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 Secretary- General, 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 Michael de Villiers Ontario Institute for Studies Faculty of Education in Education (OISE) Edgewood Campus University of Toronto University of KwaZulu-Natal Toronto, ON, Canada South Africa [email protected] [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 © The Editor(s) (if applicable) and the Author(s) 2012, corrected publication 2021 Open Access This book was originally published with exclusive rights reserved by the Publisher in 2012 and was licensed as an open access publication in March 2021 under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (http://creativecommons.org/licenses/ by-nc-nd/4.0/), which permits any noncommercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if you modified the licensed material. 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Contents 1 Aspects of Proof in Mathematics Education......................................... 1 Gila Hanna and Michael de Villiers Part I Proof and Cognition 2 Cognitive Development of Proof ............................................................ 13 David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng 3 Theorems as Constructive Visions ......................................................... 51 Giuseppe Longo Part II Experimentation: Challenges and Opportunities 4 Exploratory Experimentation: Digitally-Assisted Discovery and Proof ................................................................................ 69 Jonathan Michael Borwein 5 Experimental Approaches to Theoretical Thinking: Artefacts and Proofs ............................................................................... 97 Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Yuk Lun Leung, Maria Alessandra Mariotti, and Ian Stevenson Part III 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 Lessons from the Eastern Cultural Traditions 18 Using Documents from Ancient China to Teach Mathematical Proof ................................................................................ 423 Karine Chemla Contents vii 19 Proof in the Western and Eastern Traditions: Implications for Mathematics Education ............................................. 431 Man Keung Siu Correction to: Proof and Proving in Mathematics Education ................... C1 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]