MATHEMATICS LEARNING in Early Childhood Paths Toward Excellence and Equity Committee on Early Childhood Mathematics Christopher T Cross, Taniesha A Woods, and Heidi Schweingruber, Editors Center for Education Division of Behavioral and Social Sciences and Education THE NATIONAL ACADEMIES PRESS  500 Fifth Street, N.W.  Washington, DC 20001 NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance This study was supported by Contract No HHSP23320042509X1,TO#10 between the National Academy of Sciences and the U.S Department of Health and Human Services; by Contracts No HHSN267200700434P and HHSN267200800606P with the National Institute of Child Health and Human Development; by Contract No 20070221 with the Ewing Marion Kauffman Foundation; and by the President’s Fund of the National Research Council Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and not necessarily reflect the views of the organizations or agencies that provided support for the project Library of Congress Cataloging-in-Publication Data Mathematics learning in early childhood : paths toward excellence and equity / Christopher T Cross, Taniesha A Woods, and Heidi Schweingruber, editors ; Committee on Early Childhood Mathematics, Center for Education, Division of Behavioral and Social Sciences and Education    p cm   Includes bibliographical references and index   ISBN 978-0-309-12806-3 (hardback) — ISBN 978-0-309-12807-0 (pdf)   Mathematics—Study and teaching (Early childhood) Early childhood education I Cross, Christopher T II Woods, Taniesha A III Schweingruber, Heidi A IV National Research Council (U.S.) Committee on Early Childhood Mathematics   QA135.6.M384 2009   372.7—dc22 2009033436 Additional copies of this report are available from National Academies Press, 500 Fifth Street, N.W., Lockbox 285, Washington, DC 20055; (800) 624-6242 or (202) 334-3313 (in the Washington metropolitan area); Internet, http://www.nap.edu Copyright 2009 by the National Academy of Sciences All rights reserved Printed in the United States of America Suggested citation: National Research Council (2009) Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity Committee on Early Childhood Mathematics, Christopher T Cross, Taniesha A Woods, and Heidi ­Schweingruber, Editors Center for Education, Division of Behavioral and Social Sciences and Education Washington, DC: The National Academies Press The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters Dr Ralph J Cicerone is president of the National Academy of Sciences The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers Dr Charles M Vest is president of the National Academy of Engineering The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education Dr Harvey V Fineberg is president of the Institute of Medicine The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities The ­Council is administered jointly by both Academies and the Institute of Medicine Dr Ralph J Cicerone and Dr Charles M Vest are chair and vice chair, respectively, of the National Research Council www.national-academies.org COMMITTEE ON EARLY CHILDHOOD MATHEMATICS CHRISTOPHER T CROSS (Chair), Cross & Joftus, LLC, Danville, California OSCAR BARBARIN, School of Social Work, University of North Carolina, Chapel Hill SYBILLA BECKMANN, Department of Mathematics, University of Georgia SUE BREDEKAMP, Council for Early Childhood Professional Recognition, Washington, DC DOUGLAS H CLEMENTS, Department of Learning and Instruction, The State University of New York at Buffalo KAREN C FUSON, School of Education and Social Policy, Northwestern University YOLANDA GARCIA, E3 Institute for Advancing Excellence in Early Education, WestEd, San Jose, California HERBERT GINSBURG, Department of Human Development, Teachers College, Columbia University NANCY C JORDAN, School of Education, University of Delaware SHARON LYNN KAGAN, Teachers College, Columbia University SUSAN C LEVINE, Department of Psychology, Department of Comparative Human Development, and Committee on Education, University of Chicago KEVIN MILLER, Combined Program in Education and Psychology, University of Michigan, Ann Arbor ROBERT C PIANTA, Curry School of Education, University of Virginia TANIESHA A WOODS, Study Director PATRICIA MORISON, Interim Director, Center for Education HEIDI SCHWEINGRUBER, Deputy Director, Board on Science Education MARY ANN KASPER, Senior Program Assistant  Preface Early childhood education has risen to the top of the national policy agenda with recognition that ensuring educational success and attainment must begin in the earliest years of schooling There is now a substantial body of research to guide efforts to support young children’s learning Over the past 15 years, great strides have been made in supporting young children’s literacy This report summarizes the now substantial literature on learning and teaching mathematics for young children in hopes of catalyzing a similar effort in mathematics The need for this study was recognized and championed by the National Research Council’s (NRC’s) Mathematical Sciences Education Board following the publication in 2001 of Adding It Up: Helping Children Learn Mathematics The tireless efforts of board member Sharon Griffin and then board director David Mandel led the design of this project, which is a comprehensive examination of the evidence base that can guide mathematics education (teaching and learning) for children ages through It represents the further extension of a portfolio of NRC reports focused on mathematics learning and teaching that includes Adding It Up: Helping Children Learn Mathematics (2001); Eager to Learn: Educating Our Preschoolers (2001); How Students Learn: Mathematics in the Classroom (2005); and On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004) The majority of support for this study was provided by the U.S Department of Health and Human