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Ways of Knowing in Science and Mathematics Series RICHARD DUSCHL, SERIES EDITOR ADVISORY BOARD: Charles W. Anderson, Raffaella Borasi, Nancy Brickhouse, Marvin Druger, Eleanor Duckworth, Peter Fensham, William Kyle, Roy Pea, Edward Silver, Russell Yeany Improving Instruction in Rational Numbers and How Students (Mis-)Understand Science and Proportionality: Using Cases to Transform Mathematics: Intuitive Rules Mathematics Teaching and Learning, Volume 1 RUTH STAVY AND DINA TIROSH MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND Reading Counts: Expanding the Role of Reading in MARY KAY STEIN, WITH MELISSA BOSTON, MARJORIE A. HENNINGSEN AND AMY F. HILLEN Mathematics Classrooms RAFFAELLA BORASI AND MARJORIE SIEGEL Improving Instruction in Algebra: Using Cases to Transform Mathematics Teaching and Learning, Transforming Middle School Science Education Volume 2 PAUL DEHART HURD MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND Designing Project-Based Science: Connecting MARY KAY STEIN, WITH MARJORIE A. HENNINGSEN, Learners Through Guided Inquiry MELISSA BOSTON, AND ELIZABETH K. HUGHES JOSEPH L. POLMAN Improving Instruction in Geometry and Measurement: Using Cases to Transform Mathematics Teaching and Implementing Standards-Based Mathematics Learning, Volume 3 Instruction: A Casebook for Professional Development MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND MARY KAY STEIN, MARGARET SCHWAN SMITH, MARY KAY STEIN, WITH MELISSA BOSTON AND MARJORIE A. HENNINGSEN, AND EDWARD A. SILVER MARJORIE A. HENNINGSEN The New Science Teacher: Cultivating Good Practice Standards-Based Mathematics Assessment in Middle DEBORAH TRUMBULL School: Rethinking Classroom Practice Problems of Meaning in Science Curriculum THOMAS A. ROMBERG, EDITOR DOUGLAS A. ROBERTS AND Mathematical Development in Young Children: LEIF ÖSTMAN, EDITORS Exploring Notations Inventing Science Education for the New Millennium BÁRBARA M. BRIZUELA PAUL DEHART HURD Inside Science Education Reform: A History of Improving Teaching and Learning in Science and Curricular and Policy Change Mathematics J MYRON ATKIN AND PAUL BLACK DAVID F. TREAGUST, REINDERS DUIT, AND BARRY J. FRASER, EDITORS Connecting Girls and Science: Constructivism, Feminism, and Science Education Reform Reforming Mathematics Education in America’s Cities: ELAINE V. HOWES The Urban Mathematics Collaborative Project NORMAN L. WEBB AND THOMAS A. ROMBERG, EDITORS Investigating Real Data in the Classroom: Expanding Children’s Understanding of Math and Science What Children Bring to Light: A Constructivist RICHARD LEHRER AND LEONA SCHABLE, EDITORS Perspective on Children’s Learning in Science BONNIE L. SHAPIRO Free-Choice Science Education: How We Learn Science Outside of School STS Education: International Perspectives on Reform JOHN H. FALK, EDITOR JOAN SOLOMON AND GLEN AIKENHEAD, EDITORS Science Teaching/Science Learning: Constructivist Reforming Science Education: Social Perspectives Learning in Urban Classrooms and Personal Reflections ELNORA S. HARCOMBE RODGER W. BYBEE

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FROM TEACHERS COLLEGE PRESS A comprehensive program to improve mathematics instruction and student achievement Own the Complete Set! Improving Instruction in Rational Numbers and Proportionality Using Cases to Transform Mathematics Teaching and Learning (Volume 1) Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with Melissa Boston, Marjorie A. Henningsen, and Amy F. Hillen Improving Instruction in Algebra Using Cases to Transform Mathematics Teaching and Learning (Volume 2) Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with Marjorie A. Henningsen, Melissa Boston, and Elizabeth K. Hughes Improving Instruction in Geometry and Measurement Using Cases to Transform Mathematics Teaching and Learning (Volume 3) Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with Melissa Boston and Marjorie A. Henningsen ALSO OF INTEREST FROM TEACHERS COLLEGE PRESS Standards-Based Mathematics Implementing Standards-Based Assessment in Middle School Mathematics Instruction Rethinking Classroom Practice A Casebook for Professional Development Edited by Thomas A. Romberg Mary Kay Stein, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver www.tcpress.com

