Affect and Mathematical Problem Solving: A New Perspective PDF
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Mfect and Mathematical Problem Solving Douglas B. McLeod Verna M. Adams Editors Affect and Mathematical Problem Solving A New Perspective With 16 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Douglas B. McLeod Verna M. Adams Elementary and Secondary Education 105 Aderhold Hall Washington State University University of Georgia Pullman, Washington 99164-2122, USA Athens, Georgia 30602, USA Library of Congress Cataloging-in-Publication Data Affect and mathematical problem solving: a new perspective 1 Douglas B. McLeod, Verna M. Adams, editors. p. cm. Includes bibliographies and index. ISBN-13: 978-1-4612-8178-8 e-lSBN-13: 978-1-4612-3614-6 DOl: 10.1007/978-1-4612-3614-6 1. Problem solving-Study and teaching-Psychological aspects. 2. Mathematics-Study and teaching-Psychological aspects. 3. Mfect (Psychology) I. McLeod, Douglas B. II. Adams, Vema M. QA63.A34 1989 153.4'3-dc 19 88-39063 Printed on acid-free paper. © 1989 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1s t edition 1989 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, l75 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 dis similar methodology now known or hereafter 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. Typeset by Publishers Service, Bozeman, Montana. 9 8 7 6 5 4 3 2 1 Preface Research on mathematical problem solving has received considerable attention in recent years, not only from those who do research on mathematics learning and teaching, but also from psychologists and others who work in the cognitive sciences. Although most of this research has focused on cognitive factors, there has been a surge of interest in the role of affect in mathematical problem solving. This book explores these affective factors and their relationships to the cognitive processes involved in problem solving. The ideas are of particular interest to those who work in mathematics education, psychology, and other fields that are related to the cognitive sciences. The origins of the book can be found in George Mandler's (1984) theory of emotion and in Edward Silver's (1985) effort to integrate the multiple perspec tives that influence research on mathematical problem solving. Therefore, the interdisciplinary mix for this volume includes not only the usual perspectives of mathematics education, cognitive psychology, and artificial intelligence, but also cognitive approaches to clinical and counseling psychology. The book is divided into five parts. The first introduces the basic ideas of Man dler's theory and suggests how the theory can be applied in research on mathe matical problem solving. Part II presents five studies of learning that test and extend the theory, and the five chapters in Part III perform the same function in the context of research on teaching. Part IV includes two responses to the theory, one from the viewpoint of mathematics education, and the other from cognitive psychology. Finally, following George Polya's recommendations, Part V is spent "looking back" at the problem of how to build a theory of affective factors in mathematical problem solving. The development of an interdisciplinary volume like this one requires a sub stantial amount of time both for interaction and reflection by the participants. In this case we were fortunate to have the support of the National Science Founda tion for two meetings where the chapter authors and other researchers had a chance to debate the issues and report on their own research. The first meeting was held in May 1986, in San Diego, and focused on presenting and discuss ing Mandler's theory. Participants also outlined their plans for implementing Mandler's ideas in their own research during the next year. In June 1987, the vi Preface participants met again in San Diego and reported on the results of their efforts to incorporate affective factors in their research on mathematical problem solving. This volume grew out of the papers that were presented at the second meeting. In addition to these two meetings, subgroups of participants presented symposia at the 1987 Annual Meetings of both the American Educational Research Associ ation and the National Council of Teachers of Mathematics. Consequently, we have here not the typical report of the proceedings of a conference, but the results of two years of study, debate, and research by the participants. The preparation of this volume was supported in part by National Science Foundation Grant No. MDR-8696142, as well as by other grants from several agencies to some of the individual authors. Any opinions, conclusions, or recom mendations are those of the authors and do not necessarily reflect the views of the funding agencies. We also want to acknowledge the cooperation of both of the academic institutions that have served as the host for this project, San Diego State University and Washington State University. We want to thank all of the authors for their contributions and cooperation. We are especially grateful to George Mandler of the University of California at San Diego for his willingness to share his psychological theories and to see them tested in the domain of mathematics classrooms. Special thanks are also due to Elizabeth Fennema of the University of Wisconsin, who shared generously of her experience in research on affect and mathematics education. Fennema and Man dler both served on our interdisciplinary Project Advisory Board, along with David Carlson, a research mathematician at San Diego State University, and FJ. Crosswhite, Past-President of the National Council of Teachers of Mathematics. The Advisory Board, along with Ray Hannapel of the National Science Founda tion, played an important role in the development of the project, and we are grateful for their help. In addition, we want to thank the staff members in San Diego and Pullman who have assisted us with this project, especially