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Andreas J. Stylianides Guershon Harel Editors

Advances in Mathematics Education Research on Proof and Proving An International Perspective

ICME-13 Monographs Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, Hamburg, Germany

Each volume in the series presents state-of-the art research on a particular topic in mathematics education and reflects the international debate as broadly as possible, while also incorporating insights into lesser-known areas of the discussion. Each volume is based on the discussions and presentations during the ICME-13 Congress and includes the best papers from one of the ICME-13 Topical Study Groups or Discussion Groups.

More information about this series at http://www.springer.com/series/15585

Andreas J. Stylianides Guershon Harel •

Editors

Advances in Mathematics Education Research on Proof and Proving An International Perspective

123

Editors Andreas J. Stylianides Faculty of Education University of Cambridge Cambridge UK

Guershon Harel Department of Mathematics University of California San Diego La Jolla, CA USA

ISSN 2520-8322 ISSN 2520-8330 (electronic) ICME-13 Monographs ISBN 978-3-319-70995-6 ISBN 978-3-319-70996-3 (eBook) https://doi.org/10.1007/978-3-319-70996-3 Library of Congress Control Number: 2017958582 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

There is an international recognition of the importance of proof and proving in students’ learning of mathematics at all levels of education, and of the difficulties faced by students and teachers in this area. Also, existing curriculum materials