one  and  the  other  seems  to  depend  on  the  context  and.  on  the  relative 
size  of  the  objects  to  be  considered  :  in  Situation  5  all  the  students  except  for  2  calculated  the  ratio  between  two 
shadows  .  while  in  Situation  6  the  problem-solving  strategies  denote  a  different  perception  of  the  problem 
situation:  1 2  students  out  of  1 4  that  had  correctly  solved  also  the  previous  problem  pass  from  the  evaluation  of  how 
many  times  a  shadows  goes  into  another  to  the  evaluation  of  the  ratio  between  the  statue's  foot  and  its  height,  while 
only  one  pupil  operates  the  opposite  change,  and  a  second  one  applies  the  same  type  of  strategy  to  both  problems . 

(vi)  An  interesting  fact  emerged  in  relation  to  the  evolution  of  strategics:  several  students  combined  (Ma)  additive 
and  multiplicative  (of  "scalar"  and  "function"  type  konsiderations.  in  a  step  that  may  be  considered  as  a  transition 



toa  coherent  and  completely  multiplicative  strategy.  In  the  situation  of  the  observation  of  shadows,  the  two 
students  that  identified  a  correct  strategy,  explained  it  to  their  classmates  as  follows: 

That  is  a  difference  between  the  fence-post  and  its  shadow  as  between  the  street  lamp  and  its  shadow,  but  in  the 
street-lamp  case  the  'difference' must  be  longer,  since  the  street  lamp  is  longer.  To  make  the  street  lamp  equal  to  the 
shadow  that  it  casts,  I  most  take  away  a  greater  difference  than  in  the  posts  case.  I  see  approximately  how  many 
times  the  post  goes' into  the  street  lamp.  I  take  away  from  the  shadow  of the  street  lamp  the  difference  between  the 
post  and its  shadow as  many  times  as  the post  goes  into  the street  lamp,  so  the  slmhw  and  the  street  lamp  are  eqtial 
and  I  cm  measure  the  shadow." 

This  process  is  probably  forced  by  the  actual  experience  in  which  the  students  observe  the  "more"  that  makes  the 
shadow  different  from  the  object  that  casts  it  In  Situation  2  this  strategy  is  changed  by  two  students,  who  replace  it 
with  t  fully  multiplicative  one.  In  Situation  3,  on  the  other  hand,  where  the  conflict  with  the  arithmetical  aspects  of 
the  problem  is  strong  (the  length  to  be  determined  is  less  than  the  given  one),  the  correct  strategies  were  of  the 
type  described  above  (except  for  two  other  students).  In  Situations  5  and  6  the  correct  strategies  are  all  of  the 
completely  multiplicative  type,  save  for  one  (in  Sit.  5). 

5. Discussion 

The  analysis  that  we  have  carried  out  shows  how  problem  situations  considered  as  such  by  the  students,  and 
relevant  to  a  context  in  which  the  geometrical-physical  aspect  experienced  directly  is  paramount,  may  be  used  by 
the  teacher  to  motivate  students'  transition  from  an  additive  model  to  a  multiplicative  one.In  this  respect  the 
"experience  field"  of  the  sun  shadows,  according  to  the  analysis  of  the  protocols  produced,  seems  to  have  certain 
intrinsic  characteristics  appropriate  to  "force"  the  construction  and  the  development  of  the  students'  strategies:  the 
sun  is  the  "cause"  of  the  shadows:  the  relationship  between  the  object  that  casts  the  shadow  and  the  shadow  itself 
cannot  be  modified  by  the  observer.  The  system  comprising  the  sun,  the  objects  that  casts  the  shadows  and  the 
shadows  themselves  is  very  "rigid":  it  evokes  "contemporaneous"  and  "same-type"  relationships  between  the 
heights  of  the  objects  and  the  lengths  of  the  shadows  cast  by  them,  and  therefore  suggests  the  existence  of  the  ratio 
as  "invariant"  (compare  also  Karplus&C,  1983).  This  aspect  is  particularly  clear  in  several  texts  produced  by  the 
students  in  Situation  2: 

The  street  lamp  is  much  bigger  than  the  feme-post,  and  its  shadow  must  be  much  longer  than  the  fence-post's. 
The  difference  between  shadow  and  fence-post  cannot  be  the  same  because  the  sun  does  not  play  favourites!  Then  I 
should  know  how  many  times  the  street  lamp  is  higher  than  the  fence-post:  I  have  the  shadows  and  I  can  know  it 
dividing  the  shadow  of the  street  lamp  by  the  shadow  of the  fence-post ". 

"Everything  has  its  own  shadow  and  the  difference  between  object  and  shadow  changes  from  object  to  object.  I 
cannot  leave  from  the  street-lamp's  shadow  the  difference  fence-post  shadow  because  if  I  change  the  fence-post, 
for  example,  if  I  take  a  much  smaller  fence-post,  this  difference  changes,  and  then  also  the  height  of the  street-lamp 
changes,  and  this  is  impossible". 

The  lack  of  explicit  numerical  data  in  the  first  two  situations  seems  to  have  important  effects,  particularly  as  regards 
the  necessity  to  graphically  represent  the  problem-solving  strategies,  with  positive  consequences  on  the 
development  of  the  reasoning  based  on  "how  many  times...'  s" 
goes  into  ..."In  this  regard  very  effective  representations  ■> 
were  those  given  on  the  right,  that  were  later  taken  up  -< 
also  by  students  other  than  those  that  had  produced  them.  lc 
The  study  reinforces  the  hypothesis  (Kuchemann.1989)  that  the  presence  of  numerical  values  superimposes 
specific  difficulties  inherent  in  the  arithmetical  processing  of  the  data  to  the  difficulties  inherent  in  the 
conceptualization  of  proportionality  as  a  ratio  between  quantities.  In  particular,  a  fact  that  stands  out  is  that  the 
Tiow  many  times  a  shadow  goes"  into  another,  or  a  fence-post  into  its  shadow  .docs  not  correspond  to  the  fact  that 



the  number  of  times  may  be  determined  calculating  a  division  between  the  measurements.  All  this  is  evident  when 
the  problem  is  to  determine  the  height  of  an  object  that  is  smaller  than  the  object  that  casts  the  shadow(SiL3Xsee  ii). 
The  role  of  social  activities  in  overcoming  the  additive  model  in  the  geometrical  problems  seems  to  be  very 
important .  In  particular,  the  active  comparison  and  the  evaluation  of  strategies  permit  to  overcome  the  limits 
inherent  in  the  pure  reference  to  "viskxt":in  effect,  even  in  the  presence  of  other  students  that  suggest  recourse  to 
multiplicative  models  and  of  a  direct  experience  of 'Vision''  of  the  fence-post  and  of  the  street  lamp,  only  6  students 
in  Situation  2  produced  a  coherent  and  correct  problem-solving  strategy  (while  26  propose  a  coherent  additive 
model!).  The  situation  improves  considerably  when  the  problem  situation  and  the  problem-solving  strategies  are 
represented  and  argued,  and  the  students  have  to  follow  and  evaluate  the  reasoning  of  their  classmates.  The 
reasoning  and  representations  of  the  "best"*  students  "mediate"  the  transition  to  more  appropriate  strategies.(see  ii). 
The  analysis  also  brings  to  light  various  problems: 

-  (see  iii)For  most  students  there  is  no  transfer  of  the  multiplicative  strategy  acquired  in  the  geometrical  setting  to 
the  arithmetical  setting  of  Situation  7,  and,  when  a  transfer  seems  to  take  place,  there  is  a  change  of  strategy.  This 
may  be  due,  beside  than  to  the  lack  of  correspondence  between  arithmetical  operations  of  multiplication  and 
division  and  meanings  of  proportionality  between  quantities,  also  to  precise  characteristics  of  the  "physical- 
geometrical"  situation.  In  particular,  in  the  "arithmetical"  problem  the  reasons  that  induce  some  students  to  carry  out 
the  transition  from  the  qualitative  model  "to  grow  with..."  to  the  quantitative  model  of  "equality  of  the  number  of 
times  that..."  fail  (as  may  be  noticed  in  the  protocols  mentioned  before  with  reference  to  Situation  2,  there  appear 
extrinsic  reasons  of  "balance"  with  respect  to  the  mathematical  structure  of  the  problem  and  linked  to  the  particular 
situation  observed).  Another  failure  that  occurs  is  that  of  the  direct  perception  of  the  simultaneity  of  the 
relationships  between  the  elements  compared.  Another  diversity  is  linked  to  the  difficulty  to  produce  appropriate 
external  representations  (like  those  produced  for  geometrical  problems). 

-  Role  of  the  additive  strategies  in  overcoming  the  additive  model  and  in  approaching tne  multiplicative  model . 

Although  numerically  limited  and  not  long-lasting,  some  cases  of  interweaving  (supported  by  external 

representations)  between  additive  considerations  and  multiplicative  considerations  such  as  those  exemplified  at  the 

end  of  paragraph  4  (see  vi),  suggest  the  idea  (to  be  analysed  on  a  wider  scale  and  with  a  more  appropriate 

methodology)  that  the  multiplicative  model  in  the  geometrical  setting  may,  at  least  in  certain  instances,  result  from 

an  operation  of  "contraction"  of  additive  reasoning,  that  would  not  thus  constitute  only  an  intermediate  stage  linked 

to  the  mental  ripening  of  the  subject ,  but  a  fundamental  element  of  the  transition  to  multiplicative  strategies. 


Boero  P.(  1989),  Mathematical  literacy  for  all:  experiences  and  problems,  Proc.  PME  -XUI,  Paris,  62  -  76. 
Bondesan.M.G.;Ferrari  P.L.(  1991)The  active  comparison  of  strategies  in  problem  solving.flnxJWf-XV,  Assisi 
Douady.R.  ( 1 985),  The  interplay  between  different  settings,  Proc.  PME-9,  Noordwijkerhout,  33-52 
Grugnetti,  L. ;  Torres.C.M.  (1991),  The  "power"  of  additive  structure ...  Proc.  PME-XV,  Assisi .  vol.II,96- 100 
Harel,G.:Behr,  M.;Post,T.:Lesh,R(  1991 ),  Variables  affecting  proportionality:....flnx.  PAf£-WAssisi 
Hart,K.M.(  198 1 ),  Childr  r.  ;  Understanding ofMatttcmatics: II- 16.  J.Murray.  London 
Hoyles  C.  Noss  R.,  Sutherland  R.(  1989),  A  Logo-based  mlcroworld..,  Proc.  PME  XIII,  Paris.vol.2. 115-1 22. 
HoylesC,  Noss  R,  Sutherland  R(  1991),  Evaluatinga computer-based microworld:  Proc.  PMEXV.  Assisi 
Karplus.R.iPulos.  S.;&Stage,E.K  .(1983),  Proportional  reasoning  of  early  adolescents,  in  RLesh  &  M. Landau 
(eds.),  Acquisition  of  Mathematics  Concepts  and  Processes ,  Academic  Press,  New  York 

Kuchemann  D.(  1 989).  The  effect  of  setting  and  numerical  content  Proc  PME  XIII,  Paris,  180- 1 86 

Lester,F.:Charles,R.  (I99I),A  framework  for  research  on  problem  solving  instruction,  Proc.  NATOSeminaron 
Problem  Solving.  Viana  do  Castelo.  Springer -Verlag  (to  appear) 

Lin,  F.L.0989),  Strategics  used  by  "adders"in  proportional  reasoning,  Proc.  PME-XIII,  Paris,  234-241 
Toumiaire  F..  Pulos  S.(l  985),  Proportional  Reasoning:  A  Review  of  the  Literature,  E.S.M.  1 6, 1 8 1  -204. 
Vergnaud  G.(  1 98 1 ).  LEnfant .  la  Malhonatique  el  la  Realite.  Peter  Lang  ,Bcmc 





Collfcge  du  Vieux  Montreal 

Recent  work  suggests  that  teachers'  conceptions  of  the  nature  of 
mathematics,  its  teaching  and  learning,  are  not  always  consistent  with  their  practice. 
This  study  is  concerned  with  the  reasons  for  these  discrepancies.  Inspired  by 
Schon's  Reflective  Practitioner  (1983),  the  author  examined  her  own  teaching 
practice  while  experimenting  with  a  problem-solving  teaching  approach  with  college 
students.  Tape-recordings  of  the  teacher  in  class  and  a  teacher-journal  provide  the 
basis  for  qualitative  analysis  which  was  conducted  together  with  a  second  researcher. 
Partial  results  suggest  that  although  class  preparation  follows  teacher-conceptions,  in 
class,  spontaneous  reactions  differ.  Subsequent  analysis  will  look  for  explanations  of 
these  differences.  On  the  basis  of  these  results,  the  possibility  of  using  reflection  as 
a  way  of  improving  consistency  between  in-service  teachers'  conceptions  and  their 
practice  will  be  examined. 

(This  paper  will  be  presented  in  English  at  the  conference). 

Le  probieme 

Les  resullats  de  recherches  en  didactique  n'atteindront  la  classe  de  mathematiques 
que  si  Ton  passe  egalement  par  l'un  des  principaux  intervenants  du  systeme  didactique, 
l'enseignant.  C  est  ce  dernier  qui  en  bout  de  ligne  controle  les  choix  didactiques  et  qui  dans 
la  mesure  oil  il  est  maitre  de  ses  actes  d'enseignement  definit  le  cadre  d'apprentissage  de 
l'eleve.  Or  ces  choix,  c'est  l'avis  de  nombreux  auteurs  (Clarke,  Peterson,  1985;  Vergnaud, 
1988;  Ernest,  1989;  Thompson,  1984),  sont  commandes  par  les  conceptions  de  l'enseignant 
au  sujet  des  mathematiques,  de  leur  apprentissage  et  de  leur  enseignement. 

Certaines  etudes  (Cooney,  1985,  Thompson,  1984,  Kaplan,  1991)  laissent  voir  que  les 
conceptions  telles  que  declarees  par  l'enseignant  ne  se  transmettent  pas  toujours  dans  la 
pratique.  La  possibility  de  prendre  conscience  de  ses  conceptions  et  de  reflechir  sur  sa 
pratique  amenerait  l'enseignant  a  ameliorer  la  coherence  entre  les  conceptions  et  la 
pratique.  Plusieurs  interventions  visant  la  formation  des  maitres  particulierement,  les 
maitres  en  service  ont  experiments  divers  moyens  pour  susciter  cette  reflexion  mais,  la 
plupart  des  approches  utilisees  bien  que  fructueuses,  demandent  une  organisation 
exterieure  et  ne  se  transportent  pas  necessairement  dans  le  quotidien  des  enseignants. 

Nous  avons  done  voulu  savoir 

-Jusqu'a  quel  point  la  pratique  reflete  les  conceptions  exprimees? 
-Comment  expliquer  les  ecarts,  les  divergences? 



Et  par  la  suite  voir 

-Quels  moyens  peut-on  suggerer  aux  enseignants  pour  organiser  leur  reflexion  en 
vue  d'ameliorer  la  coherence  enrre  leurs  conceptions  et  leur  pratique? 

Notre  objectif  etait  d'ebaucher  une  methode  d'auto-analyse  des  actes 
d'enseignement  qui  permettrait  a  l'enseignant,  de  prendre  conscience  de  ses  conceptions  et 
d'observer  jusqu'a  quel  point  sa  pratique  est  coherente  avec  ses  conceptions. 

Au  point  de  depart,  nos  lectures  et  notre  experience  d'enseignante  nous 
suggeraient  certaines  r6ponses  que  nous  avons  resume  en  trois  hypotheses  que  nous 
avons  voulu  verifier. 

HI)  Certaines  realites  comme  les  contraintes  environnementales,  les  reactions 
des  Aleves  ou  encore,  les  modes  de  comportements  habituels  ou  anciens 
sont  plus  fortes  que  les  conceptions  avouees  et  gSnent  la  realisation  de 
l'enseignement  tel  que  preconcu. 

H2).  II  est  possible  pour  un  enseignant  d'analyser  sa  pratique. 

H3)  Le  fait  de  r^flechir  sur  la  pratique  de  facon  quotidienne  par  la  redaction 
d'un  journal  amene  des  modifications  a  la  pratique  et  aux  conceptions  de 
sorte  a  tendre  vers  un  equilibre  enrre  les  deux. 

L'approche  methodologique  choisie:  sujet-chercheur 

L'exploration  de  ce  probleme  demandait  une  approche  methodologique  originate. 
C'est  Interpretation  des  conceptions  qui  d'abord  etait  questionnee.  Cooney  (1985)  avait 
suggerl  que  des  differences  d'interpretations  entre  le  chercheur  et  l'enseignant  etait  peut- 
etre  ce  qui  expliquait  les  divergences  entre  les  conceptions  et  la  pratique.  En  choisissant 
d'etre  le  sujet  et  le  chercheur,  nous  pouvions  interpreter  sans  biais  nos  conceptions.  Nous 
les  avons  etablies  a  partir  d'ecrits  prealables  a  l'experimentation. 

De  plus,  le  questionnement  et  l'observation  par  un  tiers  n'est  pas  sans  influences. 
Nous  avons  voulu  limiter  autant  que  possible  ces  interferences  dues  au  cadre  de  recherche 
pour  mieux  cerner  les  interferences  entre  les  conceptions  et  la  pratique.  Nous  avons 
remplace  la  presence  d'un  observateur  exterieur  par  l'utilisation  d'enregistrements 
sonores  et  la  redaction  d'un  journal  de  bord.  La  lecture  du  journal  de  bord  pourrait  en 
plus,  faire  apparattre  les  modifications  de  nos  conceptions  en  cours  d'expenmentation. 

Pour  assurer  la  validite  de  la  recherche,  nous  avons  fait  appel  a  la  collaboration  de 
d'autres  personnes  a  differentes  eta  pes  du  travail  d 'analyse.  Une  deuxieme  chercheure  a 
travaille  en  parallele  avec  nous.  Ayant  pris  connaissance  de  nos  conceptions  par  la  lecture 
dc  nos  ecrits  prealables,  ayant  ecoute  et  code  les  enregistrements  de  l'experimentation,  elle 
pouvait  corroborer  notre  analyse  aux  diverses  e tapes.    En  dernier  lieu,  deux  autres 

Les  hypotheses 



personnes,  l'une  chercheure  et  l'autre  enseignante  ont  verifte  si  nos  interpretations 
etaient  soutenues  par  les  donnees  fournies.  Nous  etions  de  notre  c6t6  assuree  d'une 
bonne  connaissance  du  milieu  et  d'une  presence  suffisamment  longue  sur  le  terrain. 

^experimentation  avait  616  prececlee  d'une  exploration  et  d'une  pre- 
experimentation.  L'exploration  a  permis  de  mieux  cerner  le  contexte  de  notre 
experimentation.  Les  Aleves  etaient  des  Aleves  ayant  eu  des  difficultes  avec  les 
mathematiques  auparavant,  c'dtait  en  fait  leur  seul  point  commun.  Les  classes  etaient 
non-homogenes  quant  l'age,  la  provenance,  la  habilete  d'apprentissage,  les  acquis  et  la 
motivation.  Le  cours  devait  combler  les  lacunes  de  ces  eleves  en  passant  a  travers  ce  que 
Ton  convient  d'appeler  les  mathematiques  de  base:  algebre,  fonctions,  rrigonometrie. 

Forte  des  connaissances  acquises  lors  de  cette  exploration,  nous  avons  mis  sur  pied 
une  approche  pedagogique  suivant  nos  conceptions  de  l'enseignement  des 
mathematiques.  Cette  approche  misant  sur  la  participation  active  des  Aleves  demandait  la 
creation  de  materiel  didactique  particulier.  La  periode  de  pre-experimentation  nous  donne 
l'occasion  de  construire  ce  materiel  et  de  le  tester  aupres  des  Aleves  (Gattuso,  Lacasse,  1989). 

Nous  avons  pu  par  la  suite  passer  a  l'experimentation  elle-meme.  La  clientele 
etudiante  itait  sensiblement  la  meme  et  le  materiel  didactique  utilisd  a  la 
pre-experimentation  a  6t6  repris  apres  de  legers  reajustements.  Pour  les  besoins  de  la 
recherche,  nous  avons  ajoutl  l'enregistrements  sonores  des  cours  et  la  redaction  d'un 
journal  de  bord. 

L'analyse  s'est  deroulee  en  plusieurs  Stapes  dont  nous  tragons  ici  les  grandes 
lignes.  II  a  fallu  d'abord  dtablir  nos  conceptions  afin  de  pouvoir  construire  une  grille 
d'analyse.  C'est  ce  que  nous  avons  fait  en  utilisant  comme  source  de  donn6es  certaines  de 
nos  publications  anterieures  a  l'exp6rimentation. 

Nous  sommes  ensuite  passee  a  l'analyse  de  la  pratique  en  fonction  des  conceptions 
Itablies  dans  la  grille.  A  cette  etape,  les  donnees  etaient  drees  des  enregistrements  sonores 
et  du  journal  de  bord.  Apres  le  decoupage  et  le  codage  des  donnees,  nous  avons  pu  faire 
une  compilation  qui  nous  a  mcnee  aux  resultats. 

Nous  avons  par  la  suite  examine  les  reflexions  notees  en  cours  d'analyse  afin  de 
examiner  si  les  conceptions  se  modifiaient  en  cours  d'experience. 

En  dernier  lieu,  nous  avons  regard^  de  fagon  critique  le  cheminement  parcouru 
pour  en  tircr  un  cadre  de  travail  que  nous  sugg^rons  aux  enseignants  pour  organiser  leur 
reflexion  sur  leur  pratique. 





Nous  voulions  en  premier  lieu  mieux  connaitre  les  liens  entre  les  conceptions  et 
la  pratique  d'un  enseignant  de  mathematiques  et  par  la  suite  voir  s'il  etait  possible  de 
mettre  en  place  de  facon  profitable  cette  reflexion  sur  la  pratique.  Si  nous  regardons  nos 
hypotheses  de  depart,  nous  avions  prevu  que  certaines  interferences  dues  a 
l'environnement,  aux  eleves  et  aux  habitudes  de  l'enseignant  interviendraient  dans  la 
realisation  de  nos  conceptions.  Nous  pensions  egalement  qu'il  etait  possible  pour  une 
enseignant  d'analyser  sa  pratique  et  que  la  reflexion  quotidienne  ameliorerait  la  coherence 
entre  les  conceptions  et  la  pratique.  Les  conclusions  auxquelles  nous  sommes  arrivees 
nous  amenent  a  preciser  ces  hypotheses. 

Ce  travail  nous  a  permis  de  faire  un  pas  en  avant  dans  la  comprehension  des 
elements  qui  interviennent  dans  la  mise  en  pratique  des  conceptions  dans  le  cadre  de 
l'enseignement  des  mathematiques. 

Les  resultats  de  l'analyse  ont  montre  qu'il  y  avait  une  tres  bonne  coherence  (82%) 
entre  les  conceptions  telles  qu'exprimees  au  depart  et  les  actes  d'enseignement  observes. 
Ce  resultat  est  sans  signification  si  nous  ne  tenons  pas  compte  des  particularites  de  notre 
experimentation.  Nous  avions  voulu  au  point  de  depart  alleger  autant  que  possible  les 
contraintes  exterieures  qui  selon  les  auteurs  consultes  seraient  des  causes  de  discordances. 
La  grande  liberte  dont  nous  jouissions  au  moment  de  l'experimentation  a  surement  joue. 
Nous  avions  pu  definir  l'approche  pedagogique,  choisir  jusqu'a  un  certain  point  le 
contenu  du  cours  et  construire  le  materiel  didactique  en  fonction  de  nos  conceptions  au 
sujet  de  l'enseignement  des  mathematiques.  Notre  experience  en  enseignement  et  notre 
formation  premiere  en  mathematiques  nous  garantissaient  l'assurance  necessaire  pour 
entreprendre  une  telle  innovation.  Mais  il  reste  que  certains  obstacles  demeurent. 
Certains  sont  exogenes  et  d'autres  plus  personnels  a  l'enseignant  sont  endogenes. 

Nous  avons  pu  voir  que  certains  elements  dependant  de  l'organisation  scolaire 
genent.  Les  plages  horaires  extremes,  des  locaux  trop  petits  en  sont  des  exemples.  Le 
materiel  didactique  et  le  contenu  mathematique  amenent  aussi  quelques  difficultes.  Les 
protocoles  d'activites  comportaient  encore  certaines  ambigui'tes  et  le  contenu 
mathematique  etait  parfois  trop  simple  ou  trop  difficile  pour  les  eleves.  C'etait  alors 
difficile  d'aller  dans  le  sens  prevu,  c'est-a-dire,  conduire  les  eleves  a  explorer  les  concepts  et 
a  deduire  les  connaissances  a  partir  de  leurs  resultats. 

D'autres  entraves  se  trouvent  chez  les  eleves  eux-memes.  C'est  surtout  leur 
manque  de  preparation  au  niveau  des  mathematiques  et  de  la  methode  de  travail  en 
general  qui  a  contrarie  la  realisation  des  conceptions  qui  visaient  plus  a  soutenir  la 
recherche  de  solutions  qu'a  expliquer  comment  faire  le  probleme.  Le  temps  pris  par  ces 

Des  conceptions  a  la  pratique 

1  -237 

eleves  moins  prepares  etait  trop  important  et  genait  notre  gestion  du  travail  de  l'ensemble 
du  groupe. 

Enfin,  notre  systeme  de  conceptions  lui-meme  etait  en  quelque  sorte  porteur  de 
difficultes.  Ce  n'est  pas  que  les  conceptions  se  contredisaient  mais  elles  pouvaient  dans  les 
cas  limites  etre  en  conflit.  Les  conceptions  concernant  l'activite  mathematique,  ouverture, 
exploration,  autonomie  ont  pris,  sans  que  nous  nous  en  rendions  compte,  le  dessus  sur  les 
conceptions  touchant  a  l'organisation  du  cours  et  a  l'encadrement  des  eleves.  Soulignons 
ici  que  la  reflexion  et  le  bilan  qui  s'en  est  suivi  ont  permis  cette  constatation  dont  nous 
avions  jamais  pris  conscience  auparavant.  Signalons  enfin  que  nous  avons  constate  que 
notre  etat  d'esprit,  fatigue,  inquietude,  bonne  humeur  joue  egalement  sur  nos  actes 

En  resume,  nous  pouvons  conclure  que  les  liens  entre  les  conceptions  et  la 
pratique  sont  forts  et  que  s'il  y  a  prise  en  charge  consciente  des  conceptions  et  des  moyens 
pour  les  mettre  en  oeuvre,  le  transfert  des  conceptions  dans  la  pratique  se  produit. 

L'auto-analyse  comme  outil  de  reflexion 
L'auto-analyse  telle  que  nous  l'avons  pratiquee  s'est  averee  un  outil  de  reflexion 
profitable  et  realistole. 

L'auto-analyse  a  donne  lieu  a  un  bilan  professionnel  qui  a  permis  une  prise  de 
conscience  interessante  et  utile.  Les  resultats  ont  revele  certaines  de  nos  faiblesses,  ils  ont 
expose  certains  succes  encourageants  et  indique  des  modifications  dans  nos  positions.  A  la 
lumiere  de  ces  constatations,  nous  avons  pu  dans  notre  pratique  deja  distinguer  ce  qui 
concerne  l'encadrement  des  eleves  et  ce  qui  concerne  la  gestion  de  l'activite 
mathematique.  Le  fait  de  comprendre  ce  qui  se  passait  a  enormement  facilite  ces 
modifications.  C'est  un  resultat  important  qui  nous  permet  maintenant  d'etre  plus  precise 
dans  nos  demandes  aux  eleves,  ce  qui  est  profitable  pour  nous  et  pour  les  eleves. 

Ayant  realise  le  succes  de  nos  efforts  particulierement  ceux  visant  a  amener  l'eleve 
a  verbaliser  ses  demarches  et  a  evaluer  son  travail,  nous  sommes  encouragee  a  poursuivre 
dans  ce  sens  et  a  rechercher  de  nouvelles  solutions  a  d'autres  points  moins  reussis  comme 
le  travail  d'equipe  par  exemple. 

Les  hypotheses  qui  nous  avaient  conduites  au  depart  a  developper  cette  approche 
d'enseignement  se  rapportaient  beaucoup  a  l'aspect  affectif  de  l'apprentissage.  Notre 
centre  d'interfit  s'est  deplace,  nous  sommes  maintenant  beaucoup  plus  preoccupee  par 
l'activite  mathematique  elle-meme,  les  contenus,  les  activites  de  resolution  de  probleme, 
le  materiel  didactique.  En  effet,  l'activite  mathematique  se  doit  d'etre  interessante  et 
stimulante  pour  l'eleve  si  Ton  veut  qu'il  y  prenne  plaisir  et  qu'il  l'attaque  avec  confiance. 

Le  fait  d'avoir  degage  certains  obstacles  exogenes  renforcera  nos  demandes  aupres 
de  ^administration  scolaire  au  sujet  d'horaires,  de  locaux  et  de  regroupement  des  eleves 




par  exemple,  par  ce  que  nous  serons  plus  en  mesure  de  les  expliquer. 

Soulignons  finalement  que  cette  prise  de  conscience  n'aurait  pas  ete  complete  sans 
l'etape  de  l'auto-analyse.  La  lecture  du  journal  de  bord  a  expose  certaines  reactions  comme 
la  necessite  de  plus  encadrer  les  eleves.  Mais  la  reflexion  a  travers  Taction  n'etait  pas 
suffisante  et  n'amenait  pas  une  prise  de  conscience  aussi  complete.  On  peut  lire  dans  le 
journal  de  bord  des  remarques  par  rapport  aux  problemes  vecus  dans  la  classe  et  des  idees 
pour  tenter  d'y  remedier  mais,  il  n'y  a  pas  d'analyse  approfondie,  faute  de  temps  et  de 
recul,  ce  qui  fait  qu'il  n'y  a  pas  de  comprehension  de  la  situation,  on  ne  fait  que  la 
constater.  L'analyse  qui  a  suivi  a  eu  tout  autre  resultat  parce  qu'elle  a  permis  de  voir  ce  qui 
se  passait.  Nous  pensons  particulierement  aux  chevauchements  parfois  problematiques 
entre  les  conceptions  qui  ont  ete  souleves.  C'est  pourquoi  ce  qui  est  avance  dans  notre 
troisieme  hypothese  est  a  completer:  la  reflexion  quotidienne  est  necessaire  mais  il  faut 
prendre  un  certain  recul  et  faire  un  bilan  pour  arriver  a  une  meilleure  comprehension  des 
phenomenes  en  jeu. 

