Good Questions: Great Ways to Differentiate Mathematics Instruction in the Standards-Based Classroom

305 Pages·2017·2.043 MB·English

by  Marian Small

Good Questions
Great Ways to Diff erentiate
Mathematics Instruction in the
Standards-Based Classroom
Th ird Edition
Also by Marian Small
Teaching Mathematical Th inking:
Tasks and Questions to Strengthen
Practices and Processes
More Good Questions:
Great Ways to Diff erentiate
Secondary Mathematics Instruction
(with Amy Lin)
Building Proportional Reasoning
Across Grades and
Math Strands, K–8
Uncomplicating Algebra
to Meet Common Core
Standards in Math, K–8
Uncomplicating Fractions
to Meet Common Core
Standards in Math, K–7
Eyes on Math:
A Visual Approach to
Teaching Math Concepts
(Illustrations by Amy Lin)
Good Questions
Great Ways to Diff erentiate
Mathematics Instruction in the
Standards-Based Classroom
Th ird Edition
MARIAN SMALL
Foreword by Carol Ann Tomlinson
1906 Association Drive, Reston, VA 20191 nelson.com
www.nctm.org
Published simultaneously by Teachers College Press, 1234 Amsterdam Avenue, New York,
NY 10027, and National Council of Teachers of Mathematics, 1906 Association Drive,
Reston, VA 20191; distributed in Canada by Nelson Education, 1120 Birchmount Road,
Toronto, ON, Canada M1K 5G4.
Copyright © 2017 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.
For reprint permission and other subsidiary rights requests, please contact
Teachers College Press, Rights Dept.: [email protected]
Credits: Teddy bear, boat, and frame (page 148) from Nelson Math Focus 2 by Marian Small,
page 74. Copyright © 2008. Reprinted with permission of Nelson Education Limited.
Text Design: Lynne Frost
Library of Congress Cataloging-in-Publication Data
Names: Small, Marian.
Title: Good questions : great ways to diff erentiate mathematics instruction in the
standards-based classroom / Marian Small ; foreword by Carol Ann Tomlinson.
Description: Th ird edition. | New York, NY : Teachers College Press, [2017] | Subtitle
varies slightly from previous edition. | Includes bibliographical references and index.
Identifi ers: LCCN 2017007400 (print) | LCCN 2017008595 (ebook) |
ISBN 9780807758540 (pbk. : alk. paper) | ISBN 9780807775851 (ebook)
Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Individualized
instruction. | Eff ective teaching.
Classifi cation: LCC QA20.I53 S63 2017 (print) | LCC QA20.I53 (ebook) |
DDC 372.7—dc23
LC record available at https://lccn.loc.gov/2017007400
ISBN 978-0-8077-5854-0 (paper)
ISBN 978-0-8077-7585-1 (ebook)
NCTM Stock Number 15474
Printed on acid-free paper
Manufactured in the United States of America
25 24 23 22 21 19 18 17 8 7 6 5 4 3 2 1
Contents
Foreword, by Carol Ann Tomlinson ix
Preface xi
Organization of the Book xi
Changes in the Th ird Edition xii
Acknowledgments xv
Introduction: Why and How to Diff erentiate Math Instruction 1
Th e Challenge in Math Classrooms 1
What It Means to Meet Student Needs 3
Assessing Students’ Needs 4
Principles and Approaches to Diff erentiating Instruction 4
Two Core Strategies for Diff erentiating Mathematics Instruction:
Open Questions and Parallel Tasks 6
Creating a Math Talk Community 13
1 Counting & Cardinality and Number & Operations in Base Ten 17
Topics 17
Th e Big Ideas for Counting & Cardinality and for
Number & Operations in Base Ten 18
Open Questions for Prekindergarten–Grade 2 19
Open Questions for Grades 3–5 31
Parallel Tasks for Prekindergarten–Grade 2 38
Parallel Tasks for Grades 3–5 45
Summing Up 51
2 Number & Operations—Fractions 53
Topics 53
Th e Big Ideas for Number & Operations—Fractions 54
Open Questions for Grades 3–5 55
Parallel Tasks for Grades 3–5 62
Summing Up 68
v
vi Contents
3 Ratios & Proportional Relationships 69
Topics 69
Th e Big Ideas for Ratios & Proportional Relationships 70
Open Questions for Grades 6–8 70
Parallel Tasks for Grades 6–8 77
Summing Up 82
4 Th e Number System 83
Topics 83
Th e Big Ideas for Th e Number System 83
Open Questions for Grades 6–8 84
Parallel Tasks for Grades 6–8 89
Summing Up 94
5 Operations & Algebraic Th inking 95
Topics 95
Th e Big Ideas for Operations & Algebraic Th inking 96
Open Questions for Prekindergarten–Grade 2 97
Open Questions for Grades 3–5 102
Parallel Tasks for Prekindergarten–Grade 2 112
Parallel Tasks for Grades 3–5 116
Summing Up 120
6 Expressions & Equations and Functions 121
Topics 121
Th e Big Ideas for Expressions & Equations and for Functions 122
Open Questions for Grades 6–8 122
Parallel Tasks for Grades 6–8 133
Summing Up 140
7 Measurement & Data 141
Topics 141
Th e Big Ideas for Measurement & Data 142
Open Questions for Prekindergarten–Grade 2 143
Open Questions for Grades 3–5 155
Parallel Tasks for Prekindergarten–Grade 2 167
Parallel Tasks for Grades 3–5 174
Summing Up 183
8 Geometry 185
Topics 185
Th e Big Ideas for Geometry 186
Open Questions for Prekindergarten–Grade 2 187
Open Questions for Grades 3–5 194
Open Questions for Grades 6–8 202
Contents vii
Parallel Tasks for Prekindergarten–Grade 2 215
Parallel Tasks for Grades 3–5 219
Parallel Tasks for Grades 6–8 224
Summing Up 233
9 Statistics & Probability 235
Topics 235
Th e Big Ideas for Statistics & Probability 236
Open Questions for Grades 6–8 237
Parallel Tasks for Grades 6–8 246
Summing Up 257
Conclusions 259
Th e Need for Manageable Strategies 259
Developing Open Questions and Parallel Tasks 260
Th e Benefi ts of Th ese Strategies 262
Appendix A: Mathematical Content Domains and
Mathematical Practices of the Common Core State Standards 263
Appendix B: Worksheet for Open Questions and Parallel Tasks 265
Glossary 267
Bibliography 277
Index 281
Index of Subjects and Cited Authors 281
Index of Big Ideas 283
About the Author 287

