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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
I-III.
International Group for the Psychology of Mathematics
Education.
Aug 92
95Ap.
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
Curriculum
Advanced Mathematics; LOGO Programming Language;
Mathematical Communications; Mathematical Thinking;
"Mathematics Education Research; '^Psychology of
Mathematics Education; Representations (Mathematics);
Teacher Candidates; Teacher Change; Teacher
Researchers
a
ERIC
ABSTRACT
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)
IN ri 'RN V! I()NAL (iROl T FOR
ritio Fsvcrioixx^ of
^MTOM VI ICS K1>1 C VI K)K
\J11\ X JUXJl X X JLl
ME CONFERENCE
Co
I'nivei sity of Nciv Hampshire
I I)wrbani, NII (USA) '/,/,.
' Auplst 6 - 11, 1992 ;
improvement^*
FORMATION
0 S DEPARTMENT Of EDUCATION
Ott.ce ot Educational Research and
EDUCATIONAL RESOURCES INFORMATION
CENTER (ERIC)
document has Men reproduced •»
received horn the person or orflenilelion
originating it
] Minor changes nave Men mad* to improve
reproduction quality
Volume J
• Points ol view or opinions Hated in this docu
ment do not necesMrily represent ottiCial
of Ri position or policy
"PERMISSION TO REPRODUCE THIS
MATERIAL HAS BEEN GRANTED BY
TO THE ECU'" ATIONAL RESOURCES
INFORMATION CENTER lEf"' "
INTERNATIONAL GROUP FOR
THE PSYCHOLOGY OF
MATHEMATICS EDUCATION
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.
Editors:
William Geeslin and Karen Graham
Department of Mathematics
University of New Hampshire
Durham, NH 03824
USA
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1 -i
PME XVI PROCEEDINGS
Edited by WiUiam Geeslin and Karen Graham
Mathematics Department
University of New Hampshire
Durham NH
USA
PREFACE
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.
0
1 - ii
List of PME XVI Reviewers
The Program Committee wishes to thank the following people for their help during the review
process.
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
EmaYackel.USA
Michal Yerushalmy, Israel
a
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1 -iii
INTERNATIONAL GROUP FOR THE
PSYCHOLOGY OF MATHEMATICS EDUCATION
PRESENT OFFICERS OF PME:
- President KathHart (United Kingdom)
- Vice-President Gilah Leder (Australia)
- Secretary Martin Cooper (Australia)
- Treasurer Angel Gutierrez (Spain)
OTHER MEMBERS OF THE INTERNATIONAL COMMITTEE:
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)
PME XVI PROGRAM COMMITTEE:
- 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)
PME XVI LOCAL ORGANIZING COMMITTEE:
- Joan Ferrini-Mundy - Karen Graham
- William E. Geeslin - Lizabeth Yost
CONFERENCE PROGRAM SECRETARY
-William EGeeslin
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V
1 -iv
HISTORY AND AIMS OF THE P.M.E. GROUP
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
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
USA
MAdler
Department of Education
University of Witwatersrand
PO Wits 2050, Johannesburg
SOUTH AFRICA
M.C. Batanero
Escuela Universitaria del Profesorado
Campus de Cartuja
18071, Granada
SPAIN
Michael Battista
404 White Hall
Kent State University
Kent. OH 44242
USA
Nadine Bednarz
CP. 8888 - Sue a - Montreal
P.Quebec H3C3P8
CANADA
David Ben-Chaim
Weizmann Institute of Science
Rehovot, 76100
ISRAEL
Janette Bobis
University of New South Wales
POBoxl
Kensington, NSW, 2033
AUSTRALIA
PaolaBoero
Dipartimento Matematica Umversita
VULB.Alberti4
16132, Genova
ITALY
Card Brekke
Telemark Laercrhugskole
N-3670Notodden
NORWAY
Lynne Cannon
Faculty of Education
Memorial University of Newfoundland
St John's, Newfoundland A1B 3X8
CANADA
Olive Chapman
Dept of Curr. & Instruction, U. Calgary
2500 University Drive, NW
Calgary, AB
CANADA
Giampaolo Chiappini
ViaL. B. Alberti, 4
16132 Genova
ITALY
David Clarke
Australian Catholic University
17CastlebarRoad
Oakleigh, Victoria, 3166
AUSTRALIA
M.A. (Ken) Clements
Faculty of Education
Deakin University
Geelong, Victoria 3217
AUSTRALIA
JereConfrey
Dept of Education, Kennedy Hall
Cornell University
Ithaca, NY 14853
USA
Kathryn Crawford
Faculty of Education
The University of Sydney
NSW 2006
AUSTRALIA
Linda Davenport
P.O. Box 751
Portland, OR 97207
USA
Gary Davis
Institute of Mathematics Education
La Trobe University
Bundoora, Victoria 3083
AUSTRALIA
ERJC
0
1 -vi
Guida de Abreu
Dept. of Education, Trumpington St.
Cambridge University
Cambridge, CB2 1QA
UNITED KINGDOM
Linda DeGuire
Mathematics Department
California State Univerisity
Long Beach, CA 90840
USA
M. Ann Dirkes
School of Education
Purdue University at Fort Wayne
Fort Wayne, IN 46805-1499
USA
Barbara Dougherty
University of Hawaii
1776 University Avenue
Honolulu, HI 96822
USA
Laurie Edwards
Crown College
University of California
Santa Cruz, CA 95064
USA
Pier Luigi Ferrari
Dipattimento di Matematica
via L.B. Alberti
4-16132 Genova
ITALY
Rossella Garuti
Dipattimento Matematica University
via L.B. Alberti, 4
16132, Genova
ITALY
Linda Gattuso
College du Vieux Montreal
3417 Ave. de Vendome
Montreal, Quebec H4A 3M6
CANADA
J.D. Godino Escuela
Universitaria del Profesorado
Campus de Cartuja
18071, Granada
SPAIN
Susie Groves
Deakin University • Burwood Campus
221 Burwood Highway
Burwood, Victoria, 3125
AUSTRALIA
Elfriede Guttenberger
Avenida Universidad 3000
Maestria en Education Matematica
Mexico, D.F., Of Adm. 2, 1 piso
MEXICO
Lynn Hart
Atlanta Math Project
Georgia State University
Atlanta, GA 30303
USA
James Hiebert
College of Education
University of Delaware
Newark, DE 19716
USA
Robert Hunting
La Trobe University
Bundoora, Victoria, 3083
AUSTRALIA
Barbara Jaworski
University of Birmingham
Edgbaston Birmingham B15 2TT
UNITED KINGDOM
Clivc Kanes
Division of Education
Griffith University
Nathan, 4111
AUSTRALIA
TE. Kieran
Dept. of Secondary Education
University of Alberta
Edmonton T6G2G5
CANADA
Cliff Konold
Hasbrouck Laboratory
University of Massachusetts - Amherst
Amherst, MA 01003
USA
ERJC
1 - vii
Masataka Koyarna
Faculty of Education
Hiroshima University, 3-101
2-365 Kagamiyama Higashi-Hiroshima City
JAPAN
Konrad Krainer
IFF/Universitat Klagenfurt
Sterneckstrasse IS
A-9010
AUSTRIA
Gilah Lcder
Monash University
Clayton, Victoria 3168
AUSTRALIA
Stephen Lerman
103 Borough Road
London SE1 OAA
UNITED KINGDOM
Liora Linchevski
School of Education
Hebrew University
Mount Scoups, Jerusalem 91-905
ISRAEL
R.C. Lins
Shell Centre for Math Education
University Park
Nottingham, NG7 2QR
BRAZIL
Susan Magidson
EMST - 4533 Tolman Hall, School of Ed.
