Epistemological Foundations of Mathematical Experience PDF
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Recent Research in Psychology Leslie P. Steffe Editor Epistemological Foundations of Mathematical Experience With 50 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Leslie P. Steffe The University of Georgia College of Education Mathematics Education Department Athens, GA 30602 USA Library of Congress Cataloging-in-Publication Data Epistemological foundations of mathematical experience I edited by Leslie P. Steffe p. cm. Includes bibliographical references and index. ISBN -13:978-0-387-97600-6 I. Mathematics - Study and teaching - Psychological aspects Congresses. I. Steffe, Leslie P. QAIl.AIE64 1991 51O'.7'I-dc20 91-15438 Printed on acid-free paper. © 1991 Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Camera-ready copy provided by the editor using Microsoft Word. 987654321 ISBN-13:978-0-387-97600-6 e-ISBN-13:978-1-4612-3178-3 DOl: 10.1007/978-1-4612-3178-3 Dedicated to the Memory of Myron F. Rosskopf Preface On the 26th, 27th, and 28th of February of 1988, a conference was held on the epistemological foundations of mathematical experience as part of the activities of NSF Grant No. MDR-8550463, Child Generated Multiplying and Dividing Algorithms: A Teaching Experiment. I had just completed work on the book Construction ofA rithmetical Meanings and Strategies with Paul Cobb and Ernst von Glasersfeld and felt that substantial progress had been made in understanding the early numerical experiences of the six children who were the subjects of study in that book. While the book was in preparation, I was also engaged in the teaching experiment on mUltiplying and dividing algorithms. My focus in this teaching experiment was on investigating the mathematical experiences of the involved children and on developing a language through which those experiences might be expressed. However, prior to immersing myself in the conceptual analysis of the mathematical experiences of the children, I felt that it was crucial to critically evaluate the progress that we felt we had made in our earlier work. It was toward achieving this goal that I organized the conference. When trying to understand the mathematical experiences of a child, one can do no better than to interact with the child in a mathematical context guided by the intention to specify the child's current knowledge and the progress the child might make. Mathematical experiences, being relative to the particular interaction in which they occur, are always in flux and are as much a function of the adult's as they are of the child's intentions, language, and actions. Interactive mathematical activity is certainly not a neutral datum. The adult's understanding of children's interactive mathematical experiences form the foundation of any language through which children's experiences are expressed. Hence, if we, the observers, hope to communicate with each other a consensual but dynamically changing and expanding domain of interpretive constructs needs to be established. It is critical to know how the constructs that we already take as shared are being modified through their use as well as what other constructs are being developed that are yet to be shared. Piaget's reflective abstraction was one of the taken-as-shared interpretive constructs among the conferees. I was interested in how the other investigators at the conference thought about and used reflective abstraction because I have found it possible to observe children in the process of making a reflective abstraction in an experiential context. Did they view it in a functional way or did they restrict it to only the most general reorganizations that accompany discontinuities in mathematical learning? I invited Ernst von Glasersfeld to finish a paper on reflective abstraction that he had begun Preface earlier in which he interpreted reflective abstraction in the context of scheme theory. His theoretical analysis had opened up the possibility for me to view reflective abstraction as a relevant "everyday concept" and my hope was that von Glasersfeld's paper would encourage others to view it in that way. In order to explore how reflective abstraction is used by investigators in fields other than mathematics, I invited Philip Lewin to discuss his work in education in the humanities and social sciences. In his chapter, he stresses both reflective abstraction and the role of guiding images in the organization of knowing in domains that he believes are of necessity fuzzy, and where the prior experience and belief systems of the knower--even the expert- frequently have more salience than in the hard sciences and mathematics. During his presentation at the conference, he involved all of us in making reflective abstractions in the humanities and we came away with an experience that will not be soon forgotten. Ed Dubinsky shows us in his chapter just how important the prior experiences and belief systems of the knower are in advanced mathematical thinking as he discusses what he takes to be salient aspects of reflective abstraction. Along with Lewin, he provides contexts where terms like "encapsulation" and "interiorization" take on functional meaning. At the other end of the spectrum of mathematical thinking, I discuss these processes in terms of a feedback system, where the children's reflective abstractions use current sensory-motor and figurative experiences as material. The belief that mathematics is a product of reflective abstraction is developed in these four papers. No longer is reflective abstraction a theoretical construct that is used to explain only the most general and unobservable changes in the nature and organization of mathematical knowledge. Rather, it is an indispensable tool in explaining how children modify their mathematical experiences. Recasting reflective abstraction into functional forms is crucial if constructivism is to realize its potential in revitalizing the field of mathematics education. There must be a theory of construction that explains how mathematical operations are introduced by the actor as novelties through the actor's interactions with elements in its environment and how those novelties are lifted from their experiential forms. Mark Bickhard shows us how interactivism leads to a constructivism and how it accounts for reflective abstraction. This is another way of saying that reflective abstraction cannot be taken as a given in any organism nor can any of its products be taken as a given, including mathematics. Rather, reflective