1
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



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1 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 stu- dents 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 

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 MATHEMATICS LEARNING IN EARLY CHILDHOOD remarkable ability to formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities. They are positively disposed to do and to understand mathematics when they first encounter it” (p. 6). However, not much attention has been paid historically to teaching mathematics to young children before they enter the period of formal schooling. This stems, at least in part, from generally negative attitudes about mathematics on the part of the American public as well as to beliefs that early childhood education should consist of a nurturing environment that promotes social-emotional development, with academic content pri- marily focusing on language and literacy development. In fact, a majority of parents report that a positive approach to learning and language devel- opment is more important for young children than mathematics (Cannon and Ginsburg, 2008). When asked which subject was more important for her child to learn and why, one mother said (p. 249): Language. Definitely. I mean obviously they’re both [math and language] very important. But you can find people, even adults, who never learn math. I think that you could survive much better [without mathematics] than if you never learn language. I think communication is so important. If you could learn to be expressive, you could hire someone to do your math for you. Families are agents of cultural transmission, which includes conveying at- titudes about mathematics. Often, mathematics is not viewed as important to young children’s cognitive development and later academic success. Evi- dence shows, however, that learning mathematics is vital for children’s early years and for later success in mathematics as well as better overall academic outcomes in such areas as literacy, science, and technology (e.g., Duncan et al., 2007; National Association for the Education of Young Children and National Council of Teachers of Mathematics, 2002). In addition, early childhood teachers are often uncomfortable teaching mathematics (Clements and Sarama, 2007; Copley, 2004; Ginsburg et al., 2006; Lee and Ginsburg, 2007a). Many teachers avoid teaching mathemat- ics because of their own negative early experiences with mathematics. The quote below, by a pre-service teacher attending a top-ranked university, is illustrative: Overall, my personal experiences with math have not been good. . . . Throughout [my] elementary [schooling] it was either you were right or wrong. . . . As a result, I found math very boring and confusing. I am not a natural math learner. . . . I do not like the idea of teaching math to others, because I feel like I am not competent enough to teach math. I remem- ber how hard it was when I was teaching adding and subtracting to first graders, especially when some of them did not understand it. I panicked

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 INTRODUCTION when I made some mistakes myself in adding and subtracting. (Personal communication, comments by student of H. Ginsburg, Teachers College, Columbia University, September 2007.) In recent years, however, interest in mathematics as a key aspect of early childhood education has increased across both the policy and the practice communities. In 2000, the National Council of Teachers of Math- ematics (NCTM), in their revision of the 1989 standards for elementary and secondary school mathematics, included prekindergarten for the first time. Also in 2000, a conference of early childhood and mathematics edu- cators was held to focus more explicitly on standards for preschool and kindergarten children (Clements, Sarama, and DiBiase, 2004). In 2002, Good Start, Grow Smart, an early childhood-focused White House initia- tive, resulted in the linking of federal funding to the requirement that all states develop voluntary early learning guidelines in language, literacy, and mathematics. The now-suspended National Reporting System for assessing learning outcomes for children participating in Head Start programs, be- gun in 2002, originally specified four areas of focus for assessment, one of which was early mathematical skills (the other three were language-related: comprehension of spoken English, vocabulary, and letter naming) (National Research Council, 2008). Also in 2002, the National Association for the Education of Young Children and the NCTM approved a joint position statement, “Early Childhood Math: Promoting Good Beginnings,” which included recommendations to guide both policy and practice. In 2006, following on its efforts to improve language and literacy out- comes for the children it serves, the Office of Head Start turned its attention to early mathematics. It convened a mathematics working group composed of parents, local staff, researchers, and other experts in early mathematics learning and has since moved forward on developing strategies for helping Head Start and Early Head Start programs support the early mathematics learning of infants, toddlers, and preschoolers. LEARNING FROM THE RESEARCH Clearly there is growing interest in including mathematics among the learning goals for young children and in improving the teaching of mathe- matics in developmentally appropriate ways. Over the past several decades, significant investments have been made in research on early development and learning, much of which is ripe for examination and synthesis as it ap- plies to early mathematics. In the past decade, the NRC has uncovered and synthesized key aspects of the knowledge about learning and development in early childhood. In the reports From Neurons to Neighborhoods: The Science of Early Child-