University of California
Berkeley, CA 94720
USA
Enrique Castro
Departamemo Didactica de la Matemaanca
Campus de Cartuja s/n
18071 Granada
SPAIN
Amy Martinet
Center for Math, Science, & Computer Ed.
192 College Avenue
New Brunswick, NJ 08903-5062
USA
Joanna Masingila
Education 309
Indiana University
Bloomington, IN 47405
USA
S.Maury
University Montpellier U
Place Eugene Bataillon
34095 MONTPELLIER CEDEX 5
FRANCE
Luciano Meira
Mestrado em Psicologia Cognitiva
CFCH - 8" andar, Recife 50739 PE
BRAZIL
A.L Mesquita
R. Marie Brown, 7/8c
1500 Lisbon
PORTUGAL
Saburo Minato
College of Education
Akita University
Gakuencho, Tegata, Akita City
JAPAN
Michael Mitchelmore
School of Education
Macquarie University
NSW 2109
AUSTRALIA
Judit Moschkovich
4533 Tolman Hall
University of California
Berkeley, CA 94720
USA
Judith Mousley
Faculty of Education
Deakin University *
Geelong, Victoria, 3217
AUSTRALIA
Joanne Mulligan
27 King William Street
Greenwich 2065
Sydney
AUSTRALIA
1
Hanlie Murray
Faculty of Education
University of Stellenbosch
SOUTH AFRICA
Mitchell Nathan
LRDC
University of Pittsburgh
Pittsburgh, PA 15260
USA
Dagmar Neuman
Box 1010
University of GSteburg
S-43126 MOlndal
SWEDEN
F.A. Norman
Dept. of Mathematics
University of North Carolina
Charlotte, NC 28223
USA
Richard Noss
Institute of Education, U. of London
20 Bedford Way
London WC1H0AL
UNITED KINGDOM
LynneOuthred
School of Education
Macquarie University
NSW 2109
AUSTRALIA
Kay Owens
P.O. Box 555
University of Western Sydney, Macarthur
Campbelltown, NSW 2560
AUSTRALIA
Marcela Perlwitz
EMAD414
Purdue University
West Lafayette, IN 47907-1442
USA
JoSo Pedro Ponte
Av. 2y de Julmo, 134-4"
1300 Lisboa, PORTUGAL
PORTUGAL
Matthias Reiss
Stedingerstr. 40
7000 Stuttgart 31
GERMANY
Joe Relich
PO Box 10
c/o Faculty of Education
Kingswood
AUSTRALIA
Anne Reynolds
Math Education, 219 Carothers, B-182
Florida State University
Tallahassee, FL 32306
USA
Mary Rice
Deakin University
Geelong, Victoria 3217
AUSTRALIA
Naomi Robinson
Department of Science Teaching
Weizmann Institute
Rehovoth, 76100
ISRAEL
Adaiira Sienz-Ludlow
Dept of Mathematical Sciences
Northern Illinois University
DcKalb, IL 60115
USA
Victoria Sanchez
Avdo. Ciudad Jardin, 22
41005 Sevilla
SPAIN
Vinia Maria Santos
School of Education, Room 309
Indiana University
Bloomington, IN 47405
USA
Manvel Joaquim Saraiva
Universidade da Beira Interior
Rue Ferreira de Castro, 5-3
PORTUGAL
o
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1 -ix
Analucia Schliemann
Mestrado em Psicologia
8" andar, CFCH-UFPE
50739 Recife
BRAZIL
Thomas Schroeder
Faculty of Education, 212S Main Mall
University of British Columbia
Vancouver, BC, V6T 1Z4
CANADA
Yasuhiro Sekiguchi
Institute of Education
University of Tsukuba
Tsukuba-shi, Ibaraki, 305
JAPAN
Keiichi Shigematsu Takabatake
Nara University of Education
Nara630
JAPAN
Yoshinori Shimizu
4- 1 - 1 , Nukuikita-Machi
Koganei-shi
Tokyo, 184
JAPAN
Dianne Seimon
School of Education
Phillip Institute of Technology
Alva Grove, Coburg 3058
AUSTRALIA
Martin Simon
176 Chambers Building
Department of Curriculum and Instruction
Univesity Park, PA 16802
USA
Jack Smith
ColL of Education, 436 Erickson Hall
Michigan State University
East Lansing, MI 48824
USA
Judith Sowder
Ctr. for Research in Math and Science Ed.
5475 Alvarado Road, Suite 206
San Diego, CA 92120
USA
Kaye Stacey
School of Science & Math Education
University of Melbourne
Parkville, Victoria 3 142
AUSTRALIA
Rudolf StrSsser
Institut fur Didaktik dcr Mathematik
UniversitSt Bielefeld
4800 Bielefeld
GERMANY
L. Streefland
Tibcrdrcef4
3561 GG, Utrecht
NETHERLANDS
Susan Taber
717 Harvard Lane
Newark. DE 19711
USA
Cornelia Tierney
TERC
2067 Massachusetts Avenue
Cambridge, MA 02140
USA
DinaTirosh
School of Education
Tel Aviv University
Tel Aviv, 69978
ISRAEL
Maria Trigueros
Rio Hondo Num 1
ColoniaTizapan San Angel
03100, Mexico D.F.
