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PROBLEM POSING: REFLECTIONS AND APPLICATIONS This page intentionally left blank PROBLEM POSING: REFLECTIONS AND APPLICATIONS Edited by Stephen I. Brown University at Buffalo Marion I. Walter University of Oregon Copyright Ii:) 1993 by Lawrence Erlbaum Associates, Inc. All rights reserved. No part of this book may be reproduced in any form, by photostat, microform, retrieval system, or any other means, without the prior written permission of the publisher. First published by Lawrence Erlbaum Associates, Inc., Publishers 365 Broadway Hillsdale, New Jersey 07642 This edition published 2013 by Psychology Press Psychology Press Psychology Press Taylor & Francis Group Taylor & Francis Group 711 Third Avenue 27 Church Road New York, NY 10017 Hove, East Sussex BN3 2FA Library of Congress Cataloging-in-Publication Data Problem posing: reflections and applications / [edited by) Stephen I. Brown, Marion I. Walter. p. cm. Includes bibliographical references and index. ISBN 0-8058-1065-X (alk. paper) 1. Problem solving. 2. Mathematics-Study and teaching. I. Brown, Stephen I. II. Walter, Marion I., QA63.P75 1992 510-dc20 92-33746 CIP Contents PREFACE ix INTRODUCTION xili Reflections on the Power of Problem Posing xiv On the Audience for the Book xv A Note on Reading the Collection xvi A Caveat of Sorts xvii CHAPTER I. REFLECTIONS INTRODUCTORY COMMENTS 1 SECTION 1: PEDAGOGICAL FOCUS: THE DESIGN OF A COURSE EDITORS' COMMENTS 3 ESSAYS 1. In the Classroom: Student as Author and Critic Stephen I. Brown and Marion I. Walter 7 2. Problem Posing in Mathematics Education Stephen I. Brown and Marion I. Walter 16 SECTION 2: ELABORATIONS AND APPLICATIONS OF PROBLEM POSING SCHEMES EDITORS' COMMENTS 28 v Vi CONTENTS ESSAYS: 3. On Building Curriculum Materials That Foster Problem Posing, E. Paul Goldenberg 31 4. Removing the Shackles of Euclid: 8: "Strategies", David S. Fielker 39 5. "What if Not?" A Technique for Involving and Motivating Students in Psychology Courses, Werner Feibel 52 SECTION 3: RATIONALE: TOWARDS A MULTIPLISTIC VIEW OF THE WORLD EDITORS' COMMENTS 59 ESSAYS: 6. A Problem Posing Approach to Biology Education, John R. Jungck 64 7. An Experience with Some Able Women Who Avoid Mathematics, Dorothy Buerk 70 8. The Invisible Hand Operating in Mathematics Instruction: Student's Conceptions and Expectations, Raffaella Borasi 83 9. The Logic of Problem Generation: From Morality and Solving to De-Posing and Rebellion, Stephen I. Brown 92 10. Vice into Virtue, or Seven Deadly Sins of Education Redeemed, Israel Scheffler 104 CHAPTER II. ALGEBRA AND ARITHMETIC 117 INTRODUCTORY COMMENTS SECTION 1: ASKING WHY EDITORS' COMMENTS 118 ESSAYS: 11. Number Sense and the Importance of Asking "Why?" David J. Whitin 121 12. Creating Number Problems, Marian Small 130 13. Making Your Own Rules, Dan Brutlag 134 14. 1089: An Example of Generating Problems, Rick N. Blake 141 SECTION 2: MISTAKES EDITORS' COMMENTS 152 ESSAYS: 15. Mathematical Mistakes, Lawrence Nils Meyerson 153 16. Algebraic Explorations of the Error ^f = \ Raffaella Borasi 159 CONTENTS Vii SECTION 3: TINKERING WITH WHAT HAS BEEN TAKEN FOR GRANTED EDITORS' COMMENTS 164 ESSAYS: 17. Problem Stories: A New Twist on Problem Posing, William S. Bush and Ann Fiala 167 18. How to Create Problems, Stephen I. Brown 174 19. Beyond Problem Solving: Problem Posing, Barbara M. Moses, Elizabeth Bjork, and E. Paul Goldenberg 178 20. Mathematical Investigations: Description, Rationale, and Example, Barry V. Kissane 189 21. Curriculum Topics Through Problem Posing, Marion Walter 204 22. Is the Graph of y = /cx Straight? Alex Friedlander and Tommy Dreyfus 204 SECTION 4: YOUR TURN EDITORS' COMMENTS 220 ESSAY: 23. Because a Door Has To Be Open or Closed, An Intriguing Problem Solved by Some Inductive Exploration, Charles Cassidy and Bernard R. Hodgson 222 CHAPTER III. GEOMETRY: EDITORS' COMMENTS 229 INTRODUCTORY COMMENTS SECTION 1: LOOKING BACK EDITORS' COMMENTS 231 ESSAYS: 24. The Looking-Back Step in Problem Solving, Larry Sowder 235 25. Reopening the Equilateral Triangle Problem: What Happens If..., Douglas L Jones and Kenneth L Shaw 240 26. Mathematics and Humanistic Themes: Sum Considerations, Stephen I. Brown 249 SECTION 2: THE INTERTWINE OF PROBLEM POSING AND PROBLEM SOLVING EDITORS' COMMENTS 279 ESSAYS: 27. Problem Posing in Geometry, Larry Hoehn 281 28. Students Microcomputer-Aided Exploration in Geometry, Daniel Chazan 289 Viii CONTENTS SECTION 3: SOMETHING COMES FROM NOTHING EDITORS' COMMENTS 300 ESSAY: 29. Generating Problems From Almost Anything, 302 Marion Walter SECTION 4: YOUR TURN EDITORS' COMMENTS 317 ESSAY: 30. A Non-Simply Connected Geoboard-Based on the "What If Not" Idea, Philip A. Schmidt 319 AUTHOR INDEX 327 SUBJECT INDEX 330 NOTES ON CONTRIBUTORS 332 Preface Since the late 1960's, both of us have been heavily engaged in efforts to place problem posing as a central theme in mathematics education. We have published articles in professional journals, taught graduate and undergrad• uate courses on the topic at a number of institutions, and