Proof and Proving in Mathematics Education

Proof and Proving in Mathematics Education

New ICMI Study Series VOLUME 15 Published under the auspices of the International Commission on Mathematical Instruction under the general editorship of Bill Barton, President

Jaime Carvalho e Silva, Secretary-General

For further volumes: http://www.springer.com/series/6351

Information on the ICMI Study programme and on the resulting publications can be obtained at the ICMI website http://www.mathunion.org/ICMI/ or by contacting the ICMI SecretaryGeneral, whose email address is available on that website.

Gila Hanna • Michael de Villiers Editors

Proof and Proving in Mathematics Education

The 19th ICMI Study

Editors Gila Hanna Ontario Institute for Studies in Education (OISE) University of Toronto Toronto, ON, Canada [email protected]

Michael de Villiers Faculty of Education Edgewood Campus University of KwaZulu-Natal South Africa [email protected]

ISSN 1387-6872 ISBN 978-94-007-2128-9 e-ISBN 978-94-007-2129-6 DOI 10.1007/978-94-007-2129-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011940425 © Springer Science+Business Media B.V. 2012 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

1

Aspects of Proof in Mathematics Education......................................... Gila Hanna and Michael de Villiers

Part I

1

Proof and Cognition

2

Cognitive Development of Proof ............................................................ David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng

13

3

Theorems as Constructive Visions ......................................................... Giuseppe Longo

51

Part II 4

5

Experimentation: Challenges and Opportunities

Exploratory Experimentation: Digitally-Assisted Discovery and Proof ................................................................................ Jonathan Michael Borwein Experimental Approaches to Theoretical Thinking: Artefacts and Proofs ............................................................................... Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Yuk Lun Leung, Maria Alessandra Mariotti, and Ian Stevenson

Part III

69

97

Historical and Educational Perspectives of Proof

6

Why Proof? A Historian’s Perspective.................................................. 147 Judith V. Grabiner

7

Conceptions of Proof – In Research and Teaching .............................. 169 Richard Cabassut, AnnaMarie Conner, Filyet Aslı İşçimen, Fulvia Furinghetti, Hans Niels Jahnke, and Francesca Morselli

v

vi

Contents

8

Forms of Proof and Proving in the Classroom ..................................... 191 Tommy Dreyfus, Elena Nardi, and Roza Leikin

9

The Need for Proof and Proving: Mathematical and Pedagogical Perspectives ................................................................. 215 Orit Zaslavsky, Susan D. Nickerson, Andreas J. Stylianides, Ivy Kidron, and Greisy Winicki-Landman

10

Contemporary Proofs for Mathematics Education ............................. 231 Frank Quinn

Part IV

Proof in the School Curriculum

11

Proof, Proving, and Teacher-Student Interaction: Theories and Contexts ............................................................................ 261 Keith Jones and Patricio Herbst

12

From Exploration to Proof Production ................................................. 279 Feng-Jui Hsieh, Wang-Shian Horng, and Haw-Yaw Shy

13

Principles of Task Design for Conjecturing and Proving .................... 305 Fou-Lai Lin, Kai-Lin Yang, Kyeong-Hwa Lee, Michal Tabach, and Gabriel Stylianides

14

Teachers’ Professional Learning of Teaching Proof and Proving ................................................................................... 327 Fou-Lai Lin, Kai-Lin Yang, Jane-Jane Lo, Pessia Tsamir, Dina Tirosh, and Gabriel Stylianides

Part V

Argumentation and Transition to Tertiary Level

15

Argumentation and Proof in the Mathematics Classroom ................. 349 Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna S. Epp, and Denis Tanguay

16

Examining the Role of Logic in Teaching Proof................................... 369 Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay

17

Transitions and Proof and Proving at Tertiary Level .......................... 391 Annie Selden

Part VI 18

Lessons from the Eastern Cultural Traditions

Using Documents from Ancient China to Teach Mathematical Proof ................................................................................ 423 Karine Chemla

Contents

19

vii

Proof in the Western and Eastern Traditions: Implications for Mathematics Education ............................................. 431 Man Keung Siu

Appendix 1: ICMI Study 19: Proof and Proving in Mathematics Education: Discussion Document............................... 443 Appendix 2: Conference Proceedings: Table of Contents ........................... 453 Author Index.................................................................................................... 461 Subject Index ................................................................................................... 471