MEXICO
Pessia Tsamir
School of Education
Tel- Aviv University
Tel Aviv, 69978
ISRAEL
Diana Underwood
Purdue University
ENAD414
West Lafayette, IN
USA
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1 -X
Marjory Witte
OCT - University of Amsterdam
Grote Bidiersstnut 72
1013 KS, Amsterdam
NETHERLANDS
Addresses of Presenters of Plenary Sessions at PME XVI
Michele Artigue
IREM, Universe Paris 7
2 Place Jussieu
75251 Paris Cedex 5
FRANCE
M.A. (Ken) Cleme-its
Faculty of Education
Deakin University
Geelong, Victoria, 3166
AUSTRALIA
Tommy Dreyfus
Center for Technological Education
PO Box 305
Holon 58102
ISRAEL
Gontran Ervynck
Kath.Univ. Leuven
Campus Kortrijk
B-8500Konrijk
BELGIUM
Gerald A. Goldin
Center for Math, Science, & Computer Ed.
Rugers University
Piscataway, NJ 08855-1179
USA
CeliaHoyles
Institute of Education, Math
University of London
20 Bedford Way
London WC1HOAL
UNITED KINGDOM
John Mason
Open University
Walton Hall
Milton Keynes MK7 6AA
UNITED KINGDOM
Bernard Parzysz
IUFMde Lorraine
Departement de mathematiques
University deMetz
lie du Suilcy
F 57000 Metz
FRANCE
Norma Presmeg
219 Corothers Hall, B-182
Florida State University
Tallahassee, FL 32306-3032
USA
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14
1 - xi
CONTENTS OF VOLUME I
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
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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
a
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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
CONTENTS OF VOLUME II
Research Reports (continued)
Kieren, T. & Pine, S. p. 2- 1
The answer determines the question. Interventions and the growth of mathematical
understanding
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
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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
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13
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
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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
CONTENTS OF VOLUME III
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
ERIC
21
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
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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
a
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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.
mathematicians
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
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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
practices
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
problems
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
a
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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
23
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1 - 1
Working Groups
23
o
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1 -3
WORKING GROUP ON
ADVANCED MATHEMATICAL THINKING (A.M.T.)
Organisers: Gontran Ervynck, David Tall
• SESSION I: INTRODUCTION TO THE PROCESSES- OBJECTS THEME
Four initiators will present different approaches to what seems to be basically the same
theory.
- 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-
tions.
- 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.
• SESSION lit DISCUSSION
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.
• SESSION III: CONTINUATION OF THE WORK ON RIGOR AND PROOF
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 ?
• SESSION IV: PREPARING THE FUTURE
Discussion of the work of the AMT Group at PME-17, Japan 1993.
%j 0
1-4
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
group.
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)
-repeatability
-generalizability
-falsification
-objectivity
-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.
1-6
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,
Tiberdreef4
3561 GG Utrecht
The Netherlands
Tel. 31-30-611611
Fax. 31-30-660430
E-mail jan@owoc.ruu.nl
E. Language: English
3.
Source
1-7
Working group on
CULTURAL ASPECTS IN MATHEMATICS LEARNING
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
mathematics.
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
desirable.
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
EJM.HJWWBlHIJilU
1-8
GEOMETRY WORKING GROUP
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.
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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
representation?
* 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
representation?
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
RESEARCH ON THE PSYCHOLOGY OF MATHEMATICS
TEACHER DEVELOPMENT
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
39
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.
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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.
1-15
Discussion Groups
42
1 - 17
DILEMMAS OF CONSTRUCTIVIST MATHEMATICS TEACHING:
INSTANCES FROM CLASSROOM PRACTICE.
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.
43
1 - 18
MEANINGFUL CONTEXTS FOR SCHOOL MATHEMATICS
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
PARADIGMS LOST: WHAT CAN MATHEMATICS EDUCATION LEARN
FROM RESEARCH IN OTHER DISCIPLINES?
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?'
REFERENCE
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
PHILOSOPHY OP MATHEMATICS EDUCATION
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.
46
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RESEARCH IN THE TEACHING AND LEARNING OF UNDERGRADUATE
MATHEMATICS: WHERE ARE WE? WHERE DO WE GO FROM HERE?
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
BEST COPY AVAILABLE
1-22
VISUALIZATION IN PROBLEM SOLVING AND LEARNING
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?
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1 -23
Research Reports
a
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49
1-25
APPROACHES TO RESEARCH INTO CULTURAL CONFLICTS
IN MATHEMATICS LEARNING
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
50
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1-26
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
95
72
. The pupils, who performed best in school mathematics,
work in offices
. The pupils, who performed worst in
school mathematics, work in sugar cane farming
81
73
. People working in an office are schooled
. People working in sugar cane farming are unschooled
100
77
. Sugar cane workers can work out sums without being schooled
. Sugar cane workers cannot do sums without being schooled
73
27
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
51
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
contradictions:
/ (Interviewer) - Why doesn't that man (in a picture] on the tractor know
mathematics?
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
gender
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
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52
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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
understandings.
(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
examples:
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
AS
10 10
Reading the written answer: " It Is twenty. Because S plus 5 Is 10, with S plus 5 is
10."
53
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Example 2: Adding the sides of a parallelogram 3 by 4 (cm).
33
77
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)
ERIC
54
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Table 3: Different approaches to culture-conflict
Approaches
to culture
conflict
Assumptions
Curriculum
Teaching
i
Language
Culture-free
Traditional
view
No culture
conflict
Traditional
Canonical
No particular
modification
Official
Assimilation
i
i
■
Child's
V, Li 11 14 1 C
should be
useful as
examples
Some child's Caring
cultural approach
contexts ! perhaps with
included ! some pupils
■ in groups
Official, j
plus |
relevant
contrasts 1
and j
remediation
for second !
language j
learners j
Accommodation
Child's
culture
should
influence
education
Currk !um
restructured
due to
child's
culture
Teaching
style
modified as
preferred by
children
Child's home |
language •
accepted in \
class, plus |
official
language
support
Amalgamation
Culture's
adults
should
share
significantly
in education
Curriculum
jointly
organised by
teachers and
community
Shared or
team
teaching
Bl-lingual,
Li-cultural
teaching
1
Appropriation
Culture's
community
should take
over
education
Curriculum
organised
wholly by
community
Teaching
entirely by
community's
adults
Teaching in
home
community's
preferred
language
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
55
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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.
56
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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
experiences."
(More data will be presented at the conference)
Conclusion
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.
References
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
RHETORICAL PROBLEMS AND
MATHEMATICAL PROBLEM SOLVING: AN EXPLORATORY STUDY
Verna M. Adams
Washington State University
Abstract
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.
Introduction
58
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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
59
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,
60
1-36
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
mathematics.
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
problems.
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
1-37
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
62
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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.
Results
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
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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.
References
Adams, V. M. (1991). Knowledge tailing and knowledge transforming in
mathematical problem solving. Doctoral dissertation, University of
Georgia.
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
English.
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.
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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-
108.
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.