have lectured and given workshops at professional meetings in the United States and abroad. Much of our thinking on the topic culminated in our book, The Art of Problem Posing—published originally by the Franklin Institute Press in 1983, then by Lawrence Erlbaum Associates with a revised edition pub• lished in 1990 (Brown & Walter, 1990).1 It was our hope that the book as well as other articles and activities of ours would influence colleagues to (1) further explore the connections between problem posing and solving; (2) further examine the educational values of problem posing; (3) expand upon strategies for problem posing that derive from our schemes, and (4) employ these strategies in situations that we had not previously explored. Many of our colleagues did in fact take up the challenge. Articles have appeared in professional journals; talks have been given at professional meetings and bits and pieces of problem posing curriculum began to appear in several texts for students. As we promised in the 1990 edition of The Art of Problem Posing, we have created a collection of readings based upon the work of our colleagues in applying and extending the ideas that have occupied a large part of our professional lives for the past quarter of a century. We are grateful to them for having enriched the field so much more than we have dreamed when we first ventured into it. ix X PREFACE In addition, we owe much to Hollis Heimbouch, editor at Lawrence Erlbaum Associates, for offering the same confidence, good judgment, and encouragement in the creation and nourishing of this "offspring" that she provided in the "parent" publication. Shawn Vecellio, a doctoral student at The University of Buffalo, created the indexes for this collection. We are grateful to each of the publishers for having granted us permission to reproduce the articles in this collection. Below is a list of all the essays that appear in this collection together with their original citations. Two articles, the pieces by Feibel and Goldenberg, are published here for the first time. Several of the articles have been edited or excerpted from their original version, and we have occasionally made minor changes in vocabulary for reasons of clarity and in order to modify some sexist language. REFLECTIVE ESSAYS Borasi, R. (1990). The invisible hand operating in math instruction: Students conception and expectations. In T. J. Cooney (Ed.) Teaching and Learning Mathematics in the 1990's (pp. 174-182). Reston, VA: National Council of Teachers of Mathematics. Brown, S. I. (1984). The logic of problem generation: From morality and solving to de-posing and rebellion. For the Learning of Mathematics, 4, 1, 9-20. Also in L. Burton (Ed.), Girls into Maths Go (1988), London: Holt, Saunders Publishing Co., 196-222. } Brown, S. I. & Walter, M. I. (1988). Problem posing in mathematics education. Questioning Exchange, 2, 2, 123-131. Brown, S. L, & Walter, M. (1990). In the classroom: Student as author and critic. The Art of Problem Posing (pp. 119-127). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Buerk, D. (1982). An experience with some able women who avoid math. For the Learning of Mathematics, 3, 2, November, 19-24. Feibel, W. (1977). What if not: A technique for involving and motivating students in a psychology class. Unpublished talk to Western Psychological Association. Fielker, D. (1983). Removing the shackles of the euclid: 8: "strategies," Mathematics Teaching, 103, 48-53. Goldenberg, E. P. (1990). On the building curriculum materials that foster problem posing. Seeing Beauty Collections. Unpublished manuscript of Educational Development Corpo• ration for National Science Foundation. Jungck, J. R. (1985). A problem posing approach to biology education. The American Biology Teacher, 47, 264-266. Scheffler, I. (1989). Vice into virtue or seven deadly sins of education redeemed. Teachers College Record, 91, 2, 177-189. ALGEBRA AND ARITHMETIC Blake, R. N. (1984). 1089: An example of generating problems. Mathematics Teacher, 11, 14-19. xi xii CITATIONS Borasi, R. (1986). Algebraic explorations of the error ^ = \. Mathematics Teacher, 79, 246-248. Brown, S. I. (1989). How to create problems. In S. I. Brown (Ed.), Creative Problem Solving, (pp. 6-8). New York: University of State of New York; The State Education Department. Brutlag, D. (1990). Making your own rules. Mathematics Teacher, 83, 608-611. Bush, W. S., & Fiala, A. (1986) Problem stories: A new twist on problem posing. The Arithmetic Teacher, 34,4, 6-9. Cassidy, C, & Hodgson, B. R. (1982). Because a door has to be open or closed. Mathematics Teacher, 75, 155-158. Friedlander, A., & Dreyfus, T. D. (1991). Is the graph of y = kx straight: Mathematics Teacher, 84, 526-531. Kissane, B. V. (1988). Mathematical investigations: Description, rationale, example. Mathe• matics Teacher, 81, 520-528. Meyerson, L. N. (1976). Mathematical Mistakes. Mathematics Teaching, 76, 38-40. Moses, B. M., Bjork, E., & Goldenberg, E. P. (1990). Beyond problem solving: Problem posing.

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