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Improving Using Cases to Instruction Transform in Rational Mathematics Numbers and Teaching and Proportionality Learning, Volume 1 Margaret Schwan Smith Edward A. Silver Mary Kay Stein with Melissa Boston, Marjorie A. Henningsen, and Amy F. Hillen Teachers College, Columbia University New York and London

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The material in this book is based on work supported by National Science Foundation grant number ESI-9731428 for the COMET (Cases of Mathematics Instruction to Enhance Teaching) Project. Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation. Published by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027 Copyright © 2005 by Teachers College, Columbia University All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, or any information storage and retrieval system, without permission from the publisher. Library of Congress Cataloging-in-Publication Data Smith, Margaret Schwan. Improving instruction in rational numbers and proportionality / Margaret Schwan Smith, Edward A. Silver, Mary Kay Stein, with Melissa Boston, Marjorie A. Henningsen, and Amy F. Hillen. p. cm. — (Using cases to transform mathematics teaching and learning ; v. 1) (Ways of knowing in science and mathematics series) Includes bibliographical references and index. ISBN 0-8077-4529-4 (pbk. : acid-free paper) 1. Mathematics—Study and teaching (Middle school)—United States—Case studies. I. Silver, Edward A., 1948– II. Stein, Mary Kay. III. Title IV. Series. QA13.S58 2005 513'.071'2—dc22 2004055390 ISBN 0-8077-4529-4 (paper) Printed on acid-free paper Manufactured in the United States of America 12 11 10 09 08 07 06 05 8 7 6 5 4 3 2 1

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To the teachers in the QUASAR Project—although nearly a decade has passed since our work together ended, we con- tinue to draw inspiration from your work. You were true pioneers in creating instructional environments that pro- moted mathematics learning for all students. Thank you for sharing your successes and struggles with us. We continue to learn so much from you.

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Contents Acknowledgments ix Reading the Case 27 THE CASE OF MARIE HANSON 28 Introduction xi Analyzing the Case 40 Great Expectations for Middle Grades Extending Your Analysis of the Case 40 Mathematics xi Connecting to Your Own Practice 41 The Materials in This Volume xii Exploring Curricular Materials 41 This Volume and Its Companions xiv Connecting to Other Ideas and Issues 42 PART I 4. Comparing Similar Figures— THE CASE OF MARCIA GREEN 44 USING CASES TO ENHANCE LEARNING Opening Activity 44 1. Using Cases to Learn 3 Reading the Case 45 The Cases 3 THE CASE OF MARCIA GREEN 46 The Cases as Learning Opportunities 4 Analyzing the Case 54 Using the Cases 5 Extending Your Analysis of the Case 54 Connecting to Your Own Practice 55 2. Connecting Fractions, Decimals, and Exploring Curricular Materials 56 Percents—THE CASE OF RANDY HARRIS 8 Connecting to Other Ideas and Issues 56 Opening Activity 8 Reading the Case 8 5. Exploring Problems Involving THE CASE OF RANDY HARRIS 10 Ratios and Percents— Analyzing the Case 23 THE CASE OF JANICE PATTERSON 58 Extending Your Analysis of the Case 23 Opening Activity 58 Connecting to Your Own Practice 24 Reading the Case 58 Exploring Curricular Materials 24 THE CASE OF JANICE PATTERSON 60 Connecting to Other Ideas and Issues 25 Analyzing the Case 69 Extending Your Analysis of the Case 69 3. Introducing Ratios and Proportions— Connecting to Your Own Practice 70 THE CASE OF MARIE HANSON 26 Exploring Curricular Materials 70 Opening Activity 26 Connecting to Other Ideas and Issues 71 vii

