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Editors: Prof. Dr. Gabriele Kaiser, University of Hamburg Prof. Dr. Bharath Sriraman, The University of Montana

Bharath Sriraman  Lyn English Editors

Theories of Mathematics Education Seeking New Frontiers

Prof. Bharath Sriraman Dept of Mathematical Sciences The University of Montana Missoula, MT 59812-0864 USA [email protected]

Prof. Lyn English Centre for Learning Innovation Queensland University of Technology Victoria Park Road Kelvin Grove, QLD 4059 Australia [email protected]

ISBN 978-3-642-00741-5 e-ISBN 978-3-642-00742-2 DOI 10.1007/978-3-642-00742-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009938521 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Series Preface

Advances in Mathematics Education is a new and innovative book series published by Springer that builds on the success and the rich history of ZDM—The International Journal on Mathematics Education (formerly known as Zentralblatt für Didaktik der Mathematik). One characteristic of ZDM since its inception in 1969 has been the publication of themed issues that aim to bring the state-of-the-art on central sub-domains within mathematics education. The published issues include a rich variety of topics and contributions that continue to be of relevance today. The newly established monograph series aims to integrate, synthesize and extend papers from previously published themed issues of importance today, by orienting these issues towards the future state of the art. The main idea is to move the field forward with a book series that looks to the future by building on the past by carefully choosing viable ideas that can fruitfully mutate and inspire the next generations. Taking inspiration from Henri Poincaré (1854–1912), who said “To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice”, this is our attempt to create something new for the field by making useful combinations of existing ideas with those that are considered to be at the front