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ACTION RESEARCH AND THE THEORY-PRACTICE DIALECTIC: INSIGHTS
FROM A SMALL POST GRADUATE PROJECT INSPIRED BY ACTIVITY THEORY
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.
ACTION RESEARCH, THEORY-PRACTICE AND THE B ED DEGREE.
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.
66
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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.
ACTION RESEARCH AND ACTIVITY THEORY.
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
67
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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.
MARK'S RESEARCH - A BRIEF DESCRIPTION AND SOME ANALYSIS.
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)
1-44
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
that
I often asked the question "do you all agree?" without actually
confirming whether they did ... '
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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
activity...'
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1-46
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.'
REFLECTIONS
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
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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
1-48
how this rational method of noticing, analysing and acting will gat at deeply seated
social practices, e.g. gender, remains a question.
REFERENCES
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 -
221.
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.
3.
McNiff, J (1988) Action Research: Principles and Practice. Macmillan. London.
Mellin-Olsen, S (1986) The Politics of Mathematics Education. Reidel.
Dordrecht.
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 -
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STUDENTS' UNDERSTANDING OF THE SIGNIFICANCE LEVEL IN
STATISTICAL TESTS
M.A. VallecMos; M. C Batanero and J. D. Godino
University of Granada (Spain)
ABSTRACT
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.
INTRODUCTION
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
comparisons".
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
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1-50
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). • •
CONCEPTUAL ANALYSIS OF THE LEVEL OF SIGNIFICANCE
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 :
o
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
test:
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:
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75
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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:
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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
result.
EXPERIMENTAL STUDY OF MISCONCEPTIONS
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
following.
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:
AVAILABLE
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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.
RESULTS AND DISCUSSION
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
OPTIONS
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
decision:
- 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).
1-55
Misconceptions in the interpretation fif the probabilities of error. ajjd. tfiejr
relations:
- 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
2.
- 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).
CONCLUSIONS
In the analysis of the responses to the questions put forward, the existence of
o SO
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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.
REFERENCES
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.
1-57
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.
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Results
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
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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 su.ee?
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
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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.
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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.
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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
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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
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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
classroom.
Conclusions
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.
References
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
Mathcmatics/Macmillan.
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.
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ARITHMETICAL AND ALGEBRAIC THINKING
IN PROBLEM-SOLVINGi
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.
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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.
Method
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.
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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.
SOME ARITHMETICAL PROCEDURES
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
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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)
Answer:
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.
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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
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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:
Student:
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.
ALGEBRAIC PROCEDURES
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
Vieta.
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Student:
1st x 12
2nd (x+8)12 588 . x.12 + (x+8)I6
588 » 12x + 16x + 128
-12x - 16x - 128 - 588
-28x - -460
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.
CONCLUSION
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.
o
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1 -72
References
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.
199-216.
Radford, L. (a panutre). Diophante et l'algebre pnf-symbolique. Bulletin de VAssociation Mathtmatique
du Qutbec.
1-73
CONSULTANT AS CO-TEACHER:
PERCEPTIONS OF AN INTERVENTION FOR IMPROVING
MATHEMATICS INSTRUCTION
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.
Introduction
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.
ERIC
98
1-74
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
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
less-qualified.
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
99
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
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1-76
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
conference.
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
1-78
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
9
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1-79
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
worksheets.
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.
References
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.
105
1 -81
THE REDUNDANCY EFFECT IN A SIMPLE ELEMENTARY-SCHOOL GEOMETRY TASK:
AN EXTENSION OF COGNITIVE-LOAD THEORY AND IMPLICATIONS FOR TEACHING
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
Background
1-82
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
Procedure
1-83
"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
mean
time
X correct
mean
time
X correct
*ach group
non-rd
red'nt
non-rd
red'nt
non-rd
red'nt
non-rd
red'nt
369.9
463.7
66.7
33.3
277.8
4S7.6
60.0
26.7
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
108
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
cjMaiMBBimiu
1-85
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
dlag's
244.6
red'nt
396.0
% correct
dlag's
90.0
red'nt
60.0
testing phase
mean time
dlag's
108.9
red'nt
314.8
% correct
dlag's
90.0
red 'lit
60.0
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 ph.sa. 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.
Discussion
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.
110
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. Te.ch.rs ,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
d1.gr.nm.t1c) with .ss.ntl.l Information Impos.s .n .xtr.neou, cognitive led, n
may h.ve a detrimental rather than beneficial effect on learning. For optimum
effect, the usefulness of additional Information must outweigh the consequence, of
processing It.
References
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
in
ERIC
1-87
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
IODIC
\
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
side.
Figure 1: Redundant format Instructional material for Experiment 1
BEST COPY AVAILABLE
ERIC
112
1-88
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F1«ur» 2: Dlagr-tM-and-ttxt fonaat
Mttrltl for Exp*r1mnt 2
Fljura 3: Redundant fonut Instruct 1on«1
Htirlit for ExpaMaant 3
ERIC
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1 -89
ON SOME FACTORS INFLUENCING STUDENTS' SOLUTIONS IN MULTIPLE OPERATIONS PROBLEMS:
RESULTS AND INTERPRETATIONS
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:
money
needed (C)
1500
2000
maximum
admissible weight(M)
50
weight of the
envelorXE)
7
14
7
weight of each
sheet of paper (S)
(50.7.8)
(100.14.16)
(100.7.8)
(250.14.16)
2000
100
100
8
16.
8
3800
250
14
1-90
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 example.in 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:
GROUP I GROUP 2
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
116
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1-92
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).
117
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
GROUP 2/GR.VI
TABLE 2:(M.E.SM 1
ENV&SH.
PRE-ALG.
UNCLASS.
INVALID
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
GROUP 2/GR.VI
ENV&SH.
PRE-ALG.
UNCLASS.
INVALID
52 (39%)
43 (34%)
50 (38%)
34 (27%)
4 (3%)
3 (2%)
26 (20%)
46 (37%)
ENV.&SH.
PRE-ALG.
UNCLASS.
INVALID
GROUP 1 /GR.V
24 (28%)
36 (42%)
4 (5%)
21 (25%)
GROUP 2/GR.VI
25 (25%)
32 (32%)
2 (2%)
42 (41%)
ENV&SH.
PRE-ALG.
UNCLASS.
INVALID
GROUP 1 /GR.V
13 (15%*
42 (49%)
6 (7%)
25 (29%)
GROUP 1/GR.V1
11 (13%)
34 (41%)
1 (1%)
37 (45%)
GROUP 2/GR.VII1
9 (12%)
57 (75%)
0
10 (13%)
ERIC
118
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 .Howevcr.it 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
119 BEST COPY AVAILABLE
1-95
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
between:
- 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
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Reference*
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1-97
MULTIPLICATIVE STRUCTURES AT AGES SEVEN TO ELEVEN
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'
training.