A  la  suite  de  cette  experience,  nous  pouvons  dire  qu'il  est  possible  a  un  enseignant 
d'auto-analyser  sa  pratique.  Le  travail  a  parfois  ete  difficile  car  il  fallait  constamment 
trouver  des  solutions  aux  problemes  methodologiques  qui  se  presentaient.  II  fallait 
inventer  et  se  reajuster.  Nous  avons  pu  a  la  suite  de  l'examen  critique  de  notre  demarche, 
suggerer  des  moyens  que  pourrait  adopter  un  enseignant  sans  trop  perturber  sa  pratique 
reguliere  et  nous  croyons  que  la  demarche  que  nous  proposons  est  considerablement 
simplifiee  et  tout  aussi  efficace.  Nous  avons  concu  un  inventaire  de  conceptions  afin 
d'aider  I'enseignant  a  etablir  une  grille  de  conceptions  qui  lui  permettra  d'analyser  ses 
actes  d'enseignement  a  partir  de  l'enregistrements  de  ses  cours.  Nous  proposons 
egalement  une  methode  simplifiee  de  compilation  pour  faciliter  le  travail  d'analyse. 

II  faut  toutefois  se  garder  d'attendre  des  resultats  identiques  chez  toute  personne 
qui  s'engagerait  dans  une  auto-analyse.  Le  terme  l'indique,  l'analyse  est  personnelle  et  les 
resultats  seront  surement  en  fonction  du  cheminement  personnel  de  la  personne  qui 
l'entreprend.  Toutefois  le  fait  de  s'engager  dans  une  telle  entreprise  denote  une  volonte 
de  remise  en  question  qui  ne  peut  que  se  traduire  par  un  avancement  personnel. 


Bien  que  I'enseignant  soit  maitre  d'un  grand  nombre  de  choix  didactiques,  certains 
elements  sont  hors  de  sa  portee  immediate.  L'administration  et  l'organisation  scolaires 
devraient  tenir  compte  des  impacts  de  leurs  decisions  sur  I'enseignant,  l'eleve,  la  classe  et 
l'apprentissage  du  savoir.  On  devraient  egalement  apporter  plus  de  soins  aux  questions 
qui  touchent  le  regroupement  des  eleves.  Sans  viser  necessairement  l'homogeneite  des 
classes,  il  faut  tenir  compte  de  certains  facteurs,  notamment  le  nombre  d'eleves  dans  la 
classe,  le  support  didactique  dont  beneficie  I'enseignant. 


1  -239 

Encore  beaucoup  de  recherches  doivent  etre  menees  en  ce  qui  concerne  les 
mathematiques  de  l'enseignement  post-secondaire,  particulierement  en  ce  qui  est  relatif 
au  materiel  didactique.  II  y  a  peu  de  materiel  disponible  pour  l'enseignant  qui  veut 
proposer  a  ses  Sieves  des  activites  d'exploration  ou  des  problemes  allant  au  dela  de 
l'exercice  de  routine.  Nous  avons  pu  voir  que  le  materiel  didactique  joue  un  r61e 
important  dans  les  choix  de  l'enseignant,  or,  on  ne  peut  exiger  que  chacun  cree  un 
materiel  a  sa  mesure.  Prealablement,  l'6tude  d'un  point  de  vue  didactique  des 
mathematiques  enseignees  apres  le  secondaire  est  necessaire  et  ensuite,  il  faudra  faire  faire 
appel  aux  enseignants  pour  experimenter  en  classe  des  approches  nouvelles  et  en 
examiner  les  resultats. 

Cette  recherche  montre  par  ailleurs  qu'il  est  possible  d'innover  en  matiere  de 
recherche  pour  arriver  a  observer  la  classe  de  l'interieur.  II  faut  de  plus  en  plus  s'assurer 
de  la  participation  des  enseignants  a  la  recherche  et  profiter  de  ce  point  de  vue  different. 
Les  enseignants  gagneront  de  leur  c6te  une  meilleure  comprehension  des  phenomenes  en 
jeu  et  seront  plus  disponibles  pour  experimenter  les  modeles  proposes  par  les  didacticiens. 

De  plus,  les  resultats  de  l'auto-analyse  portent  a  croire  qu'il  faut  favoriser  ce  type  de 
reflexion  et  la  soutenir.  II  faudrait  poursuivre  le  present  travail  et  6tudier  les  effets  de 
l'auto-analyse  chez  d'autres  enseignants.  La  necessity  de  ce  travail  de  rdflexion  pour 
amener  une  meilleur  adequation  entre  ses  conceptions  et  sa  pratique  ne  diminue  en  rien 
le  besoin  qu'ont  les  enseignants  d'fitre  plus  informes  sur  les  recherches  specifiquement  en 
ce  qui  les  touche  de  plus  pres.  la  didactique  des  mathematiques.  Beaucoup  de  travaux  sont 
menes  sur  les  difficultes  d'apprentissage  des  eleves,  les  causes  d'echecs  entre  autres  mais 
les  enseignants  ont  peu  de  sources  d'informations  en  ce  qui  concerne  les  approches,  les 
presentations  et  les  difficultes  des  contenus  mathematiques  qui  sont  enseignes  au  niveau 

En  dernier  lieu,  on  peut  encore  se  demander  ou  commence  la  boucle  doit-on  tenter 
de  modifier  les  conceptions  des  enseignants  pour  finalement  influencer  leur  pratique  ou 
encore  essayer  de  les  inciter  a  modifier  leur  pratique  pour  susciter  des  evolutions  dans 
leurs  conceptions.  Certains  apports  exterieurs  peuvent  influencer.  Les  informations  sous 
forme  de  lecture,  de  presentations  ou  encore  de  formation  peuvent  agir  sur  les  conceptions 
de  l'enseignant  et  l'implantation  de  nouveaux  outils,  tel  que  l'ordinateur  ou  encore  des 
manuels  soutenant  une  approche  innovatrice  peuvent  amener  certaines  modifications 
dans  la  pratique  de  l'enseignant.  Cependant,  le  present  travail  montre  clairement  les 
interactions  entre  les  conceptions  et  la  pratique.  La  de  qui  selon  nous  peut  intervenir  dans 
cette  interaction  est  la  teflexion-sur-la-pratique  qui  suscite  la  confrontation  entre  les 
conceptions  de  l'enseignant  et  sa  pratique.  Nous  nous  devons  de  poursuivre  les 
experimentations  en  ce  sens. 


Nous  remercions  mesdames  Nicole  Mailloux  (Universite  du  Quebec  a  Hull)  et  Ewa 
Puchalska  (Universite  de  Montreal)  pour  leur  collaboration  lors  de  l'analyse  des  donnees. 


CLARK,  C,  PETERSON,  P.  (1985).  "Teachers'  thought  proeessess".  In  Merlin  Wittrock(Ed.): 
Handbook  of  research  on  teaching,  third  edition.  New  York:  Macmillan.  255-296. 

COONEY,  T.  (1985).  "A  beginning  teacher's  view  of  problem  solving".  Journal  for  research  in 
mathematics  education.  JUj.  (5).  324-336. 

COPA,  P.M.,  SANDMANN,  L.R..  (1987).  profile  of  excellence...or  becoming  a  more  reflective  adult 
education  practicionner.  Paper  presented  at  the  annual  meeting  of  the  American  Association  for 
Adult  and  Continuing  Education,  Washington,  DC,  October  22,  1987. 

ERNEST,  P.  (1988).  "The  impact  of  beliefs  on  the  teaching  of  mathematics".  In  Keitel,  Christine, 
Damerow,  P.,  Bisop,  A.  Gerdes,  P.  (Eds),  Mathematics.  Education,  and  Society.  Science  and 
Technology  Education,  Document  Scries  no.35.  Paris:UNESCO.  99-101 

ERNEST,  P.  (1989a).  "The  impact  of  beliefs  on  the  teaching  of  mathematics".  In  Ernest,  P.  (Ed.). 
.    nQRQI  Mathematics  Teaching  The  State  of  the  Art.  London:  Falmer.  249-254. 

ERNEST,  P.  (1989b).  "The  knowledge,  beliefs  and  attitudes  of  the  mathematics  teacher:  a  model.". 
Journal  of  Education  for  Teaching.  1$.  (1).  13-33.  .  M.,  AJZEN,  I.  (1975).  Belief,  attitude. 
intention  and  behavior.  An  introduction  to  theory  and  research.  Reading,  Massachusetts:  Addison- 

GATTUSO,  L.  ,  LACASSE,  R.  (1986).  mathophohes  nne  experience  de  reinsertion  au  niveau 
collf  gial.  Ccecp  du  Vieux  Montreal. 

GATTUSO,  L.  ,  LACASSE,  R.  .(1987).  "Les  mathophobes  une  experience  de  reinsertion  au  niveau 
coliegial".  Actes  du  onzifeme  congres  international  de  Psychology  of  Mathematics  Education. 
PME-XI.  editi  par  Jacques  C.  Bergeron,  Nicolas  Herscovics,  Carolyn  Kieran.  Juillet:  Montreal. 

GATTUSO,  L.  .LACASSE,  R.  (1988).  "Intervention  in  mathematics  course  at  the  college  level".  In 
Andrea  Borbas  (Ed.  ^Twelfth  annual  conference  of  the  international  group  for  the  Psychology  of 
Mathematics  Education.  PME-XI.  July:  Veszprem.  425-432. 

GATTUSO,  L.  .LACASSE,  R.  (1989).  1-cs  maths.  If  cnenr  et  la  raison.  tin  modcle  d'intervention 
Hant  ups  daw  de  mathemariques  au  collegia!.  Montreal:  C<gep  du  Vieux  Montreal. 

KAPLAN,  R.  (1991).  Teacher  Beliefs  and  Practice:  A  Square  Peg  in  a  Square  Hole  Willaim  Paterson 
College,  mimeographed  paper. 

SCHON,  D.  (1983).  The  reflective  practitioner.  London:  Temple  Smith. 

STENHOUSE,  L.  (1975V  An  introduction  to  curriculum  research  and  development.  London:  Heinemann. 

THOMPSON,  A.  (1984).  "The  relationship  of  teachers'  conceptions  of  mathematics  and  mathematics 
teaching  to  instructionnal  pracrice"  Educational  Studies  in  Mathematics.  IS.  105-127. 

VERGNAUD,  G.  (1988).  "Theoretical  frameworks  and  empirical  facts  in  the  psychology  of  mathematics 
education".  In  Ann  &  Keith  Hirst  (Ed  VPmrerdings  of  the  Sixt  International  Congress  on 
Mathematical  Education.  Budapest:  ICME  6. 29-47. 


1  -241 


V.  Nawro-Pelayo,  J.  D.  Godino  and  M.C.Batanero  and 

University  of  Granada  (Spain) 


The  preliminary  results  of  a  systematic  study  of  the  difficulties  and 
errors  In  solving  a  sample  of  combinatorial  problems  In  two  groups  of 
pupils  of  secondary  education  are  presented  In  this  work.  The  analysis 
of  the  task  variables  of  the  problems  constitutes  a  first  approximation 
to  the  classification  of  the  simple  combinatorial  problems  and  likewise 
enables  the  attribution  of  a  content  validity  to  the  Instrument 
developed.  In  order  to  assess  the  capacity  to  solve  this  kind  of 


In  accordance  with  Piaget  and  Inhelder  (1951),  the  development  of  the 
combinatorial  capacity  is  one  of  the  fundamental  components  of  the  formal  thinking 
and  can  be  related  to  the  stages  described  in  their  theory:  after  the  period  of 
formal  operations,  the  subject  discovers  systematic  procedures  of  combinatorial 
construction,  although  in  the  esse  of  the  permutations,  it  is  sometimes  necessary  to 
wait  until  they  are  15  years  of  age. 

However,  more  recent  results,  as  Fischbein  (1975)  indicates,  sustain  that  the 
combinatorial  problem  solving  capacity  is  not  reached  in  all  cases,  not  even  in  the 
level  of  formal  operations  without  specific  instruction.  Fischbein  and  Gazlt  (1988) 
study  the  relative  difficulty  of  the  combinatorial  problems  in  terms  of  the  type  of 
combinatorial  operation  and  nature  and  number  of  elements,  in  addition  to  the  effect 
of  the  instruction  on  the  combinatorial  capacity.  Other  authors  who  in  addition  to 
those  mentioned  have  investigated  the  difficulties  of  different  types  of 
combinatorial  problems,  are,  Mendelson  (1981),  Green  (198i).  Lecoutre  (1985)  and 
Maury  (1986). 

In  this  work,  we  describe  the  results  of  a  study  of  the  effect  on  the  relative 
difficulty  of  different  combinatorial  problems  of  several  task  variables  of  the  same. 
Although  the  study  carried  out  to  date  is  limited,  we  consider  it  to  be  of  interest 
to  describe  the  classification  carried  out  of  the  errors  and  the  differences  found  in 
this  distribution,  between  one  group  of  pupils  who  have  not  had  any  previous 
instruction  and  another  group  that  has.  As  an  additional  consequence  we  have  a  first 



version  of  a  psychometric  instrument,  available.    This   enables    us    to   measure  the 
"combinatorial  reasoning   capacity"    of    secondary    students,    and    to   diagnose  the 
intuitions  and  types  of  error  that  should  be   taken    into    account    in    teaching.  As 
Borassi  (19S7)  affirms  "errors  can  be  used  as  a  motivational  device  and  as  a  starting 
point  for  the  creative  mathematical  exploration,  involving  valuable   problem  solving 
and  problem  posing  activities"  (page  7.). 


The  test  consists  of    9    problems    some    with    several    sections,    in    total  12 
questions.  As  an  example  in  Table  1,  the  statements  of  three  of    these    problems  are 
included  that  ./ill  serve  to  describe  the  different  types  of  errors  that  the  students 
have  had  during  the  solving  process.  The  description  of  the    characteristics   of  the 
problems,  that    are    of    two   types:    of   enumeration    and    calculus,    are  presented 
schematically  in  Tabic  2. 


t.  Thrtt  boy*  ara  ««nt  to  tha  haadmaatar  for  itaallnf.  Thay  hava  to  line  up  In  a  row  outalda  tha 

haad'a  room  and  wait  for  thalr  punlahmant.  No  ona  wanta  to  ba  flnt  of  couraa! 

(a)  Suppoaa  tha  boya  ara  called  Andree.  Benito  y  Carloe  (A,  B,  C  for  ahort).  We  want  to  write  down 

all  the  poaelble  ordere  In  which  they  could  line  up. 

For  example    A  B  C        we  write  ABC  aa  ahown  below: 

I  I  I 

lit         2nd  3rd 

ANSWER:  ABC  /   //////// 

Now  write  down  all  the  other  different  ordera. 

2.  Calculate  the  number  of  different  waye  a  claee  of  10  etudenta  can  be  divided  up  Into  two  troupe, 
one  of  them  with  3  etudente  and  the  other  with  7. 

ANSWER:  There  are        different  waye. 
Briefly  explain  the  method  you  have  uead. 

3.  An  Ice  cream  lhop  eelle  five  different  of  Ice  cretm:  chocolate,  lemon,  etrawberry,  apricot 
and  vanilla.  How  many  tube  of  three  different  Iflnde  of  Ice  cream  can  be  bought? 

ANSWER:  There  are  different  tube. 

Briefly  explain  the  procedure  that  you  have  ueed. 

Inventory  Problems: 

We  give  the  name  problems  of  Inventory  to  those  problems  like  la)  taken  from 
Green's  research  (1981).  in  which  the  student  is  asked  for  an  inventory    of    all  the 

er|c  267 

1  -243 

possible  cases  produced  by  a  certain  combinatorial  operation,  in  this  case,  the 
permutations  of  three  elements.  These  problems  are  ideal  for  our  purpose  of  knowing 
the  combinatorial  capacity  of  the  students  before  the  instruction;  on  the  other  hand, 
in  Navarro-Peiayo  (19?!!  the  little  emphasis  put  on  this  type  of  exercise  in  the 
school  books,  has  been  pointed  out. 

Problems  of  calculation  of  ttje  number  of  possibilities 

In  these  statements,  as  in  problems  2)  and  3)  the  student  is  asked  the  number  of 
possibilities  without  explicitly  asking  him  for  the  inventory  of  the  same,  thus 
having  to  identify  the  combinatorial  operation.  This  is  one  of  the  difficulties 
described  by  Hadar  and  Hadass  (1981)  to  solve  combinatorial  problems. 

Task  variables  considered 

The  task  variables  that  have  been  taken  'into  consideration  for  the  choice  of 
problems  have  been  the  following: 

a)  Type  of  combinatorial  operation  (permutations,  combinations....).  This  variable 
has  been  one  of  the  determining  factors  of  the  difficulty  of  the  problems  in 
Fischbein  and  Gazit's  research  (1988). 

b)  Context.  Likewise,  the  previous  authors  distinguished  the  context  of  letters, 
numbers,  people  and  objects;  we  have  also  included  a  problem  in  which 
undistinguishable  objects  are  considered,  since  Lecoutre  (1985)  indicated  the  greater 
difficulty  in  employing  these  types  of  objects.  Likewise,  we  have  included  a 
geometrical  context,  in  item  number  5. 

c)  Value  given  to  the  parameters  m  and  n  that  have  also  been  a  factor  of  difficulty 
described  by  Fischbein  and  Gazlt  (1988). 

d)  Implicit  mathematical  model.  According  to  what  Dubois  states  (1984),  the  simple 
combinatorial  configurations  can  be  classified  in  three  models:  selections,  that 
emphasize  the  model  of  sampling,  distributions,  related  to  the  concept  of  mapping  and 
partition  or  division  of  a  set  into  subsets.  We  have  considered  these  three  models, 
in  addition  to  that  of  simple  ordering  (arrangement)  that  can  be  considered  as  a 
particular  case  in  any  of  them. 

e)  Help  provided.  To  give  an  example  or  not  in  the  statement,  and  in  the  case  of 
giving  it,  whether  a  table  or  a  tree  diagram  is  used.  This  type  of  help  was  provided 




in  items  1.  4,  5,  7  and  8. 

The  context,  model,  values  or  the  parameters  and  combinatorial  operations  used 
in  each  one  of  the  items  appear  in  Table  1.  The  numeration  of  the  problems  does  not 
correspond  to  the  order  of  presentation  in  the  questionnaire. 


For  this  first  pilot  study  we  have  preferred  to  choose  an  intentional  sample 
that  has  been  made  up  of  a  total  of  106  pupils:  57  pupils  from  the  8th  course  of 
Primary  Education  "EGB"  (14  years  of  age)  who  had  not  had  any  specific  instruction  in 
Combinatorics  when  the  test  was  carried  out  and  49  pupils  from  the  1st  course  of 
Secondary  education  "BUP"  (15  years  of  age)  after  the  period  of  Combinatorics 
teaching.  The  percentages  of  the  correct  answers  to  each  question  of  the  two  groups 
of  pupils  are  presented  in  Table  2. 


Percentage  of  correct  answers  in  the  different  problems 

Problem   Context  Combinatorial       Percentage  correct  answers 

Operation  8th  EGB  1st  BUP 

la    Arrange  people 








lc  " 




2    Partition  (people) 




3    Select  objects 




4    Throw  coi  ns 


56.  1 


5    Select  paths 

Product  Rule 



6    Select  people 




7    Distribute  objects 




8    Select  numbers 




9a  Arrange  letters 




9b  Arrange  letters 




We  can  observe  that  in  practically  all  the  items  the  percentages  of  correct 
answers  are  higher  in  pupils  of  "BUP".  There  is  an  exception  in  Item  5,  corresponding 
to  the  rule  of  the  product,  a  type  of  problem  that  in  our  study  (Navarro-  Pelayo 
(1991))  we  saw  was  little  used  In  the  text  books.  In  this  case  one  of  the  formulas 



corresponding  to  the  combinatorial  operations  cannot  be  directly  applied  because  it 
is  dealing  with  the  cartesian  product  of  the  two  different  subsets.  The  pupils  of  14 
(8th  "EGB")  who,  in  their  majority  have  tried  to  solve  the  problem  directly  using  an 
inventory,  have  obtained  better  results  than  the  pupils  of  "BUP"  who  have  tried  to 
apply  one  of  the  formulas  known  for  this  case. 

By  considering  the  magnitude  of  the  parameters  we  c?n  clearly  see  the  difference 
in  difficulty  when  the  value  of  the  number  of  elements  to  be  selected  is  2  or  3.  In 
all  these  cases  there  has  been  an  important  percentage  of  correct  answrrs,  even  in 
pupils  who  have  had  no  instruction,  from  which  a  good  combinatorial  capacity  can  be 
deduced  when  the  number  of  objects  is  small.  When  this  number  grows  the  pupil  of 
"EGB"  has  not  been  capable  of  satisfactorily  completing  a  procedure  and  as  he  did  not 
know  the  formula,  has  been  unable  to  deduce  in  many  cases.  It  is  here  where  we  can 
appreciate  a  greater  effect  of  the  instruction  and  the  age. 

Another  item  where  there  have  been  a  significant  number  of  correct  answers,  in 
spite  of  using  value  4  for  the  parameter  m,  has  been  item  7  where  a  tree  diagram  was 
given  as  help.  This  agrees  with  the  importance  that  Fischbein  and  Gazit  (1988)  give 
to  the  tree  diagram  as  a  model  to  solve  the  combinatorial  problems.  In  general, 
providing  the  pupil  with  an  example,  has  supposed  a  greater  facility  of  the  problem, 
especially  in  item  la)  where  the  percentage  of  correct  answers  has  been  surprising  In 
the  pupils  of  "EGB",  taking  into  account,  that  Piaget's  and  Fischbein's  theories 
point  out  the  permutations  as  the  most  difficult  combinatorial  operation  known  before 
instruction.  However,  this  percentage  drops  drastically  when  we  pass  to  the 
permutations  of  4  and  5  people,  and  it  even  drops  (although  not  as  drastically)  in 
the  pupils  who  have  received  instruction.  The  pupils  of  the  first  group  lack  the 
recursive  capacity  to  form  the  permutations  of  4  elements  once  those  of  3  have  been 

The  difference  of  di',"!*uity  due  to  the  type  of  combinatorial  operation  does  not 
seem  as  big  in  our  study  as  that  due  to  the  size  of  the  parameters,  since  before 
instruction  this  has  been  the  main  determinant  of  success  and  after  there  is  not  a 
very  clear  difference. 

By  considering  the  mathematical  model  under  which  the  combinatorial  operation  is 
presented,  we  do  not  observe  important  differences  in  the  model  of  selection  (Items 
3,  4,  5,  6  and  8),  arrangement  (Items  la,  lb,  lc,  9a  and  9b)  and  the  positioning  or 
application  (Item  7),  except  in  the  case  of  the  permutations  with  repetition  that 
have  turned  out  to  be  much  more  difficult.  In  this  case,  the  main  determinant  of  the 
difficulty  Is  the  fact  that  distinguishable  objects  appear  mixed  with 
Indistinguishable  ones.  However,  we  have  found  quite  an  accused  difference  In  Item  2 



referring  to  a  context  of  partition  of  a  set  Into  two  subsets  In  which  only  147.  of 
correct  answers  have  been  found  after  Instruction,  In  spite  of  being  a  typical 
combinatorial  statement. 


Error  of  order 

This  mistake,  described  by  Fischbeln  and  Gazit  (1988)  consists  of  confusing  the 
criteria  of  combinations  and  arrangements.  For  example,  in  item  3,  when  the  pupil 
considers  different  tubs  "chocolate  with  lemon  and  strawberry"  and  "chocolate  with 
strawberry  and  lemon".  This  mistake  has  been  found  In  35  of  the  total  of  the  problems 
solved  by  the  children  in  "BUP",  representing  16.57.  of  the  total  errors  in  this  group 
and  only  in  8  of  the  pupils  of  "EGB"  (2.67.  of  errors).  From  this  result  a  greater 
relative  incidence  of  this  type  of  error  can  be  Induced  in  the  pupils  who  try  to 
apply  one  of  the  combinatorial  formulas,  which  only  occurs  In  the  group  who  have 
received  instruction. 

Error  of  repetition  and  exclusion 

We  have  given  this  name  to  the  case  of  the  pupil  who  does  not  consider  the 
possibility  of  repeating  the  elements  or  when  there  is  no  possibility  to  do  so,  the 
pupil  uses  it.  For  example,  in  item  1,  when  the  pupil  uses  the  formula  of  variations 
with  repetition  or  repeats  an  element  within  the  permutation. 

In  the  case  of  item  9b),  that  deals  with  the  permutations  of  5  letters,  two  of 
them  being  the  same:  A,  B,  C,  D,  D,  another  option  followed  by  some  pupils  Is  to 
exclude  the  repeated  letter  and  form  the  permutations  of  the  remaining  ones,  thus 
taking   PR  =P  .    We   have   called   this   mistake    exclusion   error.    Letter   D  is 

5,1,1.1,2  4 

considered  to  be  fixed  and  its  permutation  with  the  remaining  ones  is  not  considered. 
This  error  has  been  committed  mainly  by  the  pupils  of  "EGB"  (17  cases;  5.67.  of  their 
errors)  and  acquires  special  Importance  since  it  has  only  been  given  when  associated 
to  a  particular  problem,  and  thus  seems  typical  of  this  type  of  problem. 

In  total  there  have  been  53  errors  of  repetition  in  "BUP",  which  represents  257. 
of  the  total  errors  as  opposed  to  8  In  EGB  (2.67.  errors),  due  to  the  fact  that  the 
first  group  prefer  the  use  of  formulas.  We  must  also  point  out  the  greater  Importance 
of  this  error  as  opposed  to  the  error  of  order,  in  the  group  of  pupils  with 

H2I1  systematic  enumeration 

This  type  of  error  described  by  Fischbeln  and  Gazit,  consists  of  trying  to  solve 


1  -247 

the  problem  by  an  enumeration  using  trial  and  error  without  a  recursive  procedure 
that  leads  to  the  formations  of  ail  the  possibilities.  It  has  been  one  of  the  most 
frequent  mistake  in  both  .groups,  24  cases  in  "BUP"  (11.37.  of  errors),  96  (31.6%)  in 
"EGB".  This  error  has  occurred  specially  before  instruction,  since  the  students  have 
used  the  enumeration  as  the  most  frequent  strategy  in  solving  the  problems. 

We  must  point  out  that  in  our  work  (Navarro-Pelayo;  1991)  we  showed  that  the 
enumeration  exercises  are  not  usually  proposed  to  the  pupils  since  it  is  considered 
this  is  an  ability  that  they  have  already  acquired.  However,  we  think  that  the 
results  of  this  first  sample  confirm  those  of  other  authors  like  Mendelsohn  (1984) 
that  many  pupils,  although  in  the  stage  of  formal  operations,  have  difficulties  with 
systematic  enumeration.  We  have  even  seen  that  these  difficulties  continue  in  some  of 
the  students  after  their  period  of  instruction  in  the  first  course  of  Secondary 
education  "BUP". 

Error  in  the  arithmetic  operations  used 

The  pupils  of  "BUP"  have  studied  combinatorics  and  in  some  cases  have  identified 
the  operation  correctly,  using  the  formula  to  solve  the  problem.  On  other  occasions 
this  operation  has  not  been  identified  -  or  at  least  it  has  not  been  indicated 
explicitly  -  and  they  try  to  deduce  the  series  of  arithmetic  opemions  necessary  for 
the  solution  using  a  direct  combinatorial  reasoning.  That  is  they  try  to  find  a 
formula,  not  valid  for  the  general  case,  but  that  can  be  used  in  the  given  problem. 
This  strategy  is  also  quite  frequent  in  the  pupils  of  8th  of  "EGB".  In  the  case  that 
a  correct  formula  has  not  been  found  with  this  procedure  we  will  say  that  there  is  an 
error  in  the  arithmetical  operations.  This  type  of  mistake  appears  in  a  total  of  20 
problems  solved  by  the  pupils  of  "BUP"  (9.47.  of  the  errors)  and  56  for  those  of  "EGB" 
(18.87.  of  the  errors). 

Mistaken  intuitive  response 

This  error  identified  by  Fischbein  (1975),  consists  of  not  justifying  the 
response,  only  giving  a  mistaken  numerical  solution.  The  frequency  of  this  type  of 
error  has  been  82  cases:  277.  of  the  errors  in  "EGB"  as  opposed  to  17.97.  in  "BUP"; 
this  type  of  response  is  still  important  in  "BUP". 

Other  errors 

-  Badly  applied  formula,  due  to  not  remembering  it,  although  the  combinatorial 
operation  has  been  correctly  Identified:  11  cases  in  secondary  school  pupils. 

-  Confusion  of  the  parameters  when  applying  the  formula:  5  cases  In  secondary  school 


1  -248 


-  Not  remembering  a  property  of  the  combinatorial  numbers. 

This  error  has  appeared  as  associated  to  item  2,  in  which  the  pupil  should  realize 
thr.t  by  considering  10  students,  the  same  number  of  groups  can  be  formed  with  3  as 
with  7,  so  once  one  of  these  groups  is  formed  the  other  one  is  determined.  The  pupils 
who  do  not  identify  this  property  add  C10.3  +  C10.7  to  give  the  solution. 

-  Incorrect  interpretation  of  the  tree  diagram  (6  pupils  of  "BUP"  and  9  of  "EGB").  In 
spite  of  the  importance  given  to  tills  didactic  resource  by  Flschbein  as  an  aid  in 
combinatorial  problem  solving,  we  have  found  ourselves  with  cases  of  bad 
interpretation  of  the  diagram  given  in  exercise  7,  even  in  pupils  who  have  been 
instructed  in  the  use  of  this  resource. 


B0RASS1,  R.  (1987).  Exploring  mathematics  through  the  analysis  of  errors.  For  the 
learning  of  mathematics.  Vol.  7,  3,  p.  2-8. 

DUBOIS,  J.  G.  (1984).  Une  systematique  des  configurations  comblnatolres  simples. 
Educational  Studies  in  Mathematics,  v.  15,  pp.  5-31. 

F1SCHBEIN,  E.  (1975).  The  Intuitive  source*  of  probabilistic  thinking  in  children. 
Reidel.  Dordrecht. 

F1SCHBEIN,  E.  y  GAZ1T,  A.  (1988).  The  combinatorial  solving  capacity  in  children  and 
adolescent.  Zentralblatt  fur  Didaktik  der  Mathemitik,  5,  pp.  193-98. 

GREEN,  D.  R.  (1981).  Probability  concepts  in  school  pupils  aged  11-16  years.  Ph.  I). 

Thesis.  Loughborough  University. 

HADAR,  N.  y  HADASS,  R.  (1981).  The  road  to  solving  a  combinatorial  problem  is  strewn 
with  pitfalls.  Eduactionai  Studies  in  Mathematics,  v.  12,  pp.  435-443. 

LECOUTRE,  M.P.  (1985).  Effect  d' information  de  nature  combinatoire  et  de  nature 
frequentlelle  sur  les  judgements  probablllstes.  Recherches  en  Didactique  des 
Mathematlques,  v. 6,  pp.  193-213. 

