Table Of Content

My Kids Can Making Math Accessible to All Learners, K–5 Edited by Judy Storeygard HEINEMANN Portsmouth, NH Heinemann 361 Hanover Street Portsmouth, NH 03801–3912 www.heinemann.com Offices and agents throughout the world © 2009 by Technical Education Research Centers, Inc. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer, who may quote brief passages in a review. “Dedicated to Teachers” is a trademark of Greenwood Publishing Group, Inc. The authors and publisher wish to thank those who have generously given permission to reprint borrowed material: Figures 18–1, 18–2, and excerpts from Investigations in Number, Data, and Space by S. J. Russell, K. Economopoulos, and L. Wittenberg. Copyright © 2008 by Pearson Education, Inc. or its affiliate(s). Used by permission. All rights reserved. This material is based upon a work supported by the National Science Foundation under Grant No. HRD-0435017. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or other funders. Library of Congress Cataloging-in-Publication Data My kids can : making math accessible to all learners, K–5 / edited by Judy Storeygard. p. cm. Includes bibliographical references. ISBN-13: 978-0-325-01724-2 ISBN-10: 0-325-01724-7 1. Mathematics—Study and teaching (Elementary). 2. Effective teaching. 3. Mathematics teachers—Anecdotes. I. Storeygard, Judy. QA135.6.M95 2009 372.7—dc22 2009000232 Editor: Victoria Merecki Production: Sonja S. Chapman Cover design: Susan Paradise Cover photograph: Christina Myren DVD production: Sherry Day Typesetter: Aptara Inc. Manufacturing: Steve Bernier Printed in the United States of America on acid-free paper 13 12 11 10 09 VP 1 2 3 4 5 To Jacob Matthew, with love. May you have teachers as dedicated and knowledgeable as those who contributed to these resources. Accompanying Video Video footage of teachers in the classroom, demonstrating key teaching moves and modeling effective classroom language, can be accessed at http://www.heinemann.com/ebooks/my_kids_can.aspx. iv Contents Acknowledgments vii Foreword by Deborah Schifter viii Introduction x Making Mathematics Explicit 1 1 Are We Multiplying or Dividing? Being Explicit in Teaching Mathematics Ana Vaisenstein 5 2 What Comes Next? Being Explicit About Patterns Laura Marlowe 18 3 You Can’t Build a Sand Castle on a Classmate’s Head: Being Explicit in Kindergarten Math Lisa Seyferth 26 4 Double or Nothing: Guided Math Instruction Michelle Perch 32 5 Focused Instruction on Quick Images: A Guided Math Group Video featuring Michael Flynn Arusha Hollister 38 6 Solving Multiplication Problems: Purposeful Sharing of Strategies Video featuring Heather Straughter Arusha Hollister 43 Linking Assessment and Teaching 47 7 Assessing and Supporting Students to Make Connections: Developing Flexibility with Counting Ana Vaisenstein 51 8 The Pieces Get Skinnier and Skinnier: Assessing Students’ Ideas About Fractions Marta Garcia Johnson 60 9 After One Number Is the Next! Assessing a Student’s Knowledge of Counting Maureen McCarty 69 10 Assessing and Developing Early Number Concepts: Working with Kristen Anne Marie O’Reilly 77 11 How Many Children Got off the Bus? Assessing Students’ Knowledge of Subtraction Video featuring Ana Vaisenstein Arusha Hollister 88 v CONTENTS 12 Get to 100: Assessing Students’ Number Sense Video featuring Michael Flynn Arusha Hollister 93 Building Understanding Through Talk 99 13 What’s Another Way to Make 9? Building Understanding Through Math Talk Christina Myren 103 14 Lightbulbs Happen: Making Connections Through Math Talk Nikki Faria-Mitchell 113 15 Talking About Square Numbers: Small-Group Discussion of Multiples and Factors Dee Watson 126 16 Kindergartners Talk About Counting: The Counting Jar Video featuring Lillian Pinet Arusha Hollister 136 17 What Do We Do with the Remainder? Fourth Graders Discuss Division Video featuring Dee Watson Arusha Hollister 140 Taking Responsibility for Learning 145 18 Becoming a Self-Reliant Learner: The Story of Eliza Kristi Dickey 149 19 Getting “Un-Stuck”: Becoming an Independent Learner Mary Kay Archer 158 20 Tasha Becomes a Learner: Helping Students Develop Confidence and Independence Candace Chick 169 Working Collaboratively 181 21 Collaborative Planning: It’s More Than One-on-One Michael Flynn 185 22 A Double Dose of Math: Collaborating to Support Student Learning Marta Garcia Johnson 194 23 Planning Guided Math Groups: A Collaboration Between Classroom Teachers and Title 1 Staff John MacDougall with Marta Garcia Johnson and Karen Joslin 202 References 217 Contributors 221 vi Acknowledgments This collection is the culmination of a long strand of work. I have been very for- tunate to work with a group of dedicated teachers and colleagues. I could never have edited these resources without their intellect and commitment to the work. My colleague, Cornelia Tierney, and I began thinking about students who struggle with mathematics ten years ago. Her vision and passion about equity shaped the projects that we initiated. Our project officer at the National Science Foundation (NSF), Dr. Larry Scadden, understood and championed our efforts. The teachers from our Accessible Mathematics project began this work with us. I am grateful to all of them: Candace