Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique PDF
Preview Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique
DIALOGUE AND LEARNING IN MATHEMATICS EDUCATION Mathematics Education Library VOLUME 29 ManagingEditor A.J. Bishop, Monash University, Melbourne, Australia Editorial Board J.P. Becker, Illinois, U.S.A. G. Leder, Melbourne, Australia A. Sfard, Haifa, Israel O. Skovsmose, Aalborg, Denmark S. Turnau, Krakow, Poland The titles published in this series are listed at the end of this volume. DIALOGUE AND LEARNING IN MATHEMATICS EDUCATION Intention, Reflection, Critique by HELLE ALRØ and OLE SKOVSMOSE Aalborg University, Denmark KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 0-306-48016-6 Print ISBN: 1-4020-0998-4 ©2003 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstore at: http://ebooks.kluweronline.com CONTENTS Acknowledgements vii Introduction 1 Qualities of Communication 1 Qualities of Learning Mathematics 7 Outline of the Content of the Book 11 Chapter 1: Communication in the Mathematics Classroom 21 Mistakes and Corrections 23 Bureaucratic Absolutism 26 Perspective 28 ‘How Much does a Newspaper Fill?’ 30 Learning as Action 41 Chapter 2: Inquiry Co-operation 45 From Exercises to Landscapes of Investigation 46 Entering a Landscape of Investigation 51 ‘What does the Danish Flag Look Like?’ 54 Inquiry Co-operation Model 62 Obstructions to Inquiry Co-operation 65 Chapter 3: Further Development of the Inquiry Co-operation Model 69 ‘Batman & Co.’ 69 The Inquiry Co-operation Model Reconsidered 100 Getting in Contact 101 Locating 101 Identifying 103 Advocating 105 Thinking Aloud 107 Reformulating 108 Challenging 109 Evaluating 109 Chapter 4: Dialogue and Learning 113 The Socratic Dialogue 114 Dialogic Qualities 115 Making an Inquiry 118 Running a Risk 122 v vi CONTENTS Maintaining Equality 124 How to Do Things with Dialogue 126 Dialogic Acts – The Inquiry Co-operation Model Reconsidered 128 DialogicTeachingandLearning – AndIts Fragility 130 DialogicTeaching and Learning – And Its Significance 132 Chapter 5: Intention and Learning 137 ‘Travel Agency’ 138 Intention 154 Intentions-in-Learning 156 Underground Intentions 158 Resources of Intentions 160 Chapter 6: Reflection and Learning 165 ‘Caramel Boxes’ 166 Reflection 184 Scope of Reflections 184 Subject of Reflections 188 Context of Reflections 191 Reflection and Intention – Explosive Concepts 193 Chapter 7: Critique and Learning 195 The Challenge of Critique 196 ‘Terrible Small Numbers’ 199 Reliability 202 Responsibility 217 Critique 230 Intention and Critique 231 Reflection and Critique 233 Dialogue and Critique 235 The Challenge of Critique – Too Difficult? 238 Chapter 8: Critical Epistemology and the Learning of Mathematics 243 Mono-logicalEpistemology 243 Dia-logical Epistemology 247 Non-critical Epistemology 251 Critical Epistemology 255 Learning Mathematics Critically 259 References 263 Name Index 277 Subject Index 281 ACKNOWLEDGEMENTS First of all, we want to thank the involved students and teachers who have made it possible for us to study details of communication in the mathematics classroom. Without their willingness to co-operate our work would not have been possible. We are grateful to every teacher and student who appear anonymous in this book, and to the teachers Bjarne Würtz Andersen, Ane Marie Krogshede Nielsen and Ib Trankjær, who provided us with the material for Chapter 1. We owe a special gratitude to Henning Bødtkjer and Mikael Skånstrøm with whom we have co- operated closely during the whole study. Their teaching has convinced us that dialogue can work in the mathematics classroom. Many persons have made comments and suggestions for the im- provement of the manuscript. We want to thank Alan Bishop, Marcelo Borba, Paul Cobb, Marit Johnsen Høines, Marianne Kristiansen, John Mason and Miriam Godoy Penteado for careful readings, inquiring dia- logues and useful comments to previous versions of the manuscript. We are also grateful to H.C. Hansen, Aage Nielsen and Paola Valero for their comments on special issues. Further, we would like to thank a group of Ph.D.- and MA-students from the State University of São Paulo at Rio Claro: Chateaubriand Nunes Amancio, Jussara de Loiola Araújo, Jonei Cerqueira Barbosa, Telma Souza Gracias, Frederico José Lopes for their important comments on different chapters. We have had the opportunity to discuss every chapter of this book at different stages within a group of researchers in the cross-institutional Centre for Research in Learning Mathematics. Thanks to Morten Blom- høj, Iben Maj Christiansen, Elin Emborg, Lena Lindenskov and Tine Wedege for valuable comments and support. Many people have helped us during the process to overcome practical, technical, and linguistic problems. We want to thank Nikolaj Hyldig, Marianne Harder Mandøe and Dana Sandstrøm Poulsen for making the transcripts, and Ebbe Klitgård for translations of transcripts. We are also grateful to Ebbe Klitgård and Marilyn Nickson for correcting our Eng- lish. Further, we want to thank Erik Nød Sørensen for computer assistance, for drawing the figures, and for setting up the manuscript. vii viii ACKNOWLEDGEMENTS Chapters 1 and 2 are based on three articles which appeared in For the Learning of Mathematics: ‘On the Right Track’, 16(1), 2-9 and 22; ‘The Students’ Good Reasons’, 16(3), 31-38; and ‘That was not the Intention! Communication in Mathematics Education’, 