Kathie Duncan. We also want to thank the people at Springer-Verlag for their help in producing this book. Finally, we thank our families for their support throughout this project. Pullman, Washington Douglas B. McLeod Athens, Georgia Verna M. Adams Contents Preface .......................... , . . . .. . .. . . . . . . .. . .. . . .. . . . v Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part I A Theory of Affect for Mathematical Problem Solving Chapter 1 Mfect and Learning: Causes and Consequences of Emotional Interactions GEORGE MANDLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2 The Role of Affect in Mathematical Problem Solving DOUGLAS B. McLEOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 3 Describing the Affective Domain: Saying What We Mean LAURIE E. HART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Part II Studies of Learning Chapter 4 Affect in Schema Knowledge: Source and Impact SANDRA P. MARSHALL .............................. 49 Chapter 5 Aesthetic Influences on Expert Mathematical Problem Solving EDWARD A. SILVER and WENDY METZGER ..... . . . . . . . . . 59 Chapter 6 Self-Confidence, Interest, Beliefs, and Metacognition: Key Influences on Problem-Solving Behavior FRANK K. LESTER, JOE GAROFALO, and DIANA LAMBDIN KROLL. . . . . . . . . . . . . . .. . . . . . . . . . 75 Chapter 7 Information Technologies and Affect in Mathematical Experiences JAMES J. KAPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 viii Contents Chapter 8 Searching for Affect in the Solution of Story Problems in Mathematics LARRY SOWDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104 Part III Studies of Teaching Chapter 9 Young Children's Emotional Acts While Engaged in Mathematical Problem Solving PAUL COBB, ERN A YACKEL, and TERRY WOOD. . . . . . . . . . .. 117 Chapter 10 Teaching Practices and Student Mfect in Problem-Solving Lessons of Select Junior-High Mathematics Teachers DOUGLAS A. GROUWS and KATHLEEN CRAMER. . . . . . . . . .. 149 Chapter 11 Mfect and Problem Solving in an Elementary School Mathematics Classroom ALBA G. THOMPSON and PATRICK W. THOMPSON ......... 162 Chapter 12 Mfective Factors and Computational Estimation Ability JUDITH THREADGILL SOWDER. . . . . . . . . . . . . . . . . . . . . . . .. 177 Chapter 13 Mfective Issues in Teaching Problem Solving: A Teacher's Perspective VERNA M. ADAMS .. '" .... " . . .. . . .. . . . . . . . . . . .. . .. 192 Part IV Responses to the Theory Chapter 14 The Study of Mfect and Mathematics: A Proposed Generic Model for Research ELIZABETH FENNEMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205 Chapter 15 Psychological Conceptions of Mathematics and Emotion BARBARA A. McDoNALD . . . . . . . . . . . . . . . . . . . . . . . . . . .. 220 Part V Looking Back Chapter 16 Mfect and Learning: Reflections and Prospects GEORGE MANDLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 237 Chapter 17 Beliefs, Attitudes, and Emotions: New Views of Mfect in Mathematics Education DOUGLAS B. McLEOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 245 Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 259 Subject Index ................................................ 265 Contributors VERNA M. ADAMS Department of Mathematics Education, University of Georgia, Athens, Georgia 30602, USA PAUL COBB Department of Education, Purdue University, West Lafayette, Indiana 47907, USA KATHLEEN CRAMER Department of Elementary Education, University of Wisconsin-River Falls, River Falls, Wisconsin 54022, USA ELIZABETH FENNEMA Wisconsin Center for Education Research, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA JOE GAROFAW Department of Curriculum Instruction and Special Education, University of Virginia, Charlottesville, Virginia 22903, USA DOUGLAS A. GROUWS Department of Curriculum and Instruction; Center for Research in Social Behavior, University of Missouri, Columbia, Missouri 65211, USA LAURIE E. HART Department of Elementary Education, University of Georgia, Athens, Georgia 30602, USA JAMES 1. KAPUT Department of Mathematics, Southeastern Massachusetts University, North Dartmouth, Massachusetts 02747; Educational Technology Center, Harvard University, Cambridge, Massachusetts 02138, USA DIANA LAMBDIN KROLL School of Education, Indiana University, Bloomington, Indiana 47405, USA x Contributors FRANK K. LESTER School of Education, Indiana University, Bloomington, Indiana 47405, USA GEORGE MANDLER Center for Human Information Processing, University of California-San Diego, La Jolla, California 92093, USA SANDRA P. MARSHALL Department of Psychology; Center for Research in Mathematics and Science Education, San Diego State University, San Diego, California 92182, USA BARBARA A. McDONALD Navy Personnel Research and Development Center, San Diego, California 92107, USA DOUGLAS B. McLEOD Elementary and Secondary Education, Washington State University, Pullman, Washington 99164-2122; Department of Mathematical Sciences, San Diego State University, San Diego, California 92182, USA WENDY METZGER Department of Mathematics, Palomar College, San Marcos, California 92069, USA EDWARD A. SILVER School of Education; Learning Research and Development Center, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA JUDITH THREADGILL SOWDER Department of Mathematical Sciences, San Diego State University, San Diego, California 92182, USA LARRY SOWDER Department of Mathematical Sciences, San Diego State University, San Diego, California 92182, USA ALBA G. THOMPSON Department of Mathematics, Illinois State University, Normal, Illinois 61761, USA PATRICK W. THOMPSON Department of Mathematics, Illinois State University, Normal, Illinois 61761, USA TERRY WOOD Department of Education, Purdue University, West Lafayette, Indiana 47907, USA ERNA YACKEL Department of Mathematical Sciences, Purdue University Calumet, Calumet, Indiana 46323, USA Part I A Theory of Affect for Mathematical Problem Solving
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