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10 MATHEMATICS LEARNING IN EARLY CHILDHOOD hood Deelopment (National Research Council and Institute of Medicine, 2000) and Eager to Learn: Educating Our Preschoolers (National Research Council, 2001b) the NRC directed its attention to early childhood institu- tions, their financing or lack of same, considerations of health and nutri- tion, and the social, emotional, and cultural components of this territory as they also focused special attention on early literacy. The report Early Childhood Assessment: Why, What, and How (National Research Council, 2008) identifies important outcomes for children from birth to age 5 and outlines the quality and purposes of developmental assessments. Although mathematics received attention to some degree in these studies, it was not a central focus of this work. The NRC study that resulted in the report How People Learn (National Research Council, 1999) drew on a large body of research in cognition to offer a set of powerful findings about teaching and learning at all levels and all subjects that, since its publication, have rippled across the research community. The most recent follow-on publication, How Students Learn: History, Mathematics, and Science in the Classroom (National Research Council, 2005a), provides several concrete examples of how this research on student learning can translate into improved practice, including one example in early childhood mathematics. Some additional examples of research in this territory also surfaced in Mathematical and Scientific De- elopment in Early Childhood (National Research Council, 2005b), which captures the discussion at an NRC workshop. The previously mentioned report, Adding It Up (National Research Council, 2001a), synthesized the research on mathematics learning in prekindergarten through eighth grade and provided advice to educators, researchers, publishers, policy makers, and parents. Taken together, these prior initiatives have helped set the stage for an in-depth examination of early learning in mathematics. THE COMMITTEE’S CHARGE In order to synthesize and distill the key lessons from the relevant research, the NRC established the Committee on Early Childhood Math- ematics in 2007. The majority of support for the study was provided by the Office of Head Start, under the auspices of the U.S. Department of Health and Human Services; supplementary funding was also provided by the National Institute of Child Health and Human Development, the Ewing Marion Kauffman Foundation, and the NRC. In recognition of the interdisciplinary nature of this work, the committee consists of experts in mathematics, psychology, neuroscience, early childhood education, and teacher education, as well as early childhood practitioners and policy mak- ers. The committee worked on the study over an 18-month period. The committee charge is as follows:

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11 INTRODUCTION To synthesize and analyze the past research on early childhood mathemat- ics from a number of disciplinary fields, draw out the implications for policy and practice affecting young children as they move through the preschool years and begin formal schooling, and provide research-based guidance to increase the number of young children, especially vulnerable children, prepared to get off to a strong start in learning mathematics dur- ing their first years of schooling. It is designed to capitalize on the research literature in the field and consider its various implications for policy mak- ers, practitioners and parents. The committee will assemble the pertinent research literature from the multiple disciplines that have focused attention on the teaching and learn- ing of mathematics by young children. They will analyze this literature in order to develop (1) appropriate mathematics learning objectives for preschool students; and (2) critical evidence-based insights related to cur- riculum, instruction, and teacher education for achieving these learning objectives. Finally, they will determine the implications of these findings for policy, practice, parent-child relations, future data collection and fur- ther research. See Box 1-1 for questions that the committee might address as part of its charge. BOX 1-1 Questions the Committee Might Address • hat does existing research tell us about what preschool children can know W about mathematics, and how they develop this knowledge? • earning of which mathematical knowledge, skills, and concepts in the pre- L school years increases the likelihood of successful mathematics learning in school and beyond? • hat do international comparisons with respect to both preschoolers and pri- W mary grades students tell us about the nature of early mathematics learning and prospects for its improvement in the United States? What approaches in other countries with respect to interventions and ongoing support could use- fully be applied here? • hat policies and practices best lay the foundation for successful mathematics W learning? • hat can parents, preschool teachers, and other adults who interact with W young children do to promote their mathematical development? • ow can we support the mathematical development of preschool teachers so H that they will be able to promote young children’s mathematical development? • ow can further research in cognitive development and preschool education H be focused to address issues that will lead to improvement in children’s math- ematical proficiency?