Services, Administration for Children and Families, Office of Head Start In particular, we thank Frank Fuentes, deputy director of the Office of Head Start, Administration for Children vii viii PREFACE and Families; Amanda Bryans, director of the Educational Development and Partnership Division, Office of Head Start, Administration for Children and Families; and Michele Plutro, education specialist, Office of Head Start, Administration for Children and Families Additional funding was provided by the Office of Planning, Research, and Evaluation in the Administration for Children and Families, under the leadership of Mary Bruce Webb Other sponsors who contributed to the project include the Ewing Marion Kauffman Foundation, under the guidance of Margo Quiriconi and Karen Norwood, and the National Institute of Child Health and Human Development, under the leadership of Daniel Berch and James Griffin In addition, the National Academies President’s Fund provided partial support for the study Our work was also advanced by the contributions of able consultants and staff and the input of outside experts Throughout the study process, the committee benefited from presentations or written input by individuals with a range of perspectives: W Steven Barnett, National Insti­ tute for Early Education Research, Rutgers, The State University of New Jersey; Linda ­ Bevilacqua, Core Knowledge Foundation; Toni Bickart, ­Creative ­ Curriculum, Teaching Strategies; Bruce D McCandliss, ­ Sackler Institute for Developmental Psychobiology, Weill Medical College of ­Cornell University; Holly Rhodes, consultant; Elisa Rosman, consultant for the ­ Georgetown ­ University Center for Child and Human Development; ­ Lawrence ­ Schweinhart, High/Scope Educational Research Foundation; ­Catherine Snow, Harvard Graduate School of Education; and Prentice Starkey, Graduate School of Education, University of California, Berkeley The committee also thanks those who wrote papers that were invaluable to our discussions: Sarah Archibald, Consortium for Policy Research in Education, University of Wisconsin-Madison; Kathryn Bouchard Chval, College of Education, University of Missouri; Jason Downer, Center for the Advanced Study of Teaching and Learning, University of Virginia; Shalom Fisch, MediaKidz Research and Consulting; Michael Goetz, University of Wisconsin, Madison; Bridget K Hamre, Curry School of Education, University of Virginia; Marilou Hyson, National Association for the Education of Young Children and George Mason University; Carolyn R Kilday, Graduate Student, Curry School of Education, University of Virginia; Pat McGuire, Graduate Student Curry Leadership Foundations and Policy, School of Education, University of Virginia; Barbara Reys, Department of Learning, Teaching, and Curriculum, University of Missouri; Catherine Scott-Little, Human Development and Family Studies Department, University of North Carolina, Greensboro; and John Switzer, Department of Learning, Teaching, and Curriculum, University of Missouri This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the Report Review Committee of the NRC The PREFACE ix purpose of this independent review is to provide candid and critical comments that will assist the institution in making its published report as sound as possible and to ensure that the report meets institutional standards for objectivity, evidence, and responsiveness to the study charge The review comments and draft manuscript remain confidential to protect the integrity of the deliberative process We thank the following individuals for their review of this report: ­Arthur Baroody, Curriculum and Instruction, University of Illinois, UrbanaChampaign; Elena Bodrova, Mid-continent Research for Education and Learning, Lakewood, CO; Karen S Cook, Department of ­Sociology, Insti­ tute for Research in the Social Sciences, Stanford University; ­ Sharon A Griffin, Department of Education, Clark University; Jacqueline A Jones, Division of Early Childhood Education, New Jersey Department of Edu­ ca­tion; Constance Kamii, Curriculum and Instruction, University of ­Alabama; Michèle M M Mazzocco, Psychiatry and Behavioral Sciences, Johns ­Hopkins School of Medicine and Math Skills Development Project, ­Kennedy Krieger West Campus, Baltimore, MD; Sally Moomaw, College of Education, Criminal Justice, and Human Services, University of Cincinnati; Donald G Saari, Institute for Mathematical Behavioral Sciences, University of California, Irvine; Maria Shea Terrell, Department of Mathematics, Cornell University; and Karen L Worth, Center for Science Education, Education Development Center, Inc., Newton, MA Although the reviewers listed above have provided many constructive comments and suggestions, they were not asked to endorse the conclusions or recommendations nor did they see the final draft of the report before its release The review of this report was overseen by Jeremy Kilpatrick, Department of Mathematics and Science Education, University of Georgia, Athens, and Charles (Randy) Gallistel, Rutgers University, Rutgers Center for Cognitive Science, The State University of New Jersey Appointed by the NRC, they were responsible for making certain that an independent examination of this report was carried out in accordance with institutional procedures and that all review comments were carefully considered Responsibility for the final content of this report rests entirely with the authoring committee and the institution We are also grateful to the work of others at the NRC, including ­Christine McShane, senior editor, Division of Behavioral and Social Sciences and Education (DBASSE), whose work greatly improved the text of the report; Kirsten Sampson Snyder, DBASSE reports officer, who worked with us through several revisions of the report; and Yvonne Wise, DBASSE production editor, who managed the report through final