Chick, Heather Straughter, Eileen Backus, Lisa Nierenberg, Karin Olson-Shannon, MaryKay Resnick, Lisa Davis, Andrea Cerda, Susan Fitzgerald, Leslie Kramer, Karen Ravin, Somchay Edwards, Lauretta Medley, Michelle Anderson, and Lauren Grace. More recently, the Educational Research Collaborative at the Technical Education Research Center (TERC) provided me with funding to pursue publication. My colleagues, Andee Rubin and Myriam Steinback, have provided unlimited support and wisdom. I am grateful for and humbled by their generosity. Arusha Hollister made major contributions to these resources, in writing the workshops and video con- text pieces and by imparting her knowledge about primary-grade mathematics. Keith Cochran, Beth Perry-Brown, and Karen Mutch-Jones have given me thoughtful ideas and comments about several of the essays. Amy Brodesky, Nancy Horowitz, and Heather Straughter were very astute reviewers. The insights of Nicole Feret were incredibly valuable in shaping the writing. David Smith lent his creativity and patience in producing the DVD. I was also privileged to have a wise group of colleagues from the Professional Development Study Group. Deborah Schifter generously offered to write the Foreword, and members of the group read several entries and provided excellent feedback and encouragement. Victoria Merecki at Heinemann has provided me with gentle, intelligent guidance and encouragement throughout. vii Foreword All teachers of mathematics face the challenge of reaching the range of students in their classroom. Many teachers feel especially daunted by the task of helping students who are struggling as learners. Faced with a student who lacks confidence, doesn’t know how to interpret a task, and loses focus, what is a teacher to do?My Kids Canis an important resource for teachers who are ready to tackle this challenge. The most important message in this book is that allchildren, given appropri- ate supports, can learn mathematics for understanding. This may come as a surprise to those who were taught that the best way to work with struggling students is to break mathematical tasks into small chunks to be memorized. Instead, the key is to find where a student is on solid ground and provide tasks that will help him or her move forward. The examples presented here demonstrate the progress students can make. My Kids Can devotes each chapter to a broad principle: assessing students’ understanding as a way of making decisions about how to proceed, making the mathematics explicit, helping students become independent learners. Within each chapter, through written narratives and video cases, individual teachers con- vey their own stories, illustrating how they worked with their students. That is, readers can see the broad principles enacted within the constraints of day-to-day classroom life: how teachers found time to conduct one-on-one interviews and to keep records of students’ progress, which assessments were particularly useful, and what kinds of accommodations were made to make the tasks accessible to students while keeping the important mathematics intact. Readers learn about the kinds of supports that made students who struggle feel confident enough to work with classmates and to speak up in whole-group discussions. We are shown the variety of representations students used to make sense of the mathematics and the connections that students were able to make. And we witness the collaborations forged between classroom teachers and special educators. Although not explicitly stated, the knowledge these teachers bring to their practice shines through each narrative and video case. They understand deeply viii FOREWORD the mathematics content they are responsible for teaching and how students learn it. They identify the central mathematical concepts, recognize how concepts are related to each other, and understand how these concepts build from content covered in earlier grades. They situate these concepts in a variety of representations and contexts familiar to students from their daily lives. They assess the cor- rectness of students’ reasoning as students explain their solutions to a problem; furthermore, if a solution is incorrect, the teachers analyze that reasoning to determine what is correct about the students’ process and where the thinking has gone awry. All of this knowledge—together with a deep sense of care and respect for their students—is applied to the goal of having each student make sense of the mathematics. A reader might ask, “How can I put so much effort into one or two or a hand- ful of students when I am responsible for so many?” The answer: The knowledge exhibited by the teachers in this volume and many of the techniques they speci- fy and elucidate enhance the learning of all of their students. When a teacher is explicit about the mathematics of a lesson, provides additional representations, and helps students figure out how to participate in whole-group discussion, it serves every student in the classroom. Indeed, the challenge of reaching the range of learners in a classroom isgreat. There is much to learn and much to do. But, as illustrated by My Kids Can, the rewards are even greater. Deborah Schifter Education Development Center, Inc. ix