MAURY,  S.  (1986).  Contribution  a  1  "etude  didactique  de  quelques  notions  de 
probability  et  de  combinatoire  a  travers  la  resolution  des  problemes.  These  d'Etat. 
Unlverslte  Montpelller  11. 

MENDELSOHN,  P.  (1981).  Analyse  procedural  et  analyse  structural  des  actlvltes  de 
permutations  d'objects.  Archives  de  Psychoiogie,  v.  49,  pp.  171-197. 

P1AGET,  J.  e  1NHELDER,  B.  (1978).  La  genese  de  l'Idee  de  hasard  chez  l'enfant. 
Presses  unlversltalres  de  France.  Paris. 

NAVARRO-PELAYO,  V.  (1991).  La  ensefianza  de  la  combinatorla  en  bachlllerato.  Memorla 
de  Tercer  Ciclo.  Dpto  de  Dldactica  de  la  Matematlca.  Universldad  de  Granada. 

1  -249 


A  random  sample  of  55  grade  3  and  4  children  from  six  schools  were  observed  while 
tackling  five  versions  of  a  real  world  problem  based  on  quotition  division.  The  children  were 
provided  with  simulated  bottles  of  medicine  (in  tablet  and  liquid  form),  which  showed  the 
total  contents  and  the  amount  to  be  taken  each  day,  and  were  asked  how  many  days  the 
medicine  would  last.  Calculators  and  concrete  materials  were  provided,  as  well  as  pencil  and 
paper.  For  all  but  the  two  most  difficult  questions,  children  overwhelmingly  chose  mental 
computation  as  their  calculating  device.  Children  predominantly  used  repeated  addition  (or 
subtraction)  rather  than  division,  which  was  almost  always  only  used  in  conjunction  with  a 
calculator.  Difficulties  encountered  by  the  children  who  used  calculators  confirm  the 
mathematical  sophistication  required  to  interpret  the  answers  thus  obtained. 


There  is  now  a  substantial  body  of  research  into  children's  understanding  of  multiplication  and 
division.  Among  major  factors  found  to  influence  children's  success  in  selecting  the  appropriate 
operations  for  word  problems  requiring  division  are  the  extent  of  familiarity  of  the  context  and  the 
structural  nature  of  the  problem,  with  partition  problems  producing  a  higher  rate  of  success  than  quotition 
or  rate  problems  (Bell,  Fischbein,  &  Greer,  1984;  Fischbein,  Deri,  Nello  &  Marino,  1985).  Prior  to  a 
study  involving  grade  5,  7  and  9  children,  Fischbein  et  al  hypothesised  that  children  have  intuitive 
models  of  division,  based  on  both  partition  and  quotition,  which  they  can  evoke  as  appropriate.  Not  only 
did  partition  problems  prove  easier  than  quotition,  but  grade  5  children  performed  considerably  worse  on 
quotition  questions  than  older  children.  For  example,  for  the  question  "The  walls  of  a  bathroom  are  280 
cm  high.  How  many  rows  of  tile  are  needed  to  cover  the  walls  if  each  row  is  20  cm  wide?",  although 
44%  of  grade  5's  correctly  chose  280  +  20  as  the  operation  required,  41%  chose  280  x  20.  The  success 
rave  for  grades  7  and  9  were  77%  and  80%,  respectively.  This  led  the  authors  to  conjecture  that  partition 
is  the  only  intuitive  primitive  model,  with  children  acquiring  the  quotition  model  with  instruction. 

*  The  interview  referred  to  in  this  paper  was  developed  and  conducted  in  collaboration  with  Ron  Welsh 
and  Kaye  Stacey  (Melbourne  University)  and  Jill  Cheeseman  (Deakin  University),  with  support  from  the 
Victoria  College  (now  Deakin  University)  Special  Research  Fund,  a  Melbourne  University  Special 
Initiatives  Grant  and  the  Melbourne  University  Staff  Development  Fund. 

Susie  Groves 
Deakin  University  •  Burwood  Campus 

1  -250 

Kouba  (1989)  analysed  the  solution  strategies  of  grade  1  to  3  children.  She  proposed  three  intuitive 
models  for  partition  -  sharing  by  dealing,  repeated  subtraction  and  repeated  addition  (using  guesses  for 
the  addend).  It  is  well  known  that  children  frequently  resort  to  informal  addition  based  strategies  for  a 
variety  of  problems  (Han,  1981,  p.47;  Bergeron  &  Herscovics,  1990,  p.  32).  Kouba  found  that  children 
employed  repeated  subtraction  and  repeated  addition  for  both  partition  and  quotition  problems,  and  hence 
questioned  the  separation  of  the  intuitive  models  for  these  types  of  division. 

Procedural  knowledge  without  conceptual  knowledge  and  the  ability  to  use  it  in  meaningful 
situations  is  of  little  use.  This  is  particularly  true  in  an  age  when  reliable  mental  methods  and  an  ability  to 
use  calculators  (together  with  an  understanding  of  the  meaning  of  the  operations  and  the  real  world 
problems  which  they  model)  are  sufficient  for  all  practical  purposes  (Han,  1981,  p.47;  Bell,  Fischbein, 
&  Greer,  1984,  p.130;  Bergeron  &  Herscovics,  1990,  p.  34).  Yet  many  children  are  being  taught  to  do 
calculations  without  being  able  to  describe  situations  in  which  they  are  applicable  and  consequently  do 
not  find  "real  world"  possibilities  reflected  in  school  mathematics  (Greer  and  McCann,  1991,  p.  85; 
Graeber  and  Tirosh,  1990,  p.583). 

Carpenter  (1986)  points  out  that  "before  receiving  instruction,  most  young  children  invent  informal 
modelling  and  counting  strategies  to  solve  basic  addition  and  subtraction  problems"  (p.  114).  Neuman 
(1991)  reports  on  children's  "original"  informally  developed  conceptions  of  division,  commenting  that 
"young  children  who  have  not  been  formally  taught  division  seem  to  believe  that  it  is  possible  to  solve  all 
problems  in  some  way"  (p.76).  Children  were  again  found  to  use  repeated  addition  and  repeated 
subtraction  for  both  partition  and  quotition  problems.  She  questions  the  early  introduction  of  the  division 
algorithm  as  opposed  to  the  elaboration  of  children's  own  informally  developed  thoughts. 

Results  obtained  from  a  large  sample  of  grade  5  and  6  children,  using  a  pencil  and  paper  test  of 
problem  'olving  (Stacey,  Groves,  Bourke  &  Doig,  in  press),  indicate  that  most  upper  primary  children 
do  not  use  learned  multiplication  and  division  skills,  with  large  numbers  of  these  children  still  using 
repeated  addition  to  solve  a  problem  based  on  quotition  (Stacey,  1987,  p.  21). 

While  there  is  no  centralised  curriculum  in  Victoria  (Australia),  most  primary  schools  base  their 
mathematics  policy  on  the  state  guidelines  (Ministry  of  Education,  Victoria,  1988).  At  grade  3,  children 
are  expected  to  learn  number  facts  including  division  by  2,  3,  4,  5  and  10,  as  well  as  use  calculators  for 


1  -.251 

computation,  while  at  grade  4  they  are  usually  introduced  to  division  of  2  and  3  digit  numbers  by  1  digit 
numbers  (p.97-8).  Pencil  and  paper  "long  division"  has  not  been  included  in  guidelines  for  over  6  years. 

This  paper  reports  on  the  extent  to  which  55  grade  3  and  4  children,  who  were  observed  while 
tackling  a  "real  world"  problem  based  on  quotition  division,  as  part  of  a  longer  interview,  were  able  to 
find  correct  or  reasonable  answers,  the  calculating  devices  they  chose  to  use  and  the  extent  to  which  they 
made  sensible  and  efficient  use  of  calculators. 

For  the  past  three  years,  as  part  of  the  Victoria  College  Calculator  Project  and  the  University  of 
Melbourne  Calculator-Aware  Program  for  the  Teaching  of  Number,  children  entering  six  schools  have 
been  given  "their  own"  calculator  to  use  at  all  times.  Teachers  have  been  provided  with  a  program  of 
professional  support  to  assist  them  in  using  calculators,  not  just  as  "number  crunchers",  but  also  as  a 
means  to  create  a  rich  mathematical  environment  for  children  to  explore  (see,  for  example.  Groves,  1991; 
Groves,  Cheeseman,  Clarke,  &  Hawkes,  1991,  Welsh,  Rasmussen  &  Stacey,  1990). 

In  1991,  as  pan  of  an  investigation  of  the  long-term  learning  outcomes  of  the  projects,  over  430 
grade  3  and  4  children  at  these  six  schools  were  given  a  written  test  and  a  test  of  calculator  use.  These 
children,  who  have  not  been  involved  in  the  calculator  projects,  form  the  control  group  for  the  study.  In 
addition  to  the  tests,  a  random  sample  of  55  of  the  grade  3  and  4  children  were  given  a  25  minute 
interview,  designed  to  test  their  understanding  of  the  number  system;  their  choice  of  calculating  device, 
for  a  wide  range  of  numerical  questions;  and  their  ability  to  solve  "real  world"  problems  amenable  to 
multiplication  and  division,  with  or  without  calculators.  Throughout  the  interviews,  children  were  free  to 
use  whatever  calculating  devices  they  chose.  Unifix  cubes  and  multi-base  arithmetic  (M  AB)  blocks  were 
provided  as  well  as  pencil  and  paper  and  calculators.  Many  of  the  questions  were  expected  to  be 
answered  mentally.  The  tests  and  interviews  will  be  carried  out  again  at  grade  3  and  4  level  in  1992  and 
1993.  Among  the  hypotheses  for  the  long-term  study  is  an  expectation  that  children  involved  in  the 
calculator  projects  will  perform  better  on  the  "real  world"  problems,  selecting  appropriate  processes  more 
frequently  and  making  better  use  of  calculators. 

This  paper  focuses  on  interview  results  from  the  "real  world"  problem  amenable  to  division  -  a 
simplified  version  of  the  question  from  the  problem  solving  test  referred  to  earlier  (Stacey,  Groves, 



Bourkc  &  Doig,  in  press).  The  question  consists  of  five  parts,  each  with  the  same  structure.  In  the  first 
two  parts,  children  were  presented  with  clear  bottles  containing  the  appropriate  number  of  white, 
medicine-like  tablets  (actually  sweets).  The  bottles  were  attractively  labelled  with  the  contents  and  the 
amount  to  be  taken  each  day  -  for  example,  in  Ml  the  label  clearly  displayed  "15  tablets  take  3  each  day", 
as  well  as  the  distractor  "$7.43".  For  the  remaining  three  parts,  accurate  volumes  of  coloured  liquid  were 
used  with  information  such  as  "  120  ml  take  20  ml  each  day"  and  a  price.  For  this  example  (the  first  using 
liquid  "medicine"),  20  ml  was  poured  from  the  bottle  into  a  clear  medicine  measure.  In  each  case, 
children  were  asked  how  many  days  the  medicine  would  last.  (For  further  details,  see  Welsh,  1991 .) 

As  well  as  their  answers,  children's  choice  of  calculating  device  were  recorded.  These  were 
classified  as  calculator,  written  algorithm,  Unifix  or  MAB,  mental  (which  was  further  sub-classified  to 
indicate  automatic  response  and  use  of  fingers)  and  other  (such  as  drawing  or  the  use  of  non-standard 
algorithms).  Wherever  possible,  the  mathematical  processes  used  were  also  recorded.  Original 
classifications  of  the  processes  included  division,  counting  on  (repeated  addition),  repeated  subtraction, 
multiplication  and  other  less  frequently  used  processes. 

Frequencies  of  correct  and  incorrect  answers,  use  of  calc  lating  devices  and  solution  processes  for 
each  of  pans  Ml  to  M5  of  the  medicine  question  are  shown  on  the  "double-sided"  Table  1 .  The  left  side 
shows  choice  of  calculating  device  against  correctness  of  answer,  while  the  right  side  shows  solution 
processes.  In  those  parts  of  the  question  where  remainders  occur  (M2,  M4  and  M5),  an  extra  category  of 
answer  is  included  to  indicate  answers  which,  while  incorrect,  give  the  correct  number  of  whole  days 
(or,  in  the  case  when  the  answer  is  7.5  days,  give  8  days).  The  categories  for  choice  of  calculating  device 
and  solution  processes  have  been  collapsed.  Categories  rarely  used  are  included  under  "other"  -  e.g. 
standard  written  algorithms  (which  were  never  used  successfully)  and  the  (rare)  attempts  to  use  an 
incorrect  algorithm  (such  as  a  single  subtraction). 

Correctness  of  answers.  Table  1  shows  the  dramatic  decrease  in  correct  answers  when 
remainders  are  involved.  Even  for  the  relatively  easy  problem  of  "21  tablets,  4  per  day",  less  than  45% 
of  children  give  a  correct  answer  such  as  "5      days"  or  "5  days  with  1  tablet  left  over",  although  a 




Table  t:  Frequencies  of  correct  and  incorrect  answers, 
use  of  calculating  devices  and  solution  processes 



M.    C     O  NA 


AR  CO    DI  UM    O  NA 



IS  tablets, 
3  per  day 

NA  3 

38      2      6  0 
7      0  10 
0     0      0  1 


3     17      2    17      7  0 
0      10      6  10 
0      0     0     0      0  1 




45      2      7  1 


3     18      2    23      8  1 


21  tablets, 
4  per  day 

*  ■» 

17  4  3  0 
5      3      3  0 

10  0  3  0 
0      0     0  7 


3     12      4      4      1  0 
0      3      2      2      4  0 
0      4      0      6      3  0 
0      0     0     0      0  7 




32      7      9  7 


3     19      6    12      8  7 


120  ml, 
20  ml  /day 



26      5      1  0 
11      2      3  0 
0     0      0  7 


5  13  6  7  1  0 
0  2  0  9  5  0 
0      0     0     0      0  7 



37      7      4  7 


5     15      6    16      6  7 


300  ml, 
40  .rd  /day 


6      8  10 
5      5      0  0 
18      2      4  0 
0      0     0  6 


0  6  7  2  0  0 
0  5  5  0  0  0 
0  5  1  11  7  0 
0      0      0     0      0  6 




29     15      5  6 


0     16    13     13      7  6 


375  ml, 
24  ml  /day 


*  6 


0    13      0  0 
17      0  0 
7     10      8  0 
0     0     0  9 



0  0  13  0  0  0 
0  1  7  0  0  0 
0  5  4  7  9  0 
0      0     0      0      0  9 



8    30      8  9 


0      6    24      7      9  9 


'  M  -  mental;  l.  -  calculator,  vj  -  ouicr  i  e.g.  drawing,  oiw-its;;        ■  uu  «io«h  jhvu 

2  AR  -  automatic  response;  CO  -  counting  on/back  (repeated  addition/subtraction);  DI  -  division; 
UM  -  unknown  mental  process;  O  -  other  (  e.g.  multiplication,  single  subtraction) 

3  *J  -  correct  answer;  X  -  incorrect  answer;  NA  -  no  answer  given 
incorrect  answer  with  integer  part  correct  (e.g.  5,  5+,  5  remainder  3,  5  remainder  25) 
incorrect  answer  with  integer  part  correct  (e.g.  7, 7+,  7  remainder  80, 7  remainder  2)  or  8 
incoirect  answer  with  integer  part  correct  (e.g.  15, 15+,  15  remainder  ?,  15  3/4) 

4  * 

5  * 

6  * 




1  -254 

further  20%  give  an  answer  involving  5  days.  Incorrect  answers  for  M2  range  from  2  to  15  days,  with  5 
children  giving  answers  of  over  10  days.  Given  the  provision  of  concrete  models  and  the  familiarity  of 
the  situation,  these  results  confirm  that  many  children  find  it  difficult  to  relate  school  mathematics  to  real 
world  problems.  While  the  liquid  medicine  problem  involving  a  whole  number  answer,  M3,  produced  a 
high  rate  of  success,  it  was  anticipated  that  M4  and  M5  would  be  much  more  difficult,  as  children  would 
be  unlikely  to  have  the  skills  to  find  a  solution  except  by  using  a  calculator.  (In  fact,  these  parts  were 
included  specifically  to  determine  the  extent  to  which  children  can  successfully  use  calculators  to  solve 
such  problems.)  The  results  from  these  parts  confirm  our  expectations,  with  over  half  of  the  children 
who  achieve  a  correct  or  reasonable  answer  for  M4  using  a  calculator,  and  only  one  child  succeeding 
without  a  calculator  for  the  more  difficult  MS. 

Lhe  of  calculating  devices.  For  all  but  M5,  children  showid  an  overwhelming  preference  for 
mental  computation.  Even  for  M4,  where  we  had  expected  children  to  use  calculators,  29  of  the  49 
children  who  attempted  the  problem  chose  to  do  it  mentally.  For  all  pans,  approximately  half  of  the 
children  who  used  mental  computation  augmented  it  with  the  use  of  fingers.  Only  three  children  gave 
responses  automatically  to  any  questions  -  including  one  child  who  gave  immediate  correct  answers  to 
the  first  four  parts  and  then  used  his  calculator  to  incorrectly  read  the  answer  to  M5  as  "15  remainder 
point  625".  Four  children  successfully  used  drawings  or  concrete  materials  for  some  or  all  of  Ml  to  M3, 
but  only  one  of  these  was  successful  in  either  of  the  other  two  pans  -  M4  using  a  calculator. 

Processes  used.  Only  a  small  handful  of  children  were  observed  giving  "automatic  responses" 
to  the  first  three  pans.  For  Ml,  a  large  number  of  children  gave  the  correct  answer  after  pausing  to 
calculate  mentally.  It  was  often  impossible  to  determine  the  mental  processes  used  as  time  constraints  did 
not  allow  for  extended  probing  -  hence  the  classification  "unknown  mental  process".  For  the  first  three 
parts,  counting  on  or  counting  back  (repeated  addition  or  subtraction)  outnumbered  all  other  known 
processes  almost  two  to  one.  Children  only  began  to  use  division  when  the  numbers  dictated  the  use  of  a 
calculator.  In  fact  for  M5  the  only  correct  answers  were  obtained  using  division  on  a  calculator. 

Effective  use  of  calculators.  While  the  first  requirement  for  effective  calculator  use  in 
problems  such  as  these  is  to  recognise  the  operation  as  division,  it  is  also  necessary  to  be  able  to  make 


1  -255 

sense  of  the  answer  displayed.  The  difficulties  were  particularly  apparent  in  M5,  where  30  of  the  46 
children  who  attempted  the  problem  used  a  calculator,  but  only  13  found  the  correct  answer.  Of  the 
remaining  16  children.  9  were  unable  to  read  the  number  displayed  correctly.  Such  difficulties  are  further 
highlighted  by  an  earlier  question  on  the  interview.  Children  were  shown  278  +  39  and  "the  answer 
found  by  someone  using  a  calculator"  -  i.e.  7- 1282051.  They  were  asked  firstly  to  read  the  number  and 
then  to  say  "about  how  big"  it  is  or  give  a  "number  close  to  it".  Only  14  children  were  able  to  read  the 
number  correctly  (i.e.  say  the  words  "seven  point  one  two  ..."),  with  16  passing  the  question  and  the 
remaining  25  giving  answers  like  "7  point  12  million       In  response  to  the  size  of  the  number,  15 
answered  in  the  range  5  to  9,  with  20  passing  and  18  giving  very  large  answers  (e.g.  7  million).  While 
this  level  of  understanding  is  to  be  expected,  it  highlights  the  fact  that  calculator  use  will  only  increase 
children's  facility  with  division  if  it  is  accompanied  by  considerable  change  in  children's  mathematical 
sophistication  and  overall  number  sense. 


These  results  confirm  the  fact  that  children  are  able  to  devise  their  own  means  of  solving  problems 
based  on  quotition  -  provided-  the  numbers  are  not  too  difficult  to  handle.  Their  methods  are 
predominantly  based  on  repeated  addition  or  subtraction.  Wherever  possible,  children  used  mental 
computation  in  preference  to  calculators,  with  almost  no  attempt  to  use  pencil  and  paper,  except  to  draw 
diagrams.  Nevertheless,  the  fact  that  several  children  consistently  used  completely  inappropriate 
operations,  such  as  a  single  subtraction,  to  arrive  at  blatantly  incorrect  answers  such  as  351,  days, 
indicates  the  extent  to  which  school  mathematics  is  seen  as  completely  divorced  from  the  real  world.  Bell 
et  al  had  predicted  that  calculators  would  allow  a  wider  range  of  numbers  to  be  considered  earlier  in 
primary  school,  but  warned  that  "this  still  leaves  thJ  question  of  what  meanings  the  pupils  can  attach  to 
the  operations"  (Bell,  Fischbcin.  &  Greer,  1984,  p.  130).  The  results  obtained  here  confirm  not  only  the 
importance  of  attaching  meaning  when  using  calculators,  but  also  the  necessity  to  develop  skills  such  as 
estimation  and  approximation  and  a  strong  intuitive  understanding  of  aspects  of  the  number  system  such 
as  decimals.  Future  results  to  be  obtained  from  children  with  long-term  experience  of  calculators  will 
hopefully  demonstrate  the  extent  to  which  this  can  realistically  be  achieved. 





Bell,  A.,  Fischbein,  E.,  &  Greer,  B.  (1984).  Choice  of  operation  in  verbal  arithmetic  problems:  The 
effects  of  number  size,  problem  structure  and  context.  Educational  Studies  in  Mathematics,  15, 

Bergeron,  J.  C,  &  Herscovics,  N.  (1990).  Psychological  aspects  of  learning  early  arithmetic.  In  P. 
Nesher  &  J.  Kilparrick  (Eds.).  Mathematics  and  cognition:  A  research  synthesis  by  the  International 
Group  for  the  Psychology  of  Mathematics  Education  (pp.  31-52).  Cambridge:  Cambridge 
University  Press. 

Carpenter,  T.  (1986).  Conceptual  knowledge  as  a  foundation  for  procedural  knowledge.  In  J.  Hiebert 
(Ed.).  Conceptual  and  procedural  knowledge:  The  case  of  mathematics  (pp.  1 13-132).  Hillsdale, 
NJ:  Lawrence  Ertbaum  Associates. 

Fischbein,  E.,  Deri,  M.,  Nello,  M.  S.,  &  Marino,  S.  M.  (1985).  The  role  of  implicit  models  in  solving 
verbal  problems  in  multiplication  and  division.  Journal  for  Research  in  Mathematics  Education,  16, 

Graeber,  A.  O.,  &  Tirosh,  D.  (1990).  Insights  fourth  and  fifth  graders  bring  to  multiplication  and 
division  with  decimals.  Educational  Studies  in  Mathematics,  21,  565-588. 

Greer,  B.,  &  McCann,  M.  (1991).  Children's  word  problems  matching  multiplication  and  division 
calculations.  In  F.  Furinghetti  (Ed.).  Proceedings  of  the  Fifteenth  International  Conference  on  the 
Psychology  of  Mathematics  Education  (Vol.  2,  pp.  80-87).  Assisi:  International  Group  for  the 
Psychology  of  Mathematics  Education. 

Groves.  S.  (1991).  Calculators  as  an  agent  for  change  in  the  teaching  of  primary  mathematics:  The 
Victoria  College  Calculator  Project  Unpublished  paper  presented  at  Annual  Conference  of 
Australian  Association  for  Research  in  Education. 

Groves,  S.,  Cheeseman,  J.,  Clarke,  C,  &  Hawkes,  J.  (1991)  Using  calculators  with  voung  children  In 
Jill  O'Reilly  &  Sue  Wettenhall  (Eds.)  Mathematics:  Ideas  (pp.  325-333).Melbou'rne:  Mathematical 
Association  of  Victoria,  Twenty-eighth  Annual  Conference. 

Hart,  K.  M.  (1981).  Children's  understanding  of  mathematics:  11-16.  London:  John  Murray. 

Hiebert,  J.,  &  Lefevre,  P.  (1986).  Conceptual  and  procedural  knowledge  in  mathematics:  An 
introductory  analysis.  In  J.  Hiebert  (Ed.).  Conceptual  and  procedural  knowledge:  The  case  of 
mathematics  (pp.  1-27).  Hillsdale,  NJ:  Lawrence  Erlbaum  Associates. 

Kouba.  V.  L.  (1989).  Children's  solution  strategies  for  equivalent  set  multiplication  and  division  word 
problems.  Journal  for  Research  in  Mathematics  Education,  20,  147-158. 

Ministry  of  Education,  Victoria  (1988).  The  mathematics  framework:  P-10.  Melbourne:  Ministry  of 
Education  (Schools  Division),  Victoria. 

Neuman,  D.  (1991).  Early  conceptions  of  division:  A  phenomenographic  approach.  In  F.  Furinghetti 
(Ed.).  Proceedings  of  the  Fifteenth  International  Conference  on  the  Psychology  of  Mathematics 
Education  (Vol.  3,  pp.72-79).  Assisi:  International  Group  for  the  Psychology  of  Mathematics 

Stacey,  K.  (1987).  What  to  assess  when  assessing  problem  solving.  Australian  Mathematics  Teacher, 
43(3),  21-24. 

Stacey,  K..  Groves,  S.,  Bourke,  S.,  &  Doig,  B.  (in  press).  Profiles  of  problem  solving.  Melbourne: 
Australian  Council  for  Educational  Research. 

Welsh,  R.,  Rasmussen,  D.,  &  Stacey,  K.  (1990).  CAPTN:  A  Calculator-Aware  Program  for  the 
Teaching  of  Number  .Unpublished  paper  presented  at  Twelfth  Annual  Conference  of  Mathematics 
Education  Research  Group  of  Australasia. 

Welsh,  R.  (1991)  The  CAPTN  Project  -  The  long  term  effect  of  calculator  use  on  primary  school 
children's  development  of  mathematical  concepts.  Unpublished  paper  presented  a:  Thirteenth 
Annual  Conference  of  Mathematics  Education  Research  Group  of  Australasia. 

UML  281 

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T.ynn  C.  Hart 
Deborah  H.  Najee-ullah 
Georgia  State  University 

As  teachers  change  their  pedagogical  practices  to  reflect 
current 'research  on  teaching  and  learning,  the  mathematics 
education  research  community  has  a  unique  opportunity  to 
study  this  process  of  change.    This  paper  will  present 
results  from  one  teacher,  Margaret,  in  one  project,  the 
Atlanta  Hath  Project  (AMP),  for  one  year,  1990-1991,  as  she 
attempts  to  modify  her  instructional  environment  to  reflect 
current  recommendations  for  reform.    This  pilot  work  lays 
the  groundwork  for  future  research  on  teacher  change  in  AMP. 

Learning  environments  are  emerging  that  are  quite  different  from 
the  lecture  dominated  mathematics  classroom  that  many  teachers  and 
students  have  experienced.    The  Atlanta  Hath  Project  (AMP),  a  four- 
year  National  Science  Foundation  sponsored  project  in  the  second  year 
of  operation,  is  implementing  a  research-based  teacher  education  model 
which  assists  teachers  in  constructing  new  knowledge  about  the 
teaching  and  learning  of  mathematics.    AMP  is  studying  how  these 
teachers  change  their  instructional  practices  over  four  years. 

(von  Glasersfeld,  in  press;  Wertsch  i  Toma,  1991)  and  of  metacognition 
(Flavell,  1975).     A  more  thorough  discussion  of  the  theoretical 
perspective  of  AMP  and  the  framework  for  studying  teacher  change  can 
be  found  elsewhere  (Hart,  1991). 

This  paper  will  explore  aspects  of  change  in  the  learning 
environment  and  in  teacher  knowledge  for  one  teacher,  Margaret,  now  in 
her  second  year  with  AMP.     In  particular  we  will  discuss  the  following 
questions  about  change  in  classroom  discourse  and  beliefs; 

Theoretical  Orientation 

The  theoretical  orientation  of  the  Atlanta  Math  Project  is 
grounded  in  the  theories  of  constructivism  and  social  constructivism 

Studying  Teacher  Change 


Whose  ideas  are  being  explored  in  Margaret's  classroom? 

What  types  of  questions  are  being  asked? 

How  is  conflict  resolved? 

How  is  student  thinking  encouraged? 

Is  mathematical  thinking  modeled? 

Who  are  students  talking  to? 

How  do  Margaret's  beliefs  about  learning  mathematics  and  teaching 
mathematics  change  in  year  one? 

How  do  Margaret's  beliefs  about  mathematical  tasks  and  content 

The  data  chosen  for  this  report  are  two  videotapes  of  Margaret 
teaching  her  grade  6  class  in  September  and  May  of  her  first  year  with 
the  project  and  responses  to  a  project  instrument  completed  by 
Margaret  before  and  after  year  one.     A  research  team  composed  of  the 
two  project  directors,  the  assistant  project  director,  a  research 
associate  and  three  graduate  students  have  met  regularly  during  year 
two  to  analyze  and  discuss  the  process  for  studying  change.     We  have 
used  Margaret  as  a  first  attempt  to  refine  our  methods.    We  will  use 
this  paper  as  an  opportunity  to  share  our  struggles  and  achievements 
and  to  solicit  feedback  on  our  work. 

Margaret  is  a  sixth  grade  teacher  in  a  rural  middle  school  near 
Atlanta,  Georgia.     Identified  by  the  mathematics  coordinator  for  her 
school  system  as  a  strong  teacher  with  leadership  ability,  she 
attended  five  days  of  staff  development  with  AMP  during  the  summer  of 
1990.     She  was  introduced  to  the  theories  of  constructivism  and 
metacognition,  the  positions  on  reform  set  forth  by  the  National 
Council  of  Teachers  of  Mathematics,  and  she  experienced  planning, 
teaching  and  reflecting  from  these  perspectives. 






1  -259 

Margaret  is  fairly  new  to  teaching.    Her  first  year  with  AMP 
(1990/91)  was  her  third  year  of  teaching.    A  mature  woman  with  a 
family  of  her  own,  she  turned  to  teaching  later  in  her  life.  Margaret 
showed  a  great  deal  of  maturity  and  intuitiveness  during  the  summer 
workshops.     She  demonstrated  caution  in  accepting  without  question  the 
ideas  we  explored,  but  displayed  a  willingness  to  learn  and  try  new 
approaches . 

snapshot  one:  July  31.  199Q. 

At  the  beginning  of  the  AMP  summer  workshop,  Hargaret  was  asked 
to  respond  to  a  set  of  statements  designed  to  elicit  her  current 
beliefs.     She  was  asked  to  describe  a  good  mathematics  teacher,  a  good 
mathematics  supervisor,  a  good  mathematics  student,  and  a  good  math 
problem.     Finally  she  was  asked,  how  do  middle  school  children  learn? 
Her  responses  provided  an  opportunity  to  gain  insight  into  some  of  her 
professed  beliefs  about  teaching  mathematics,  about  learning 
mathematics,  and  about  worthwhile  mathematical  tasks. 