publication As well, we are thankful to those who assisted committee members with literature searches or background research, including Patricia Harvey, Julie Shuck, and Matthew Von Hendy, at the National Academies The committee appreciates the support provided by the Center for  PREFACE Education, under the leadership of Patricia Morison Taniesha Woods, the study director, provided invaluable support and guidance to the committee throughout the study We could not have asked for a better colleague Senior program assistant Mary Ann Kasper masterfully handled all the logistical aspects of this project, including our four committee meetings We are also grateful for the leadership and support of Heidi Schweingruber, deputy director of the Board on Science Education, who provided much thoughtful counsel throughout this process and contributed substantially to editing the report in the final stages Christopher T Cross, Chair Committee on Early Childhood Mathematics Contents Summary PART I: INTRODUCTION AND RESEARCH ON LEARNING Introduction Foundational Mathematics Content Cognitive Foundations for Early Mathematics Learning 4 Developmental Variation, Sociocultural Influences, and Difficulties in Mathematics 21 59 95 PART II: TEACHING-LEARNING PATHS 5 The Teaching-Learning Paths for Number, Relations, and Operations 6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement 127 175 PART III: CONTEXTS FOR TEACHING AND LEARNING Standards, Curriculum, Instruction, and Assessment 8 The Early Childhood Workforce and Its Professional Development 225 289 PART IV: FUTURE DIRECTIONS FOR POLICY, PRACTICE, AND RESEARCH Conclusions and Recommendations xi 331 xii CONTENTS APPENDIXES A Glossary B Concepts of Measurement C Biographical Sketches of Committee Members and Staff 351 359 363 Index 371 Summary Mathematics education has risen to the top of the national policy agenda as part of the need to improve the technical and scientific literacy of the American public The new demands of international competition in the 21st century require a workforce that is competent in and comfortable with mathematics There is particular concern about the chronically low mathematics and science performance of economically disadvantaged students and the lack of diversity in the science and technical workforce Particularly alarming is that such disparities exist in the earliest years of schooling and even before school entry Recognizing the increasing importance of mathematics and encouraged by a decade of success in improving early literacy, the Mathematical Sciences Education Board of the Center for Education at the National Research Council established the Committee on Early Childhood Mathematics The committee was charged with examining existing research in order to develop appropriate mathematics learning objectives for preschool children; providing evidence-based insights related to curriculum, instruction, and teacher education for achieving these learning objectives; and determining the implications of these findings for policy, practice, and future research The committee found that, although virtually all young children have the capability to learn and become competent in mathematics, for most the potential to learn mathematics in the early years of school is not currently realized This stems from a lack of opportunities to learn mathematics either in early childhood settings or through everyday experiences in homes and in communities This is particularly the case for economically disad  MATHEMATICS LEARNING IN EARLY CHILDHOOD vantaged children, who start out behind in mathematics and will remain so without extensive, high-quality early mathematics instruction In fact, well before first grade, children can learn the ideas and skills that support later, more complex mathematics understanding There is expert consensus that two areas of mathematics are particularly important for young children to learn: (1) number, which includes whole number, operations, and relations; and (2) geometry, spatial thinking, and measurement A rich body of research provides insight into how children’s proficiency develops in both areas and the instruction needed to support it The committee used this evidence to develop research-based teaching-learning paths to guide policy and practice in early childhood education Examination of current standards, curricula, and instruction in early childhood education revealed that many early childhood settings not provide adequate learning experiences in mathematics The relative lack of high-quality mathematics instruction, especially in comparison to literacy, reflects a lack of attention to mathematics throughout the childhood education system, including standards, curriculum, instruction, and the preparation and training of the teaching workforce For example, many widely used early childhood curricula not provide sufficient guidance on mathematics pedagogy or content When early childhood classrooms have mathematics activities, they are often presented as part of an integrated or embedded curriculum, in which the teaching of mathematics is secondary to other learning goals Emerging research indicates, however, that learning experiences in which mathematics is a supplementary activity rather than the primary focus are less effective in promoting children’s mathematics learning than experiences in which mathematics is the primary goal Finally, education and training for most teachers typically places heavy emphasis on children’s social-emotional development and literacy, with much less attention to mathematics In fact, academic activities such as mathematics can be a context in which social-emotional development and the foundations of language and literacy flourish As noted, opportunities to experience high-quality mathematics instruction are especially important for low-income children These children, on