Introduction
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
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1-98
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
be?
(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).
Tablet
Percentage of correct answers and wrong additive strategies for the
four-number problems
AL2
BL2
AL5
BL5
AL12
Correct
26.6
19.2
30.8
15.9
9.3
Wrong additive strategy
1.4
1.9
12.1
32.2
10.7
AU2
BU2
AU6
BU6
AU11
Correct
59.6
49.3
19.3
44.7
33.9
Wrong additive strategy
1.3
9.6
56.4
28.8
8.3
123
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
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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
contexts.
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
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FRUIT AND VEGETABLE PRICES
WOEI3KEST10U
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.
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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)
Class
1
5
7
2
6
8
3
4
9
10
Style
A
A
A
B
B
B
C
C
C
c
Pre Mean
10.7
7.8
8.0
5.7
8.7
9.7
4.7
10.8
7.4
14.7
Mean Gain
2.3
5.5
3.1
1.9
3.7
3.6
.9
.6
2.7
13
p value
*
.160
.255
.612
*
.010
(* 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.
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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.
References
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
Nottingham.
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.
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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.
129
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MIDDLE GRADE STUDENTS' REPRESENTATIONS OF LINEAR UNITS
DR. P. LYNNE CANNON
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
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
UUB
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
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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.
Results
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.
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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
Test
Interview
Response
U nl Li U nl Li
n Mi r n Mi r
U nl Li U nl Li
D Mi L
D
Mi L
Dominant: Iv
Discrete
yfameg
M B
B B
M
. ? . . ?
. . ? M
M .
M
Lolande
M B
B B
. ?
M
Conn ie
NR . . M --> B
? B
M .
M
B B
B B
. ?
M
? . . B
B , MB
M
M
Dominant lv
Line Seoments
B B
. . B . . B
. M
M
Dahl ia
B B
. . B . MB
M
M
Tammy
B B
. . B . . B
M
M
Lara
B B
. . B . . B
M
M
Kasey
B B
. . B B
M
M
Line seament
Brock
. B . . B
. . B B
•>
M
UX.
. B . . B
. . B . . B
. M
M
Coran
. B . . B
. . B . . B
M
M
Pete
B . . B
. . B B
. M
M
Note.
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
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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.
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1 . (James, test)
I 1 2 3 4 5 6
LJuBS.
2. (Kasey, test) I I I t {
'I
I 1 2 3 4 5 6
Hubs.
3. (Edwin, interview)
I' I I I I II
1 2 3 4 5 6
fluas
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. ■
Discussion
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
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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.
References
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.
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CHOICE OF STRUCTURE AND INTERPRETATION OF RELATION
IN MULTIPLICATIVE COMPARE PROBLEMS
Enrique Castro Martinez
Luis Rico Romero
Encarnaci6n Castro Martinez
Departamento de Didactica de la Matematica
Universidad de Granada. Espafia.
ABSTRACT
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,
p.16).
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
press).
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.
f
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
expression:
"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*.
Exempli;
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".
HYPOTHESIS
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
relation.
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.
METHOD
Subjects
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.
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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
»1
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
«2
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
"3
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.
RESULTS
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
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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
ones.
Table 2
Number of different processes
leading to vrong answers
Table 3
Somber of different processes
leading to right answers
Rl
R2
R3
R4
Rl
R2
R3
R4
6
6
5
5
"»1
2
1
2
2
92
5
6
7
6
92
3
3
4
5
03
6
9
7
U
03
1
1
1
1
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
Rl
R2
R3
R4
Qi
8
7
7
7
Q2
8
9
11
11
Q3
7
10
8
5
DISCUSSION
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?.
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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.
*1
R2
*3
R4
Qi
Change of
structure
Change of
structure
Ch. of str.
5% (50%)
Reversal
of relation
9% (60%)
(*) (**)
20% (61%)
Rev. of rel.
3% (30%)
11% (70%)
Q2
Change of
structure
Change of
structure
Change of
structure
Change of
structure
59% (92%)
60% (90%)
32% (80%)
32% (91%)
Q3
Ch. of str.
15% (46%)
Ch. of str.
23% (56%)
Reversal
of relation
Reversal
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.
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CONCLUSIONS
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.
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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-
ERIC/SMEAC.
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:
NCTM.
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.
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PERSONAL EXPERIENCE IN MATHEMATICS LEARNING AND PROBLEM SOLVING
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.
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A DIFFERENT VIEW OF PROBLEM SOLVING
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
save?
Solution:
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 ...
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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
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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
stage.
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.
Onewayinwhichthesequalitiesweremanifestedinthebusinessproblemstheysolved
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
reality.
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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.
A DIFFERENT VIEW OF MATHEMATICS
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
emergedportraysmathematics,notasadehumanizedprocessorsldIltobemastered,butasituation
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
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151
about mathematics as
experiences.
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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.
CONCLUSION
The outcome of this study is suggesting recognition of a dimension of mathematics
thattend^tobeunder-representedintheschoolcuniculumthusdenyingstudentswhounderstands
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,
thewritingandcooperativelearningmovementsinmathematicseducationhavenotgonefarenough
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.
152
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REFERENCES
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,
Cc*b.P.1986.Con«exU,go,Is,be.iefsand.e,r„ingmathem.ti«.Eoi^^
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.
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ummwrnntrnm
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INTERPRETATION AND CONSTRUCTION OF COMPUTER-MEDIATED GRAPHIC
REPRESENTATIONS FOR THE DEVELOPMENT OF SPATIAL GEOMETRY SKILLS
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
knowledge.
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.
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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.
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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
trihedral.
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.
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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
side).
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
sufficient.
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
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intend to evaluate a posteriori the students' btUviour in relation to the visual feedback offered by the
computer.
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
strategy.
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
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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
thefrcomrnunication.
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
characteristics.
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.
References
[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.
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RESPONSES TO OPEN-ENDED TASKS IN MATHEMATICS:
CHARACTERISTICS AND IMPLICATIONS
David J. Claike and Peter A. Sullivan
Mathematics Teaching and Learning Centre
Australian Catholic University (Victoria) - Christ Campus
Summary
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.
INTRODUCTION
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.
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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 STUDY
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:
Subtraction
Last night I did a subtraction task. I can remember some of the numbers.
What might the missing numbers have been?
Rounding
A number has been rounded off to 5,8 What might the number have been?
Area
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)
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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".
RESULTS
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.
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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
responses?
Does question format affect the number of pupils who give multiple or general
responses?
Stage 3
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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
format.
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.
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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.
167
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
contexts;
• 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
acquisition.