Beliefs  about  teaching  mathematics.     Margaret  stated  that  a 
teacher  should  be  flexible  in  her/his  thinking,  creative,  open,  and 
display  an  enjoyment  of  mathematics.     She  felt  having  a  background 
strong  in  content  and  knowing  a  variety  of  instructional  strategies 
were  important.     She  said  teaching  should  be  organized  and  relevant. 
Teachers  should  demonstrate  respect  for  students  and  their  ideas  and 
should  exhibit  joy  and  interest  for  mathematics  to  students. 

Beliefs  ahnut  learning  mathematics.     Mairgaret  gave  a  description 
of  the  learner  which  included  inquiry,  thoughtful,  creativity,  and 
enjoyment  for  mathematics.     She  said  a  "good"  learner  has  a 
recognition  for  the  relevance  of  mathematics.     She  said  learners  need 
to  engage  in  both  individual  and  group  problem  solving  that  relate  to 
common,  everyday  experiences.     Learning  occurs  through  listening  and 


1  -260 

discussing  mathematics  with  others  and  when  students  reflect  and 
organize  their  knowledge  and  use  their  knowledge  about  math  in 
different  ways. 

Beliefs  about  worthwhile  mathematical  tasks.     Margaret  felt  tasks 
should  be  relevant  and  require  students  to  inquire.    Tasks  require 
discussion.    Tasks  should  provoke  thought,  require  reflection  and 
synthesis  of  math  knowledge,  and  tasks  should  cause  students  to  think 
about  and  solve  problems  in  more  than  one  way. 
Snapshot  two:  September  20.  1990 

The  first  videotape  of  Margaret  is  of  her  teaching  a  lesson  on 
estimation.     The  students  are  sitting  in  double-wide  desks  facing  the 
front  of  the  room  in  rows  and  columns.     The  class  discusses  the 
problem  with  Margaret  at  the  overhead  and  students  responding  to 
questions  by  raising  their  hands  and  being  called  on  one  at  a  time. 

The  nature  of  the  mathematical  task.     Margaret  uses  an  experience 
of  renting  lockers  the  children  had  engaged  in  the  day  before.  She 
poses  the  following  questions,  "Yesterday,  about  700  students  rented 
lockers.     The  lockers  cost  $2  each.    How  did  that  make  you  feel?"  The 
students  expressed  strong  feelings  (e.g.,  it  was  too  crowded,  it  was 
confusing,  etc.)  and  this  dialogue  opened  the  floor  for  further 
discussion.     Margaret  continued,  "How  much  money  did  the  school  take 
in?"    Since  lockers  could  be  shared  and  exact  numbers  of  students  had 
not  been  determined,  there  were  numerous  opportunities  for  estimation. 

Th«  nature  of  classroom  discourse.     The  direction  of 
communication  was  always  the  same.    Margaret  would  ask  a  question,  a 
student  would  respond  to  the  question,  and  Margaret  would  respond  to 
the  student.     If  a  student  disagreed  with  a  statement  made  by  another 
student  the  opposition  was  directed  toward  Margaret — not  the  other 


Margaret's  questions  were  a  mixture  of  one-right-answer  questions 
and  more  probing  questions  such  as  "Why?"  or  "Anything  else?"  Thought 
provoking  questions  were  also  raised  by  students.    Very  little  ti»e 
was  given  to  think  about  questions.    Margaret  did  not  explore  any  of 
these  questions  in  depth,  but  accepted  partial  thoughts  and  did  not 
require  justification,  elaboration  or  explanation.     The  students  did 
not  talk  to  one  another  and  were  not  asked  to  listen  to  the  responses 
of  others.    Although  presented,  alternative  ideas  were  not  explored 
and  as  a  result  conflict  was  not  explored  or  resolved.     Some  students 
were  called  upon  more  frequently  than  others.     Not  s.U  members  of  the 
class  were  included  in  the  discussion. 
Snapshot  threg:     May  3.  1991 

Margaret's  classroom  had  changed  during  the  school  year.  The 
double-wide  desks  were  now  arranged  with  two  desks  facing  each  other 
forming  small  seating  groups  of  four.     The  overhead  remained  at  the 
front  of  the  room. 

Tho  nature  of  the    mathematical  task.     The  tasks  presented  in 
this  second  lesson  explored  division  of  fractions.    The  children  were 
first  asked  to  contrast  the  division  problem,  1/2  divided  by  1/4, 
written  with  a  division  symbol  and  as  a  complex  fraction.    This  was 
followed  by  four  word  problems  that  involved  dividing  with  fractions. 
They  were  instructed  to  solve  these  by  writing  mathematical  sentences 
Finally  the  students  were  asked  to  count  the  number  of  "halves"  and 
"fourths"  in  two  inches,  three  inches,  five  and  one-half  inches,  etc. 
to  determine  the  pattern  and  "discover"  the  algorithm  for  division  of 

ThP  naturp  of  classrnnm  discourse.     Margaret  used  a  cooperative 
learning  technique  of  th ink-pair-share  to  begin  each  phase  of  the 
lesson.     Students  were  asked  to  think  about  the  problera(s),  to  share 


1  -262 

their  thinking  with  a  partner,  and  then  to  share  their  thinking  with 
the  class.     Margaret's  questioning  was  still  a  mixture  of  one-right- 
answer  and  aore  thought-provoking  questions.     Students  still  directed 
their  responses  and  questions  to  her,  but  Margaret  had  begun  to  act 
■ore  as  a  facilitator  of  the  discussion.     Consider  the  following 
dialogue  about  contrasting  the  two  division  problems. 

Brian  If  you  reverse  the  order  of  the  numbers  you  get  the 

same  thing. 
MARGARET     (repeats  his  comment) 
Brian  yeah 
Matt  Mo  you  won't 

MARGARET    You  don't  agree  Matt? 
Matt  Mo  you  won't 

MARGARET    Brian  says  if  you  reverse  the  order  you  get  the  same 
thing,  Matt  says  no  you  won't.    What  do  you  think? 

Margaret  did  not,  however,  explore  conflict  in  depth.    As  soon  as  a 
third  student,  Jennifer,  suggested  that  you  cu-ii.':  not  reverse  whole 
number  division,  Margaret  seemed  satisfied  that  the  argument  was 
settled.     Brian  was  not  convinced  and  suggested  fractions  might  be 
different.     A  comment  was  then  made  by  John  that  you  can  reverse 
addition  and  multiplication,  but  not  subtraction  and  division. 
Margaret  disregarded  these  conflicting  positic n*  and  simply  turned  to 
Brian  and  said  "These  properties  [John  just  mentioned]  knock  this 
out."    That  was  the  end  of  the  discussion. 
Snapshot  four;  June  15.  1991 

At  the  beginning  of  the  second  summer  staff  development,  Margaret 
was  asked  to  respond  to  the  instrument  described  in  snapshot  One. 
Following  is  a  summary. 

Beliefs  about  teaching  mathematics.     Margaret  stated  that  a  good 
teacher  is  flexible  in  approach  and  content,  open  to  new  ideas, 
methods  and  challenges,  creative,  a  good  planner,  and  efficient  user 
of  time.     They  are  well  versed  in  current  teaching  strategies  and 

Er|c  287 


1  -263 

naterials  and  seek  new  avenues  for  exploring  their  own  knowledge  of 
teaching  strategies  and  content.    A  good  teacher  is  responsive  to  the 
needs  of  students  by  providing  feedback.     Good  teaching  requires  being 
skilled  at  diagnosing  student  abilities  and  levels  of  Mathematical 
knowledge . 

Beliefs  about  learning  mathematics.     Margaret  described  the 
learner  as  being  confident  in  ability,  not  afraid  to  fail,  able  to  see 
relationships,  prepared  for  class,  open  to  new  ideas  from  others,  and 
motivated  by  questions  or  problems.     She  felt  that  students  learn  by 
doing  mathematics,  solving  problems,  listening  to  others,  talking 
about  mathematics.    They  need  to  find  the  mathematics  relevant  to 
their  lives  and  investigate  situations  requiring  mathematics. 

Beliefs  about  worthwhile  mathematical  tasks.     Good  problems 
require  creativity  and  may  stimulate  an  extension  in  thinking.  They 
should  have  a  variety  of  strategies  possible  for  finding  the  solution. 
They  may  have  more  than  a  simple  solution  and  may  prompt  connections 
to  other  problems  and/or  life  situations. 

An  analysis  of  the  four  snapshots  of  Margaret  reveal  many 
consistencies  in  her  behaviors  and  her  reflections.    While  the 
consistencies  are  of  interest  and  necessary  as  we  attempt  to  interpret 
the  data,  e.g.,  her  consistent  reference  to  mathematics  needing  to  be 
relevant  to  real  life,  the  brief  space  allowed  here  will  only  permit 
some  discussion  of  change. 

The  beliefs  Margaret  expressed  in  Snapshot  One,  e.g,  teachers 
need  to  respect  student  ideas,  students  need  to  work  in  groups,  tasks 
need  to  provide  opportunities  for  reflection  and  synthesis  by  the 
student,  were  not  consistent  with  the  classroom  discourse  observed  in 
Snapshot  Two.     On  the  contrary,  Margaret  had  the  students  working 




alone.     She  listened  courteously,  but  did  not  explore  student 
thinking.    And,  while  the  task  certainly  provided  the  opportunites 
described,  they  were  not  pursued.    Margaret  appeared  to  be  focusing 
primarily  on  the  lesson  and  on  her  teaching  behaviors. 

What  is  of  interest,  then,  is  the  careful  analysis  of  Snapshots 
Three  and  Four  at  the  end  of  the  year.     It  is  here  we  notice  nore 
careful  attention  to  sturiant  thinking  and  student  organization. 
Margaret  is  displaying  more  of  the  characteristics  she  described  at 
the  beginning  of  the  year.     She  is  respecting  students  ideas  and  is 
open  to  their  thinking.     Students  are  nore  frequently  working  in 
groups.    They  are  listening  and  discussing  ideas,  albeit  they  are 
still  passing  through  the  teacher. 

This  careful  analysis  of  Margaret  has  confirmed  informal 
observations  the  research  team  has  made.     Initially  teachers  acquire 
knowledge  about  alternative  ways  of  teaching.    As  they  put  this 
knowledge  into  practice  they  focus  on  themselves  and  their  behaviors. 
Overtime,  they  are  more  able  to  direct  their  attention  to  the  student 
and  student  thinking.    They  begin  to  consider  alternative  solution 
paths.    The  quality  of  classroom  discourse  improves. 


Flavell,  J.  (1975).  Metacogntive  aspects  of  problem  solving.    In  L.B. 

Resnick  (Ed.),  The  nature  nf  tnfcalligence  (pp.  231-235). 

Hillsdale,  NJ:  Erlbaum. 
Hart,  L.  (1991).  Assessing  teacher  change.    In  R.  Underhill  6  C.  Brown 

(Eds.),  Th»  Ei«v«n»h  Annual  Heating  of  the  North  American  Chapter 

of  th»  Psychology  of  HatheaatJ R«  Education.  Vol.  II,  pp.  78-84. 

Virginia  Tech. 

von  Glaserfeld,  E.  (1983).     Learning  as  a  constructive  activity.  In 
J.  Bergeron  6  N.  Herscovics  (Eds.)  Proceedings  of  the  Fifth 
Annual  Heating  of  the  Worth  *i«r<r»n  Chapter  of  the  International 
ftroup  for  the  Psychology  of  Hathemat<o«  Education  (pp. 41-70). 
Montreal,  Canada:    University  of  Montreal. 

Wertsch,  J.V.  6  Toma,  C.  (in  press).  Discourse  and  learning  in  the 
classroom:    A  sociocultural  approach.  In  L.  Steffe  (Ed.), 
r?on«tructivi«»  in  Education.    Hillsdale,  N.J.:  Erlbaum. 




1  -265 


Nicolas  Herscovics.  Concordia  Uaiveraity,  Montreal 
Liora  Linctwvski,  The  Hebrew  Uaiveraity,  Jerusalem 

5017105  i  Eirit  degree  equities  io  thich  the  unknotn  appear*  on  both  tides  of  the 
equal  sign  by  forial  Mthodi  wolves  tio  lajor  cognitive  obitadei.  !be  first 
one  is  tbe  students'  inability  to  operate  spontaneously  on  or  litb  the  uninon: 
tie  second  one,  ptrhapi  tien  lore  coiplu,  it  tbe  indents'  difficolty  nth 
operating  on  an  equation  at  a  latheiatical  object.  Tbe  objective  of  tbe  teaching 
experuent  reported  here  tai  to  omeoie  tat  second  obstacle  by  a  procedure  based 
on  iecoi position  of  a  ten  into  a  sm  or  a  difference  of  tern  (e.g.  Sn  *  41  =  In 
t  5  ;  5n  Ml  :  3i  ♦  5a  *  51  folloied  bf  cancellation  of  identical  tens  on 
both  sides  of  the  equal  sign.  Stole  thii  procedure  ns  adequate  then  a  ten 
i as  replaced  br  a  sol,  ujor  obitadei  tere  found  in  tbe  case  of  decoiposiug  a 
ten  into  a  difference. 

In  a  previous  paper  (Herscovics  &  Linchevski,  1991(a))  we  have  tried  to  trace 
the  upper  limits  of  solution  processes  used  by  seventh  graders  prior  to  any 
fornal  instruction  in  algebra.  Our  investigation  has  shown  that  the  students 
were  able  to  solve  successfully  most  of  the  first  degree  equations  in  one 
unknown,  However  the  solution  Methods  they  used  clearly  showed  that  the  notion 
of  the  didactic  cut  is  valid  (Filloy  i  Rojano,  1984). 

The  solution  procedures  which  the  students  used  were  based  exclusively  on 
operating  with  the  numerical  terms  ,  therefore  when  given  equations  in  which 
the  unknown  appeared  twice  on  one  side  of  the  equation  or  on  both  sides  (ax  + 
bx  =  c  ;  ax  +  b  =  cx)  the  preferred  node  of  solution  was  that  of  systematic 
substitution.  Therefore  we  proposed  viewing  the  didactic  cut  in  terms  of 
cognitive  obstacle  (Herscovics,  1989)  and  defining  it  as  the  students' 
inability  to  operate  spontaneously  with  or  on  the  unknown. 

Viewing  the  didactic  cut  as  a  cognitive  obstacle  led  us  to  consider  various 
ways  to  overcome  it.  Vtoile  students  can  develop  meanings  for  an  equation  and 
for  the  unknown  siaply  in  terms  of  numerical  relationship,  this  does  not 
extend  to  operating  with  or  on  the  unknown.  Such  operations  have  to  be  endowed 
with  specific  meanings  of  their  own.  This  is  what  is  achieved  when  the 
classical  balance  model  is  introduced  to  represent  an  equation.  One  can  add  or 
take  away  specific  numerical  quantities  as  well  as  quantities  involving  the 
unknown.  Another  model  based  on  the  equivalence  of  rectangular  areas  has  been 
proposed  by  Filloy  and  Rojano  (1989) .  The  authors  have  pointed  out  that  all 
physical  models  contain  inevitable  intrinsic  restrictions  regarding  their 
applicability  to  various  types  of  numerical  operations  on  the  unknown  (Filloy 
&  Rojano,  1989).  One  cannot  represent  5n  -  3  =  27  on  the  balance  because  of 
the  subtraction.  The  area  model  representation  of  7n  -  12  =  9n  -  24  becomes 
quite  sophisticated. 

Vhile  obviously  lacking  the  relevance  of  physical  models,  numerical  models  do 
not  have  such  restrictions  (Herscovics  I  Kieran,  1980).  In  a  teaching 
experiment  based  on  the  use  of  arithmetic  identities,  Kieran  (1988)  found  that 
students  who  tended  to  focus  on  inverse  operations  had  difficulties  in 
accepting  the  notion  of  an  equivalent  equation  obtained  by  operating  on  both 
sides  of  the  initial   equation.    Perhaps    these   difficulties   are   even  more 

Research   funded"  by     Quebec   Ministry  of    Education-   (Fonda   FCAR  EQ2923) 

Preliminary  considerations. 


complex  than  those  identified  with  the  didactic  cut,  as  operating  on  an 
equation  implies  keeping  track  of  the  entire  numerical  relationship  expressed 
by  the  equation  while  it  is  being  subjected  to  a  transformation.  These 
considerations  led  us  to  design  an  individualized  teaching  experiment  which 
would  give  us  the  opportunity  to  study  the  cognitive  potential  of  an 
alternative  approach  (Herscovics  S  Linchevski,  1991  (b)).  We  prepared  a 
sequence  of  three  lesf>ons,  each  lesson  was  semi-standardized.  The  lessons,  as 
well  as  the  pre-test  and  the  post-test,  were  videotaped,  and  an  observe"  with 
a  detailed  outline  recorded  all  students  responses.  We  chose  six  seventh 
graders,  as  described  in  the  introduction,  from  three  levels  of  mathematical 
ability:  Andrew  and  Daniel  were  the  top  students,  Andrea  and  Robyn  average, 
and  Joel  and  Audrey  were  weakest.  The  first  lesson  was  aimed  at  overcoming 
the  students'  inability  to  spontaneously  group  terms  involving  the  unknown  on 
the  same  side  of  the  equal  sign.  In  the  paper  "Crossing  the  didactic  cut  in 
algebra  :  grouping  like  terms"  (Herscovics  &  Linchevski,  1991  (b))  we  gave  a 
detailed  description  of  this  lesson.  The  teaching  intervention  was  based  on 
the  students  natural  tendency,  which  had  been  found  in  our  previous 
investigation,  to  group  terms  involving  the  unknown  without  any  coefficient  (n 
+  n  =  76,  n  +  5  +  n  =  55) .  We  assumed  that  this  tendency  can  be  exploited  by 
increasing  the  number  of  terms  (e.g.  n  +  n  +  n  +  n  =  68)  and  relating  this 
string  of  terms  to  the  multiplicative  term  (e.g.  4n  =  68).  The  teaching 
experiment  was  successful.  However,  a  problem  of  an  arithmetic  nature 
occurred.  In  jumping  over  terms  in  order  to  group  ,  students  were  influenced 
by  the  operation  following  the  term  they  started  with. 

In  this  paper  we  will  describe  and  analyze  lessons  2  and  3  which  deal  with 
equations  in  which  the  unknown  appears  on  both  sides  of  the  equal  sign. 

The  Cancellation  principle. 

The  notion  of  grouping  like  terms  can  be  extended  to  decomposition  of  a  term 
into  a  sum  or  a  difference.  Grouping  and  decomposition  can  then  be  used  to 
introduce  a  relatively  simple  solution  procedure  based  on  tranformation- 
within-the-equation.  For  instance  terms  in  5n  +  17  =  7n  +  3  can  be 
decomposed  into  5n  +  14  +  3  =  5n  +  2n  +  3.  One  can  then  appeal  to  a 
cancellation  principle  to  simplify  this  to  read  14  -  2a.  an  equation  that  can 
easily  be  solved  by  all  the  students.  Of  course,  the  decomposition  of  terms 
can  also  deal  with  difference.  For  instance  13n  -  22  =  6n  +  41  can  be 
expressed  as  7n  +  6n  -22  =  6n  +  63  -22  and  cancellation  reduces  the  equation 
to   7n  =  63. 

From  a  cognitive  perspective,  the  cancellation  procedure  which  we  refer  to  as 
"Cancel lation-Within-thB-Equation"  might  prove  to  be  easier  than  the  other 
procedures  since  the  transformations  are  local,  terms  are  grouped  or 
decomposed  into  equivalent  sums  or  differences  without  any  operation  on  the 
equation  as  a  whole. 

Preliminary  assessment  of  comparison  and  cancellation  procedures 

Prior  to  introducing  cancellation  procedures  in  an  explicit  form,  we  wanted  to 
verify  the  existence  of  a  pre- requisite  procedure,  the  comparison  of 
corresponding  terms.  The  first  three  questions  given  to  the  students  were 
similar  in  form  to  those  found  in  Filloy  and  Rojano  (1984).  The  3tudents  were 
asked  :  "Just  by  looking  at  this  equation  can  you  tell  me  something  about  the 

1)  n  +  25  =  17  +  25 
2a)    n  +  19  =  n  +  n 

2b)    Do  you  think  that  the  other  n  must 

have  the  same  value  or  can  it  be  different? 

1  -267 

3a)  n  +  24=  n  +  2a  3b) .  Do  you  think  that  the  other  n  must  have  the  sane 

value  or  can  it  be  different? 
For  equation  (1)  all  the  students  except  Audrey  compared  corresponding  terms 
to  conclude  that  n  =  17.  In  (2a)  4  out  of  the  6  equated  n  to  19  and 
indicated  that  all  the  occurrences  of  n  must  be  19.  This  is  in  contrast  to  the 
results  obtained  by  Filloy  and  Rojano.  As  for  equation  (3a)  it  was  solved  by 
comparison  by  5  students. 

In  order  to  assess  whether  the  students  would  use  comparison  to  avoid 
unnecessary  arithmetic  operations  we  asked  them:  "What  do  you  think  would  be 
a  fast  way  of  checking  if  the  two  sides  of  :  82  +  27  +  79  -  57  =  82  +  27  + 
37  -  15  are  equal?"  All  six  compared  the  two  sides  by  simply  performing  the 
last  indicated  operation.  This  provides  some  evidence  that  the  students  can 
use  comparison  and  develop  procedural  shortcuts. 

The  last  two  equations  preceding  the  instruction  were  aimed  at  verifying  if 
the  presence  of  identical  terns  on  each  side  of  an  equation  might  induce 
spontaneous  cancellation. 

The  students  were  asked:  "If  you  read  the  left  side  and  then  the  right  side 
of  the  equal  sign  what  is  the  first  thing  you  would  do  to  solve  the  equation?" 
1)  7n  +  29  =  to  +  36  +  29  2)  3n  +  4n  +  21  =  3a  +  57 

Andrew  spontaneously  cancelled  the  identical  terms  in  both  equations.  Joel 
cancelled  29  in  equation  (1).  The  other  four  grouped  the  numerical  terms  in 
(1)  and  the  terms  in  the  unknown  in  (2).  Hence  we  can  conclude  that  apart  from 
Andrew,  the  cancellation  process  had  not  yet  been  acquired. 

Lesson    2    -    Cancellation  of  additive  terms. 

Part  1:     Introduction  of  the  balance  model. 

Vie  first  presented  the  students  with  the  equation  5n  +  3a  +  11  =  5n  +  11  +  39 
and  asked  them  if  they  could  think  of  an  equation  as  one  side  balancing  the 
other.  We  then  introduced  little  cutouts  of  each  part  of  the  equation  which 
were  put  on  the  respective  arms  of  a  scale  drawn  on  a  worksheet.  Students 
were  then  asked  if  relieving  the  same  weight  on  each  side  would  leave  it 
balanced,  and  if  the  same  would  be  true  with  numbers.  We  used  this  model  to 
introduce  the  principle  that  "Equals  taken  away  from  Equals  leave  Equals" .  We 
then  suggested  that  they  look  at  the  scale  and  asked  if  they  noticed  any  equal 
terms  on  both  sides.  They  pointed  at  11  and  5n.  Vhen  asked  if  these  could  be 
taken  away,  5  out  of  the  6  removed  both  11  and  5n,  while  Joel  removed  only  11. 
The  students  were  left  with  3n  and  39  and  "solved"  this  "equation".  Then  the 
question  of  whether  or  not  the  solution  they  found  (n  =13)  would  also  be  the 
solution  of  the  initial  equation  was  raised.  All  six  were  convinced  that  it 

This  introductory  model  had  the  distinct  advantage  of  condensing  the  whole 
cancellation  procedure  and  of  offering  the  students  a  type  of  "inactive"  mode 
of  representation.  However,  as  mentioned  in  the  introduction,  we  did  not  want 
to  build  on  this  model  because  of  its  restriction  (Filloy  &  Rojano  1989). 
Hence  we  proceeded  to  justify  the  whole  process  of  cancellation  on  the  basis 
of  an  "arithmetic"  model  (Herscovics  &  Kieran,  1980). 

Part    2  :     Introduction  of  the  arithmetic  model. 

We  showed  the  students  the  equation  7  x  9  +  11  =  74  and  constructed  from  this 
arithmetic  equation  an  algebraic  equation  by  hiding  a  number  in  turn  by 
finger,  place  holder  and  finally  by  letter  as  in  Herscovics  and  Kieran  (1980). 
We  repeated  this  transformation  with  the  number  13  in  8  x  13  +  11  =  5  x   13  + 



1  -268 

50  in  order  to  obtain  an  algebraic  equation  with  the  unknown  on  both  sides: 
8n  +  11  =  Sn  +  50.  After  pointing  out  that  none  of  the  procedures  they  knew 
could  efficiently  solve  this  type  of  equation,  we  told  them  that  we  would 
develop  a  new  procedure  and  verified  each  step  in  this  development  by 
operating  simultaneously  on  the  algebraic  equation  and  the  arithmetic  equation 
which  we  rewrote  as  :  8  x  13  +  11  =  5  x  13  +  50  to  remind  ourselves  that 
we  have  to  imagine  that  the  solution  was  hidden. 

Part    3  :    The  cancellation  procedure. 

In  introducing  the  cancellation  procedure,  we  had  to  choose  between  starting 
with  the  cancellation  of  the  numerical  terms  or  the  terms  in  the  unknown.  The 
advantage  of  the  latter  is  that  the  equation  obtained  can  be  solved  by  inverse 
operations  (e.g.  3n  +  11  =  50).  The  disadvantage  is  that  cancelling  the 
terms  in  the  unknown  might  seem  arbitrary  since  it  meant  cancelling  a 
generalized  number  before  justifying  the  procedure  with  a  specific  number. 
Rather  than  creating  the  possibility  of  such  a  cognitive  problem  we  decided  on 
the  longer  process  of  starting  with  the  first  choice. 

Cancellation  of  identical  numerical  terms. 

We  started  by  asking  the  following  questions:  "When  I  look  at  the  equation 
8n  +  11  =  5n  +  50  can  I  write  it  as  8n  +  11  =  5n  +  39  +  11  ?  Is  this 
equation  still  balanced  out  ?  Will  the  solution  be  the  same  ?" 
We  wish  to  point  out  that  for  ail  Che  transformations  we  introduced  in  the 
cancellation  process,  each  one  was  accompanied  by  questions  regarding  the 
maintenance  of  numerical  equilibrium  and  the  invariance  of  the  solution. 
These  were  always  followed  by  a  verification  of  the  corresponding 
transformation  on  the  arithmetic  equation,  whether  the  students  agreed  and 
responded  affirmatively  to  each  of  the  questions,  or  thought  that  the  equality 
or  the  solution  would  be  affected  by  the  tranformations,  or  were  not  sure. 
Then  we  asked  the  following  question:  "What  if  I  take  away  11  on  both  sides, 
do  you  think  that  both  sides  will  still  be  equal  ?...  What  is  the  new 
equation  we  get?. . .  Do  you  think  the  solution  is  the  3ame  for  both  equations?" 
Five  out  of  the  6  thought  that  removing  11  on  both  sides  would  maintain  the 
equality.  Regarding  the  invariance  of  the  solution  only  two  were  sure.  At 
this  point  with  the  help  of  Andrea  we  realized  that  they  referred  to  another 
interpretation  of  the  word  "solution",  the  one  usually  used  in  arithmetic,  the 
"answer"  on  the  right  aide  of  an  arithmetic  equation. 

We  justified  the  transformation  by  showing  the  steps  on  the  algebraic 
equation:  8n  +  11  =  5n  +  11  +  39 

The  students  could  check  the  validity  of  their  operation  by  verifying  it  on 
the  arithmetic  equation. 

After  the  justification  and  the  verification  we  suggested  a  shortcut  saying  : 
"Let   me   show   you  a  short  way  of  doing  what  we  just  did.     We  start  with  the 
equation     8n  +  11  =  5n  +  50     split    50   and  replace  it  by     11  +  39.  We  get 
8n   +    11  =  5n  +  11  +  39  and  we  siaply  cross  out  11  on  both  sides  : 
8n  +  11  =  5n  +  11  +  39.     We  called  it    "Cancelling   11   on  both  aides"  or 
"Cancelling  the  addition  of  11  on  both  sides" . 

Cancelling  the  terms  in  the  unknown. 

After  reducing  the  initial  equation  to  8n  =  5n  +  39  we  repeated  the  steps  and 
the  questions  described  above  regarding  the  replacement  of  8n  by  5n  +  3n  and 
subtracting    En    from  both  sides.  Two  students  felt  that    splitting    8n  would 

1  -269 

change  the  balance.  Verifying  their  assumption  on  the  corresponding  arithmetic 
equation  caused  a  change  in  their  initial  conception.  Vfe  again  suggested  the 
shortcut  5a  +  3a  =  5n  +39  calling  it  "cancellation  of  the  same  term  on 
both  sides".  A  summary  of  this  lengthy  introduction  provided  the  opportunity 
to  put  together  all  the  steps  and  to  ask  the  students  how  would  they  choose 
the  terns  to  be  split  up  for  eventual  cancellation.  They  all  used  the 
criterion  of  "bigger"  term  to  indicate  their  choice. 

Following  this  reflection  on  the  cancellation  procedure  we  asked  each  student 
to  solve  I2n  +  79  =  7n  +  124.  All  of  our  students  except  of  Audrey  solved  it 
without  any  problem,  three  started  by  replacing  124  by  79  +  45  and 
cancelling  79.  They  fh»n  rewrote  the  equation  and  split  12a  into  5n  +  7n, 
cancelled  7n,  rewrote  5n  =  45  and  divided  45  by  5.  The  others  started  by 
replacing  the  unknown,  cancelling  and  then  splitting  up  the  numeric  term. 
Audrey,  the  weakest  student,  had  to  be  shown  the  introductory  example  again, 
following  which  she  rewrote  the  given  example  by  decomposing  12n  to  5n  +  7n, 
cancelled  7n,  rewrote  5n  +  79  =  124,  and  then  used  inverse  operations.  The 
next  equation  was  12a  +  109  =  18n  +  67.  All  of  the  students  used  the  same 
procedure  they  had  used  before.  Audrey  had  to  be  guided  in  how  to  re-insert 
12a  +  6n   into  the  eqution. 

Flexibility  in  the  choice  of  sub  procedures. 

Ip  order  to  verify  if  the  student  could  solve  the  equation  using  other  sub 
procedures,  and  in  order  to  raise  the  question  of  the  in variance  of  the 
solution,  we  asked  the  students  to  solve  the  same  equation  (12a  +  109  =  18n  + 
67) ,  but  to  start  by  decomposing  a  term  other  than  the  term  they  started  with 
before.  All  of  them  were  able  to,  and  stated  their  conviction  that  the  order 
of  cancellation  did  not  affect  the  solution.  When  asked  to  solve  109  =  6n  + 
67  using  another  procedure,  they  used  inverse  operations. 