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viii Contents PART II Appendix A: Sample Responses to FACILITATING LEARNING FROM CASES THE CASE OF RANDY HARRIS 109 Teacher-Generated Solutions to the 6. Using Cases to Support Learning Opening Activity 109 About Teaching 75 Facilitator-Generated List of Supporting Why Cases? 75 and Inhibiting Factors 110 Learning from Cases 76 Teacher-Generated List of Supporting What Can Be Learned from Our Cases 77 and Inhibiting Factors 110 Preparing for and Facilitating Case Discussions 78 Appendix B: Sample Responses to THE CASE OF MARIE HANSON 114 Putting the Pieces Together 80 Teacher-Generated Solutions to the 7. Facilitating Learning from Opening Activity 114 THE CASE OF RANDY HARRIS 81 Facilitator- and Teacher-Generated List of Case Analysis 81 Ways in Which Marie Hanson Supported Her Students’ Learning of Mathematics 116 Facilitating the Opening Activity 85 Facilitating the Case Discussion 86 Appendix C: Sample Responses to Extending the Case Experience 87 THE CASE OF MARCIA GREEN 119 Teacher-Generated Solutions to the 8. Facilitating Learning from Opening Activity 119 THE CASE OF MARIE HANSON 88 List of Marcia Green’s Questions 120 Case Analysis 88 Teacher-Generated Criteria for Facilitating the Opening Activity 93 Good Questions 120 Facilitating the Case Discussion 94 Extending the Case Experience 95 Appendix D: Sample Responses to THE CASE OF JANICE PATTERSON 123 9. Facilitating Learning from Teacher-Generated Solutions to the THE CASE OF MARCIA GREEN 96 Opening Activity 123 Case Analysis 96 Facilitator-Generated Responses to the Facilitating the Opening Activity 100 Question, “Did Janice Patterson Really Facilitating the Case Discussion 101 Teach?” 127 Extending the Case Experience 102 Teacher-Generated Responses to the Question, “Did Janice Patterson 10. Facilitating Learning from Really Teach?” 127 THE CASE OF JANICE PATTERSON 103 Case Analysis 103 References 131 Facilitating the Opening Activity 106 About the Authors 135 Facilitating the Case Discussion 107 Extending the Case Experience 108 Index 137

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Acknowledgments The ideas expressed in this book grew out of our work Gilberto Cuevas, Beatrice D’Ambrosio, Skip Fennell, on the QUASAR Project and have developed over the past Linda Foreman, Susan Friel, Judith Jacobs, Jeremy decade through our interactions and collaborations with Kahan, Rebecca McGraw, Jack Moyer, Kathy Pfaendler, many teachers, teacher educators, and mathematicians. Elizabeth Phillips, and Judith Zawojewski, who piloted We would like to thank mathematicians and mathemat- early versions of the cases, provided helpful feedback, ics teacher educators Hyman Bass, John Beem, Nadine and expanded our view regarding the possible uses of Bezuk, Kathleen Cramer, George Bright, Victoria Kouba, the cases and related materials. John Moyer, John P. Smith III, Judith Roitman, and Orit Finally, we would like to acknowledge the contribu- Zaslavsky, who provided feedback on early versions of the tions of Cristina Heffernan, who developed the COMET cases. Your thoughtful comments helped ensure that the website, provided feedback on early versions of the cases cases were both sound and compelling. and facilitation materials, and identified tasks in other We are indebted to Victoria Bill whose varied and curricula that corresponded to the cases; Michael Steele, frequent use of the cases over the past 5 years has helped who provided technological assistance in preparing the us recognize the flexibility and power of the cases to final manuscript; and Kathy Day, who provided valu- promote learning in a range of situations; and to our able assistance in preparing figures, locating data, and colleagues Fran Arbaugh, Cathy Brown, Marta Civil, copying materials. ix

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