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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 ..............