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12 MATHEMATICS LEARNING IN EARLY CHILDHOOD The committee cast its net widely to examine as much of the rel- evant research as possible. For some issues, the evidence base was limited: Throughout the report, we attempt to recognize and acknowledge the limitations of the evidence base and, at the end of the report, suggest some areas in which the scope and quality of research can be strengthened. The committee was not able to pursue in depth the entire array of possible issues related to mathematics education during early childhood; for example, we lacked the time, resources, and expertise to do a comprehensive interna- tional comparative analysis of early childhood education in mathematics. We do discuss the literature on the role of language as a shared cultural experience that shapes children’s mathematical learning. In addition, nei- ther program evaluation nor accountability, both of which are important to children’s early mathematics education programs, is discussed at length in the report. In addressing the charge, although the committee did examine research related to the development of number and space concepts for the very early years (i.e., infancy through age 3), our focus was on children ages 3 through 6 and early mathematics education—which includes learning, teaching, teacher education, and curriculum. The committee paid special attention to the learning and teaching practices that underscore mathemati- cal development in children from age 3 through the end of kindergarten. This age range was chosen as the focus because it provides children with key cognitive and social development opportunities associated with success- ful entry into formal schooling. Evidence demonstrates that preschool-age children are excited about learning and enjoy activities that develop their mathematics competencies (Gelman, 1980; Ginsburg et al., 2006; National Research Council, 2001b; Saxe et al., 1987); this period is thus critical for maintaining and enhancing motivation to learn, especially for children from disadvantaged backgrounds, because enriching early learning experiences can enable them to begin kindergarten on a more level footing with their more advantaged peers. The committee has put particular emphasis on the need to translate research on early childhood mathematics into practice for all children. Still, young children from disadvantaged backgrounds show lower levels of mathematics achievement than children from middle-class and higher status backgrounds (Clements and Sarama, 2007; Ginsburg and Russell, 1981; Hughes, 1986; Jordan, Huttenlocher, and Levine, 1994; Saxe et al., 1987; Starkey and Klein, 2000; Starkey, Klein, and Wakeley, 2004). The committee paid particular attention to issues of equity in early mathemat- ics education throughout the report because of evidence indicating that, whereas all young children can benefit from intentional mathematics in- struction, children who are at risk because of particular life circumstances

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1 INTRODUCTION (e.g., low socioeconomic status) will fall further behind their more affluent peers over the course of their schooling if they do not receive more intensive mathematics teaching (Starkey and Klein, 2000). The committee held four meetings, which provided opportunities for discussions with practitioners, researchers, and other experts in the field of early childhood education. These discussions helped committee members develop a better understanding of the history and positions in the various stakeholder communities as well as the reasoning behind their positions. Our analyses draw on a variety of sources. The committee examined rel- evant summary data produced by government agencies and professional organizations. We reviewed a wide body of interdisciplinary research and commissioned a number of research synthesis papers by experts. Often, practitioners and policy makers state that the research community is too far removed from what is actually happening in the classroom, causing researchers to make recommendations that cannot be realistically imple- mented. The committee is keenly aware of this concern, and thus we attempt to put forth here policy recommendations that are grounded in research as well as the action steps necessary to implement them. THE EARLY CHILDHOOD EDUCATION AND CARE DELIVERY SYSTEM One important issue that influenced the committee’s thinking about recommendations for policy and practice is the multifaceted and complex nature of the early childhood education “system.” Before the beginning of formal schooling, children spend their days in a wide variety of settings. If they are not cared for at home by their parents or relatives, children typically receive care through the country’s early education and child care system, which consists of a loosely sewn-together patchwork of different kinds of programs and providers that vary widely in their educational mis- sion and whether they are explicitly designed to provide education services. Data from the nationally representative Early Childhood Longitudinal Study, Birth (ECLS-B) cohort show that about 60 percent of preschool-age children are in center-based care (including Head Start settings), about 21 percent of children are in home-based care arrangements, and about 20 percent have no formal child care arrangements (see Table 1-1) (Jacobson Chernoff, McPhee, and Park, 2007). In addition, about 43 percent of children younger than age 6 live in low-income families (Chau and Douglas-Hall, 2007). The high cost of high-quality early education and care is unaffordable for many low- and middle-income families (Zigler, Gilliam, and Jones, 2006). For example, the average annual cost for full-day center-based care for preschool-age