average, demonstrate lower levels of competence with mathematics prior to school entry, and the gaps persist or even widen over the course of schooling Providing young children with extensive, high-quality early mathematics instruction can serve as a sound foundation for later learning in mathematics and contribute to addressing long-term systematic inequities in educational outcomes The committee found that although the research to date about how young children develop and learn key concepts in mathematics has clear implications for practice, the findings are neither widely known nor imple-  SUMMARY mented by early childhood educators or those who teach them To ensure that all children enter elementary school with the mathematical foundation they need for success requires that individuals throughout the early childhood education system—including the teaching workforce, curriculum developers, program directors, and policy makers—transform their approach to mathematics education in early childhood by supporting, developing, and implementing research-based practices and curricula RECOMMENDATIONS Recommendation 1: A coordinated national early childhood mathematics initiative should be put in place to improve mathematics teaching and learning for all children ages to A number of specific recommendations for action follow from this overarching recommendation The specific steps and the individuals or organization that must be involved in enacting them are outlined below We provide further guidance about how to enact these steps in Chapter Recommendation 2: Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics The mathematical process goals should be integrated in these content areas Children should understand the concepts and learn the skills exemplified in the teaching-learning paths described in this report Recommendation 3: All early childhood programs should provide high-quality mathematics curricula and instruction as described in this report Recommendation 4: States should develop or revise their early childhood learning standards or guidelines to reflect the teaching-learning paths described in this report Recommendation 5: Curriculum developers and publishers should base their materials on the principles and teaching-learning paths described in this report Recommendation 6: An essential component of a coordinated national early childhood mathematics initiative is the provision of professional development to early childhood in-service teachers that helps them (a)  MATHEMATICS LEARNING IN EARLY CHILDHOOD to understand the necessary mathematics, the crucial teaching-learning paths, and the principles of intentional teaching and curriculum and (b) to learn how to implement a curriculum Recommendation 7: Coursework and practicum requirements for early childhood educators should be changed to reflect an increased emphasis on children’s mathematics as described in the report These changes should also be made and enforced by early childhood organizations that oversee credentialing, accreditation, and recognition of teacher professional development programs Recommendation 8: Early childhood education partnerships should be formed between family and community programs so that they are equipped to work together in promoting children’s mathematics Recommendation 9: There is a need for increased informal programming, curricular resources, software, and other media that can be used to support young children’s mathematics learning in such settings as homes, community centers, libraries, and museums Part I Introduction and Research on Learning Introduction For centuries, many students have learned mathematical knowledge— whether the rudiments of arithmetic computation or the complexities of geometric theorems—without much understanding Of course, many students tried to make whatever sense they could of procedures such as adding common fractions or multiplying decimals No doubt many students noticed underlying regularities in the computations they were asked to perform Teachers who themselves were skilled in mathematics might have tried to explain those regularities But mathematics learning has often been more a matter of memorizing than of understanding Today it is vital that young people understand the mathematics they are learning Whether using computer graphics on the job or spreadsheets at home, people need to move fluently back and forth between graphs, tables of data, and formulas To make good choices in the marketplace, they must know how to spot flaws in deductive and probabilistic reasoning as well as how to estimate the results of computations Public policy issues of critical importance hinge on mathematical analyses (pp 15-16) These words are from an earlier National Research Council (NRC) report called Adding It Up: Helping Children Learn Mathematics (National Research Council, 2001a) It focused on examining the evidence about school mathematics and outlining what it means to be mathematically proficient from prekindergarten to eighth grade The report offers much to guide current policy and practice in elementary and middle schools across the nation Yet the report also draws attention to the importance of what happens before children enter formal schooling: “Young children show a  ... teaching (Early childhood) Early childhood education I Cross, Christopher T II Woods, Taniesha A III Schweingruber, Heidi A IV National Research Council (U.S.) Committee on Early Childhood Mathematics. .. this report Recommendation 6: An essential component of a coordinated national early childhood mathematics initiative is the provision of professional development to early childhood in-service teachers... to mathematics education in early childhood by supporting, developing, and implementing research-based practices and curricula RECOMMENDATIONS Recommendation 1: A coordinated national early childhood