Mare  equations:    The  students  were  asked  to  solve  : 

(1)    19n  =  13n  +  72  (2)    57  +  8n  =  6n  +  71       (3)    12n  +  30  =  13n  +  19 

(4)    6n  +  23  =  n  +  88  (5)    71  +  12n  +  38  =  13n  +  67  +  5n 

Equation  (1)  was  solved  by  all  the  students.  In  equation  (2)  Robyn 
spontaneously  decomposed  into  sums  both  8n  and  71  and  used  double 
cancellation.  Audrey  split  up  71  but  did  not  know  where  to  replace  it,  so  we 
used  an  arrow  to  help  her  to  remember  the  term  3he  wanted  to  replace .  In 
equation  (3)  to  our  surprise,  all  of  the  students  split  13n  into  12a  +  In,  and 
Andrea  joined  Robyn  in  double  cancellation.  Audrey,  when  ending  up  with  11  = 
In  stated  that  it  did  not  make  sense.  She  had  to  be  shown  that  In  was  the  same 
as  n  just  as  1x3  is  the  same  as  3.  In  equation  (4)  Andrea  and  Audrey 
got  to  =  n  +  45,  and  were  perplexed  by  the  presence  of  a  singleton.  They 
overcame  this  obstacle  when  asked  to  write  6n  as  a  string  of  additions 
(Herscovics  &  Linchevski  1991  (b) ) . 

The  last  equation  was  intended  to  verify  if  students  would  first  group  and 
then  decompose  or  would  start  by  splitting  up.  Andrea  and  Audrey  started  by 
splitting  up  followed  by  innediate  cancellation,  while  the  others  grouped 
first.  Andrea,  Robyn  and  Joel  used  a  double  cancellation  in  the  solution 
process . 

We  ended  lesson  2  with  a  short  review.  We  presented  the  students  with  some 
equations,  asking  them  to  indicate  which  procedure  should  be  used  to  solve. 
Lesson  3  started  in  the  same  way  but  this  time  we  asked  them  also  to  solve 
the  equations.  Only  Audrey  had  difficulties  regarding  rewriting  the  equation 
after  transformations. 


Lesson    3 . 

Preliminary  considerations. 

As  lit  lessen  2  we  chose  to  start  by  introducing  the  cancellation  of  the 
numerical  term.  In  the  pretest  we  had  found  that  our  students  experienced 
some  difficulty  with  the  composition  of  consecutive  subtractions.  Some  of  the 
students  did  not  perceive  that  189  -  50  -  50  was  the  sate  as  189  -  100.  Thus 
we  decided  to  steer  our  students  toward  numerical  situations  which  avoided 
this  problem.  We  tried  to  achieve  this  by  focusing  their  attention  on  the 
decomposition  of  a  numerical  tern  that  was  added.  This  also  brought  us  to 
limit  the  scope  of  this  teaching  experiment  to  forms  involving  subtraction 
only  on  one  side  of  the  equation  of  either  a  numerical  term  or  a  term  in  the 
unknown  as  in   190  -  8n  =  18n  -  18. 

Part    1  :    Decomposition  of  a  numerical  term. 

As  in  lesson  2  we  built  on  an  arithmetic  model.  The  student  constructed  an 
algebraic  equation  from  an  arithmetic  equation  by  hiding  a  number,  but  this 
tine  with  subtraction  on  one  side  :  6n  +  17  =  8n  -  11.  We  used  decomposition 
of  17  into  28-11.  The  stages  of  instruction  were  exactly  as  in  lesson  2. 
This  enabled  us  to  highlight  the  basic  principle  "Equals  added  to  Equals  give 
Equals" .  We  called  this  principle  "Cancelling  the  subtraction  of  11  on  both 
sides" . 

During  the  suntiary  review  of  this  procedure  we  discussed  with  the  students  how 
to  choose  the  term  to  be  expressed  as  a  difference.  To  assess  how  well  our 
students  had  grasped  our  instructions  we  asked  them  to  solve  : 
19n  +  23  =  24n  -  22.  The  two  top  students  figured  out  mentally  the 
decomposition  and  immediatelly  wrote  :  19n  +  45  -  22  =  24n  -  22  and  solved 
the  equation  using  two  cancellations.  However  the  other  4  students  needed 
soma  guidance  as  3  of  t*«m  decomposed  23  into  22+1.  The  next  equation  was 
solved  by  4  students,  and  the  two  others  needed  some  help  in  splitting  a  term 
into  a  difference.  At  this  stage  we  presented  our  students  with  the  equation 
17n  -  48  =  13n.  We  wondered  if  after  cancellation  of  13n  they  would  experience 
any  problem.  The  results  confirmed  our  conjecture,  as  four  of  the  6,  after  the 
cancellation  of  13n,  did  not  know  how  to  re-write  the  equation.  Andrew, 
looking  at  4n  -  48  =  stated  :  "All  the  weight  is  on  one  side  and  you  don' t 
have  a  solution" .  We  reminded  them  that  cancellation  was  justified  by  the 
subtraction  of  13n  from  both  sides. 

Part    2  :    Restrictions  on  cancellation. 

In  order  to  verify  if  the  students  perceived  the  importance  of  not  only 
"cancelling  out"  the  same  number  but  also  the  same  operation,  we  presented  the 
equation  15n  +  18  =  17n  -  18  asking  if  we  could  cancel  18  on  both  sides. Five 
out  of  the  6  explained  "If  you  want  to  cancel  out,  you  must  make  sure  it's  the 
same  operation" .  We  recall  that  in  the  first  equation  when  they  were  asked  to 
solve  19n  +  23  =  24n  -22  three  of  them  split  23  into  22+1  and  at  that  time 
we  pointed  out  to  them  that  one  could  justify  cancellation  only  if  the  sens 
operation  on  the  term  is  involved. 

Part    3  :    Decomposition  of  •  term  involving  the  unknouwn. 

At  this  stage  we  began  to  observe  some  ot  the  foundation  problems  we  had 
observed  at  the  beginning  of  the  teaching  experiment,  which  we  previously 
called  "a  detachment"  of  an  operation  sign  from  the  ten  (Barscovics  & 
Linchewki  1991  (a)  (b)).  W»  gave  them  the  equation  155  -  6n  =  3n  +  11, 
which  4  of  the  6  re-wrote  as  155  -3n  +  3n  =  3a  +  11  in  order  to  cancel  3n. 
Our    teaching    intervention  was   based  on  numerical  examples,  and  on  pointing 


explicitly  at  3a  to  be  split  up.  Only  in  the  third  equation  of  that  type  were 
all  6  students  able  to  express  a  term  in  the  unknown  as  a  difference.  The 
last  equation  to  be  solved  was  rather  complex  :  77  -  8a  +  113  =  13a  -  18  +  5n. 
Andrew,  Robyn,  Joel  and  Audrey  first  grouped  and  then  decomposed.  Robyn  and 
Joel  reordered  the  equation  before  grouping.  Robyn  and  Audrey  used  inverse 
operation  when  obtaining  an  equation  with  only  a  numerical  tern  on  one  side. 

Post    test . 

The  post  test  took  place  one  month  after  the  last  meeting.  The  students  had 
not  done  any  algebra  since  the  last  lesson,  and  therefore  we  thought  that  some 
of  the  procedures  would  not  come  spontaneously  to  their  Bind.  We  thus  had 
prepared  two  triggers,  in  order  to  jolt  their  memory  and  place  them  again  in 
the  framework  needed  for  the  solution,  to  be  used  qlIx  if  necessary.  The 
first  trigger  was  a  list  with  the  procedures'  names,  and  the  second  one  was  a 
ready-made  right  and  wrong  cancellation  procedures. 

We  will  discuss  only  the  items  of  the  post-  test  which  are   directly  relevant 
to   lessons  2  and  3. 

Comparison  of  algebraic  equations  : 
We  gave    the    same  items  as  in  the  preliminary  assessment  of  comparison.  This 
time  all  of  the  students  mentioned  cancellation  as  the    first   procedure  they 
would   use   except  for  Joel.  In  :    7n  +  29  =  4n  +  36  +  29  he  first  grouped  the 
numerical  terms. 

Solvino  equations. 
Due  to  space  limitations  we  will  not   go   into   detailed   description  of  sub 
procedures   used   by   the   students,    but   vill    comment  that  many  interesting 
individual  differences  have  been  found.     The  equations  were  given  one   at  a 

Al     Single  occurrence  of  the  unknown. 
(1)    13n  +  196  =  391       (2)  16n  -  215  =  265       (3)  12n  -  156  =  0 
All  the  students  solved  by  using  inverse  operations. 

We  must  note  how  stable  this  procedure  has  remained  over  a  period  of  7  months 
(ftsrscovics  and  Linchevski,  1991(a)).  Even  after  learning  the  decomposition 
of  numbers  into  a  sum  or  a  difference,  this  new  method  did  not  interfere 
with  the  inversing  procedure. 

Results  from  parts  (B)  and  (C)  "grouping  like  terms"  are  given  in 
Herscovics  and  Linchevski  1991(b). 

Dl   IMmown  on  both  sides  of  the  equal  sign,  involving  only  addition. 

(1)     to   +    39 =   7n        (2)    5n  +  12  =  32a  +  24       (3)  12a  +  79  =  7a  +  124 

(4)     71  +  12a  +  38  =  13a  +  67  +  5a 

la  equation  (1)  (2)  and  (3)  Andrew,  Daniel,  Joel  and  Robyn  iamediately 
decomposed  terms  and  solved  successfully  all  of  the  equations  while  Andrea  and 
Audrey  had  to  be  shown  triggers  (1)  and  (2).  Evidenced  by  the  comparison  part 
of  the  post-test,  both  of  them  remembered  cancellation,  so  probably  they  had 
forgotten  the  decomposition  part. 

The  students  had  not  lost  their  mastery  of  notation  and  could  efficiently 
write  down  their  steps.  Also  they  were  taught  to  cancel  one  term  at  a  time. 
In  the  post-test  Daniel,  Andrea  and  Joel  used  double  cancellation  and  Robyn 
and  Audrey,  after  cancelling  the  term  in  the  unknown,  used  inverse  operations. 
Only  Andrew  stayed  with  the  procedure  we  taught.  As  for  equation  (4)  we  saw 
grouping    first  and  then  double  cancellation,  as  well  as  splitting  up  from  the 



very  beginning.  The  students  who  started  by  splitting,  after  being  prompted, 
willingly  solved  by  first  grouping  and  then  splitting. 

EJ  Unknown  on  both  sides  involving  also  subtraction. 

(1)  19n  +  23  =  24a  -  22  (2)  155  -  to  =  3a  +  11  (3)  17n  -  48  =  13n 
(4)    89    -  5n  =  7n  +  5  (5)    77  -  8n  +  113  =  13a  -  18  +  5n 

It  is  in  this  part  that  mat  difficulties  emerged.  Also  same  of  the  students 
avoided  the  need  to  decompose  a  numerical  tent  into  a  difference  by  cancelling 
first  the  tent  in  the  unknown  and  then  using  inverse  operations.  All  the  basic 
problem  mentioned  previously,  the  detachment  of  the  minus  sign  and  jumping 
off  with  the  posterior  operation  were  observed. 

The  students  tended  to  decompose  "the  bigger  number"  regardless  of  its  sign 
(e.g.  in  equation  (2)  155  -  3n  +  3a  =  3n  +  11)  in  order  to  obtain 
cancellation,  or  split  a  number  into  two  numbers  when  the  operation 
proceeding  the  new  numerical  term  on  the  left  was  not  the  sane  as  the 
operation  preceeding  the  corresponding  number  on  the  right. 


"The  Cancellation  Within  the  Equation"  was  accepted  by  the  students  as  a 
smooth  extension  of  their  spontaneous  ability  to  use  comparison  in  the  context 
of  some  specific  mathematical  equalities.  This  tendency  was  supported  by  both 
the  balance  model  and  the  arithmetic  model  for  justifying  cancellation.  The 
decomposition  of  a  number  was  a  natural  complementary  process  to  that  of 
grouping  like  terms.  Moreover,  it  was  evident  that  the  students  were  able  to 
go  beyond  the  instruction  by  themselves,  inventing  more  efficient  procedures. 
However,  when  a  decomposition  into  a  difference  was  involved,  the  cognitive 
obstacles  we  have  mentioned  in  previous  papers  were  found;  the  detachment  of 
the  minus  sign  and  jumping  off  with  the  posterior  operation.  For  some  students 
expressing  a  number  as  a  difference  when  the  subtrahend  is  a  given  constraint 
was  not  a  trivial  problem.  Although  this  procedure  was  addressed  during  the 
lesson,  in  the  post-test  they  experienced  the  same  difficulties.  Seme  of  them 
kept  splitting  the  "bigger"  number  into  two  "smaller"  ones.  This  seems  to  put 
in  question  the  benefit  of  extending  the  cancellation  procedure  beyond 
replacing  terms  by  equivalent  sums. 


Filloy,  E.  &  Rojano,  T.  (1989),  Solving  equations:  the  transition  from 

arithmetic  to  algebra.  For  the  Learning  of  Mathematics,  9,  2,  19-25 
Filloy,  E.  &  Rojano,  T.  (1984),  From  an  arithmetical  thought  to  an  algebraic 

thought.  Proceeding  of  PME-NA  VI,  Mbser,  J.  (Ed.),  Madison,  Wisconsin,  51-56 
Herscovics,  N.  (1989),  Cognitive  obstacles  encountered  in  the  learning  of 

algebra.  Research  issues  in  the  learning  and  teaching  of  algebra,  Wagner,  S. 

&  Kieran,  C.  (Eds),  Res too,  Virginia:  NCTM,  and  Hillsdale,  N.J.:  Erlbaum, 


Herscovics,  N.  &  Kieran,  C.  (1980),  Constructing  meaning  for  the  concept  of 
equation.  The  Mathematics  Teacher,  vol  73,  no.  8,  572-580 

Herscovics,  N.  &  Linchevski,  L.  (1991),  Pre  -  algebraic  thinking:  Range  of 
equations  and  informal  solution  processes  used  by  seventh  graders  prior  to 
any  instruction,  P.  Boero  (Ed.),  Proceeding  of  PME  XV,  Assisi  Italy, 
pp.  173-181 

Herscovics,  N.  &  Linchevski,  L.  (1991),  Crossing  the  didactic  cut  in  algebra, 
R.  J.  Underbill  (Ed.),  Proceeding  of  PME  -  NA  Virginia,  V.  II  pp.  196-202 

Kieran,  C.  (1988),  Two  different  approaches  among  algebra  learners.  In  A. 
Coxford  (Ed.),  The  ideas  of  algebra,  K-12  (1988  Yearbook,  pp.  91-96). 
Reston,  Va:  National  Council  of  Teachers  of  Mathematics 

Er|c  297 




Jimi  Hiebert  and  Diana  Hearne 

University  of  Delaware 

In  this  atudy,  we  examined  relationahipa  between  instruction,  students' 
under standing ,  and  students'  performance  as  they  begen  to  acquire  computational 
strategies  in  multidigit  addition  and  aubtrectlon.    We  were  interested  in  now 
conceptual  underatandlng  infracted  with  akillful  performance  es  students  racaivad 
instruction  on  addition  and  subtraction  with  ragrouping,  and  in  how  these 
interactiona  were  intluancad  by  different  kinda  of  instruct .tonei  activities.  The 
results  indiceted  thet  instructional  ectivities  which  amphaaijad  mathematical 
connections  through  the  discussion  of  problems  and  aiternetive  solution  strategies 
were  more  cloaaly  related  to  the  development  of  both  understanding  and  akilled 
performance  than  were  activitiea  that  emphasized  procedural  skills  through  paper- 
end-pencil  practice.     However,  the  relationships  are  not  straightforward  and 
several  clusterings  of  individual  cases  are  presented  to  reveal  some  of  the 
complexities . 

The  current  reform  movement  in  mathematics  education  in  the  United  States  is 
based,  at  least  in  part,  on  the  belief  that  instruction  should  be  redesigned  to 
facilitate  a  higher  level  of  conceptual  understanding  and  to  decrease  the  emphaeis 
on  drill-and-practice.    Although  such  alternative  approachee  are  widely  preeumed 
to  promote  a  more  flexible  use  of  knowledge  and  better  problem  solving  skills,  we 
still  have  little  evidence  on  the  way  in  which  understanding  and  performance 
interact  and  on  the  way  in  which  alternative  instructional  approaches  influence 
theee  interactione. 

The  notion  of  understanding  has  a  rich  history  in  mathematice  education. 
Many  of  the  psychological  descriptions  of  understanding  mathematics  (e.g., 
Brownell,   1947)  are  based  on  the  Idea  of  establishing  relationships  between  facts, 

1  -274 

procedure*,  representation*,  and  to  on.     Our  view  of  understanding  i*  consistent 
with  this  perspective  and  with  more  recent  discussions  of  building  cognitive 
connections  (e.g.,  Hiebert  £  Carpenter,  in  press).     We  believe  that  understanding 
develops  as  students  establish  connection*  of  many  kinds:     between  familiar  ideas 
and  new  material,  between  different  forms  of  representation  (e.g.,  physical  and 
written),  between  procedures  and  underlying  principles. 

In  this  study,  we  followed  students  during  the  first  three  years  of  school 
and  examined  the  development  of  understanding  and  performance  in  multidigit 
arithmetic.     He  focused  on  the  way  in  which  different  instructional  approaches 
influenced  this  development.     We  were  especially  interested  in  the  influence  of 
approachee  that  emphaeized  the  construction  of  connections. 

Sample.     Data  were  collected  from  an  initial  cohort  of  about  ISO  students 
during  their  first,  second,  and  third  years  of  school.     Many  new  students  entered 
the  classrooms  during  the  three  years  and  some  left,  so  the  number  of  students  and 
their  instructional  history  depend  on  the  time  of  assessment.     The  students  attend 
suburban-rural  public  schools. 

Instruct ion.     Several  different  instructional  approaches  for  place  value  and 
addition  and  subtraction  were  observed.     During  the  first  year,  two  of  the  six 
classrooms  followed  the  textbook  using  relatively  conventional  instruction.  The 
other  four  classrooms  implemented  an  alternative  approach  during  the  five  weeks  of 
place  vplue  and  addition/subtraction  topics.     The  alternative  approach  was 
characterized  by  greater  student  use  of  physical  representations,  increased 
emphasis  on  translating  between  different  kinds  of  representations  (e.g., 
physical-verbal-written  symbols),  greater  use  of  story  problem  situations,  and 
fewer  problems  covered  but  increased  time  spent  per  problem  during  class 
discussions.     Class  disruptions  usually  involved  analyses  of  problems  and  sharing 
alternative  solution  strategies.     (See  Hiebert  4  Wearne,   in  press,   for  a  more 
complete  description  of  these  classrooms.) 

1  -  275 

Th.  following  ye.r  th.  .tud.nt.  w.r.  r....ign.d  to  .lx 
cli,ltoM..     Four  of  th.  .IX  folio--  th.  t.xtboo*  In  .  r.l.tivtly 
conv.ntion.l  w.y  .nd  two  cl...«oo-  u..d  th.  .Itern.tive  Th. 
.lt.rn.tlv.  induction  cl...roo„.  only  .tud.nt.  who  h.d  received 
.lt.rn.tiv.  induction  In  y..r  1-    Th.  .lt.rn.tlv.  .n  ext.n.ion  of  th.t 
u..d  during  flr.t  gr.d..  «»ph..i*.d  «th-*i«X  conn.ction.  through  cl...„.  of  problem  .nd  .alution  .tr.tegie.  .nd  through  th.  u..  of  different 
for™,  of  In  flr.t  problem  w.r.  .itu.t.d  in  .tory 

cont.xt..     Both  .ppro.che.  devoted  .bout  X2  we.,,  to  pl.c.  .nd 
«ddit ion/.ubtr«ction  in.truct ion. 

Ouring  th.  third  ye.r.  the  initial  cohort  plu.  .bout  75  n.w  .tud.nt.  w.r. 
...ign.d  to  nin.  cU..«-..     Three  cL-«—  »-  the  .lt.rn.tiv.  .nd 
.U  el...«~  used  v.rying  textboo*  .ppro.che..    The  ^ority  of  .tud.nt.  who 
t.c.ived  .lt.rn.tiv.  in.truction  during  ye.r.  1  .nd  2  were  in  th.  .lt.rn.tiv. 
in.truction  cl.....  in  year  3.     The  cl.-.roo*.  devoted  10-14  wee,,  of  In.truction 
to  pl.ce  v.lue  .nd  addition/ subtract ion. 

Ml  -tudent.  were  given  written  te.t.  thre.  ti»e.  y.«~ 
n..r  th.  b.ginning.  middle,  and  end  of  th.  .chool  ye.r.  *t  the  _  tU..  -bout 
h.lf  th.  .tud.nt.  w.r.  interviewed  individually.    The  int.rvlew.e.  randomly 

f  „».r  1—12  .tudent.  from  each  of  the  «ix  cl.saroomB-- 
..l.cted  «t  the  beginning  of  ye.r  1-  1-  atuaen 

.nd  th.  -tudent.  were  interviewed  throughout  year.  2  .nd  3. 

The  t..t.  .nd  interview,  were  con.tructed  to  »e..ur.  ,1,  .tud.nt. • 
und.r.t.ndlng  of  grouplng-by-ten  ide..  .nd  of  the  po.itlon.l  n.tur.  of  th.  written 

not.tion  ,»  — ■*»  -  —  -  '^"^  ^  ^ 

..grouping,  .nd.   ,3,  -tudent..  understanding  of  the  computational  procedure,  they 


OUS.T^M^.    *urin9  »  "d  2'  *11  °f  tta  Cl"t0°m' 

ob..rv.d  once  or  twice  .  wee,  during  in.truction  on  pl.c.  .nd 
.ddition/.ubtr.ction.     Durin,  year  3.  .11  cl..—  ob.erv.d  for  three 


1  -276 

con..cutiv.  d.y.  during  r#1.v.nt  in.truction.  pield  note.  war#  ^  Qn  claistoom 
•ctivitie.  .„d  th  ion.  were  audiot.ped  .nd  tr.n.cribed. 

W.  will  focu.  on  th.  data  from  year,  j  .nd  2|  .t  fch„  time  Qf  ^ 
d.t.  from  y..r  3  h.d  not  been  completely  gathered  nor  analysed.     v..r  3  data  will 

included  in  th.  conference  prestation.    Given  .p.ce  limitation.,  w.  „U1 
.»»rl..  th.  re.ult.;  mor.  defiled  pr...nt.tion.  ,re  available  £rom  the  author.. 

Bgtween-oro„P  r.rform.nce  different.,     m  general,  .tud.nt.  who  engaged  in 
th.  alternative  in.truction  for  two  year,  perform.d  better  on  .11  typo,  of  written 
test  it™,     specifically,  they  .cored  higher  on  iteme  me..uring  (1)  knowledge  of 
Pl.ce  .nd  the  tene-.tructure  of  written  notation,    (2)  computation  On 
in.tructed  problem.,    ,3,  computation  on  nonin.truct.d  or  novel  problem.,  .nd  ,4, 
•tory  problem..     For  most  item.,  the  difference,  in  percentage  c  .rr.ct  between  th. 
two  group,  at  th.  end  of  th.  ..cond  year  ranged  from  10*  -  40*. 

PtofUe,  of  emerging  compftpnre.     Within-.ubject  profile,  helped  to 
ch.ract.ri,.  the  nature  of  the  between-group  difference,  in  performance  and  probed 
further  into  the  rel.tion.hip.  between  under.t.nding  and  performance  under 
different  inetruction.l  condition..     For  purpo.e.,  we  can  con.ider 
two  very  different  group,  of  .tud.nt.-tho.e  who  entered  the  .econd  y..r  with  .  rich  under.t.nding  of  grouping-by-t.n  idea,  .nd  how  th...  conn.ct 
with  th.  po.ition.1  .ymbol  .y.tem  .nd  tho.e  who  .till  und.r.tood  littl.  .bout 
th...  id....     Nin.  of  th.  65  .tudent.  interviewed  at  th.  beginning  of  th.  ..cond  „.r.  rel.tiv.  expert.,  performing  .ucc.fully  .nd  giving  m..ningful 
explanation..     ,hey  all  were  highly  .ucc.ful  on  mo.t  .ddition,  .ubtr.ction,.nd  .ddend  .tory  problem,  during  th.  ..cond  ye.r,  but  their  con.truction  and 
choico  of  comput.tio„  .tr.t.gie.  .how.d  ..v.r.l  di.tinct  p.ttern..     FOur  .tud.nt. 
cr..ted  decompo.ition  .tr.t.gie.  in  which  th.y  dealt  „ith  th.  l.rg.r  digit.  (..g., 

hundr.d.,   fir.t,  r.g.rdl...  of  r.grouping  demand.,  ,nd  u..d  th...  n 

.ft.r  th.y  h.d  been  expo..d  ,.t  horn,  or  school,  to  th.  algorithm..     Three  .tud.nt. 


1  -277 

developed  the  same  decomposition  strategies  but  twitched  to  the  standard 
algorithms  one*  they  had  been  exposed  to  them  end  ueed  them  eucceeefully  on  all. 
problem*.    Three  etudente  showed  less  evidence  of  using  self-generated  strategies 
consistently,  switched  to  algorithms  as  soon  as  they  saw  them,  and  made  some  of 
the  classic  regrouping  errors  on  the  more  difficult  problems. 

In  contrast,  23  students  began  second  grade  with  very  little  understanding  of 
grouplng-by-ten  and  place  value  ideas.     Again,  several  different  patterns  of 
performance  and  understanding  emerged.     Some  students  were  uniformly  unsuccessful 
throughout  the  year,   some  students  showed  a  sharp  rise  in  computational 
performance  after  learning  the  algorithms  (independent  of  understanding),  and  a 
third  group  showed  more  gradual  Increase  in  performance,  based  on  Invented 
strategies,  that  seemed  to  keep  pace  with  their  increasing  understanding. 

Interestingly,  cases  of  these  profile  patterns  occurred  in  both  kinds  of 
instruction.     However,  their  frequency  of  occurrence  differed.     Hore  students  in 
the  alternative  instruction  classes  constructed  and  used  their  own  computation 
strategies  and  depended  less  on  the  standard  algorithms.     For  example,  at  the 
middle  of  the  second  year,  before  instruction  on  the  standard  algorithm  for 
addition  with  regrouping,  81*  of  the  correct  responses  of  the  alternative 
instruction  interviewees  were  generated  by  self-constructed  strategies  compared  to 
39*  for  the  textbook  instruction  interviewees.     Standard  algorithms  (learned  at 
home  according  to  their  users)  accounted  for  most  of  these  students',  correct 
responses.     Fewer  students  in  the  conventional  classes  used  the  understanding  they 
possessed,  even  if  it  was  substantial,  to  develop  their  own  strategies  or  adjust 
taught  procedures  to  solve  new  problems. 

Linkc  between  instruction,  understanding,  and  performance.     In  order  to  link 
learning  with  instruction,  we  were  interested  in  the  observed  differences  in 
instruction  that  might  explain  these  different  learning  profiles.     Both  content 
and  pedagogical  differences  were  investigated.     Content  differences  were  not  found 
in  the  scope  of  the  curriculum  but  rather  in  the  nature  of  the  activities.  More 


1  -278 

of  the  activities  in  the  alternative  instruction  involved  connecting  procedure* 
with  conceptual  underpinning*  and  connecting  different  waye  of  solving  problem*. 
For  example,  a  great  deal  of  time  wae  epant  in  year  2  asking  students  to  share 
invented  strategies  and  then  asking  them  to  explain  why  the  proceduree  worked  and 
how  they  were  the  same  as  or  different  than  other  procedures. 

Pedagogical  differences  are  more  difficult  to  summarise.     In  year  1,  the 
alternative  instruction  (compared  with  the  more  conventional  Instruction)  used 
fewer  materials  and  used  them  more  consistently  as  tools  for  solving  problems 
rather  than  for  demonstration;  solved  fewer  problems  but  devoted  more  time  to 
solving  each  problem;  and  delivered  more  coherent  lessons.     Details  of  these 
results  are  presented  in  Hiebert  and  Wearne  (in  press).     In  year  2,  differences 
were  found  again  in  use  of  materials  and  the  time  spent  per  problem.     The  same 
material  (base-10  blocks)  was  used  consistently  in  the  alternative  instruction 
classrooms  and  was  always  available;  a  few  different  materials  were  used  in  the 
more  conventional  classrooms  but  only  for  one  or  two  lessons  each.     During  40 
minute  lessons,  the  two  alternative  instruction  classes  averaged  12  problems  per 
lesson  in  one  class  and  14  problems  per  lesson  in  the  other  class.     The  four  more 
conventional  classes  averaged  24,  29,  36  and  38  probleme  per  leeson.    Finally,  in 
the  alternative  instruction  classrooms,  students  talked  much  more  relative  to  the 
teacher  and  the  teachers  asked  many  more  questions  that  requested  analyses  of 
probleme,  description  of  alternative  solution  stratagiee,  and  explanations  of  why 
procedures  worked. 

Relationships  between  teaching,  understanding,  and  performance  are  excremely 
complex.     Neverthelees,  this  brief  summary  of  data  hints  at  several  links.  First, 
the  development  of  understanding  seems  to  affect  performance  through  the 
construction  of  robust  strategies  that  are  applied  successfully  acroas  a  range  of 
problems •     That  is,  understanding  doee  not  tranalata  automatically  into  improved 
performance;  the  Impact  of  understanding  on  performance  ie  mediated  by  the  kinds 



of  strategies  students  use  to  perform  tasks.     If  students  arc  encouraged  to  invent 
and  analyze  strategies,  it  is  likely  their  understanding  and  perforrasnce  will  be 
closely  linked.    This  appears  to  be  true  for  high  and  low  achievers  alike. 

Second,  instruction  may  be  related  most  importantly  to  learning  in  terms  of 
whether  it  affords  opportunities  for  students  to  use  their  understandings  to 
develop  and  modify  procedures.     It  is  clear  that  the  relationship  between 
understanding  and  performance  can  be  fragile  in  the  face  of  instructional  demands. 
The  data  indicate  that  understanding  does  not  necessarily  translate  into,  or  even 

inform,  procedural  skill.     Further,  taught  procedures  can  take  students  well 
beyond  their  level  of  understanding.    If  students  are  to  engage  in  productive 

interactions  between  understanding  and  procedural  skill,  instruction  may  need  to 

focus  on  supporting  students  efforts  to  construct,  analyze,  and  modify  a  variety 

of  procedures. 

