Improving Instruction in Algebra

169 Pages·2004·1.58 MB·English

by  Margaret Schwan Smith, Edward A. Silver, Mary Kay Stein, Melissa Boston, Marjorie A. Henningsen

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
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
Using Cases to
Improving Transform
Instruction Mathematics
in Algebra Teaching and
Learning, Volume 2
Margaret Schwan Smith,
Edward A. Silver,
Mary Kay Stein
with Marjorie A. Henningsen,
Melissa Boston,
and Elizabeth K. Hughes
Teachers College, Columbia University
New York and London
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 algebra / Margaret Schwan Smith,
Edward A. Silver, Mary Kay Stein, with Marjorie A. Henningsen, Melissa Boston, and
Elizabeth Hughes
p. cm. — (Using cases to transform mathematics teaching and learning ; v. 2) (Ways of
knowing in science and mathematics series)
Includes bibliographical references and index.
ISBN 0-8077-4530-8 (pbk. : acid-free paper)
1. Algebra—Study and teaching (Middle school)—United States—Case studies. I.
Silver, Edward A., 1948– II. Stein, Mary Kay. III. Title IV. Series.
QA159.S65 2005 2004055361
ISBN 0-8077-4530-8 (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
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.
Contents
Acknowledgments ix 3. Examining Nonlinear Growth
Patterns—THE CASE OF ED TAYLOR 32
Introduction xi Opening Activity 32
Great Expectations for Middle-Grades Reading the Case 32
Mathematics xi THE CASE OF ED TAYLOR 34
The Materials in This Volume xii Analyzing the Case 47
Why Patterns and Functions in Algebra? xiv Extending Your Analysis of the Case 47
This Volume and Its Companions xv Connecting to Your Own Practice 48
Exploring Curricular Materials 48
Connecting to Other Ideas and Issues 48
PART I:
USING CASES TO ENHANCE LEARNING 4. Comparing Linear Graphs—
THE CASE OF EDITH HART 50
1. Using Cases to Learn 3
Opening Activity 50
The Cases 3
Reading the Case 50
The Cases as Learning Opportunities 5
THE CASE OF EDITH HART 52
Using the Case Materials 6
Analyzing the Case 60
Extending Your Analysis of the Case 60
2. Examining Linear Growth Patterns— Connecting to Your Own Practice 61
THE CASE OF CATHERINE EVANS AND
Exploring Curricular Materials 61
DAVID YOUNG 8
Connecting to Other Ideas and Issues 62
Opening Activity 8
Reading the Case 8 5. Interpreting Graphs of Time
THE CASE OF CATHERINE EVANS 10 versus Speed—THE CASE
THE CASE OF DAVID YOUNG 19 OF ROBERT CARTER 64
Analyzing the Case 28 Opening Activity 64
Extending Your Analysis of the Case 28 Reading the Case 64
Connecting to Your Own Practice 29 THE CASE OF ROBERT CARTER 66
Exploring Curricular Materials 29 Analyzing the Case 79
Connecting to Other Ideas and Issues 30 Extending Your Analysis of the Case 79
vii
viii Contents
Connecting to Your Own Practice 80 10. Facilitating Learning from
Exploring Curricular Materials 80 THE CASE OF ROBERT CARTER 115
Connecting to Other Ideas and Issues 80 Case Analysis 115
Facilitating the Opening Activity 119
Facilitating the Case Discussion 120
PART II: Extending the Case Experience 121
FACILITATING LEARNING FROM CASES
Appendix A. Sample Responses to
6. Using Cases to Support Learning
THE CASE OF CATHERINE EVANS
About Teaching 85
AND DAVID YOUNG 123
Why Cases? 85
Sample Solutions to the Opening Activity 123
Learning from Cases 86
Teacher-Generated Solutions to the
What Can Be Learned from Our Cases 87
“Case Analysis” Activity 125
Preparing for and Facilitating Case
Discussions 88
Appendix B. Sample Responses to
Putting the Pieces Together 90
THE CASE OF ED TAYLOR 126
Sample Solutions to the Opening Activity 126
7. Facilitating Learning from
Teacher-Generated Solutions to the
THE CASE OF CATHERINE EVANS
“Case Analysis” Activity 131
AND DAVID YOUNG 91
Case Analysis 91
Facilitating the Opening Activity 95 Appendix C. Sample Responses to
THE CASE OF EDITH HART 133
Facilitating the Case Discussion 98
Sample Solutions to the Opening Activity 133
Extending the Case Experience 99
Teacher-Generated Solutions to the
“Case Analysis” Activity 135
8. Facilitating Learning from
THE CASE OF ED TAYLOR 100
Case Analysis 100 Appendix D. Sample Responses to
Facilitating the Opening Activity 103 THE CASE OF ROBERT CARTER 137
Facilitating the Case Discussion 105 Sample Solutions to the Opening Activity 137
Extending the Case Experience 106 Teacher-Generated Solutions to the “Case
Analysis” Activity 139
9. Facilitating Learning from
THE CASE OF EDITH HART 107 References 141
Case Analysis 107
Facilitating the Opening Activity 110 About the Authors 145
Facilitating the Case Discussion 112
Extending the Case Experience 114 Index 147
Acknowledgments
The ideas expressed in this book grew out of our work Linda Foreman, Susan Friel, Judith Jacobs, Jeremy
on the QUASAR Project and have developed over the past Kahan, Rebecca McGraw, Jack Moyer, Kathy Pfaendler,
decade through our interactions and collaborations with Elizabeth Phillips, and Judith Zawojewski who piloted
many teachers, teacher educators, and mathematicians. early versions of the cases, provided helpful feedback,
We would like to thank mathematicians and mathemat- and expanded our view regarding the possible uses of
ics teacher educators Hyman Bass, John Beem, Nadine the cases and related materials.
Bezuk, Kathleen Cramer, George Bright, Victoria Kouba, Finally we would like to acknowledge the contribu-
John Moyer, John P. Smith III, Judith Roitman, and Orit tions of Cristina Heffernan who developed the COMET
Zaslavsky, who provided feedback on early versions of the website, provided feedback on early versions of the
cases. Your thoughtful comments helped ensure that the cases and facilitation materials and identified tasks in
cases were both sound and compelling. other curricula that corresponded to the cases; Michael
We are indebted to Victoria Bill whose varied and Steele who updated the COMET website and created
frequent use of the cases over the past 5 years has helped final versions of the figures; Amy Fleeger Hillen who
us recognize the flexibility and power of the cases to assisted in editing the final manuscript; and Kathy Day
promote learning in a range of situations; and to our who provided valuable assistance in preparing initial
colleagues Fran Arbaugh, Cathy Brown, Marta Civil, versions of the figures, locating data, and copying
Gilberto Cuevas, Beatrice D’Ambrosio, Skip Fennell, materials.
ix