Contributors

Ferdinando Arzarello Department of Mathematics, University of Torino, Torino, Italy, [email protected] Maria Giuseppina Bartolini Bussi Department of Mathematics, University of Modena and Reggio Emilia (UNIMORE), Modena, Italy, [email protected] Paolo Boero Dipartimento di Matematica, Università di Genova, Genova, Italia, [email protected] Jonathan Michael Borwein Centre for Computer-Assisted Research Mathematics and its Applications, CARMA, University of Newcastle, Callaghan, NSW 2308, Australia, [email protected] Richard Cabassut LDAR Laboratoire de Didactique André Revuz, Paris 7 University, Paris, France. IUFM Institut Universitaire de Formation des Maîtres, Strasbourg University, Strasbourg, France, [email protected] Karine Chemla CNRS, Université Paris Diderot, Sorbonne Paris Cité, Research Unit SPHERE, team REHSEIS, UMR 7219, CNRS, F-75205 Paris, France, [email protected] Ying-Hao Cheng Department of Mathematics, Taipei Municipal University of Education, Taipei, Taiwan, [email protected] AnnaMarie Conner Department of Mathematics & Science Education, University of Georgia, Athens, GA, USA, [email protected] Michael de Villiers School of Science, Mathematics & Technology Education, University of KwaZulu-Natal, Durban, South Africa, [email protected] Nadia Douek Institut Universitaire de Formation des Maîtres, Université de Nice, Nice, France, [email protected] ix

x

Contributors

Tommy Dreyfus Department of Mathematics, Science and Technology Education, School of Education, Tel Aviv University, Tel-Aviv, Israel, [email protected] Viviane Durand-Guerrier Département de mathématiques, I3M, UMR 5149, Université Montpellier 2, Montpellier, France, [email protected] Susanna S. Epp Department of Mathematical Sciences, DePaul University, Chicago, IL, USA, [email protected] Fulvia Furinghetti Dipartimento di Matematica, Università di Genova, Genova, Italy, [email protected] Judith V. Grabiner Department of Mathematics, Pitzer College, Claremont, CA, USA, [email protected] Gila Hanna Ontario Institute for Studies in Education, University of Toronto, Toronto, Canada, [email protected] Patricio Herbst School of Education, University of Michigan, Ann Arbor, MI, USA, [email protected] Wang-Shian Horng Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, [email protected] Feng-Jui Hsieh Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, [email protected] Filyet Aslı İşçimen Department of Mathematics and Statistics, Kennesaw State University, Kennesaw, GA, USA, [email protected] Hans Niels Jahnke Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany, [email protected] Keith Jones Mathematics and Science Education Research Centre, School of Education, University of Southampton, Highfield, Southampton, UK, [email protected] Ivy Kidron Applied Mathematics Department, Jerusalem College of Technology (JCT), Jerusalem, Israel, [email protected] Boris Koichu Department of Education in Technology and Science, Technion – Israel Institute of Technology, Haifa, Israel, [email protected] Margo Kondratieva Faculty of Education and Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada, margo. [email protected] Kyeong-Hwa Lee Department of Mathematics Education, Seoul National University, Seoul, South Korea, [email protected] Roza Leikin Department of Mathematics Education, Faculty of Education, University of Haifa, Haifa, Israel, [email protected]

Contributors

xi

Allen Yuk Lun Leung Department of Education Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong, [email protected] Fou-Lai Lin Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, [email protected] Jane-Jane Lo Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA, [email protected] Giuseppe Longo CNRS – École Normale Supérieure et CREA, École Polytechnique, Paris, France, [email protected] Maria Alessandra Mariotti Department of Mathematics and Computer Science, University of Siena, Siena, Italy, [email protected] Francesca Morselli Dipartimento di Matematica, Università di Genova, Genoa, Italy, [email protected] Elena Nardi School of Education, University of East Anglia, Norwich, UK, [email protected] Susan D. Nickerson Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA, [email protected] Judy-anne Osborn Centre for Computer Assisted Mathematics and Its Applications, School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia, [email protected] Frank Quinn Virginia Tech, Blacksburg, VA 24061, USA, [email protected] Annie Selden Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, USA, [email protected] Haw-Yaw Shy Department and Graduate Institute of Mathematics, National Changhua University of Educatio, Changua, Taiwan, [email protected] Man Keung Siu Department of Mathematics, University of Hong Kong, Hong Kong SAR, China, [email protected] Ian Stevenson Department of Education and Professional Studies, King’s College, London, UK, [email protected] Andreas J. Stylianides Faculty of Education, University of Cambridge, Cambridge, UK, [email protected] Gabriel Stylianides Department of Education, University of Oxford, Oxford, UK, [email protected] Michal Tabach School of Education, Tel-Aviv University, Tel-Aviv, Israel, [email protected] David Tall Mathematics Education Research Centre, University of Warwick, Coventry, UK, [email protected]

xii

Contributors

Denis Tanguay Département de mathématiques, Université du Québec à Montréal (UQAM), Montreal, QC, Canada, [email protected] Dina Tirosh School of Education, Tel-Aviv University, Tel-Aviv, Israel, [email protected] Pessia Tsamir School of Education, Tel-Aviv University, Tel-Aviv, Israel, [email protected] Greisy Winicki-Landman Department of Mathematics and Statistics, California State Polytechnic University, Pomona, CA, USA, [email protected] Walter Whiteley Department of Mathematics and Statistics, York University, Toronto, Canada, [email protected] Kai-Lin Yang Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, [email protected] Oleksiy Yevdokimov Department of Mathematics & Computing, University of Southern Queensland, Toowoomba, Australia, [email protected] Orit Zaslavsky Department of Teaching and Learning, New York University, New York, NY, USA Department of Education in Technology and Science, Technion – Israel, Institute of Technology, Haifa, Israel, [email protected]