A  third  conclusion,  of  a  somewhat  different  kind,  is  based  on  the  finding 
that  routine  procedural  skills  developed  just  as  well  or  better  in  the  alternative 
classes  as  in  the  more  conventional  drill  and  practice  environments.    Even  though 
students  in  the  alternative  classes  spent  less  time  practicing  routine  skills  on 
fewer  problems,  their  performance  did  not  suffer.    This  may  be  the  most  salient 
finding  for  immediate  classroom  application  because  it  frees  teachers  to  try  their 
own  alternative  approaches,  even  if  they  are  still  accountable  for  high 
performance  on  routine  tasks. 

Brownell,  W.  A.  (1947).    The  place  of  meaning  in  the  teaching  of  arithmetic. 

Elementary  School  Journal.  47,  2S6-26S. 
Hiebert,  J.,  6  Carpenter,  T.  P.   (in  press).    Learning  and  teaching  with 

understanding.     In  D.  h.  Grouws  (Ed.),  Handbook  of  research  on  mathematics 

teaching  and  learning.     New  York!  Macmillan. 




Hiebert,  J.,  &  Wearne,  0.  (in  press).    Links  batwaan  teaching  and  learning  place 

value  with  understanding  in  first  grade.    Journal  for  Research  in  Mathematics 





1  -281 


Rnhrrt  P.  Hunting.  Kristine  L.  Pepper,  &  Sandra  J.  Gibson 
The  Institute  of  Mathematics  Education 
La  Trobe  University 

We  wanted  to  know  what  enabled  young  children  to  solve  challenging  dealing  tasks  in  which 
perceptual  cues  were  restricted.  A  sequence  of  partitioning  tasks  designed  to  progressively  limit  children  s 
access  to  percental  cues  was  administered  to  30  preschool  children  aged  three  to  five  years.  An  analysis  of 
strategies  used  by  both  successful  and  unsuccessful  children  suggested  that  development  of  a  stable 
pattern  of  operations  having  an  iterative  structure  is  critical.  Further,  reliance  on  sensory  feedback  as  a 
means  of  monitoring  commencement  of  internally  regulated  cycles  seemed  to  constrain  solution  success. 

Young  children  have  considerable  informal  and  intuitive  mathematical  concepts  before  entering  school 
(Gelman  &  Gallistel,  1978;  Irwin,  1990;  Miller,  1984;  Resnick,  1989;  Wright,  1991).  One  particular 
cognitive  skill  is  an  ability  to  equally  divide  a  set  of  discrete  items  (Davis  &  Pitkethly,  1990;  Hunting  & 
Sharpley,  1988;  Pepper,  1991).  A  common  task  used  with  preschoolers  is  a  collection  of  12  items  •- 
sometimes  food  such  as  jelly  beans  -  which  are  to  be  shared  equally  between  three  dolls.  Various  names 
have  been  used  to  describe  the  process  observed  or  infered  from  the  behavior  of  the  subjects  studied: 
partitioning,  sharing,  dealing,  or  distributive  counting.  A  feature  of  successful  efforts  to  distribute  items 
equally  is  a  powerful  algorithm  leading  to  the  creation  of  accurate  equal  fractional  units.  Three  nested 
actions  comprise  the  basic  algorithm:  (1)  allocation  of  item  to  a  recipient,  (2)  iteration  of  the  allocation  act 
for  each  recipient  to  complete  a  cycle,  and  (3)  if  items  remain,  repetition  of  the  cycle  (Hunting  &  Sharpley, 
1988).  The  ability  of  young  children  to  solve  tasks  of  this  kind  is  important  for  mathematics  education 
because  such  actions  can  form  a  meaning  base  for  the  notation  and  symbolism  of  division,  and  for 
fractions  and  ratios.  As  Saenz-Ludlow  (1990)  says,  "It  seems  that  fraction  schemes  spring  out  of  iterating 
schemes  that  lead  to  partitioning  schemes"  (p.  51). 

Subsequent  examination  of  children's  partitioning  behavior  showed  that  some  young  children  who 
used  systematic  methods  varied  the  order  in  which  items  were  allocated  for  each  cycle  of  the  procedure  • 
(Hunting,  1991).  Also,  some  children  were  able  to  maintain  the  dealing  procedure  as  they  were  carrying 
out  a  conversation  with  the  interviewer,  or  re-establish  the  order  of  allocation  after  being  distracted  or 
interrupted.  Pepper  (1991)  found  that  preschoolers'  ability  to  succeed  with  dealing  tasks  was  not  related  to 
their  counting  competence.  In  a  follow  up  study,  Pepper  (1992)  attempted  to  limit  young  children's  use  of 
pre-numerical  skills  such  as  subitizing  (Kaufman,  Lord,  Reese,  &  Volkmann,  1949),  visual  height 
comparisons,  and  pattern  matching,  by  including  a  task  called  Money  Boxes,  in  which  items  to  be 
allocated  --  coins  --  became  hidden  from  view  once  placed  in  opaque  containers.  Of  a  sample  of  25  four 
and  five  year  old  children  studied,  16  succeeded  with  the  Money  Box  task;  and  were  evenly  distributed 
across  three  categories  of  counting  competence  (rho=0. 1 1 ,  p=0.58).  The  most  commonly  observed 
strategy  was  a  systematic  dealing  procedure  where  each  cycle  began  with  the  same  doll  and  money  box. 
Two  strategics  were  suggested  by  these  results.  First,  a  particular  position  of  doll  and/or  money  box 


served  as  a  sign  post  to  mark  the  commencement  of  a  new  cycle.  Second,  mental  records  of  lots  of  three 
were  used  to  regulate  items  as  they  were  being  distributed. 

In  summary,  when  given  sharing  tasks  involving  discrete  items,  in  which  the  items  are  visible  at  all 
stages  of  the  solution  process,  young  children  seem  able  to  use  different  schemes  as  they  work  towards 
creating  equal  shares.  These  schemes  include  comparing  heights  of  stacked  piles,  placing  items  in  lines 
and  comparing  lengths  or  matching  one-to-one  across  shares,  successive  comparison  of  items  in  each 
share  using  subidzing  as  items  are  allocated,  counting,  using  one  recipient  as  a  marker,  and  mental  records 
of  lots  corresponding  to  the  number  of  recipients.  Table  1  lists  these  schemes.  Of  interest  was  whether 
young  children,  if  denied  access  to  perceptual  cues  needed  to  use  a  particular  scheme,  could  adapt  by 
using  a  different  scheme  which  relied  on  internal  regulation  of  actions.  We  also  wanted  to  explore  more 
deeply  by  what  means  children  succeeded  with  tasks  such  as  Money  Boxes  where  schemes  seeming  to 
depend  on  perceptual  feedback,  such  as  pattern  matching,  could  not  be  used.  We  decided  to  examine  in 
more  detail  the  strategies  used  by  successful  and  unsuccessful  children  on  the  Money  Box  task,  and 
compare  these  with  their  methods  for  solving  tasks  that  preceded  and  followed  the  Money  Box  task. 

Cognitive  Scheme 

Behavioral  Indicator 

Pattern  matching/subitizing 

organized  display  of  items  replicated 
across  recipients 

Measurement  of  height 

stacks  items,  lowers  heads  to  visually 
compare,  moves  stacks  together 

Measurement  of  length 

sets  out  items  in  corresponding  lines 


counts  in  process,  able  to  say  how  many 
in  each  pile  at  the  end  as  verification 

Mental  grouping  and  monitoring  of 
represented  items 

begins  cycle  at  different  points, 
organizes  number  of  items  for  each  cycle 
in  advance,  pauses  in  process/tolerance 
of  distraction 

Recipient  as  sign-post  or  marker 

begins  cycle  at  the  same  place 

Table  1:  Possible  schemes  used  to  solve  partitioning  tasks 


Thirty  children  attending  the  La  Trobe  University  Child  Care  Centre  were  individually  interviewed 
during  November  1991.  Children  interviewed  were  from  three  age  groups.  Six  children  were  from  a  three 
year  old  group,  with  mean  age  three  years  two  months  (3.2),  median  age  3.3,  mode  3.3,  and  range  3.0- 
3.4  years.  Fifteen  children  were  from  a  group  of  four  year  olds,  with  mean  age  3.1 1,  median  age  3.10, 

ERIC  307 

1  -283 

mode  3. 10,  and  age  range  3.6-4.4  years.  Nine  children  were  from  a  group  of  five  year  olds.  Their  mean 
age  was  4.1 1.  median  age  4.11,  mode  5.1.  and  their  age  range  was  4.6-5.2  years.  Children  were  selected 
on  the  basis  of  parents'  consent  to  having  their  children  participate.  The  children's  parents  were  either 
students  or  academic  staff  of  the  University. 

A  set  of  partitioning  tasks  involving  distribution  of  discrete  items  was  administered.  These  differed  in 
difficulty  according  to  particular  schemes  it  was  thought  children  might  employ  in  the  course  of  their 
solutions.  All  interviewed  children  were  given  an  initial  task,  called  "Stickers",  followed  by  the  first 
Money  Box  task  involving  15  coins  and  three  dolls.  The  sequence  of  tasks  which  followed  varied  for  each 
individual  according  to  whether  that  child  was  successful  or  not.  Figure  1  shows  the  flowchart  governing 
task  administration.  The  triplet  of  numerals  in  brackets  represents  the  numbers  of  children  from  each  age 
group  who  succeeded  for  each  task  -  the  first  numeral  represents  the  number  of  youngest  children. 


-No(5,?,l)-M  TeaforTvo 

Yes  (1,8,8) 

(  Money  Boxes 

-No  (1,6,4)-*-^ 

-No (3,2,0)  -*■ 
-Yes  (2,4,0)  -*■ 


Tea  for  Three 

V  No  (0,3,1)  - 
-Yes  (1,0,3)- 


Yes  (0,2,4) 

^MoWyBBOX")- No  (0,2,1) 
Yes  (0,0,3) 


fMoa*y  80X88  J- No  (0,0,1)- 


Yes  (0,0,2) 

Money  Boxes 

^       D  . 

-No  (0,0,2)-* 
-Yes  (0,0,0)-* 


■  Figure  1 :  Flowchart  of  interview  tasks 

The  interviews  were  conducted  in  a  small  room  adjacent  to  activity  rooms  that  the  four  and  five  year  old 
groups  used.  All  interviews  were  video  taped  for  later  analysis. 
A  description  of  the  tasks  of  interest  in  this  report  now  follows. 

Stickers.  The  child  is  invited  to  observe  a  sock  puppet,  operated  by  the  interviewer,  distribute  12 
monochromatic  stickers  between  two  dolls.  After  preliminary  discussion  designed  to  put  the  child  at  ease, 
the  child  is  asked  to  observe  Socko  give  out  the  stickers  to  the  dolls  "so  each  doll  gets  the  same."  The 





interviewer  says,  "Socko  isn't  very  clever  at  sharing  out.  I  want  you  to  watch  what  Socko  does,  and  tell 
me  if  each  doll  gets  a  fair  share."  The  puppet  gives  four  stickers  to  one  doll  and  eight  stickers  to  the  other 
in  a  non-systematic  way.  The  child  is  asked  if  the  dolls  get  the  same  each,  and  whether  the  dolls  would  be 
happy  with  their  share.  Regardless  of  the  child's  responses  to  these  questions,  she  is  then  asked  to  teach 
Socko  how  to  distribute  all  the  stickers  so  each  doll  gets  the  same.  After  the  distribution  concludes  the 
child  is  asked  if  the  dolls  got  the  same  each,  and  for  a  justification. 

Money  Boxes  (A I  Three  identical  opaque  money  boxes  are  placed  in  a  row  on  the  table  in  front  of  each 
of  three  dolls.  A  stack  of  IS  twenty  cent  coins  are  positioned  near  the  money  boxes.  The  child  is  told: 
"Mum  wants  all  the  pocket  money  shared  out  evenly  so  each  doll  gets  the  same.  Can  you  share  the  money 
into  the  money  boxes  so  each  doll  has  the  same?  How?  Show  me."  The  child  is  encouraged  to  distribute 
all  the  coins  into  the  money  boxes,  and  when  the  task  is  completed  is  asked:  "Has  each  doll  got  an  even 
share?  How  can  you  tell?" 

Money  Boxes  (BV  A  similar  task  to  Money  Boxes  (A)  except  for  this  task  17  coins  are  to  be 

Money  Boxes  (CX  Five  identical  opaque  money  boxes  are  placed  in  a  circle  on  the  table.  Nineteen 
coins  are  to  be  distributed. 

Money  Boxes  (D1.  Five  identical  opaque  money  boxes  are  placed  on  n  circular  rotating  platform  known 
as  a  "lazy  susan".  The  child  is  shown  how  the  tray  works,  and  it  is  explained  that  the  tray  will  be  rotated 
sometime  during  the  allocation  process.  Again,  19  coins  are  placed  in  a  stack  for  distribution. 

Tea  for  Three.  On  a  table  are  placed  18  items  to  be  distributed  between  three  dolls.  On  an  adjacent  table 
is  a  toy  cook  top  with  pot  and  spoon.  In  the  pot  are  12  white  "crazy  daisy"  plastic  items.  The  interviewer 
says  the  dolls  are  going  to  have  their  dinner,  and  indicates  the  items  in  the  pot  on  the  toy  stove.  The 
interviewer  then  says:  "The  meal  is  cooked  and  the  dolls  are  very  hungry.  Can  you  serve  out  the  food  so 
all  the  food  is  given  out  and  each  doll  gets  the  same  amount?"  If  the  child  stops  before  all  the  items  are 
distributed,  the  interviewer  says:  "Has  all  the  food  been  given  out?  Remember  the  dolls  are  very  hungry". 
The  child  is  then  asked  to  consider  the  allocation  outcome  with  the  question  "Do  you  think  each  ooll  has 
the  same  amount?  How  do  you  know?"  If  the  child  disagrees  she  is  asked:  "Can  you  fix  it  up?" 

Tea  for  Two.  Children  unsuccessful  with  the  Stickers  task  are  invited,  in  pairs,  to  set  a  table  for  two 
dolls,  and  distribute  12  items  of  "food".  Results  not  reported  in  this  paper. 

Seventeen  of  the  30  children  succeeded  with  the  Stickers  task.  Of  these,  six  succeeded  with  the  first 
Money  Box  task.  We  first  consider  solution  strategies  observed  for  children  who  succeeded  with  the  first 
Money  Box  task,  and  their  behavior  on  subsequent  variations.  Then,  we  will  examine  solution  strategies 
typified  by  children  unsuccessful  with  the  first  Money  Box  task,  and  compare  these  with  their  solution 
strategies  on  the  Tea  for  Three  task,  where  items  remained  perceptually  accessible. 


1  -285 

Solution  strategics  of  successful  ch"dren 

The  first  Money  Box  task  required  children  to  share  15  coins  equally  between  three  dolls.  Two  solution 
strategies  were  obsirved.  The  first  is  exemplified  by  Sharlene  (3.7)  and  is  represented  by  the  following 
table,  in  which  A,  i\  and  C  represent  each  of  the  money  boxes,  and  the  bullets,  ♦,  represent  coins.  The 
flow  of  action  proceeds  frrm  lei":  to  righ*. 

Sharlene's  strategy  shows  (1)  cycles  of  three,  in  which  each  box  is  visited  just  once,  and  (2)  the  same 
money  box  used  at  the  commencement  of  each  cycle.  The  second  solution  strategy  is  exemplified  by  Jim 
(5. 1).  His  strategy  also  involves  cycles  of  three,  but  different  money  boxes  mark  the  commencement  of 
each  cycle.  Sharlene's  strategy  is  cyclic  and  regular,  Jim's  strategy  is  cyclic  and  irregular. 

Of  the  six  successful  children,  three  showed  cyclic  and  regular  strategies  (Sharlene  (3.7),  Aaron  (4.9), 
Elise  (4.7));  the  other  children  cyclic  but  irregular  strategies  (Jim  (5.1),  Kalhara  (5.1),  Sophie  (4.2)). 
Kalhara  and  Sophie  showed  at  least  three  regular  cycles  of  allocation. 

For  Money  Boxes  (B),  in  which  17  coins  were  to  be  distributed  to  three  boxes,  three  children,  after 
distribution,  said  the  dolls  did  not  receive  a  fair  share  (Jim  (5.1),  Elise  (4.7),  Kalhara  (5.1)).  Elise  and 
Kalhara  used  the  cyclic  regular  method:  Jim  used  a  cyclic  irregular  method  as  before.  Sharlene  (3.7)  and 
Aaron  (4.9)  showed  cyclic  regular  solutions  but  said  the  dolls  got  the  same.  Sophie  (4.2)  was 
unsuccessful.  Her  allocations  followed  a  cycle  of  C,  B,  A,  except  for  the  second,  which  was  C,  C,  B. 
She  chattered  to  the  interviewer  during  the  allocations.  Hesitation  in  fluency  of  her  actions  seemed  to 
coincide  with  the  onset  of  utterances. 

Money  Boxes  (C)  involved  five  identical  boxes  arranged  in  a  circle.  Nineteen  coins  were  to  be  shared. 
Jim  (5. 1)  and  Elise  (4.7)  succeeded  on  this  task;  Elise  on  the  second  attempt.  She  was  very  uncertain  on 
her  first  attempt,  repeatedly  asking  the  interviewer  if  she  had  placed  a  coin  in  a  particular  money  box.  She 
used  a  different  approach  the  second  time,  taking  piles  of  six,  seven,  three,  and  three.coins  from  the  stack 
in  her  left  hand  as  she  proceeded.  The  interviewer  also  advised  her  not  to  talk  while  she  was  working. 
Kalhara  (5. 1 )  said  all  the  money  boxes  got  the  same.  All  children  showed  a  one  coin-one  box,  one  coin- 


B  • 
C  • 

Figure  2:  Sharlene's  cyclic  and  regular  solution  strategy 

A  • 
B  • 


Figure  3:  Jim's  cyclic  and  irregular  solution  strategy 


next  box  sequential  strategy  beginning  at  th»  box  nearest  them  on  the  table,  thus:  A,  B,  C  D,  E,  A,  B,  C, 

D.  E.... 

Jim  and  Elise  were  given  Money  Boxes  (D)  where  five  money  boxes  were  placed  on  a  "lazy  susan". 
The  lazy  susan  was  spun  2.4  times  after  the  tenth  coin  had  been  placed.  Jim  commenced  placing  the  1 1th 
coin  in  the  "right"  box  and  continued  sequentially,  at  the  end  saying  "there's  only  four  more  left."  Elise 
placed  two  coins  in  the  third  box  even  before  the  tray  was  rotated.  After  rotation  she  changed  direction, 
using  a  one  coin-one  box  sequential  strategy.  She  was  not  successful. 

A  follow-up  interview  was  given  to  Jim  in  which  a  task  similar  to  Money  Boxes  (D)  was  given.  The 
tray  was  rotated  1.8  times  after  the  ninth  coin  was  posted.  There  were  22  coins  in  all.  Jim  placed  the  tenth 
coin  in  the  next  box,  despite  the  intervening  rotation.  He  indicated  the  boxes  did  not  receive  the  same 
number  of  coins,  saying,"because  this  one  didn't  have  any"  -  as  he  touched  the  next  box  in  the  sequence 
after  the  last  coin  had  been  posted 

Solution  strategies  of  unsuccessful  children 

Three  patterns  of  response  were  observed  in  the  children  who  were  unsuccessful  with  the  Money 
Boxes  (A)  task.  The  first  pattern  was  cyclic  like  that  observed  in  the  successful  children.  However, 
children  who  did  this  were  not  consistent  in  its  use  (Joshua  (4.4),  Vanessa  (5.2)).  The  second  pattern  was 
to  piace  a  sequence  of  three  or  more  coins  in  the  same  box.  Six  children  did  this  (Leo  (3.3),  Anton  (3.10), 
Blake  (4.1),  Julian  (4.2),  Tim  (4.10),  Carta  (4.1 1)).  Carta  was  the  only  child  whose  solution  was 
exclusively  of  this  sort  (see  Figure  4).  A  third  pattern  was  to  place  a  coin  in  the  box  adjacent  to  the  box 
previously  visited,  like  a  zig-zag  (see  Figure  5).  Three  children's  responses  were  predominantly  of  this 
sort  (Justin  (3.10),  Tess  (4.3),  Brian  (5.0)).  Other  responses  were  non-cyclic  and  irregular. 

Figure  5:  Tess's  solution  strategy 
Tea  for  Three  was  given  to  children  who  were  unsuccessful  solving  the  first  Money  Box  task.  A 
significant  degree  of  consistency  of  response  was  observed  across  these  two  tasks.  Table  2  summarizes 
strategies  used  for  each  task.  Tim  and  Brian  were  the  only  children  who  had  success  with  Tea  for  Three. 
Tim  counted  the  items  onto  the  dishes.  He  knew  each  dish  contained  six  items.  Since  he  did  not  count  all 
the  items  before,  his  estimate  for  the  first  dish  was  a  good  one.  Brian  was  successful  because  his  strategy 
was  wholly  systematic. 



Figure  4:  Carta's  solution  strategy 





Money  Boxes  (A) 

Tea  for  Three 

Leo  (3.3) 

Non-cyclic,  irregular 

Non-cyclic,  irregular:  not 

nnwfl  v  J.  »u/ 

Nf*l-/*vrlir  imp  pill  ar 

Task  not  given 

Justin  (3.10) 

Adjacent  box  strategy 

Adjacent  dish  strategy 
piedorninantly:  not  successful 

Blake  (4.1) 

Sequence  of  coins  in  each  box, 

nnvinmi  n  a  nf  1  v 


Sequence  of  items  for  each  dish: 
not  successful 

Julian  (4.2) 

Sequence  of  coins  in  each  box 

Task  not  given 

Tess  (4.3) 

Adjacent  box  strategy 

Task  not  given 

Joshua  (4.4) 

First  three  cycles  irregular 

First  four  cycles  irregular:  not 

Tim  (4.10) 

Sequence  of  coins  in  each  box, 

exclusively:  successful 


Sequence  of  coins  in  each  box, 

Placed  nandfuls  of  items  on  each 
dish:  not  successful 

Brian  (5.0) 

Cvclic  with  adjacent  box  strategy 

Cyclic  and  regular  successful 

Vanessa  (5.2)  Cyclic  predominantly 

Cyclic  mixed  with  adjacent  dish 
strategy:  not  successful 

Table  2:  Relationship  between  responses  across  Money  Boxes  (A)  and  Tea  for  Three  tasks 


A  set  of  tasks  involving  Money  Boxes  was  used  to  study  partitioning  schemes  used  by  young  children. 
These  tasks  restricted  children's  use  of  schemes  dependent  on  perceptual  cues  such  as  comparison  of 
heights  of  shared  items,  comparison  of  lengths,  one  to  one  matching  across  shares,  and  successive 
comparison  using  subitizing.  What  internal  regulations  of  actions  made  it  possible  to  succeed  in  these 
circumstances?  The  first  requirement  would  seem  to  be  a  mechanism  for  monitoring  "lots"  or  units  of 
multiple  allocations.  Internally  constructed  units  consisting  of  a  temporal  sequence  of  discrete  counts  or 
tallies  replayed  again  and  again  could  be  needed.  Alternatively,  ability  to  visualize  a  spatial  configuration 
corresponding  to  the  number  of  Money  Boxes,  which  can  be  "scanned"  iteratively.  Such  internal 
constructions  can  be  considered  empirical  abstractions  (von  Glasersfeld,  1982).  Empirical  abstractions 
occur  "when  the  experiencing  subject  attends,  not  to  the  specific  sensory  content  of  experience,  but  to  the 
operations  that  combine  perceptual  and  proprioceptive  elements  into  more  or  less  stable  patterns.  These 
patterns  are  constituted  by  motion,  either  physical  or  attentional,  forming  "scan  paths"  that  link  particles  of 
sensory  experience.  To  be  actualised  in  perception  or  representation,  the  patterns  need  sensory  material  of 
some  kind,  but  it  is  the  motion,  not  the  specific  sensory  material  used,  that  determines  the  pattern's 
character"  (p.  196).  The  difference  between  Jim's  and  Sharlene's  scheme  for  solving  the  first  Money  Box 
task  was  Sharlene's  use  of  one  box  exclusively  as  a  marker.  This  behavior  indicates  she  relied  more  on 
sensory  feedback  located  in  the  physical  presence  of  the  three  boxes.  In  contrast,  Jim's  irregular  starting 
points  for  his  allocation  cycles  suggests  greater  confidence  in  a  represented  cycle  and  some  cycle  counter 
independent  of  the  boxes.  Jim's  performance  on  more  challenging  tasks  involving  five  boxes  arranged  in 




a  circle  showed  he  was  capable  of  keeping  track  of  the  next  box  to  receive  a  coin  and  monitor  position 
reached  in  a  cycle  consisting  of  five  elements. 

The  critical  difference  between  successful  and  unsuccessful  children  on  the  fust  Money  Box  task  was 
the  development  of  a  sable  pattern  of  operations  having  an  iterative  structure.  The  role  played  by  temporal 
or  spatial  representations  of  perceptual  lots  is  unclear,  as  is  the  interaction  between  representational  and 
direct  sensory  experience  in  the  process  of  solving  these  kinds  of  tasks. 


Davis,  G.  (1990).  Reflections  on  dealing:  An  analysis  of  one  child's  interpretations.  In  C.  Booker,  P. 
Cobb,  &  T  N.  de  Mendicuti  (Eds.),  Proceedings  of  the  Fourteenth  PME  Conference  Vol.  3  (pp.  1 1- 
1 8).  Mexico  City :  Program  Committee  of  the  1 4th  PME  Conference. 

Davis,  G.  E.,  &  Pitkethly,  A.  (1990).  Cognitive  aspects  of  sharing.  Journal  for  Research  in  Mathematics 

Edmanan,21(2),  145-153. 

Gelman,  R.,  &  Gallistel,  C.  R.  (1978).  The  child's  understanding  of  number  Cambridge  MA:  Harvard 
University  Press. 

Hunting,  R.  P.  (1991).  The  social  origins  of  pre-fraction  knowledge  in  three  year  olds.  In  R.  P.  Hunting 
&  G.  Davis  (Eds.),  Farlv  fraction  learning  (pp.  55-72).  New  York:  Springer- Verlag. 

Irwin,  K.  C.  (1990).  Children's  understanding  of  compensation,  addition,  and  subtraction  in  part/whole 
relationships.  In  G.  Booker,  P.  Cobb,  &  T  N.  de  Mendicuti  (Eds.),  Proceedings  of  the  Fourteenth 
PME  Conference  Vol.  3  (pp.  257-264).  Mexico  City:  Program  Committee  of  the  14th  PME 

Kaufman,  E.  L.,  Lord,  M.  W.,  Reese,  T.  W.,  &  Volkmann,  J.  (1949).  The  discrimination  of  visual 
number.  American  Journal  of  Psychology.  £2, 489-525. 

Miller,  K.  (1984).  Child  as  the  measurer  of  all  things:  Measurement  procedures  and  the  development  of 
quantitative  concepts.  In  C.  Sophian  (Ed.),  Origins  of  cognitive  skills  (pp.  193-228).  Hillsdale  NJ: 

Pepper,  K.  L.  (1991).  Preschoolers'  knowledge  of  counting  and  sharing  in  discrete  quantity  settings.  In 
R.  P.  Hunting  &  G.  Davis  (Eds.),  Early  fraction  learning  (pp.  103-129).  New  York:  Springer-Verlag. 

Pepper,  K.  L.  (1992).  A  study  of  preschoolers'  pre-fraction  knowledge.  Masters  thesis  in  preparation,  La 
Trobe  University. 

Resnick,  L.  B.  (1989).  Developing  mathematical  knowledge.  American  Psychologist.  44.  162-169. 

Saenz-Ludlow,  A.  (1990).  Michael:  A  case  study  of  the  role  of  unitizing  operations  with  natural  numbers 
in  the  conceptualization  of  fractions.  In  G.  Booker,  P.  Cobb,  &  T  N.  de  Mendicuti  (Eds.), 
Proceedings  of  the  Fourteenth  PME  Conference  Vol.  3  (pp.  51-58).  Mexico  City:  Program  Committee 
of  the  14th  PME  Conference. 

von  Glasersfeld,  E.  (1982).  Subitizing:  The  role  of  figural  patterns  in  the  development  of  numerical 
concepts.  Archives  de  Psychologic.  5J2, 191-218. 

Wright,  R.  J.  (1991).  What  number  knowledge  is  possessed  by  children  entering  the  kindergarten  year  of 
school?  Mathematics  Education  Rest-arch  Journal.  3(11  1-16. 


1  -289 

The  Emancipatory  Nature  of  Reflective  Mathematics  Teaching 
Barbara  Jaworski,  University  of  Birmingham,  U.K. 

Critical  reflection  on  the  act  of  teaching  may  be  seen  to  be  liberating  for  the  teacher,  who. 
as  a  result,  has  greater  knowledge  and  control  of  the  teaching  act.  This  paper  supports 
such  contention  where  the  teaching  of  mathematics  is  concerned  by  drawing  on  research 
with  one  teacher  who  might  be  seen  to  engage  in  critical  reflective  practice.  It  considers 
also  how  the  researcher  might  influence  the  liberating  process  through  which  teacher- 
emancipation  occurs. 


If  the  term  emancipation  -  a  state  of  being  set  free  from  bondage  (Chambers'  English  Dictionary)  - 
is  applied  to  teachers,  it  might  be  inferred  that  the  teacher  who  is  not  emancipated  remains  in 
some  form  of  bondage  -  for  example,  the  constraints  of  an  imposed  curriculum. 

Anecdote  abounds  to  support  the  frequency  of  statements  from  mathematics  teachers  in  the  vein 
of  "I  have  taught  them  blank  so  many  times  and  they  still  can't  get  it  right",  or  "I  should  like  to 
teach  more  imaginatively,  but  if  I  did  I  should  never  have  time  to  complete  the  syllabus".  Such 
statements  typically  come  from  teachers  who  are  bound  by  tradition,  convention  or  curriculum, 
and  who  fail  to  perceive  their  own  power  to  tackle  constraints.  The  result  for  pupils  is  likely  to 
be  a  limited  or  impoverished  mathematical  experience. 