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1 MATHEMATICS LEARNING IN EARLY CHILDHOOD TABLE 1-1 Children Participating in Regular Nonparental Education and Early Care, 2005-2006 (percentage) Center- Home-Based Based Non- No Regular Relative Nonrelative Head Head Multiple Nonparental Characteristic Care Care Start Start Arrangements Arrangement Total 13 8 45 13 2 20 Child Race/Ethnicity White, 11 9 53 7 2 18 non-Hispanic Black, 14 4 37 25 3 16 non-Hispanic Hispanic 16 6 31 19 1 27 2a Asian, 16 3 55 6 18 non-Hispanic 1a American 14 5 29 31 20 Indian and Alaska Native, non-Hispanic 2a Other, 19 9 40 12 18 non-Hispanic Socioeconomic Statusb Lowest 15 5 22 25 2 31 20 percent Middle 15 7 44 13 2 20 60 percent Highest 6 11 71 1 2 10 20 percent NOTE: Percentages do not sum to 100 because of rounding error. aStandard error is more than one third as large as estimate. bSocioeconomic status (SES) is a measure of social standing. This SES variable reflects the socioeconomic status of the household at the time of the preschool parent interview in 2005. The components used to create the measure of SES were as follows: father/male guardian’s education; mother/female guardian’s education; father/male guardian’s occupation; mother/ female guardian’s occupation; and household income. SES was collapsed first into quintiles, then into a 20/60/20 percent distribution by collapsing the middle three quintiles. SOURCE: Jacobson Chernoff, McPhee, and Park (2007).

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15 INTRODUCTION children ranges from $3,794 in Mississippi to $10,920 in the District of Columbia (National Association of Child Care Resource and Referral Agencies, 2007). An increase in women’s participation in the workforce has also contributed to the demand for high-quality preschool and child care (National Research Council, 2001b). Over the past four and a half decades, women’s participation in the workforce has grown from 38 per- cent in 1960 to 60 percent in 2002 (U.S. Census Bureau, 2003), with 59 percent of mothers of 4-year-olds working outside the home (Jacobson Chernoff, McPhee, and Park, 2007). Head Start is a large, federally funded early childhood program that promotes school readiness for economically disadvantaged children and families; the program provides comprehensive child development services (education, health, nutritional, social, and other services). In fiscal year 2007, the program served 908,412 children, most of whom were 3- and 4-year-olds (87 percent). The reach of the program is large—in 2007 there were over 49,000 Head Start classrooms located in over 18,000 centers—which makes its policies and practices influential in early child- hood education. Of the 60 percent of children in the United States who attend center- based care, approximately 22 percent are enrolled in state-funded preschool, which is the largest source of public prekindergarten (Barnett et al., 2007). Increasingly, states are moving toward state-funded preschool education to provide early education and care for children, particularly for those whose families would otherwise not be able to afford it. Georgia and Oklahoma, for example, have public preschool programs that enroll (if parents choose) 4-year-olds across the state (Barnett et al., 2007). Voluntary universal pre- school is one policy option that has been suggested as a way to provide opportunities for all children, regardless of family income, to receive high- quality early education and care (Zigler et al., 2006). However, some have argued against voluntary universal preschool in favor of programs that target low-income children (e.g., Ceci and Papierno, 2005; Fuller, 2007). Ceci and Papierno (2005), for example, suggest that targeted programs are more effective in terms of financial and educational benefits because they use (often limited) early education funds to help the most disadvantaged children. Revisions to legislation and new policy initiatives have also shaped early childhood education policy in recent years. For example, beginning with the National Education Goals of 1990, the No Child Left Behind (NCLB) Act of 2001, and continuing through the 2007 reauthorization of the Head Start Act, interest in young children’s preparation for school has increased. Central aims of these pieces of legislation are to support young children’s development and learning so that they make a successful transi-