Foreword
THAT I WOULD BE writing a Foreword for a book on teaching math seems at
once both ironic and absolutely appropriate.
Th e irony stems from my long and generally unhappy life as a math student.
In elementary school, math was neither easy nor hard for me. It was simply some-
thing I did. It evoked neither pleasure nor pain. I suppose it was a bit like household
chores—something necessary to get through the day, but certainly nothing to be
coveted. Th rough those years, math was as rote as household chores. I memorized
the required number sets, counting, and multiplication, absorbed algorithms I
watched my teacher scribe on the blackboard in the front of the room, and repeated
what I saw. Th ere was no joy in it, but it was doable.
Once I entered the world of algebra, however, math became a sinister thing.
Letters invaded the numbers. Equal signs took on a super power. While I could
occasionally duplicate what my teacher chalked in front of us, I could no longer
commit the strings to memory. Th ere was no reason to them, and worse, I didn’t
even really grasp the notion that they were all about reasoning. I simply knew that
I could no longer succeed with math. Th e conclusion I drew, of course, was that I
was no good at math.
I received a one-year reprieve from my self-imposed verdict when geometry
intervened between Algebra I and Algebra II—and before trigonometry had its way
with me. In that one year, math was glory. In that one year, math was about reason-
ing. Th ere was order to it, like the order in the universe. Th ere were words that
marched along with the fi gures and problems. I loved it in the way that I loved good
writing. It revealed beauty and it dignifi ed my possibilities as a learner rather than
eroding them.
But the interlude didn’t last long enough to erase the damage done earlier or
to reconfi gure my sense of myself as outpriced by math. It took many years for me
to realize that the dismay I felt in the presence of math was neither a fault in me
nor an indictment of math as a content area. Rather, I came to see myself as one
among a legion of students who had abandoned a content area—or even school—
because the way a subject was taught drained all the life from it, even as it drained
life from the learners.
Th e appropriateness of my writing this Foreword is, of course, not completely
separate from the irony. I have spent my career as a teacher (an English teacher, to
be clear) and then as a professor with an evolving belief, then a conviction, and
ix