Reflective  practice 

Many  educationalists  have  advocated  reflective  practice  as  a  means  of  emerging  from  such 
shackles.  I  must  make  clear  that  the  term  reflection  as  I  use  it  here  has  a  critical  dimension  and  is 
more  than  just  'contemplative  thought'.  Van  Manen  (1977)  defines  reflection  at  three  different 
levels,  the  third  of  which,  critical  reflection,  concerns  the  ethical  and  moral  dimensions  of 
educational  practice.  Boud,  Keogh  and  Walker  (1985)  speak  of  "goal-directed  critical  reflection" 
which  concerns  reflection  which  is  "pursued  with  intent".  Smyth  (1987)  advocates  "a  critical 
pedagogy  of  schooling  which  goes  ccnsiderably  beyond  a  reflective  approach  to  teaching", 
suggesting  that  the  reflective  approach  is  not  itself  critical.  However,  Kemmis  (1985)  brings  these 
two  elements  very  firmly  together,  as  in 

Wc  arc  inclined  to  think  of  reflection  as  something  quiet  and  personal.  My  argument  here  is  that 
reflection  is  action-oriented,  social  and  political.  Its  product  is  praxis  (informed,  committed 
action)  the  most  eloquent  and  socially  significant  form  of  human  action  (p  141) 



1  -290 

It  is  reflection  in  Kemmis'  sense  which  1  address  in  this  paper.  I  will  make  the  case  that  reflective 
practice  in  matnematics  teaching,  which  is  critical  and  demands  action,  is  a  liberating  force,  and 
that  teachers  engaging  in  such  reflection  are  emancipated  practitioners. 

Teachers'  voice 

The  emancipated  teacher  may  be  seen  to  be  in  theoretical  control  of  the  practice  of  teaching.  This 
implies  that  the  teacher  explicates  theories,  or  gives  them  'voice'. 

Cooney  (1984)  refers  to  teachers'  "implicit  theories  of  teaching  and  learning  which  influence 

classroom  acts",  saying  further, 

I  believe  that  teachers  make  decisions  about  students  and  the  curriculum  in  a  rational  way 
according  to  the  conceptions  they  hold.  (My  italics) 

Although  the  classroom  act  itself  may  be  seen  as  an  explication  of  theory,  teachers'  thinking  is 
often  not  explicitly  articulated,  and  it  is  left  to  researchers  outside  the  classroom  to  give  voice  to 
teachers'  conceptions.  Elbaz  (1990)  suggests  that  it  has  become  important  that  researchers  into 
teachers'  thinking  "redress  an  imbalance  which  had  in  the  past  given  us  knowledge  of  teaching 
from  the  outside  only"  by  encouraging  expression  of  teachers'  own  voice. 

Having  'voice'  implies  that  one  has  a  language  in  which  to  give  expression  to  one's  authentic 
concerns,  that  one  is  able  to  recognise  those  concerns,  and  further  that  there  is  an  audience  of 
significant  others  who  will  listen. 

Smyth  (1987)  goes  further  in  speaking  of  teacher  emancipation,  that  only  by  exercising  and 
'intellectualising'  their  voice,  will  teachers  be  empowered  in  their  own  profession. 

To  reconceptualise  the  nature  of  teachers'  work  as  a  form  of  intellectual  labour  amounts  to 
permitting  and  encouraging  teachers  to  question  critically  their  understandings  of  society, 
schooling  and  pedagogy. 

These  notions  pose  a  dilemma  for  theorists,  researchers  or  teacher-educators  proposing  teacher 
emancipation,  because  to  be  truly  emancipated  teachers  themselves  must  be  their  own  liberators. 

My  experience  as  a  teacher,  and  in  working  with  teachers,  suggests  that  critical  reflective  practice 
(which  I  discuss  further  below)  can  be  a  liberating  process,  but  that  it  is  actually  very  difficult  to 
sustain  if  working  alone.  In  my  research  with  teachers  I  believe  that  I  have,  to  some  extent, 
facilitated  their  reflective  practice  by  being  there  and  by  asking  questions.  I  propose,  therefore, 
that  researchers  working  with  teachers  can  be  catalysts  for  liberation,  through  their  encouraging 
of  questioning  of  practice,  and  provision  of  opportunity  for  teachers  to  exercise  their  voice. 


The  role  of  the  researcher 

Elbaz  (1987),  while  acknowledging  the  "large  gap  between  what  researchers  produce  as 
reconstructions  of  teachers'  knowledge  ...  and  teachers'  accounts  of  their  own  knowledge", 
nevertheless  expresses  the  hope, 

I  would  like  to  assume  that  research  on  teachers'  knowledge  has  some  meaning  for  the  teachers 
themselves,  that  it  can  offer  ways  of  working  with  teachers  on  the  elaboration  of  their  own 
knowledge,  and  that  it  can  contribute  to  the  empowerment  (of]  teachers  and  the  improvement  of 
what  is  done  in  classrooms,  (p  46) 

The  purpose  of  my  own  research  with  teachers  was  to  attempt  to  elicit  the  deep  beliefs  and 
motivations  which  influenced  their  teaching  acts.  My  methodology  involved  talking  extensively 
■with  the  teacher  both  before  and  after  a  lesson  which  I  observed.  Fundamental  to  any  success  I 
might  have  had  in  this  was  the  development  of  a  level  of  trust  between  teacher  and  researcher 
which  would  allow  sensitive  areas  to  be  addressed.  For  the  teacher  cooperating  in  my  research, 
and  attempting  seriously  to  tackle  the  questions  I  asked,  a  consequence  was  a  making  explicit  of 
theories  of  teaching  which  could  then  be  used  to  influence  future  practice.  It  was  not  part  of  my 
research  aims  to  influence  the  practice  of  the  teachers  with  whom  I  worked,  it  was  an  inevitable 
consequence  that  it  did.  However,  change  was  effected  by  the  teacher,  and  in  this  respect  the 
researcher  acted  as  a  catalyst. 

An  example  of  developing  practice  related  to  teacher-researcher  discussion 

The  teacher  was  about  to  teach  a  lesson  on  vectors  to  follow  up  his  introduction  of  vectors  to  his 
year-10  class  (15  year  olds)  in  a  previous  lesson  I  had  asked  him  to  tell  me  what  he  would  do  in 
the  coming  lesson,  and  he  replied  that  he  wanted  to  "recap  what  a  vector  AB  is".  He  referred  to 
notes  which  he  had  prepared  with  plans  for  the  lesson.  The  following  piece  of  transcript  records 
part  of  my  conversation  with  the  teacher  before  that  lesson.  (T  -  teacher,  R  -  researcher,  myself) 



What  do  you  mean  by  '  recap"?  You  recap? 



Me  recapping  -  well  -  me  asking  questions.  "Now,  what's  meant  by  adding  vectors? 
What's  meant  by  taking  away  a  vector,  or  a  minus  vector?"  And  then  asking,  "what's  the 
difference  between  those  three  -  the  vcclor-AB,  AB,  and  BA?" 



Now  is  it  your  expectation  that  by  asking  appropriate  questions  you  will  get  all  that 
information  from  them? 



Hopefully  yes.  And  what  1  wanted  to  do  today  was  not  really  concentrate  too  much  on 
vectors,  but  say,  "If  we've  got  a  vector  (3,4)  can  we  find  the  length  of  AB?  Hello  -  we're 
back  into  Pythagoras  •  ha  ha  ha!" 














And  then  give  them  some  questions,  and  then  get  them  to  check  over  their  homework  after 
they've  sorted  out  -  oh  -  one  bit  I've  missed  on  here  (referring  to  his  notes)  I  want  them  to 
say  what  2AB  is  -  something  else  we  talked  about,  and  I  want  to  talk  about  AB  and  BA  as 
vectors,  and  AB  and  BA  as  lines. 






We're  really  talking  about  notation,  aren't  we,  now? 



Right.  -  You  said,  give  them  ten  questions.  What  sort  of  questions? 



They're  going  to  be  quite  straightforward. 



To  do  what? 



Find  the  lengths  of  vectors. 



So,  for  example,  "Find  the  length  of  a  vector ... 



...  AB,  if  AB  is  3.4." 
(slight  digression  here  on  what  the  3,4  notation  looks  like) 



Or  you  could  get  them  to  invent  some  for  themselves. 



Yes,  that  would  be  quite  interesting,  wouldn't  it. 

The  conversation  proceeded  to  more  general  aspects  of  teaching  and  learning,  and  at  one  point 
the  teacher  offered  an  anecdote  from  one  of  his  lessons  with  lower-attaining  pupils  in  which  they 
had  been  invited  to  invent  'think  of  a  number*  games  for  each  other. 



I  started  playing  the  game,  like,  "I'm  thinking  of  a  number.  I  double  it  and  add  three,  and 
my  answer  is  seven.  What  was  the  number  I  started  with?"  After  a  bit  of  practice,  no 
problems.  But  the  things  they  were  asking  each  other  were  out  of  this  world.  If  I'd  asked 
them  they'd  have  gone  on  strike! 



How  do  you  mean? 



Well,  they  were  saying  "I'm  thinking  of  a  number,  I've  halved  it,  I've  added  three  to  it, 
I've  multiplied  by  three,  I  take  two  away,  I  divide  it  by  seven  and  my  answer  is  twenty  one. 
What  number  did  I  start  with?"  And  they  could  actually  solve  them.  Now  if  I  went  in  and 
put  that  on  the  board  for  a  bottom  ability  group  they  would  go  on  strike. 



Yes,  right. 



And  when  you  actually  got  back  to  it,  they  had  this  inverse  relationship  all  sorted  out.  They 
couldn't  write  it  down,  but  they  had  it  all  sorted  out.  That's  what,  yeah,  it's  there  isn't  it, 
them  setting  their  own  levels.  I  don't  do  it  often  enough.  I  must  do  it  more  often. 
You  can  have  that  for  what  it  is! 



Thank  you!  How  about  doing  it  there  (I  pointed  to  his  notes  for  the  vectors  lesson)? 



What,  getting  them  to  set  their  own? 



Get  them  to  set  their  own. 



(Pause)  I'll  trv. 




1  -293 

28  R       It'll  be  interesting  to  see  if  they  only  come  up  with  questions  of  a  particular  type,  because 
that  will  tell  you  something  about  the  way  they  arc  thinking. 

30  T       Can  1  say,  "Be  inventive?" 

31  R  Sure! 

32  T        OK,  We'll  do  that.  

At  statement  8,  the  teacher  said  he  would  give  the  class  some  questions.  He  then  returned  to 
talking  again  about  his  general  lessons  plans.  1  was  interested  in  what  the  questions  would  look 
like  and  so  I  asked  him  (statement  11).  His  reply  that  they  would  be  straightforward,  was 
followed  by  a  digression  into  forms  of  notation.  1  brought  him  back  to  the  questions  again  with 
my  statement  (17)  that  he  could  get  them  to  invent  some  for  themselves.  He  acknowledged  this, 
but  little  more  at  that  point. 

I  was  interested  In  what  his  questions  would  be,  because  1  wondered  what  they  would  contribute 
to  the  pupils'  perceptions  of  vectors.  My  remarks  were  a  focuMng  device  where  our 
consideration  of  these  questions  were  concerned.  If  1  had  not  pursued  them,  the  teacher  may  not 
have  provided  any  more  information.  1  had  great  power  to  focus  in  this  way,  although  I  did  not 
at  the  time  select  explicitly  this  focus  in  preference  to  others.  My  suggestion  was  spontaneous.  It 
was  not  my,  or  our,  pre-planned  intention  to  focus  on  pupils'  inventing  of  their  own  questions. 
It  arose  in  and  from  the  context  of  the  conversation,  which  was  about  the  teacher's  concerns. 

As  part  of  the  continuing  conversation,  the  teacher  came  up  with  the  anecdote  about  pupils  in 
another  class  setting  their  own  challenges,  and  the  value  that  he  saw  in  this.  It  is  my  speculation 
that  this  was  triggered  by  my  suggestion,  and  that  certain  associations  were  set  up  in  response  to 
our  talk.  This  analysis  came  some  time  after  my  work  with  the  teacher,  so  I  was  not  able  to  check 
its  validity  with  him.  However,  his  telling  of  the  anecdote  gave  me  opportunity  to  reiterate  my 
suggestion  (statement  24),  and  for  the  teacher  to  agree  to  try  it  out  (statement  27).  His  questions  at 
the  end  recognise  that  this  is  a  suggestion  from  me,  and  seek  in  some  way  my  clarification  of  the 
extent  of  invention  1  envisage. 

Thus,  1  influenced  the  teacher's  planning  and  execution  of  the  vectors  lesson  more  overtly  than 
had  been  my  intention.  However,  1  feel  that  he  was  able  to  set  pupils  an  open  task  of  inventing 
their  own  questions  because  he  could  see  this  in  the  context  of  other  open  tasks  which  he  had  set, 
and  which  had  been  successful.  Moreover,  his  style  of  working  with  the  pupils  was  such  that  an 
activity  of  this  kind  was  not  unfamiliar  territory  to  them.  It  is  interesting  to  consider  the  extent 
to  which  my  suggestion  depended  on  my  knowledge  of  his  practice,  and  the  extent  to  which  his 
acceptance  of  it  depended  on  his  reciprocal  knowledge.  The  developing  trust  between  us  made  a 
significant  contribution  to  our  joint  understanding  of  what  was  possible  in  his  classroom. 

1  -294 

In  the  lesson  itself,  he  introduced  that  task  with  the  words:  "I  would  like  you  to  make  your  own 
questions  up  and  write  your  own  answers  out  and  then  "hare  your  questions  with  a  neighbour. 
Could  you  be  inventive  please.  Don't  put  up  a  whole  series  of  boring  questions".  I  discussed 
aspects  of  this,  and  pupils'  responses  to  it,  in  Jaworski  (1991  b)  in  another  context,  so  I  shall  not 
repeat  those  details  here.  However,  the  outcome  in  terms  of  some  pupils'  questions  and 
responses  was  very  satisfactory.  It  opened  up  areas  which  the  class  had  not  yet  addressed:  for 
example,  the  special  nature  of  parallel  vectors,  and  the  related  notations  for  vectors  of  equal 
length  albeit  of  different  directions  in  different  positions,  both  arose  from  pupils'  own 
investigations.  It  provided  the  teacher  with  opportunity  to  address  such  questions  in  a  way 
meaningful  to  pupils  because  they  had  arisen  from  the  pupils'  own  thinking.  Our  retrospective 
reflection  on  this  lesson,  acknowledged  the  value  and  success  of  the  activity. 

Critical  reflection  influencing  the  teaching  act 

1  believe  that  this  episode  charts  a  stage  in  this  teacher's  own  development  as  a  teacher.  For  him, 
in  this  case,  critical  reflection1  involved  making  explicit  the  value  of  occasions  where  he  asked 
pupils  to  be  inventive  in  setting  their  own  challenges.  It  resulted  in  his  becoming  more  aware  of 
opportunities  where  he  could  encourage  pupils  in  this.  Three  weeks  later  I  saw  a  lesson  in 
which  he  returned  to  pupils  some  tests  which  they  had  done  and.  he  had  marked.  Rather  than 
present  a  set  of  correct  solutions  for  them  to  compare,  he  offered  a  set  of  'answers'  of  his  own,  all 
of  which  had  errors  in  them.  Their  task  was  to  spot  the  errors,  and  to  explain,  in  discussion  with 
neighbours,  what  would  be  correct.  In  this  way  he  hoped  to  challenge  them  to  work  dynamically 
on  their  own  solutions  and  errors,  rather  than  passively  to  accept  the  teacher's  'correct'  solutions. 

I  believe  that  enabling  the  pupils  to  take  more  responsibility  for  their  work  and  thinking  through 
setting  their  own  challanges  was  an  aspect  of  this  teacher's  philosophy  and  operation  which 
developed  during  the  time  that  I  was  working  with  him.  I  propose  that  this  speaks  to  the 
emancipation  of  this  teacher,  in  that  he  was  actively  seeking  ways  of  enhancing  pupils'  learning, 
which  brought  him  into  a  more  acute  knowledge  and  control  of  the  teaching  situation,  and  thus 
of  his  own  direction  and  purpose.  In  this  he  was  engaged  in  a  process  of  self-liberation. 

Our  conversation  often  focused  on  the  liberating  process  itself.  On  one  occasion  we  discussed 
different  sorts  of  decisions  which  the  teacher  had  made  in  various  lessons  which  I  had  seen,  and 
the  difference  between  responding  to  a  pupil  instinctively,  and  making  a  more  informed 

1  I  have  explicated  in  some  detail  the  stages  of  critical  reflection  which  formed  pan  of  my  analysis  of 
covcrsations  with  teachers  in  my  wider  study.  This  is  included  in  a  paper  "Reflective  practice  in 
mathematics  teaching"  which  is  currently  submitted  for  publication. 


1  -295 

response  or  judgment.  The  teacher  commented,  "I  feel  that  responses  are  judgments  that  have 

proved  right  in  the  past  and  been  taken  on  board."  He  went  on. 

You've  been  through  a  lot  of  these  situations  before  your  responses.  Don't  they  actually  come 
from  things  which  happen  in  the  past  and  you're  saying,  1  made  a  judgment  then  that  was  a  good 
one,  or  saw  someone  do  something  that  was  good.  And  you  actually  take  that  on  board.  Isn't 
that  what  developing  as  a  teacher  is  all  about?" 

Some  manifestation  of  this  general  principle  might  arise  after  the  vectors  lesson  and  the  asking 
of  pupils  to  invent  their  own  questions.  Perhaps  in  some  other  lesson  later,  the  teacher  would 
recall  aspects  of  this  activity,  and  our  subsequent  analysis  of  it,  and  it  would  influence  his 
teaching  at  that  instant. 

I  have  suggested  (Jaworski,  1991a  and  b)  that  it  is  such  in-the-moment  recognition  of  choice  of 
reponse,  based  on  previous  experience  made  explicit,  that  is  the  action  outcome  of  critical 
reflection.  I  go  further  here  in  suggesting  that  this  is  the  essence  of  the  liberating  process.  The 
more  critical  such  reflection  is,  in  being  disciplined  about  identifying  the  issues  in  a  particular 
lesson,  the  choices  taken,  the  decisions  made,  and  their  effect  on  learning  and  teaching,  the  more 
able  the  teacher  is  likely  to  be  to  act  appropriately  to  what  arises  on  a  subsequent  classroom 
occasion.  Developing  as  a  teacher  is  the  result  of  such  action.  Such  development  is  dynamic, 
and,  if  recognised  and  used  deliberately,  it  can  be  liberating  and  empowering. 

Teacher  emancipation 

Teacher  emancipation,  according  to  sources  quoted  at  the  beginning  of  this  paper,  arises 
consciously  from  teachers  becoming  aware  of  their  own  knowledge  and  purpose  through  critical 
enquiry  into  their  practice.  Emancipation  seems  to  be  a  state  within  the  liberating  process  of 
action-oriented  critical  enquiry.  In  the  case  of  mathematics  teaching  this  involves  questioning 
both  pupils'  perceptions  of  the  mathematics  on  which  a  lesson  is  based,  and  the  pedagogy  to  be 
employed  in  developing  this  mathematics.  Teachers  have  to  know  what  they  hope  to  achieve  in 
terms  of  the  mathematical  content  of  a  lesson  and  their  pupils'  constructions  of  this 
mathematics,  and  also  in  terms  of  the  teaching  acts  which  will  be  employed.  Although  this 
content  and  these  acts  will  be  designed  to  fit  some  prescribed  curriculum,  they  do  not  need  to  be 
conditioned  or  bound  by  it.  The  curriculum  to  which  the  above  teacher  worked  required  pupils' 
understanding  of  the  elements  of  vectors  which  were  being  addressed  in  the  lesson.  It  did  not 
prescribe  the  means  by  which  such  content  would  be  made  available  to  the  pupils,  and  it  did  not 
preclude  the  pupils  coming  to  aspects  of  that  content  through  their  own  investigations.  The 
teacher's  overt  knowledge  of  mathematics  and  pedagogy,  based  on  his  own  developing 
experience,  as  well  as  a  confidence  in  his  own  ability  to  make  appropriate  choices  and  judgments, 
enabled  him  to  construct  suitable  teaching  acts.   This  meant  that  the  teacher  himself  was  in 



control  of  the  learning  environment  of  pupils  in  his  classroom,  and  moreover  that  he  could  take 
responsibility  for  what  occurred  rather  than  blaming  defects  on  pupils'  inability  to  remember  or 
retain,  or  on  constraining  effects  of  the  curriculum.  His  awareness  of  this  level  of  responsibility  2 
and  his  overt  exercise  of  control  were  indicators  of  his  emancipated  position.  For  the  teacher  1 
have  described,  encouraging  pupil-emancipation  might  be  seen  as  an  element  of  his  control. 

I  have  indicated,  where  the  above  teacher  was  concerned,  that  my  research  presence  had  some 
effect  on  his  developing  practice.  How  does  teacher  development  and  subsequent  emancipation 
depend  on  such  presence,  and  how  far  is  it  possible  for  a  teacher  to  achieve  this  alone? 

I  have  no  research  evidence  to  present  in  order  to  address  this  question.  The  teacher  group 
working  together,  perhaps  in  small-scale  action  research,  to  support  and  encourage  such  practice 
can  be  an  effective  sustaining  medium  (see  for  example,  Kemmis  1985,  Gates,  1989,  Mathematical 
Association  1990).  However,  further  research  is  needed  into  the  development  of  the 
emancipated  teacher  through  a  liberating  process  of  action-oriented  critical  enquiry,  particularly 
where  the  teaching  of  mathematics  and  its  effect  on  pupils'  learning  is  concerned. 


Boud,  D.,  Keogh,  R.  and  Walker,  D.  (1985)  Reflection:  turning  experience  into  learning.  London:  Kogan 

Cooney,  T.J.  (1984)  The  contribution  of  theory  lo  mathematics  teacher  education'  in  H.G.  Steiner  et  al 
Theory  of  mathematics  education  (TME)  Bielefeld,  Germany:  Universitat  Bielefeld/IDM 

Elbaz,  F.  ( 1987)  "Teachers'  knowledge  of  teaching:  strategies  for  reflection',  in  J.  Smyth  (ed.)  Educating 
teachers.  London:  Falmer 

Elbaz,  F.  (1990)  'Knowledge  and  discourse:  the  evolution  of  research  on  teacher  thinking,'  in  C.  Day,  M. 

Pope  and  P.  Denicolo  (eds.)  Insight  into  teachers'  thinking  and  practice.  London:  Falmer 
Gates,  P.  (1989)  'Developing  conscious  and  pedagogical  knowledge  through  mutual  observation',  in  P. 

Woods  (ed.)  Working  for  teacher  development.  London:  Peter  Francis 
Jaworski,  B.  (1991a)  Interpretations  of  a  Constructivist  Philosophy  in  Mathematics  Teaching. 

Unpublished  PhD  Thesis.  Milton  Keynes,  England:  Open  University 
Jaworski,  B.  (1991b)  "Some  implications  of  a  constructivist  philosophy  of  mathematics  teaching  for  the 

teacher  of  mathematics",  in  Proceeding  of  PME  XV,  Assisi,  Htaly. 
Kemmis,  S.  (1985)  'Action  Research  and  the  politics  of  reflection'  in  Boud,  D.,  Keogh,  R.  and  Walker, 

D.(eds.)  Reflection:  turning  experience  into  learning.  London:  Kogan  Page 
Mathematical  Association,  (1991)  Develop  your  Teaching,  Cheltenham,  UK:  Stanley  Thomes 
Smyth,  J.  (1987)  'Transforming  teaching  through  intellectualising  the  work  of  teachers.'  in  J.  Smyth  (ed.) 

Educating  teachers.  London  Falmer 
Van  Manen,  M.  (1977)  'Linking  ways  of  knowing  with  ways  of  being  practical'  Curriculum  Inquiry 

6(3),  205-228 

~  Evidence  of  this  may  be  found  in  Jaworski  (1991a,  Chapter  7,  and  1991b,  page  219)  Adetailed 
account  of  the  'vectors'  lesson  is  provided  in  Jaworski  (1991  a),  and  a  curtailed  account,  more  specifically 
related  to  constructivist  aspects  of  the  teacher's  thinking,  in  Jaworski  ( 1991b) 


1  -297 


Clive  Kanes 

Institute  for  Learning  ih  Mathematics  and  Language 
Griffith  University,  Brisbane 

This  paper  starts  by  discussing  a  number  of  paradoxes  to  have  recently  emerged  in  theories  of 
learning  and  teaching  mathematics.  These  are  found  to  make  similar  assumptions  about  the 
nature  of  mathematical  knowledge  and  its  epistemology.  A  detailed  analysis  of  a  transcript, 
recording  the  linguistic  interaction  between  the  researcher  and  a  number  of  senior  high  school 
students,  follows.  This  analysis  traces  the  breakdown  of  a  didactic  contract  (Brousseau)  and 
its  subsequent  re-establishment;  it  also  studies  how  the  pedagogic  sequencing  facilitates 
learning  the  attainment  of  learning  goals.  The  transcript  is  also  used  to  exemplify  the  occurence 
of  paradox  in  pedagogic  situations.  The  paper  concludes  by  adapting  a  model,  drawn  from  the 
field  of  genre  studies,  in  order  to  provide  a  theoretical  account  of  linguistic  utterances 
constitutive  of  pedagogic  interactions  and  their  epistemological  implications. 

§1  Introduction 

In  Plato's  Meno  (80c),  Socrates  presents  a  paradox  which  shows  that  a  student  cannot 
learn  what  he  or  she  does  not  already  know:  For  if  the  student  had  the  knowledge  there  would 
be  no  need  to  seek  it,  and  if  the  student  lacked  knowledge,  then  how  would  the  student  even 
know  what  to  look  for?  The  standard  rebuttal  of  this  paradox  points  to  an  apparent  confusion 
about  the  meaning  of  words,  for  instance,"having  knowledge".  Nevertheless,  paradoxes  of 
this  kind  -  the  learning  paradox  (Bereiter,  1985)  is  almost  exactly  similar  -  bedevil  modem 
theories  of  teaching  and  learning  mathematics  (Brousseau,  1986). 

A  motivating  question  for  this  paper  therefore  is:  How  may  paradox  free  models  of 
teaching  and  learning  be  constructed?  The  paper  presents  an  interim  report  of  a  study  into  the 
meaning  and  use  of  words  in  pedagogical  interactions  in  mathematics  classrooms. 

§2   Paradoxes  in  mathematics  education  and  some  remarks  on  epistemology 

The  constructivist  view  that  learning  is  a  process  in  which  the  learner  is  actively 
engaged  in  a  process  of  restructuring  or  organising  knowledge  schemata  is  widely  held  in 
information  processing  psychology  (Resnick,  1983;  Bereiter,  1985).  This  model  for  learning 
is  ,  however,  prone  to  paradox.  Bereiter  (1985),  for  instance,  refers  to  the  learner's  paradox 

...if  one  tries  to  account  for  learning  by  means  of  mental  actions  carried  out  by  the 
learner,  then  it  is  necessary  to  attribute  to  the  learner  a  prior  cognitive  structure  that  is  as 
advanced  or  complex  as  the  one  to  be  acquired.(o.  202) 


More  significantly  for  educators,  the  move  to  develop  instructional  procedures  consistent  with 
constructivist  learning  theory  has  also  run  into  difficulties  (Cobb,  1988, 1992:  Kanes,  1991). 
Cobb  argues,  for  instance,  that  Resnick's  notion  of  "instructional  representation",  violates  the 
autonomy  of  the  individual  constructor  at  a  key  stage  in  the  pedagogic  interaction.  Resnick's 
procedure  therefore  erroneously  reinstates  as  constructivist,  a  variant  of  the  absorption  model 
for  learning. 

Occurrence  of  paradox  has  also  been  noted  in  work  proceeding  on  more  general 
pedagogic  grounds.  Brousseau  (1986)  for  instance  argued  that  teacher  and  student  enter  a 
didactic  contract  in  which  the  teacher  must  ensure  that  the  student  has  an  effective  means  of 
acquiring  knowledge  and  in  which  the  student  must  accept  responsibility  for  learning  even 
though  not  being  able  to  see  or  judge,  beforehand,  the  implication  of  the  choices  offered  by  the 
teacher.  Brousseau  argued  that  the  contract  is  driven  into  crisis  and  ultimately  fails,  for 

all  that  he  [the  teacher]  undertakes  in  order  to  get  the  pupil  to  produce  the  expected 
patterns  of  behaviour  tends  to  deprive  the  pupil  of  the  conditions  necessary  to 
comprehend  and  learn  the  target  notion:  if  the  teacher  says  what  he  wants  he  cannot 
obtain  it.  (p.  120) 

Similarly,  Steinbring  (1989)  observes  that  the  teaching  process  of  making  all  meanings  explicit 

leads  to  the  effect  that  by  the  total  reduction  of  the  new  knowledge  which  is  to  be 
learned  to  knowledge  already  known,  nothing  really  new  can  be  learned,  (p.  25) 

Obviously,  in  a  few  short  sentences  one  is  not  able  to  treat  the  issues  represented  by 
these  paradoxes  exhaustively.  However,  it  is  interesting  to  note  that  the  context  within  which 
each  of  these  arise  provides  a  similar  epistemological  stance.  Each  assumes  that  mathematical 
knowledge  is  essentially  a  matter  of  content  and  that,  as  such,  is  capable  of  being  made,  in 
principle  at  least,  totally  explicit  For  instance,  in  the  learner's  paradox,  knowledge  is 
individually  constructed  as  a  representation  of  a  knowledge  target,  and  therefore,  in  a  sense,  is 
actually  derived  by  the  individual.  It  follows  that  this  kind  of  knowledge  can,  and  in  pedagogic 
episodes  should,  be  made  explicit.  To  illustrate  by  a  metaphor  Constructing  a  clock  means 
being  able,  in  principle  at  least,  to  make  explicit  each  of  the  parts  of  the  clock.  When  teaching 
clock-making  the  detailing  of  the  clock's  mechanism  may,  for  the  benefit  of  the  apprentice, 
need  to  take  place  at  a  fine  level.  In  that  case,  the  clock  maker  is  actually  engaged  in  a  process 
of  re-presenting  the  clock  to  the  apprentice  as  an  articulation  of  its  parts.  In  the  same  way, 
constructed  mathematical  knowledge  is  re-presentational  and  explicatable.  This  view, 
however,  asserts  an  epistemology  of  reference  over  intention  or  transaction.  For  instance,  the 

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representational  clockmaker  is  unable  to  convey  the  overall  coherence  of  the  form  of  the  clock 
or  the  degree  and  manner  in  which  its  structure  complements  a  certain  aesthetic  economy  or 
style  (expressive  or  intentional  characteristics);  likewise,  she  is  unable  to  convey  the  actual 
experience  of  the  actions  of  making  a  watch  (transactional  or  pragmatic  characteristics). 