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
CONCLUSIONS
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.
REFERENCES
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.
ACKNOWLEDGEMENTS
1 - 145
OVER-EMPHASISING PROCESS SKILLS IN SCHOOL Matofmitipc
NEWMAN ERROR ANALYSIS DATA FROM FIVE COUNTWES
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.
170
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
I I PROCERUS rCrV^^f CARELESSNESS
. L_ ^ >s
QUESTION
FORM
| TRANSFORMATION
I READUG
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)
ERIC
171
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
A
B
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
1.
Table 1 „ ..
Background Details of the Asian and PNG Studies
Study
Country Grade Sample Number Language of test
level size of errors & of Newman
analysed
interview
6
95
1851
English
5
23
327
English
7
18
382
Bahasa
Malaysia
9
72
220*
Thai
Was the interview in
student's language
of instruction?
Clarkson (1983) PNG
Kaushil et al. (1985) India
Marinas &
Clements (1990)
Singhatat (1991)
Malaysia
Thailand
Yes
Yes
Yes
Yes
• 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.
174
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
Study
Error
Type
Clarkson
(1983)
(n = 1851 errors)
%
Kaushil
etal.(1985)
(n = 329 errors)
%
Marinas &
Clements (1990)
(n = 382 errors)
%
Singhatat
(1991)
(n = 220 errors)
%
Overall
%
Reading
12
0
0
0
8
Comprehension
21
24
45
60
28
Transformation
23
35
26
8
24
Process Skills
31
16
8
15
25
Encoding
1
6
0
0
1
Careless
12
18
21
16
14
Discussion
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
ERIC
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 = □
%
75
273 + 7 = □
76
55
A shop is open from 1 pm :o 4 pm. For how
many hours is it open?
44
87
It is now 5 o'clock. What time was it 3 hours ago?
47
88
Suniti has 3 less shells than Aarthi. If Suniti has 5 shells,
how many shells does Aarthi have?
42
73
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.
References
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
Centre.
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.
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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
University.
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
Jovanovich.
Ora, M. (1992). An investigation into whether senior secondary physical science students in
Indonesia relate their practical work to their theoretical studies. Unpublished manuscript,
SEAMEO-RECSAM, Penang.
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
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177
1 - 153
REVISED ACCOUNTS OF THE FUNCTION CONCEPT USING MULTI-
REPRESENTATIONAL SOFTWARE, CONTEXTUAL PROBLEMS AND STUDENT
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.
PATHS
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
situations.
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'.
179
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
X
y»xz
ay
4<AU)
0
0
1
1
> i
> 2
2
4
> 3
> 2
3
9
> 5
Fig. 2
X
y-2X
Ay
A<*y)
y-2*
®y
y-2*
1+Iy
0
1
1
> 2
1
2
1
2
> 1
> 1
2
> 2
2
4
2
4
> 2
> 4
> 2
4
> 2
4
8
3
8
> 4
8
> 2
8
16
4
16
> 8
16
16
32
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
exponential.
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180
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
concept
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
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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
representations.
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.
184
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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
Bibliography
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
Utrecht
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
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APPLYING THEORY IN TEACHER EDUCATION: CHANGING PRACTICE IN
MATHEMATICS EDUCATION.
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.
Introduction
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
1-162
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
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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
education.
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.
188
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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
school.
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
learners.
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
groups.
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
cjMaiMBBimiu
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.
References
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 Education.london, 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
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SCHOOL MATH TO INQUIRY HATH:
novum FROM HERE TO THERE
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
problematic.
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
a
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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
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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
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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
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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
there?
SA&SB: (Mumbling about 550's and 3)
T*1- What were you trying to do there? What was the purpose for doing that?
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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
known?
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
(mumble).
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
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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.
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CmcIiisIm
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
Academic.
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.
REFERENCES
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CUTTING THROUGH CHAOS: A CASE STUDY IN MATHEMATICAL PROBLEM SOLVING
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.
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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.
more.
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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
disc.
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
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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
it!
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.
REFLECTIONS ON THE PROBLEM-SOLVING PROCESS
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
tradiction
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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
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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
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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
mathematics.
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
208
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
faculties.
REFERENCES
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
THE DEVELOPMENT OF PROBLEM-SOLVING ABILITIES:
ITS INFLUENCE ON CLASSROOM TEACHING
by
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
Method
ERIC
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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.
Subjects
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.)
ERIC
211
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Results
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:
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[ 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
development:
[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.
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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.
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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."
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Conclusion
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.
References
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.
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SELF-DIRECTED PROBLEM SOLVING:
IDEA PRODUCTION IN MATHEMATICS
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.
AUTONOMY
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
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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,
1989).
A LEARNING BASE
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)
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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.
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A STUDY AND A RESPONSE
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
contexts.
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
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respond to social situations and, when needed, divert their efforts to skill
development.
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.
ATTITUDES AND THINKING POWER
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
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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.
REFERENCES
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(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),
134-141.
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
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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,
13,5-15.
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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PROJECT DELTA: TEACHER CHANGE IN SECONDARY CLASSROOMS
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)
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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
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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.
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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
workshop.
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.
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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
problems.
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
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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
comments.
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.
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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'
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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.
References
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
Education.
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:
Harvard
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REASONING AND REPRESENTATION IN FIRST YEAR
HIGH SCHOOL STUDENTS
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.
Introduction
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)
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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
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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.
Methodology
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
Boxer).
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:
1+3=4
1+3+5=9
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
here.
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Results
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.
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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: ...um, 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
empirical.
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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
problem.
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
pairs).
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.
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239
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.
Conclusions
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
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order to provide a better understanding of how more sophisticated and powerful kinds of mathematical
reasoning might be learned by students in secondary school.
References
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
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PROBLEM-SOLVING IN GEOMETRICAL SETTING: INTERACTIONS
BETWEEN FIGURE AND STRATEGY
Pier Luigi Ferrari
Dipartimento di Matematica - Universita di Genova
Summary
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.
INTRODUCTION
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).
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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
procedure)
= 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).
2. THE ORGANIZATION OF THE RESEARCH
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
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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
triangle;
» 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
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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).
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who almost never «re able to design some strategy to solve complex problems and often meet with
difficulties even when solving simple problems.
3. SOME FINDINGS
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
figure.
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.
ERIC
246
1-222
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 ("...now 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
4. FINAL REMARKS
=» 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).
REFERENCES
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-
251.
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,
1-8.
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.
42-46.
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.
1-225
A SEQUENCE OF PROPORTIONALITY PROBLEMS: AN EXPLORATORY STUDY
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.
1-226
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
setting).
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.
