Table Of ContentLearning and Teaching Early Math
In this important book for pre- and in- service teachers, early math experts Douglas Clements and
Julie Sarama show how “learning trajectories” help diagnose a child’s level of mathematical under-
standing and provide guidance for teaching. By focusing on the inherent delight and curiosity behind
young children’s mathematical reasoning, learning trajectories ultimately make teaching more
joyous. They help teachers understand the varying levels of knowledge exhibited by individual
students, which, in turn, allows them to better meet the learning needs of all children. Using straight-
forward, no-n onsense language, this book summarizes the current research about how children
learn mathematics, and how teachers can build on what children already know to realize more effec-
tive teaching.
T his second edition of Learning and Teaching Early Math remains the defi nitive, research-b ased
resource to help teachers understand the learning trajectories of early mathematics and thus to
become quintessential professionals. Updates to the new edition include:
• Explicit connections between “learning trajectories” and the Common Core State Standards;
• New coverage of patterns and patterning;
• The incorporation of hundreds of recent research studies.
Douglas H. Clements is the Kennedy Institute’s Endowed Chair in Early Childhood Learning,
Executive Director of the Marsico Institute of Early Learning and Literacy, part of the Morgridge
College of Education, and a Professor in the Department of Educational Research, Policy, and
Practice at the University of Denver, USA.
Julie Sarama is the Kennedy Institute’s Endowed Chair in Innovative Learning Technologies and a
Professor in the Department of Educational Research, Policy, and Practice at the University of
Denver, USA.
Studies in Mathematical Thinking and Learning
Alan H. Schoenfeld, Series Editor
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A Guide for Observations and Self-Assessment ( 2nd ed.).
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High School Mathematics .
Clements , D. H. & Sarama , J. ( 2014 ). L earning and Teaching Early Math: The Learning Trajectories
Approach ( 2nd ed.).
Clements , D. H. , Sarama , J. , & DiBiase , A.-M. ( 2003 ). Engaging Young Children in Mathematics:
Standards for Early Childhood Mathematics Education .
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Classroom Cultures .
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Bringing Understandings to Light .
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English , L. D. ( 2004 ). Mathematical and Analogical Reasoning of Young Learners .
Fennema , E. & Romberg , T. A. ( 1999 ). Mathematics Classrooms that Promote Understanding .
Fennema , E. & Nelson , B. S. ( 1997 ). Mathematics Teachers in Transition .
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Teaching and Learning .
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Mathematics Education .
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Geometry and Space .
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Sociohistorical Context, Community Forces, School Infl uence, and Individual Agency ( 2nd ed.).
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Children .
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Classroom .
Reed , S. K. ( 1999 ). Word Problems: Research and Curriculum Reform .
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Work: Connecting Curriculum Materials and Classroom Instruction .
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Achievement: An Example of a Summative Evaluation of a Standards-Based Curriculum .
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Science Matters .
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Representation of Functions .
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Trajectories for Young Children .
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Learning and Teaching Early Math
The Learning Trajectories Approach
Second edition
Douglas H. Clements and Julie Sarama
Second edition published 2014
by Routledge
8th Floor, 711 Third Avenue, New York, NY 10017, USA
and by Routledge
2 Park Square, Milton Park, Abingdon, Oxford, OX14 4RN, UK
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2014 Taylor & Francis
The right of Douglas H. Clements and Julie Sarama to be identifi ed as
authors of this work has been asserted by them in accordance with
sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilized in any form or by any electronic, mechanical, or other means,
now known or hereafter invented, including photocopying and recording,
or in any information storage or retrieval system, without permission in
writing from the publishers.
Trademark notice : Product or corporate names may be trademarks or
registered trademarks, and are used only for identifi cation and
explanation without intent to infringe.
First edition published in 2009 by Taylor & Francis
Library of Congress Cataloging-in-Publication Data
Clements, Douglas H .
Learning and teaching early math : the learning trajectories approach /
Douglas H. Clements, Julie A. Sarama. – Second edition.
pages cm. – (Studies in mathematical thinking and learning series)
Includes bibliographical references and index. 1. Mathematics–Study and
teaching (Early childhood) 2. Educational psychology. 3. Child
development. 4. Curriculum planning. I. Sarama, Julie. II. Title.
QA135.6.C55 2014
372.7’049–dc23
2013037230
ISBN: 978–0–415–82851–2 (hbk)
ISBN: 978–0–415–82850–5 (pbk)
ISBN: 978–0–203–52057–4 (ebk)
Typeset in Minion
by Refi neCatch Limited, Bungay, Suffolk, UK
Contents
Preface ix
Acknowledgments xii
1. Young Children and Mathematics Learning 1
2. Quantity, Number, and Subitizing 9
3. Verbal and Object Counting 22
4. Comparing, Ordering, and Estimating 50
5. Arithmetic: Early Addition and Subtraction and Counting Strategies 68
6. Arithmetic: Composition of Number, Place Value, and Multidigit Addition
and Subtraction 93
7. Spatial Thinking 124
8. Shape 141
9. Composition and Decomposition of Shapes 170
10. Geometric Measurement: Length 186
11. Geometric Measurement: Area, Volume, and Angle 198
12. Other Content Domains 214
13. Mathematical Processes and Practices 230
14. Cognition, Affect, and Equity 237
15. Early Childhood Mathematics Education: Contexts and Curricula 266
16. Instructional Practices and Pedagogical Issues 291
References 334
Index 363
vii
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Preface
Who dares to teach must never cease to learn.
(John Cotton Dana, 1856–1929)
T hink of the biggest number you can. Now add fi ve. Then, imagine if
you had that many Twinkies. Wow, that’s fi ve more than the biggest
number you could come up with!
(Child, age 6)
Everyone knows that effective teaching involves “meeting the students where they are” and helping
them build on what they know. But that’s easier said than done. Which aspects of mathematics
are important, which less so? How do we diagnose what a child knows? How do we build on that
knowledge—in which directions, and in what ways?
W e believe that “learning trajectories” answer these questions and help teachers become more
effective professionals. Just as importantly, they open up windows to seeing young children and math
in new ways, making teaching more joyous because the mathematical reasoning of children is
impressive and delightful.
L earning trajectories have three parts: (a) a specifi c mathematical goal, (b) a path along which
children develop to reach that goal, and (c) a set of instructional activities that help children move
along that path. So, teachers who understand learning trajectories understand the math, the way
children think and learn about math, and how to help children learn it better.
L earning trajectories connect research and practice. They connect children to math. They connect
teachers to children. They help teachers understand the level of knowledge and thinking of their
classes a nd the individuals in their classes as key in serving the needs of all children. (Equity issues
are important to us and to the nation. This entire book is designed to help you teach all children, but
equity concerns are discussed specifi cally and at length in C hapters 14, 1 5, and 1 6. ) Learning and
Teaching Early Math will help you understand the learning trajectories of early mathematics and so
to become a quintessential professional.
L earning and teaching, of course, take place in a context. For the last decade, we have had
the honor and advantage of working with several hundred early childhood teachers, who have
worked with us creating new ideas for teaching and invited us into their classrooms to test these
ideas with the children in their charge. We wish to share with you a bit about this collaborative
work.
ix