18(2), 42-51. We are grate- ful to the editor of FLM for agreeing to the re-use of this material here. This study is carried out as part of the research initiated by the Centre for Research in Learning Mathematics, which is based on a co-operation between the Danish University of Education, Roskilde University Centre and Aalborg University. The Centre is supported financially by the Dan- ish Research Council for the Humanities. Aalborg, August 2002 Helle Alrø and Ole Skovsmose INTRODUCTION “Today we really learnt something!” Mary exclaimed after she, together with Adam, had concentrated for almost two hours on setting up a spreadsheet. Something significant seems to have happened for Mary, something that should be considered when theorising about the learning of mathematics. In this study we are going to meet with Mary and Adam and many other students in the mathematics classroom. The main purpose of this meeting is to gather empirical resources to gain a better understanding of the role of communication in learning mathematics. The initial idea that guides our investigations can be condensed in the following hypothesis: The qualities of communication in the classroom influence the qualities of learning mathematics. This is not a very original statement and certainly very general. If the statement is to be provided with meaning it is important to clarify at least the two expressions: ‘qualities of communication’ and ‘qualities of learning mathematics’. In this introduction, as well as during the rest of this book, we are going to struggle with clarifying in what sense communication and learning can be connected, and how to conceptualise this connection. QUALITIES OF COMMUNICATION In many different contexts, both inside and outside school, special attention is paid to communication. Thus, companies organise workshops and courses on communication in order to improve the way they operate (see, for instance, Isaacs, 1999a; Kristiansen and Bloch-Poulsen, 2000). The improvement of communication is expected not only to have an influence on the atmosphere of the workplace, but also on the way the company operates in terms of business, as expressed in figures and budgets. Communication becomes related to the idea of the ‘learning organisation’. Qualities of communication can be expressed in terms of interpersonal relationships. Learning is rooted in the act of communicating itself, not 1 2 INTRODUCTION just in the information conveyed from one party to another. Thus, communication takes on a deeper meaning. In Freedom to Learn, first published in 1969, Carl Rogers (1994) considers interpersonal relationships as the crucial point in the facilitation of learning. Learning is personal, but it takes place in the social contexts of interpersonal re- lationships. Accordingly, the facilitation of learning depends on the quality ofcontact in the interpersonal relationship that emerges from the communication between the participants. In other words, the context in which people communicate affects what is learned bybothparties. This brings forward the idea that some ‘qualities of communication’ could be clarified in terms of dialogue. The word ‘dialogue’ has many everyday descriptive references but the important factor common to all is that they involve at least two parties. For instance, it is possible to talk about the dialogue between East and West and about the breakdown of the dialoguebetweenPalestine andIsrael. Suchreferencesto dialogue are not strictly part of our concern. In philosophical contexts the notion of dialogue occurs in many places. Plato presented his ideas as dialogues; in 1632 Galileo Galilei wrote The Dialogue Concerning the two Chief World Systems (which brought him close to the Inquisition), and Imre Lakatos (1976) presented his investigation of the logic of mathematical discovery in the form of a dialogue taking place in an imaginary classroom. Such uses of ‘dialogue’ refer first of all to analytical forms and presentations of inquiries and of ‘getting to know’. As soon as we enter the field of ‘getting to know’, dialogue becomes relevant to epistemology. However, although our concept of dialogue is also related to epistemology in this way, it will diverge from the traditional philosophical use of the term by being related to ‘real’ dialogues and not to in-principle dialogues. We use the word ‘dialogue’ for a conversation with certain qualities, and the specification of ‘dialogue’ is one of the tasks awaiting for us as part of this study. In talking of qualities related to conversation, we recognise that the notion of quality may have a double meaning. On the one hand, quality may refer to properties of a certain entity. Thus, we can talk (almost in Aristotelian terms) about the quality of a cup as being different from the quality of a glass. In this sense quality refers to descriptive aspects of an entity. However, quality may also contain a normative element. Thus, we can talk about one glass being of a better quality that another glass. Maintaining the distinction between descriptive and normative references to quality is not simple. For instance, we may prefer the quality of a glass to the quality of a cup when drinking wine. In a similar way, we may prefer a dialogue when we think of certain forms of learning, bearing in mind that dialogue refers to certain properties of an interaction.
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