ED 383 538

SE 055 811






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

International Group for the Psychology of Mathematics


Aug 92


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

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




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

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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





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


1 -i

PME XVI PROCEEDINGS Edited by WiUiam Geeslin and Karen Graham Mathematics Department University of New Hampshire Durham NH USA


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

The academic program of PME XVI includes:

92 research reports

4 plenary addresses

1 plenary panel

11 working groups

6 discussion groups

2 featured discussion groups

31 short oral presentations

40 poster presentations.

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

The review process

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


1 - ii

List of PME XVI Reviewers

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


Michal Yerushalmy, Israel



1 -iii



- President KathHart (United Kingdom)

- Vice-President Gilah Leder (Australia)

- Secretary Martin Cooper (Australia)

- Treasurer Angel Gutierrez (Spain)


Mich61e Artigue (France) - Frank Lester (USA)

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

BernadetteDenys (France) - Carolyn Maher (USA)

Claude Gaulin (Canada) - Nobuhiko Nohda (Japan)

Gila Hanna (Canada) - Joao Ponte (Portugal)

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


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

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

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

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

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


- Joan Ferrini-Mundy - Karen Graham

- William E. Geeslin - Lizabeth Yost




1 -iv


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

The major goals of the Group are:

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

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

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


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

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

The subscription can be paid together with the conference fee.

1 -V

Addresses of Authors Presenting Research Reports at PME XVI

Vcma Adams

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


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


David Ben-Chaim Weizmann Institute of Science Rehovot, 76100 ISRAEL

Janette Bobis

University of New South Wales POBoxl

Kensington, NSW, 2033 AUSTRALIA


Dipartimento Matematica Umversita


16132, Genova


Card Brekke Telemark Laercrhugskole N-3670Notodden NORWAY

Lynne Cannon

Faculty of Education

Memorial University of Newfoundland

St John's, Newfoundland A1B 3X8


Olive Chapman

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


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



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


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


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


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



1 - vii

Masataka Koyarna

Faculty of Education

Hiroshima University, 3-101

2-365 Kagamiyama Higashi-Hiroshima City


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


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


Joanna Masingila Education 309 Indiana University Bloomington, IN 47405 USA


University Montpellier U

Place Eugene Bataillon



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




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


Purdue University

West Lafayette, IN 47907-1442


JoSo Pedro Ponte

Av. 2y de Julmo, 134-4"

1300 Lisboa, PORTUGAL


Matthias Reiss Stedingerstr. 40 7000 Stuttgart 31 GERMANY

Joe Relich PO Box 10

c/o Faculty of Education



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


Adaiira Sienz-Ludlow

Dept of Mathematical Sciences

Northern Illinois University

DcKalb, IL 60115


Victoria Sanchez Avdo. Ciudad Jardin, 22 41005 Sevilla SPAIN

Vinia Maria Santos

School of Education, Room 309

Indiana University

Bloomington, IN 47405


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



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


Martin Simon

176 Chambers Building

Department of Curriculum and Instruction

Univesity Park, PA 16802


Jack Smith

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


West Lafayette, IN




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


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

John Mason Open University Walton Hall


Bernard Parzysz

IUFMde Lorraine

Departement de mathematiques

University deMetz

lie du Suilcy

F 57000 Metz


Norma Presmeg

219 Corothers Hall, B-182

Florida State University

Tallahassee, FL 32306-3032




1 - xi


Preface P- l-»

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

History and Aims of the PME Group p. Mv

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

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

Working Groups

Advanced mathematical thinking P- 1-3

Organizers: G. Ervynck & D. Tall

Algebraic processes and structure P- 1-4

Organizer R. Sutherland

Classroom Research P- 1-5

Organizer F.J. van den Brink

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

Geometry P- 1-8 Organizer: H. Mansfield

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

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

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

Representations P- 1-11 Organizer G. Goldin

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

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

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


x o

1 - xii

Discussion Groups

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

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

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

Philosophy of mathematics education Organizer P.Ernest

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

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

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

Research Reports de Abreu, Guida

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

Rhetorical problems and mathematical problem solving: An exploratory study

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

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

Students' understanding of the significance level in statistical tests

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

Students' cognitive construction of squares and rectangles in Logo Geometry

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

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

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

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

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

p. 1-20 p. 1-21

p. 1-22

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

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

1 - xiii

Boero, Shapiro, L. P> !'89

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

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

Multiplicative structures at ages seven to eleven

Cannon, P.L. P- 1-105

Middle grade students' representations of linear units

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

Choice of structure and interpretation of relation in multiplicative compare problems

