Proof and Proving in Mathematics Education: The 19th ICMI Study

467 Pages·2012·7.685 MB·English

by  Gila Hanna, Michael de Villiers (auth.), Gila Hanna, Michael de Villiers (eds.)

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 (cid:129) 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
© 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, microfi lming, recording or otherwise, without written
permission from the Publisher, with the exception of any material supplied specifi cally 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......................................... 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
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 , S outh Africa, [email protected]
Nadia Douek Institut Universitaire de Formation des Maîtres , Université de Nice ,
Nice , France, [email protected]
ix