Transformation—A Fundamental Idea of Mathematics Education Sebastian Rezat · Mathias Hattermann Andrea Peter-Koop Editors

Transformation—A Fundamental Idea of Mathematics Education Sebastian Rezat · Mathias Hattermann Andrea Peter-Koop Editors Transformation—A Fundamental Idea of Mathematics Education 1 3 Editors Sebastian Rezat Andrea Peter-Koop EIM - Institut für Mathematik Fakultät für Mathematik - IDM Universität Paderborn Universität Bielefeld Paderborn Bielefeld Germany Germany Mathias Hattermann Fakultät für Mathematik - IDM Universität Bielefeld Bielefeld Germany ISBN 978-1-4614-3488-7 ISBN 978-1-4614-3489-4 (eBook) DOI 10.1007/978-1-4614-3489-4 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013955389 © Springer Science+Business Media, LLC 2014 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) For Rudolf Sträßer Introduction This book intends to open up a discussion on fundamental ideas in didactics of mathematics as a scientific discipline. We want to introduce fundamental ideas as a possi- ble answer to the diversity of theories in the field. Instead of providing a solely theoretical contribution, we suggest entering into this discussion by focusing on the “idea of transformation” that we regard as being fundamental to didactics of mathematics. Transformation is a matter of interest in many areas of didactics of mathematics conceived of as “the sum of scientific activities to describe, analyze and better un- derstand peoples’ joy, tinkering and struggle for/with mathematics” (Sträßer 2009, p. 68): transformations of representations of mathematics and related transformations of mathematics, transformations of artifacts into instruments, transformations of mathematical knowledge, transformation of practice, transformation of solving strategies, and transformation of acquired heuristics to new similar problems, just to name a few. Accordingly, many theoretical approaches aim to conceptualize and grasp transformations: semiotics, the instrumental approach (Rabardel 1995), transposition didactique (Chevallard 1985), and the nested epistemic actions model (Schwarz et al. 2009). By looking at these theories as being related to the same fundamental idea, we can ask further questions such as: How do we approach transformations research in didactics of mathematics? How is transformation conceptualized in each of these theories? What do we know/ learn about transformations related to the teaching and learning of mathematics? In the following section, we will elaborate on the theoretical origins of our approach. Theoretical Background Our approach is embedded in the debate about the diversity of theories in didactics of mathematics. The diversity of theories has been an issue of discussion ever since the foundation of the discipline. This is documented in the Theory of Mathematics Education Group (TME) founded by Steiner and regular study groups at the Inter- national Congress on Mathematics Education (ICME) and the annual conference vii viii Introduction Fig. 1 A landscape of strategies for connecting theoretical approaches (Bikner-Ahsbahs and Pre- diger 2010, p. 492) of the International Group for the Psychology of Mathematics Education (IGPME). The current significance of this issue as well as the controversy about it can be seen in the comprehensive volume Theories of Mathematics Education (Sriraman and English 2010). The tenor of the contributions is that diversity of theories is an inevitable and even welcome hallmark of didactics of mathematics. The theoretical manifoldness is traced back to the vast variety of goals and research paradigms by many researchers, which are recorded in volumes such as “Didactics of mathematics as a scientific discipline” (Biehler et al. 1994) or the Study of the International Commission on Mathematical Instruction (ICMI) “What is research in mathematics education, and what are its results” (cf. Sierpinska and Kilpatrick 1998). Critics such as Steen (1999) argue that a lack of focus and identity pervades the foundations of the discipline: there is no agreement among leaders in the field about goals of research, important questions, objects of study, methods of investigation, criteria for evaluation, significant results, major theories, or usefulness of results (Steen 1999, p. 236). This observation even leads him to question the scientific nature of the field which he describes as a field in disarray, a field whose high hopes for a science of education have been over- whelmed by complexity and drowned in a sea of competing theories (Steen 1999, p. 236). This criticism is often encored by the call for a grand theory of mathematical think- ing. Although a growing number of convincing arguments is presented to support the necessity of multiple theories (e.g., Bikner-Ahsbahs and Prediger 2010; Lerman 2006), the related problems of the discipline’s missing focus and identity persist. The questions are how we deal with this variety and if there are other ways to pro- mote the development of focus and identity of the discipline than a grand theory of mathematics education. Bikner-Ahsbahs and Prediger (2010) argue that “the diversity of theories and theoretical approaches should be exploited actively by searching for connecting strategies” in order to “become a fruitful starting point for a further development of the discipline” (p. 490). Based on a meta-analysis of case studies about connecting theories, they suggest different strategies for connecting theories, which they call “networking strategies” (Bikner-Ahsbahs and Prediger 2010, p. 492). These networking strategies are organized according to their degree of integration between the two extremes “ignoring other theories” and “unifying globally” as shown in Fig. 1. Introduction ix Although this overview of strategies for networking theories in didactics of mathematics provides a fruitful approach to deal with multiple theories, it seems hardly capable of contributing to the discipline’s search for focus and identity, be- cause it does not say anything about the phenomena these theories are related to. The networking strategies can be understood as heuristics to connect given theories. However, how to find theories that are worthwhile connecting? Which theories re- late to a certain phenomenon? In order to answer these questions, we suggest reflecting upon fundamental ideas of didactics of mathematics as a scientific discipline. Pointing out fundamental ideas could help to focus on the core issues of the discipline and could provide a means to organize theories in terms of being related to a similar idea. Fundamental Ideas In his seminal book “The Process of Education” (1960), Bruner introduced fundamental ideas as a means for curriculum development. For him they provide an answer to the basic problem that learning should serve us in the future which is at the heart of the educational process and therefore a fundamental problem of curriculum development. Students only have limited exposure to exemplary materials they are to learn. How can they learn something that is relevant for the rest of their lives? He argues that this “classic problem of transfer” can be approached by learning about the structure of a subject instead of simply mastering facts and techniques. “To learn structure” for Bruner means “to learn how things are related” (Bruner 1960, p. 7). According to him, transfer is dependent upon the mastery of the structure of a subject matter in the following way: in order for a person to be able to recognize the applicability or inapplicability of an idea to a new situation and to broaden his learning thereby, he must have clearly in mind the general nature of the phenomenon with which he is dealing. The more fundamental or basic is the idea he has learned, almost by definition, the greater will be its breadth of applicability to new problems. Indeed,

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