Chapman, O. p. 1-121

Personal experience in mathematics learning and problem solving

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

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

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

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

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

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

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

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

Crawford, K. P- 1-161

Applying theory in teacher education: Changing practice in mathematics education

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

School math to inquiry math: moving from here to there

Davis, G. p. 1-177

Cutting through Chaos: A case study in mathematical problem solving

DeGuire,L.J. p. 1-185

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

Dirkes, M.A. P- M93

Self-directed problem solving: Idea production in mathematics

Dougherty, B.J. P- 1-201

Project DELTA: Teacher change in secondary classrooms

Edwards, L.D. P- 1'209

Reasoning and representation in first year high school students

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

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




1 - xiv

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

A sequence cf proportionality problems: An exploratory study

Gattuso, L. p. 1-233

Discrepancies between conceptions and practice: A case study

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

Analysis of students' errors and difficulties in solving combinatorial problems

Groves, S. p. 1-249

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

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

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

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

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

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

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

Jaworski, B. p. 1-289

The emancipatory nature of reflective mathematics teaching

Kanes, C. p. 1-297

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

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


1 - XV

Letter, G.C. P- 2-33

Measuring attitudes to mathematics

Lerman, S. P- 2-40

The Junction of language in radical constructivism: A Vygotskian perspective

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

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

Lins, R. P- 2-56

Algebraic and non-algebraic algebra

Magidson. S. ^ P- 164

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

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

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

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

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

Meira,L. P- *96

The microevolution of mathematical representations in children s activity

Mesquita, AJL. P- 2-104

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

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

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

Mitchelmore, M. P- 2"120

Children's concepts of perpendiculars

Moschkovich, J. P- 2-128

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

Mousley, J. ^ P- 2-136

Teachers as researchers: Dialectics cf action and reflection

Mulligan, J. P- 2-144

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

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

The development of young students' division strategies



1 -xvi

Nathan, M.J. p. "160

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

Neuman, D. p. 2- no

The influence cf numerical factors in solving simple subtraction problems

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

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

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

Logo mathematics and boxer mathematics: Some preliminary comparisons

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

Representation of area: A pictorial perspective

Owens, K. p. 2-202

Spatial thinking takes shape through primary-school experiences

Perlwitz, MD. p. 2-2 10

The interactive constitution of an instructional activity: A case study

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

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

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

Kasimir: A simulation of learning iterative structures

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

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

Rice, M. p. 2-250

Teacher change: A constructivist approach to professional development

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

Connectedness in teaching algebra: A novice-expert contrast

Saenz-Ludlow, A. p. 2-266

Ann's strategies to add fractions

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

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


1 - xvii

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

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

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

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

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

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

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



1 - xviii

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

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

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

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

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

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

Short Oral Presentations

Abcle, A. . p. 3-125

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

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

Achievement and thinking strategies on "reversed items"

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

The processing loads and relations between counting and place value

Cantoral-Uriza, R. p. 3-127

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

Ernest, P. p. 3-127

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

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

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

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

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

Kynigos, C. p. 3-129

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

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

Understanding mathematics class discussions

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

Cognitive origins of students' errors in writing equations


1 - xix

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

Towards applied problem solving

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

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

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

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

Nevile. L. P- 3-132

Teaching recursion as shifts of attention

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

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

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

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

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

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

Perks. P. P- 3-134

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

Quintal, C. P- 3-135

Hierarchies of cognitive difficulty in early algebra

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

Pupil strategies for solving algebra word problems with a spreadsheet

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

Children's developing concepts of landmarks in the number system

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

Equations and inequalities Processes without objects?

Shiu. C. p. 3-137

Assumptions and intentions in distance learning materials for mathematics

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

Writing in mathematics: Is it always beneficial?

Teppo.A.R. P-3'138

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




1 - XX

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

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

Ikonic and early concrete symbolic responses to two fraction problems

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

Conceptual splatter and metaphorical noise: The case of graph continuity

Yoshida, M.

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

Zazkis, R.

Inverse of a product - A theorem out of action

Poster Presentations

Barocio Quijano, R. & Brefia Sanchez, J.

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

Becker, G.

Analogical reasoning: Basic component in problem solving activities

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

Berenson, S.B.

Race and gender interactions and constructivist teaching

Bergsten, C.

Schematic structures of mathematical form

Borba, M. & Confrcy, J.

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

Carraher, D.W.

Relational thinking and rational numbers

Chazan, D.

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

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

The other consequences of a problem-based mathematics curriculum

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

Design of a Logo environment for elementary geometry

p. 3-138

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

p. 3-141

p. 3-145

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

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

1 - xxi

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

Multimodal functioning in mathematLal problem solving

Coidero-Osorio, F.

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

DeFranco, T.C.

The role ofmetacognition in mathematical problem solving among PhD. 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


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



1 - xxiv

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

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



1 - 1

Working Groups




1 -3


Organisers: Gontran Ervynck, David Tall

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