§3    Analysis  and  discussion  of  a  pedagogic  episode 

In  order  to  provide  an  illustrative  focus  for  the  theoretical  statements  of  the  last  section  and 
those  to  come  later  in  the  paper,  this  section  will  present  a  linguistic  analysis  of  a  pedagogic 
episode.  The  sequence  studied  is  drawn  from  a  stimulated  recall  (Keith,  1988;Parsons  et  al, 
1983).  This  method  involved  video  recording  a  lesson  in  a  naturalist  context  and,  immediately 
after  this,  replaying  the  tape  to  a  teacher-chosen  subset  of  students.  Students  were  asked  to 
respond  freely  to  the  tape,  the  researchers  reserved  the  right  to  ask  probing  questions.  Those 
present  include  6  students  chosen  from  the  class  by  the  teacher,  together  with  2  researchers. 
The  classroom  teacher  was  not  present.  The  discussion  between  students  and  the  researcher  is 
on  the  application  of  'dummy  variables'  as  indices  in  expressions  involving  complex  algebraic 

In  analysing  this  episode,  it  has  been  assumed  that  in  order  to  recover  the  shifting 
epistemological  positions  of  the  Researcher  and  of  the  participating  students,  each  utterance 
would  need  to  be  individually  scrutinised  for  evidence  of  fine  grain  structure.  The  presumption 
has  been,  that  only  as  the  fruit  of  such  an  endeavour,  would  nuances  indicative  of  the  shifts 
sought,  show  themselves. 

Note:  In  the  following  transcript  'R'  represents  the  Researcher,  Ms  X  is  the  regular  classroom 
teacher;  L8,  L9  etc  refer  to  lines  8  and  9  etc  of  the  transcript  s  shown. 



Now  the  very  first  step  here,  where  you've  got  arg(zi/z2),  Ms  X  wants  you 


to  focus  on  z\lzi.  Now  the  first  thing  that  she  did  was  to  write  that  out  in  a 


trigonometric  form,  or  a  polar  form.  And  she  wrote  on  the  top  line,  what  did 


she  write? 



ri ...  {inaudible) 



Outside  of? 



Inside  the  brackets,  I  think  it's  cos9]  +  isinG] 

Assertions  in  Ll-3  are  followed  by  a  single  question  in  L4:  in  these,  the  researcher  announces 
the  theme  of  the  inquiry.  Primary  focus  is  set  on  the  structure  of  the  mathematical  steps  Ms  X 
performs,  not  their  meaning  or  reference,  nor  any  possible  function  they  may  perform. 
Further,  in  these  opening  utterances  the  Researcher  is  both  signalling  the  attention  the  students 
give  to  Ms  X  as  Teacher  as  well  as  displacing  the  Teacher  in  this  triangular  relationship  of 



1  -300 

power.  The  utterance  "Ms  X  wants  you  to  focus  ..."  evidently  means  "I  -  the  Researcher  - 
want  you  to  focus 

8        R:     Why  did  she  say  n  and  9 1? 

In  this  utterance  the  Researcher  inaugurates  the  main  body  of  the  episode.  An  inquiry 
concerning  Ms  X's  intentions  is  opened.  There  are  two  parts  related  to  this  task:  content 
(What  is  her  meaning  for  n  and  6i?)  and  function  (How  do  the  nominated  subscripts  function 
in  the  mathematical  procedures  implied?). 

9Students:    (Several  students  exclaim  at  once)  Because  that's  the  modulus  and  argument 

Interestingly,  a  large  proportion  of  the  students  answered  immediately  in  this  way.  This 
response,  however,  only  picks  up  the  content  aspects  of  the  utterance  (L8).  That  is,  the 
students  have  only  adopted  the  level  at  which  the  meaning,  or  reference  of  the  symbols  t\  and 
6i  is  signalled.  A  study  of  the  relationship  between  this  semantic  content,  and  transactional 
elements  which  could  permit  the  capture  Ms  X's  intentions  is  not  considered,  or  if  considered, 
not  pursued.  The  emphatic  tones  &.<d  chorus  like  response  of  the  students  may  also  indicate 
growing  resistance,  even  annoyance,  on  the  part  of  the  students.  The  Researcher  has  usurped 
the  role  of  the  teacher  (we  saw  this  in  Ll-7),  but  now  seems  unable  or  unwilling  to  take  over 
the  didactical  contract  (see  above)  originally  forged  in  class  between  Ms  X  and  her  students. 
Once  having  gained  admittance  to  the  code,  the  Researcher  seems  to  be  consciously  attempting 
to  disturb  it,  threatening  to  bring  about  its  collapse.  Apparently,  with  Ms  X,  it  is  part  of  the 
didactical  contract  that  teacher  questions  solicit  information  and  that  valid  responses  assume  the 
referential  mode.  But  the  Researcher,  by  asking  such  an  apparently  straightforward  question, 
now  rejects  in  advance  not  only  the  answer  presented  by  the  students,  but  even  the  referential 
form  the  answer  takes.  Crisis  in  the  contract  is  deepened  further  by  the  apparent  lack  of 
guidance  to  the  students  as  to  what  alternative  form  a  valid  answer  would  take. 

10      R:     Sorry,  just  explain?  Sorry  who's  talking? 

The  two  questions  here  reveal  a  great  deal.  Both  acknowledge  the  impasse  in  which  the 
students  have  been  placed  by  the  Researcher,  and  hence  each  begins  with  "sorry".  The 
repetition  of  "sorry",  however,  raises  the  ironic  questions:  Who  is  sorry?  Who  ought  to  be 
sorry?  These  signs  also  serve  to  reinforce  a  consensus  view  that  the  usurped  contract  has 
collapsed;  and  they  herald  a  new  phase,  described  by  Brousseau  (1986,  p.  1 1 3)  as  the 
interactive  process  of  searching  for  a  contract.  In  Brousseau's  theory,  knowledge  arises 

for  zi! 



precisely  as  the  resolution  of  crises  such  as  those  described  here.  And  indeed,  in  constructing  a 
new  contract  the  Researcher  has  already  taken  a  lead:  In  each  question  the  Researcher  begins  to 
suggest  a  new  basis  for  interaction.  In  the  first,  "Sorry,  just  explain?",  students  are 
encouraged  to  treat  the  symbols  as  a  prompt  to  perform  an  action  of  some  kind  rather  than  as  a 
cue  to  passively  provide  information.  Thcsecond  suggested  premise  of  a  new  contract  relates 
to  the  form  of  admissible  interactions  within  the  social  space  of  the  episode:  Researcher  - 
Student  interactions  are  to  be  one-to-one. 

11  Alan:    Because,  well  we've  got  subscript  one,  for  zi,  we  sort  of  use  the  same 

12  subscript,  probably. 

Alan,  identified  by  the  teacher  prior  to  this  intervention  as  a  quiet,  co-operative  student,  is  the 
first  student  to  attempt  to  work  within  these  shifting  terms.  However,  as  the  colloquial 
expression  ("sort  of)  and  the  terminating  ("probably")  would  indicate,  Alan  is  not  certain  of 
his  ground,  nor  of  the  social  topography  defining  the  interaction.  Alan  has  been  very  accurate, 
however,  in  picking  up  the  clue  provided  in  L10  as  to  what  might  constitute  a  successful 
response  to  the  motivating  question  asked  in  L8.  In  his  response,  he  switches  away  from  the 
semantic  or  referential  content  of  the  symbols,  and  attempts  to  focus  on  the  mathematical 
operations  (actions)  implied  or  controlled  by  the  symbols.  Nevertheless,  Alan  has  only 
glimpsed  the  choice  between  reference  and  action  which  has  just  offered  by  the  Researcher,  and 
almost  certainly  has  not  yet  grasped  the  consequences.  The  role  of  the  Researcher  in 
confirming  or  disconfirming'the  validity  of  Alan's  gesture  is  now  crucial  in  the  process  of 
establishing  a  new  order  in  which  this  pedagogical  crisis  may  be  resolved. 

13  R:    Would  it  have  mattered  what  subscript?  If  she'd  written '2',  would  that  have 

14  been  wrong?  If  she  had  written  n  would  that  have  been  wrong? 

Alan's  contribution  is  implicitly  accepted  by  the  Researcher  as  valid.  The  first  question  plays  a 
double  role  of  reinforcing  and  extending  the  fledgling  contract.  Reinforcement  is  accomplished 
by  verbal  cues  such  as  adopting  Alan's  reference  to  "subscript",  the  use  of  which  had  not 
hitherto  become  explicit.  Non  verbal  cues  such  as  tone  of  voice  and  the  absence  of  a  wait  time 
(neither  shown  in  the  transcript)  also  implied  Alan  and  the  Researcher  may  be  reaching 
common  ground.  Extension  of  the  contract  is  also  achieved  by  this  question.  This  is  done  by 
switching  attention  from  the  action  performed  when  using  a  given  subscript,  replacing  this  with 
a  question  directly  relating  the  intention  lying  behind  the  choice  of  a  subscript:  "Would  it  have 
mattered  what?"  is  made  to  read  "Would  it  have  been  against  her  intention  if?".  This  step  in  the 
pedagogic  sequencing,  if  accepted  by  the  students,  represents  a  final  transformation  of  focus 
which  has  travelled  from  reference  to  action  and  now  settles  on  intention.  Indeed,  this  is 

1  -302 

precisely  where  the  Researcher  wishes  to  end  up,  for  this  is  the  perspective  sought  from  the 
students  in  L8  and  either  rejected  or  not  observed  by  them  in  L9. 

Note  that  the  researcher  is  not  modelling  the  'correct'  answer  to  the  question  posed  in  L8, 
instead  however,  the  student's  responses  are  being  scaffolded  with  respect  to  their 
epistemological  focus.  Could  the  Researcher  have  been  more  direct  here  and  merely  asked 
"Would  this  have  violated  her  intentions?"?  By  this  time  it  should  be  clear  why  the  answer  is 
'no'.  Some  students  may  have  still  thought  that  this  question  sought  information  about  her 
meaning,  whereas  the  question  reaches  much  further  than  that,  towards  grasping  the  balance 
between  the  knowledge  of  what  the  relevant  signs  mean  and  the  knowledge  of  what  their 
functional  significance  in  a  mathematical  procedure  is  ie  a  balance  between  epistemologies  of 
reference  and  action.  Such  a  misunderstanding  would  provide  an  example  of  Brousseau's  so- 
called  paradox  of  the  'devolution  of  situations'  by  virtue  of  which,  the  anxiety  of  the  teacher  to 
give  the  students  what  they  appear  to  want  -  need  -  forecloses  the  possibility  of  them  being  able 
to  directly  obtain  it.  Instead,  by  shifting  the  pedagogy  through  an  epistemology  of  action  the 
teacher  gains  leverage'  which  may  be  employed  to  refocus  student  attention. 

The  second  and  third  questions  could  be  construed  as  attempts  to  lead  the  students  through 
a  thought  experiment  consisting  of  actions  premised  on  a  hypothetical  condition.  Note  that 
question  two  is  more  general  (suppose  '2'  is  the  nominated  subscript)  than  question  three  (it 
would  then  follow  that  Ms  X  would  have  to  write  n)-  Each  question  requires  the  students  to 
consider  the  consequences  for  the  mathematical  procedure  if  Ms  X  had  nominated  '2'  as  a 
subscript.  By  asking  whether  or  not  this  hypothetical  choice  would  be  wrong  the  Researcher  is 
asking  whether  or  not  the  transactions  implied  by  the  symbols  would  disrupt  the  relationship 
between  the  form  and  content  of  the  underlying  mathematics,  in  other  words:  Would  they  be 
syntactically  correct? 

15  Alice:    Only  if  she  had  have,  it  would  have  been  confusing,  because  you've  got  z\ 

16  and  Z2,  and  then  you've  got,  it  would  be  easier  to  have  t\  and  6i  then  they've 

17  got,  it  makes  a  link  there  so  you  have,  you  say  that  it's  with  the  same,  the 

18  same  problem. 

Alice,  reported  by  the  teacher  to  be  a  strong  student,  provides  a  relatively  sophisticated 
response  to  L13-14.  Her  first  utterance,  "Only  if  she  had  have"  clearly  implies  her  answer  to 
the  second  and  third  questions  is  in  the  negative.  However,  she  does  not  leave  the  matter  there. 
Her  attention  has  been  focussed  on  function,  whether  such  an  operation  would  be  facilitated  by 
a  co-operation  between  the  knowledge  of  form  and  content,  as  would  be  required  by 
unproblematic  mathematical  transaction.  At  this  point  she  identifies  a  disjunction  which  would 
arise  between  the  form  or  structure  of  the  mathematical  statement  and  its  content  or  reference: 
"it  would  have  been  confusing".  An  optimal  match,  facilitating  action  (eg  mathematical 
manipulation)  -  "it  makes  a  link"  -  is  obtained  by  matching  subscripts.  Alice  has  arrived  at  a 





defence  of  Ms  X's  use  of  dummy  variables  which  employs  both  her  knowledge  of  the  form  or 
overall  structure  of  the  mathematics  together  with  her  knowledge  of  it  as  an  event.  Thus,  she  is 
able  to  express  Ms  X's  intention  to  obtain  or  maximise  clarity.  Both  the  form  and  the  intention 
of  the  utterance  are  co-incident  A  response  such  as  this  was  sought  in  L8. 

§4  Reference,  Structure  and  Action  in  mathematics  pedagogy 

Working  within  theories  of  genre  (Bakhtin,  1986:  Holquist,  1990;  Smales,  1990), 
Ongstad  (1991)  has  provided  a  model  which  affords  a  starting  point  in  understanding  the 
theoretical  relationships  apparent  in  the  transcript  analysis  conducted  above.  Seminal  in  this 
model  is  Ongstad's  observation  that  in  making  an  utterance 

"you  are  doing  three  things  all  at  once,  you  refer,  act  and  structure." 

(italics  added,p.l3) 

In  the  accompanying  table,  terms  relating  to  the  analysis  of  utterances  are  arranged  in  a  3x4 
grid.  Read  in  columns,  the  terms,  taken  pairwise,  are  contrastive.  Read  in  rows,  the  table  sets 
out  terms  which  correlate. 

(?rid.  setting  nut  the  key  analytic  categories  in  an  arianted  version  of 

Ongstad's  model 












Since  in  Ongstad's  model  every  utterance  can  be  analysed  in  terms  of  structure, 
reference  and  action  (2nd  column),  each  of  the  12  terms  set  out  in  this  grid  can  be  brought  to 
bear  on  the  analysis  of  any  single  utterance.  The  richness  of  this  model  allows  us  to  trace  the 
shift  in  emphasis  of  these  terms  amongst  utterances  which  constitute  any  given  linguistic 
interaction.  Such  an  analysis  was  illustrated  in  the  previous  section. 

Alternatively,  the  grid  can  be  thought  of  as  map  on  which  may  be  traced  'pathways'  for 
the  development  (both  effective  and  ineffective)  of  mathematical  knowledge.  Each  of  the  three 
sets  of  correlational  terms  might  be  said  to  support  an  epistemological  viewpoint.  Learner's 
normally  need  to  have  access  to  at  least  these  three.  For  example,  Cobb  (1991)  traces  how  the 
reflexivity  between  the  syntax  and  semantics  is  obtained  or  mediated  by  the  pragmatics  inherent 
in  consensual  knowledge. 



Ongstad  emphasises  (p.  12)  that  contradictions  and  paradoxes  arise  when  the  multi- 
dimensional character  of  utterances  is  denied.  Analyses  offered  in  this  paper  amply 
substantiates  this  point.  This  does  not  mean,  however,  that  utterances  equally  emphasise  all  the 
elements  capable  of  influencing  them.  On  the  contrary,  the  selective  emphasis  a  sender  or 
receiver  places  on  utterances  lends  a  particular  character  to  the  interaction.  Where,  however, 
the  task  of  the  interlocutor  is  to  alter  or  direct  the  interaction,  as  is  the  case  for  a  teacher,  the 
full  range  of  perspectives  is  open  in  order  to  facilitate  the  development  of  a  pedagogic  strategy 
and  student  learning.  Once  again,  the  analysis  of  §3  provides  a  rich  example  of  such  a  process. 

§5  Conclusion 

In  the  last  decade  it  has  become  more  common  for  research  to  emphasise  the  consensual  aspect 
of  mathematical  learning  and  teaching  processes.  Matching  this  has  been  a  growing  sensitivity 
towards  epistemological  questions.  This  paper  firmly  endorses  both  these  developments.  At 
the  heart  of  the  present  work  has  been  the  suggestion  that  within  the  dynamics  of  microsocial 
interaction,  epistemological  shifts,  as  indicated  by  linguistic  utterances,  critically  determine  the 
character  of  pedagogic  interactions.  Arising  from  this,  the  presence  of  paradox  in  certain 
theories  of  learning  and  instruction  may  indicate  an  'epistemological  cramping'  -  or  an  over 
reliance  on  one  view  about  what  qualifies  as  mathematical  knowledge.  Paradox  free  theories  of 
mathematical  pedagogy  depend,  it  would  seem,  on  the  disposition  to  retain,  foster  and  protect  a 
certain  epistemological  dynamism. 


Bakhttn,  M.  (1986/1953).  The  problem  of  speech  genres.  In  Emerson,  C.  and  M.  Holquist  (Eds.)  Speech 

Genres  and  Other  Late  Essays.  Austin,  TX:  University  of  Texas  Press 
Berciter,  C.  (1985).  Toward  a  solution  of  the  teaming  paradox.  Review  of  Educational  Research,  55(2),  201-226 
Brousseau,  Guy  (1986).  Basic  theory  and  methods  in  the  didactics  of  mathematics,  In  P.F.L.  Verstappen  (ed) 

Second  conference  on  systematic  co-operation  between  theory  and  practice  in  mathematics  education.  2- 

7  November  1986.  Lochem.  The  Netherlands 
Cobb,  Paul  (1988).  The  tension  between  theories  of  teaming  and  instruction  in  mathematics  education. 

Educational  Psychologist,  23(2),  87- 103. 
Cobb,  Paul  (1991).  Some  thoughts  about  individual  learning,  group  development,  and  social  interaction.  Paper 

presented  to  PMEXV.  Astissi:  Italy 
Cobb,  Paul  (1992).  A  construe tivijt  alternative  to  the  representational  view  of  mind  in  mathematics  education. 

Journal  for  Research  in  Mathematics  Education,  January. 
Holquist,  M.  (1990).  Dialogism:  Bakhttn  and  his  world.  Routledge. 
Keith,  M  J.  ( 1988)  Stimulated  Recall  and  Teachers'  Thought  Processes:  a  critical  review  of  the 

methodology  and  an  alternative  perspective.  Paper  presented  at  the  Annual  Meeting  of  the  Mid  South 

Educational  Research  Association.  Louisville,  Kentucky. 
Kanes,  C  (1991,  November).  Language  and  text  in  mathematics  education:  Towards  a  poststructuralist  account. 

Paper  presented  to  the  1991  Annual  Conference  of  the  Australian  Association  for  Research  in 

Education.  Surfers  Paradise,  Queensland. 
Ongstad,  S.  (1991,  November).  What  is  Genre?  Paper  presented  to  the  LERN-Conference,  Sydney,  Australia 
Parsons,  J.M.,  Graham,  N.,  and  Hones* ,  T.  (1983).  A  Teachers'  Implicit  Model  of  How  Children  Leant.  British 

Educational  Research  Journal.  9(1),  pp  91-100. 
Plato  (1987).  Mem.  In  Hamilton,  E.  and  H.  Cairns  (eds.),  The  Collected  Dialogues  of  Plato, 

Bollingen  Series  LXXI,  Princeton  University  Pret* 
Resnick,  L.B.  (1983).  Towards  a  cognitive  theory  of  instruction.  In  S.O.  Paris,  G.M.  Olson  and.H.  Stevenson 

(Eds.),  Learning  and  motivation  in  the  classroom.  Lawrence  Eribaum  Associates. 
Steinbring,  H.  (1989).  Routines  a  meaning  in  the  mathematics  classroom.  For  the  Learning  of  Mathematics, 

9(1).  24-33 

Er|c  329 



ofjhe  t  . 



UnivQrsitv^of  New  Hampshire 
,   Eiurham,  NH  (USA) 
Augusf)  -11, 19^ 

;  Volume  II 





of  the 






University  of  New  Hampshire 
Durham,  NH  (USA) 
August  6-  11,  1992 

Volume  II 




Published  by  the  Program  Committee  of  the  16th  PME  Conference,  USA. 

All  rights  reserved. 

William  Geeslin  and  Karen  Graham 
Department  of  Mathematics 
University  of  New  Hampshire 
Durham.  NH  03824 



Research  Reports  (continued) 

Kicren,  T.  &  Piric,  S.  P- 2-1 

The  answer  determines  the  question.  Interventions  and  the  growth  of  mathematical 

Konold,  C.  &  Falk,  R.  p.  2-9 

Encoding  difficulty:  A  psychological  basis  for  'misperceptions'  of  randomness 

Koyama,  M.  p.  2-17 

Exploring  basic  components  of  the  process  model  of  understanding  mathematics 
for  building  a  two  axes  process  model 

Kraincr,  K.  p.  2-25 

Powerful  tasks:  Constructive  handling  of  a  didactical  dilemma 

Leder,  G.C.  p.  2-33 

Measuring  attitudes  to  mathematics 

Lemian,  S.  p.  2-40 

The  function  of  language  in  radical  constructivism:  A  Vygotskian  perspective 

Linchcvsky,  L.,  Vinncr,  S.,  &  Karsemy,  R.  p.  2-48 

To  be  or  not  to  be  minimal?  Student  teachers'  views  about  definitions  in  geometry 

Lins,  R.  P-  2-56 

Algebraic  and  non-algebraic  algebra 

Magidson,  S.  P-  2-64 

What's  in  a  problem?  Exploring  slope  using  computer  graphing  software 

Martino,  A.M.  &  Mahcr,  C.A.  p.  2-72 

Individual  thinking  and  the  integration  of  the  ideas  of  others  in  problem 
solving  situations 

Masingila,  J.  p.  2-80 

Mathematics  practice  in  carpet  laying 

Mauiy,  S.,  Lerougc,  A.,  &  Bailie,  J.  P-  2-88 

Solving  procedures  and  type  of  rationality  in  problems  involving  Cartesian 
graphics,  at  the  high  school  level  (9th  grade) 

Mcira,  L.  P-  2-96 

The  microevolution  of  mathematical  representations  in  children's  activity 

Mcsquita,  A.L.  p.  2-104 

Les  types  d' apprehension  en  geometrie  spatiale:  une  etude  clinique  sur  le 
developpement-plan  du  cube 

Minato,  S.  &  Kamada,  T.  p.  2- 1 12 

Results  of  researches  on  causal  predominance  between  achievement  and  attitude 
in  junior  high  school  mathematics  of  Japan 

er|c  333 


Mitchelmorc,M.  p.  2-120 

Children's  concepts  of  perpendiculars 

Moschkovich,  J.  p.  2-128 

Students'  use  of  the  x-intercept:  An  instance  of  a  transitional  conception 

Mousley.J.  p.  2-136 

Teachers  as  researchers:  Dialectics  of  action  and  reflection 

Mulligan, 'J.  p.  2-144 
Children's  solutions  to  multiplication  and  division  word  problems: 
A  longitudinal  study 

Murray,  H.,  Olivier,  A.,  &  Human,  P.  p.  2-152 
The  development  of  young  students'  division  strategies 

Nathan,  M.J.  p.  2-160 

Interactive  depictions  of  mathematical  constraints  can  increase  students' 
levels  of  competence  for  word  algebra  problem  solving 

Neuman,  D.  p.  2-170 

The  influence  of  numerical  factors  in  solving  simple  subtraction  problems 

Norman,  F.A.  &  Prichard,  M.K.  p.  2-178 

A  Krutetskiian  framework  for  the  interpretation  of  cognitive  obstacles: 
An  example  from  the  calculus 

Noss,  R.  &  Hoyles  C.  p.  2- 1 86 

Logo  mathematics  and  boxer  mathematics:  Some  preliminary  comparisons 

Outhred,  L.  &  Mitchelmore,  M.  p.  2- 1 94 

Representation  of  area:  A  pictorial  perspective 

Owens,  K.  P-  2-202 

Spatial  thinking  takes  shape  through  primary-school  experiences 

Perlwitz,  M.D.  P-  2-210 

The  interactive  constitution  of  an  instructional  activity:  A  case  study 

Ponte,  J.P.,  Matos,  J.  F.,  Guimaries,  H.M.,  Leal,  L.C.,  &  Canavarro,  A.P.  p.  2-218 

Students'  views  and  attitudes  towards  mathematics  teaching  and  learning: 
A  case  study  of  a  curriculum  experience 

Reiss,  M.  &  Reiss,  K.  p.  2-226 

Kasimir:  A  simulation  of  learning  iterative  structures 

Relich,  J.  P-  2-234 

Self-concept  profiles  and  teachers  of  mathematics:  Implications  for  teachers 
as  role  models 

Reynolds,  A.  &  Wheatley,  G.H.  p.  2-242 

The  elaboration  of  images  n  the  process  of  mathematics  meaning  making 

er|c  334 


Rice,  M. 

Teacher  change:  A  construcdvist  approach  to  professional  development 

Robinson,  N.,  Even,  R.,  &  Tirosh,  D. 

Connectedness  in  teaching  algebra:  A  novice-expert  contrast 

Sienz-Ludlow,  A. 

Ann's  strategies  to  add  fractions 

Sanchez,  V.  &  Llinaies,  S. 

Prospective  elementary  teachers' pedagogical  content  knowledge  about 
equivalent  fractions 

Santos,  V.  &  Kroll,  D.L. 

Empowering  prospective  elementary  teachers  through  social  interaction, 
reflection,  and  communication 

Saraiva,  M.J. 

Students'  understanding  of  proof  in  a  computer  environment 

Schliemann,  A.,  Avelar,  A.P.,  &  Santiago,  M. 

Understanding  equivalences  through  balance  scales 

Schroeder,  T.L. 

Knowing  and  using  the  Pythagorean  theorem  in  grade  10 

Sekiguchi,  Y. 

Social  dimensions  of  proof  in  presentation:  From  an  ethnographic  inquiry 
in  a  high  school  geometry  classroom 

Shigematsu,  K. 

Metacognition:  The  role  of  the  "inner  teacher" 

Shimizu,  Y. 

Metacognition  in  cooperative  mathematical  problem  solving:  An  analysis 
focusing  on  problem  transformation 

p.  2-250 
p.  2-258 
p.  2-266 
p.  2-274 

p.  2-282 

p.  2-290 
p.  2-298 
p.  2-306 
p.  2-314 

The  Answer  Detetiwwes  the  Que  Man.  btovenOonsandfheGnoiwthcrf 

MatfMEnraSU£aH  UlOEnDnVig 

Tnm  Kteren       and  Susan  Flllg 

University  of  Alberta        University  of  Oxford 

AJnfrtrt  Our  work  over  the  past  four  years  has  looked  at  the  growth  of  mathematical  understanding 
as  a  dynamic,  levelled  but  not  linear,  process.  An  outline  of  our  theory  and  Its  features  Is  given  in  this 
paper  before  it  goes  on  to  address  the  question  of  how  a  teacher  can  Influence  an  environment  for  such 
growth.  We  identity  three  kinds  of  intervention:  provocative,  invocaOve  and  validating  and  use  these 
concepts  in  analysing  interactions  between  a  teacher  and  two  students.  Our  contention  Is  that  for  the 
promotion  of  growth  the  teacher  needs  to  believe  that  It  Is  the  student  response  which  determines  the 
nature  of  the  question. 

"The  task  of  education  becomes  a  task  of  first  Inferring  models  of  the  students'  conceptual  constructs  and 
then  generating  hypotheses  as  to  how  the  students  could  be  given  the  opportunity  to  modify  their 
structures  so  that  they  lead  to  mathematical  actions  compatible  with  the  Instructor's  expectations  and 
goals."  (1) 

"an  organism  has  somehow  to  acquire  the  capacity  to  turn  around  on  Its  own  schemata  and  to  construct 
them  afresh  It  Is  what  Rives  consciousness  Its  most  prominent  function.  I  wish  I  knew  exactly  how 
thlsisdone.:  (Bartlettlntt)) 

Over  the  past  four  years  we  have  been  building  and  testing  a  theory  of  the  growth 

of  mathematical  understanding  which  views  mathematical  understanding  not  as  an 

acquisition  (e.g.  3),  nor  as  a  developmental  phase  (e.g.  4),  but  as  a  dynamic  process.  Using 

this  theory  we  have  attempted  to  show  that  the  growth  of  a  person's  understanding  of 

any  topic  can  be  mapped  on  a  model 

comprising  eight  embedded  levels  of 

understanding  moving  from  initial 

primitive  knowing  through  to 

inventising,  (  fig  1)  We  maintain  that 

growth  through  such  levels  or  modes 

of  understanding  is  not  in  any  way 

monotonic  but  involves  multiple  and 

varied  actions  of  folding  back  to  inner, 

less  formal  understanding  in  order  to 

use  that  understanding  as  a 

springboard  for  the  construction 



of  more  sophisticated  outer  level  understanding.  We  are,  of  course,  still  in  the  ongoing 
process  of  elaborating  the  elements  in  such  understanding.  Our  own  understanding  of 
the  emerging  theory  is,  itself,  subject  to  constant  acts  of  folding  back  with  a  view  to 
gaining  greater  insight  into  the  phenomenon  of  mathematical  understanding. 

The  quotation  from  Bartlett,  above,  prompts  us,  too,  to  ask  the  questions,  "how 
might  such  re-construction  happenr  and  "what  roles  might  teachers  play  in  bringing  it 
about  for  their  students?".  In  this  paper  we  wish  to  consider  the  nature  of  some  teacher 
interventions  and  their  impacts  on  student  understanding.  More  broadly,  we  also 
illustrate  that  such  interventions  do  not  have  to  originate  with  the  teacher,  although  it 
seems  likely  that  only  the  teacher  is  in  a  position  to  create  such  interventions 
deliberately.  Theoretical  descriptions  of  such  interventions ,  which  we  call  provocative 
innovative  and  validating  will  be  followed  by  analysis  of  an  incident  in  terms  of  the 
effect  of  certain  questions  on  the  growth  of  understanding  of  a  single  student. 

We  do  not  claim  to  be  alone  in  the  field,  attempting  to  answer  the  questions  posed 
above,  but  to  be  taking  a  different  stand  point  from  which  to  analyse  the  phenomenon  of 
growth  of  mathematical  understanding.  For  example,  Maher  et  al  (5)  consider  the 
mathematical  behaviour  of  one  child  sampled  over  four  years  and  indicate  in  global  or 
macroscopic  ways  the  nature  of  change  in  sophistication  of  such  behaviour.Our  work 
differs  from  theirs  in  that  it  is  driven  by  a  particular,  albeit  developing,  theory  and  tries  to 
comprehend  the  dynamics  of  growth  as  they  occur  in  local  situations.  It  allows  us  to 
examine  teaching  strategies,  interventions  and  effects  in  day  to  day  classroom 
environments.  Edwards  and  Mercer  (2)  do  look  in  detail  at  interactions  between  teachers 
and  students  and  their  impact  on  un