2.Method
1-227
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
1
Ma
Via
D
■ Ma
RV
M
M
Mu
1
A
D
A
R
M
M
A
3
A
A
A
A
A
4
A D
Ma
RV
M
M
A
5
A
B
M
M
*
6
B
B
R
A
M
A
7
A
A
A
A
A
8
A
D
Mb
RV
M
M
Mu
9
A
D
Mb
R
B
A
Mb
10
Ma
D
Ma
RV
M
M
Mb
11
A
1)
A
M
M
A
12
A
D
A
R
M
M
Mu
13
A
Mb "
R V
M
M
A
14
A
A
B
A
A
IS
Ma
D
Ma
RV
Ma
M
Mb
16
A
A
R
A
M
17
A
A
A
A
*
18
A
b
Mb
R
M
A
A
19
A
MB
R
A
A
A
20
A
D
MB
R
M
A
Mu
21
A
!>
Mb
R
M
A
A
22
B
B
ft
B
A
23
A
li
A
M
M
A
24
M
D
MB
RV
M
M
Mb
25
B
Mb
R
A
B
Mu
26
B
B
A
B
Mb
27
A
A
A
M
A
28
Ma
Ma
D
M
RV
M
M
B
29
B
11
A
M
A "
30
A
A
R
A
A
A
31
A
MB
R
A
A
A
32
A
M
B
A
A
33
M
D
Ma
R V
M
M
A
34
A
1!
R
A
■ A
A
35
A
A
R
A
B
A
16
A
MB
R
M
A
A —
yi
A D
Mb
A
A
A
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
254
o
ERIC
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 model.no 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
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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
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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.
REFERENCES
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
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DISCREPANCIES BETWEEN CONCEPTIONS AND PRACTICE: A CASE STUDY
lindaOattuso
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?
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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
miiaiMBBiwjjij
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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.
L'experimentation
'analyse
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Conclusions
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
d'enseignement.
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
ERIC
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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.
Implications
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.
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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
post-secondaire.
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.
1-240
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.
r£f£rences
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-
Wesley
GATTUSO, L. , LACASSE, R. (1986). l.es 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.
265
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ANALYSIS OF STUDENTS' ERRORS AND DIFFICULTIES IN SOLVING
COMBINATORIAL PROBLEMS
V. Nawro-Pelayo, J. D. Godino and M.C.Batanero and
University of Granada (Spain)
ABSTRACT
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
problems.
INTRODUCTION
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
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1-242
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.).
DESCRIPTION OF THE PROBLEMS PROPOSED
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.
TABLE 1
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 Mr.de 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
268
ERIC
1-244
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.
RELATIVE DIFFICULTY OF THE PROBLEMS: EFFECT OF THE TASK VARIABLES
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.
TABLE 2
Percentage of correct answers in the different problems
Problem Context Combinatorial Percentage correct answers
Operation 8th EGB 1st BUP
la Arrange people
P3
89.5
95.9
lb
P4
17.5
51.0
lc "
P5
7.0
46.9
2 Partition (people)
C10,3
0.0
6.1
3 Select objects
C5.3
8.8
20.4
4 Throw coi ns
VR2.2
56. 1
67.3
5 Select paths
Product Rule
43.9
26.5
6 Select people
V4.3
1.8
49.0
7 Distribute objects
V4.2
38.6
67.3
8 Select numbers
V5.2
38.6
51.0
9a Arrange letters
P5
7.0
53.1
9b Arrange letters
PR5,1,1,1,2
0
12.2
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
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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
formed.
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
ummmmmim
1-246
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.
TYPES OF ERRORS IDENTIFIED
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
instruction.
H2I1 systematic enumeration
This type of error described by Fischbeln and Gazit, consists of trying to solve
ummmmmim
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
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pupils.
- 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.
REFERENCES
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
PROCESSES AND STRATEGIES OF THIRD AND FOURTH GRADERS
TACKLING A REAL WORLD PROBLEM AMENABLE TO DIVISION
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.
Introduction
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
ummmmmim
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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,
Method
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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
Results
ummmmmim
1-253
Table t: Frequencies of correct and incorrect answers,
use of calculating devices and solution processes
Question
Device1
M. C O NA
Total
AR CO DI UM O NA
j-
Process
Ml
IS tablets,
3 per day
How
many
>l3
X'
NA 3
38 2 6 0
7 0 10
0 0 0 1
46
8
1
3 17 2 17 7 0
0 10 6 10
0 0 0 0 0 1
V3
X3
NA3
days?
Total
45 2 7 1
55
3 18 2 23 8 1
Total
M2
21 tablets,
4 per day
How
many
days?
* ■»
X
NA
17 4 3 0
5 3 3 0
10 0 3 0
0 0 0 7
24
11
13
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
*4
X
NA
Total
32 7 9 7
55
3 19 6 12 8 7
Total
M3
120 ml,
20 ml /day
How
many
days?
X
NA
26 5 1 0
11 2 3 0
0 0 0 7
32
16
7
5 13 6 7 1 0
0 2 0 9 5 0
0 0 0 0 0 7
X
NA
Total
37 7 4 7
55
5 15 6 16 6 7
Total
M4
300 ml,
40 .rd /day
How
many
days?
*5
X
NA
6 8 10
5 5 0 0
18 2 4 0
0 0 0 6
15
10
24
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
*5
X
NA
Total
29 15 5 6
55
0 16 13 13 7 6
Total
M5
375 ml,
24 ml /day
How
many
days?
4
* 6
X
NA
0 13 0 0
17 0 0
7 10 8 0
0 0 0 9
13
8
25
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
*s
X
NA
Total
8 30 8 9
55
0 6 24 7 9 9
Total
' 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 *
o
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278
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
jjMaiMBBimiu
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.
Conclusion
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.
ERIC
280
1-256
References
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,
129-147.
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,
3-17.
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
Education.
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
1 -257
PICTURES IN AN EXHIBITION: SNAPSHOTS OF A
TEACHER IN THE PROCESS OF CHANGE
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
1-258
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.
change?
Methods
Margaret
CJMaiMBBlMJjlJ
OC1
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
ERIC
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
student.
1-261
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
fractions.
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
286
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
UMmmmmm
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
Discussion
288
1-264
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.
References
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.
o
ERIC
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1 -265
•CANCELLATION WITHIN-THE-EQUATION " AS A SOLUTION PROCEDURE
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.
1-266
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
solution?
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
was.
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 +
ERIC
292
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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
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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.
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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
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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
time.
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
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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.
Conclusion.
"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.
References
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,
60-86
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
1-273
EMERGING. RELATIONSHIPS BETWEEN TEACHING
AND LEARNING ARITHMETIC DURING THE PRIMARY GRADES
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 .
Background
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.