abstraction is itself introduced by the actor as a novelty and consists of combinations of mental operations. How these operations, once introduced, become available to the actor in any particular context is a critical research problem whose resolution would be of great benefit to education in mathematics. My current hypothesis is that reflective abstraction is not ubiquitous nor is it independent of current mathematical knowledge or of viii Preface particular environments. In view of Bickhard's work on rationality, we should expect our models of reflective abstraction to contain what he calls principles of selection ( critical principles), and because of the similarity of critical principles and negative feedback in scheme theory, the work of von Glasersfeld and that of Bickhard are reciprocally informing. Interactivism fmds an expression in the paper by Robert G. Cooper, where the idea of repeated experience is explored (repeated interaction of an organism with its environment). When I invited him to the conference I had no idea how solidly his thinking would connect with a functional view of reflective abstraction and with the work on recursion discussed by Thomas Kieren and Susan E.B. Pirie. As Kieren and Pirie explain it, recursion is based on an interactivism. So, along with feedback in the context of scheme theory, it can be used to broaden our understanding of repeated experience as advanced by Cooper. Repeated interaction of an organism with its environment, when applied to us as researchers, is a crucial part of constructing a conceptual map of children's mathematical experiences. I have found learning children's mathematics to be a very protracted process with the child as my only teacher. Although I am always interested in the modifications that children make in their mathematical knowledge as they work with me, without a model of that knowledge the modifications could be only described, not explained. Jere Confrey, in her paper, captures the essence of the models in a comment that I find particularly elegant in its simplicity--"Through the process of the interview, my own conception of exponential functions was transformed, elucidated, and enriched" (p. 129). This comment highlights the relativistic nature of our understanding of the mathematics of another person--we understand that mathematics using our own conceptual constructs. So if we take learning the mathematical knowledge of another person seriously, we should expect modifications in our knowledge to occur. Confrey uses the operation of unitizing--encapsulation in the language of Dubinsky and Lewin--throughout her paper to explain a student's concept of multiplication. We also fmd recursion as explained by Kieren and Pirie re emerging as a way to interpret the student's work with exponential growth or decay. These examples serve as confirmations of the functional nature of reflective abstraction and contribute to the emergence of a new learning theory in mathematics education. Larry Hatfield introduces the idea of a high quality mathematical experience. He identilles emotional states and enquiry states--states of feeling and thinking--as inter-related but distinguishable in his explorations of what it means to have high quality mathematical experiences. These two states of feeling and thinking are what Bickhard calls critical principles. Bickhard believes, as does Hatfield, that rationality emerges out of creative engagement with problems. They both develop the thesis that the standard IX Preface opposition between the emotions and the thought of an individual is a false opposition. Without motivation, interactivism ceases to be a viable theory of knowing and the individual necessarily is viewed as a passive organism waiting for the world to impress itself on his or her mind. Such a regressive view of mind is discussed in the introductory chapter written by Clifford Konold and David K. Johnson. The introductory and the fInal discussion chapter written by Patrick W. Thompson were not available at the time of the conference in any form. Konold, Thompson, and von Glasersfeld served as discussants of the other papers in this volume and it was only after the conference that they graciously accepted my invitation to write their respective chapters. As editor of the volume, I wish to express my appreciation to them and to David K. Johnson for coauthoring the introductory chapter. Several people served as discussion moderators and contributed immensely to the quality of the conference through their commentaries and syntheses of the discussions. As conference director, I express my gratitude to Dr.'s Ben Blount, Paul Cobb, Pat Kyllonen, John Olive, Neil Pateman, George Stanic, and Patricia Wilson. rmally, I would like to thank the staff of the Institute for Behavioral Research and the Department of Mathematics Education for their help in organizing and conducting the conference and in preparing the manuscript for this book. Special thanks are due to Ms. Gilda A. Ivory for her cheerful and tireless help in typing the many revisions of the papers and in preparing a camera-ready copy of the manuscript. Leslie P. Steffe Athens, Georgia x Acknowledgments The conference on which this book is based was supported by the National Science Foundation under Grant No. MDR-85504063, and the Department of Mathematics Education and the Institute for Behavioral Research both of the University of Georgia. All opinions and findings are those of the authors' and are not necessarily representative of the sponsoring agencies. Contents Preface ..... . VII Acknowledgments Xl Contributors ... xvii 1 Philosophical and Psychological Aspects of Constructivism. 1 Clifford Konold and David K. Johnson Foundationalism and Constructivism 2 Psychological Constructivism . 5 Foundations Revisited . 9 Overview of Chapters . . . . . 10 2 The Import of Fodor's Anti-Constructivist Argument 14 Mark H. Bickhard Fodor's Argument ....................... . 14 Reconstructing the Argument ................. . 16 What's Wrong with Contemporary Models of Representation? 17 Why Should We Care? ...... . 19 Interactivism: Outline of a Solution 20 Conclusions . . . . . . . . . . . . . 25 3 The Learning Paradox: A Plausible Counterexample 26 Leslie P. Steffe A Guiding Analogy . . . . . . . . . . . 26 An Example of the Learning Paradox .. 27 A Weak Form of the Innatist Hypothesis 28 Learning the Initial Number Sequence . 30 A More General Reformulation of Learning 37 Final Comments .............. . 42
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