Method
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 ..cpnd-gr.de
cli,ltoM.. Four of th. .IX cl...room. folio-- th. t.xtboo* In . r.l.tivtly
conv.ntion.l w.y .nd two cl...«oo- u..d th. .Itern.tive .ppro.ch. Th.
.lt.rn.tlv. induction cl...roo„. -nt.in.d only .tud.nt. who h.d received
.lt.rn.tiv. induction In y..r 1- Th. .lt.rn.tlv. .ppro.ch. .n ext.n.ion of th.t
u..d during flr.t gr.d.. «»ph..i*.d «th-*i«X conn.ction. through cl...
dl.cu..io„. of problem .nd .alution .tr.tegie. .nd through th. u.. of different
for™, of r.pr...nt.tion. In flr.t gr.de. problem w.r. .itu.t.d in .tory
cont.xt.. Both .ppro.che. devoted .bout X2 we.,, to pl.c. v.lu. .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. .ppro.ch .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. ..ch 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. w.re 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
u..d.
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. v.lu. .nd
.ddition/.ubtr.ction. Durin, year 3. .11 cl..— w.re ob.erv.d for three
300
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 v.lu. .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 illu.tr.tion purpo.e., we can con.ider
two very different group, of .tud.nt.-tho.e who entered the .econd y..r with .
comp.r.tiv.ly 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
gr.de „.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
mi..ing .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... .tr.t.gi n
.ft.r th.y h.d been expo..d ,.t horn, or school, to th. algorithm.. Three .tud.nt.
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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
BEST COPY AVAILABLE
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.
Conclusions
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
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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.
References
ummmmmim
1-280
Hiebert, J., & Wearne, 0. (in press). Links batwaan teaching and learning place
value with understanding in first grade. Journal for Research in Mathematics
Education.
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PRESCHOOLERS' SCHEMES FOR SOLVING PARTITIONING TASKS
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
1-282
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
Counting
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
Method
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,
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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.
Stickers
-No(5,?,l)-M TeaforTvo
Yes (1,8,8)
*
( Money Boxes
I
-No (1,6,4)-*-^
-No (3,2,0) -*■
-Yes (2,4,0) -*■
Stop
Tea for Three
V No (0,3,1) -
-Yes (1,0,3)-
Stop
Yes (0,2,4)
^MoWyBBOX")- No (0,2,1)
Yes (0,0,3)
Stop
fMoa*y 80X88 J- No (0,0,1)-
Stop
Yes (0,0,2)
Money Boxes
^ D .
-No (0,0,2)-*
-Yes (0,0,0)-*
Stop
■ 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
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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
distributed.
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.
Results
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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-
A
B •
C •
Figure 2: Sharlene's cyclic and regular solution strategy
A •
B •
C
Figure 3: Jim's cyclic and irregular solution strategy
1-286
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.
A
B
C
Figure 4: Carta's solution strategy
A
B
C
1-287
Child
Money Boxes (A)
Tea for Three
Leo (3.3)
Non-cyclic, irregular
Non-cyclic, irregular: not
successful
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
\Jl vUUI UM JCU IUJ
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
successful
Tim (4.10)
Sequence of coins in each box,
predominantly
exclusively: successful
Carla(4.11)
Sequence of coins in each box,
exclusively
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
Discussion
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
312
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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.
References
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
Conference.
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:
Erlbaum.
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.
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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.
Bondage
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)
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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.
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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)
1
R
What do you mean by ' recap"? You recap?
2
T
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?"
3
R
Now is it your expectation that by asking appropriate questions you will get all that
information from them?
4
T
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!"
5
R
Right
ft
T
Okav?
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7
R
Yes
8
T
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.
9
R
Right
10
T
We're really talking about notation, aren't we, now?
11
R
Right. - You said, give them ten questions. What sort of questions?
12
T
They're going to be quite straightforward.
13
R
To do what?
14
T
Find the lengths of vectors.
15
R
So, for example, "Find the length of a vector ...
16
T
... AB, if AB is 3.4."
(slight digression here on what the 3,4 notation looks like)
17
R
Or you could get them to invent some for themselves.
18
T
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.
19
T
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!
20
R
How do you mean?
21
T
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.
22
R
Yes, right.
23
T
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!
24
R
Thank you! How about doing it there (I pointed to his notes for the vectors lesson)?
25
T
What, getting them to set their own?
26
R
Get them to set their own.
27
T
(Pause) I'll trv.
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o
ERIC
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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.
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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.
ummmmmim
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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
320
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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.
References
Boud, D., Keogh, R. and Walker, D. (1985) Reflection: turning experience into learning. London: Kogan
Page
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)
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REFERENCE, STRUCTURE AND ACTION: ELIMINATING
PARADOXES IN LEARNING AND TEACHING MATHEMATICS
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
whereby
...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)
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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
manipulations.
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.
1
R:
Now the very first step here, where you've got arg(zi/z2), Ms X wants you
2
to focus on z\lzi. Now the first thing that she did was to write that out in a
3
trigonometric form, or a polar form. And she wrote on the top line, what did
4
she write?
5
Sarah:
ri ... {inaudible)
6
R:
Outside of?
7
Alice:
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
ERIC
324
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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!
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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
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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
ERJC
a—
327
1-303
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
SYNTACTIC
STRUCTURE
INTENTIONAL
(EXPRESSIVE)
FORM
SEMANTIC
REFERENCE
INFORMATIONAL
(INDICATIVE)
CONTENT
PRAGMATIC
ACTION
FUNCTION
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.
328
1-304
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.
References
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
INTERNATIONAL GROUP FOR
THE PSYCHOLOGY OF
M ATH EM ATICS EDUCATION '
PROCEEDINGS
ofjhe t .
SIXTEENTH
PME CONFERENCE
UnivQrsitv^of New Hampshire
, Eiurham, NH (USA)
Augusf) -11, 19^
; Volume II
ERIC
INTERNATIONAL GROUP FOR
THE PSYCHOLOGY OF
MATHEMATICS EDUCATION
proceeding:
of the
i
ME CONI
H
jLs.
H
N<
University of New Hampshire
Durham, NH (USA)
August 6- 11, 1992
Volume II
o
ERIC
331
Published by the Program Committee of the 16th PME Conference, USA.
All rights reserved.
Editors:
William Geeslin and Karen Graham
Department of Mathematics
University of New Hampshire
Durham. NH 03824
USA
332
2-i
CONTENTS OF VOLUME n
Research Reports (continued)
Kicren, T. & Piric, S. P- 2-1
The answer determines the question. Interventions and the growth of mathematical
understanding
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
2-ii
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
2-iii
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
336 BEST COPY AVAILABLE
2-2
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 
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