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16 MATHEMATICS LEARNING IN EARLY CHILDHOOD tion into kindergarten and to provide equitable educational opportunities for all students. With the implementation of NCLB, many school districts began to place a major emphasis on the academic success of students in the early elementary grades. NCLB testing requirements do not begin until children reach third grade, but implications from the law exist for lower grades and preschool programs. The emphasis that has been placed on accountability for early childhood learning has caused concern among researchers, parents, and early education stakeholders because of the strong focus on academic development rather than the combination of academic and social-emotional development. This tension is not new; the early childhood education com- munity has grappled with the notion that preschool programs should be more focused on academics, in contrast to the idea that they should focus instead on children’s social-emotional development. The consequences of accountability systems have brought an increased emphasis and disagree- ments about what should be the focus of early education and care. In addition to NCLB, Good Start, Grow Smart, President Bush’s plan to strengthen early learning (White House, n.d.), promoted accountability for preschool children’s learning outcomes in literacy and mathematics and also called for program improvements in language and literacy develop- ment. A major premise of this initiative was to close the achievement gap between socioeconomic and racial/ethnic groups. Until recently, the focus of these efforts was targeted at improving literacy and language development (e.g., Reading First and Early Reading First). However, with recent research clearly demonstrating the importance of early childhood mathematics to later success in reading and mathematics, policy makers are beginning to see the value of investing in early childhood mathematics. As discussed more fully in Chapter 8, policies aimed at changing or improving the education and learning of 3- to 6-year-olds still need to consider the diverse range of settings and characteristics of those who will do the teaching in these settings. ORGANIZATION OF THE REPORT The report is organized into four parts. Part I focuses on the research on learning and summarizes the nearly 30 years of research demonstrating that young children are able to learn foundational mathematics. As these chapters show, preschool-age children possess a well-developed understand- ing of informal mathematics (Ginsburg, Klein, and Starkey, 1998), and they are able to learn complex mathematics before school entry (Clements and Sarama, 2007; Ginsburg et al., 2006). Chapter 2 provides an overview of the important mathematical thinking processes and mathematical ideas for the early childhood period, summa- rizing the areas in which children need foundational learning opportuni-

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1 INTRODUCTION ties. Chapter 3 reviews the evidence about how young children’s everyday mathematics learning begins in infancy with the proximal environments in which they develop. More specifically, it focuses on cognitive development and includes a discussion of the research on infancy. Chapter 4 examines individual variation in children’s mathematics learning and performance, with particular attention to mathematics learning disabilities. The chapter also considers sources of individual variation, such as familial practices, and group variation, such as socioeconomic status and race/ethnicity. Part II focuses on a sequence of milestones for children in the core areas of number (including whole number, relations, and operations) and geometry and measurement. Chapter 5 focuses on number and operations, and Chapter 6 on geometry and measurement. In Part III the committee turns to topics of implementation of math- ematics learning and teaching in the classroom context. Chapter 7 covers the research concerning standards, curriculum, teaching, and formative assessment. Chapter 8 focuses on the early childhood workforce and exam- ines issues of teacher education and professional development. Part IV contains the committee’s synthesis of its major conclusions and outlines the recommendations that flow from these conclusions, focusing particularly on what changes are needed to improve the quality of math- ematics learning for young children. The committee also lays out an agenda for future research. Appendix A is a glossary that defines terminology used throughout the report and Appendix B supplements Chapter 6. Appendix C presents biographical sketches of committee members and staff. REFERENCES AND BIBLIOGRAPHY Barnett, W.S., Hustedt, J.T., Friedman, A.H., Boyd, J.S., and Ainsworth, P. (2007). The State of Preschool 200: State Preschool Yearbook. New Brunswick: Rutgers, The State Uni- versity of New Jersey, The National Institute for Early Education Research. Available: http://nieer.org/yearbook/pdf/yearbook.pdf#page=6 [accessed August 2008]. Cannon, J., and Ginsburg, H. (2008). “Doing the math”: Maternal beliefs about early math- ematics versus language learning. Early Education and Deelopment, 1(2), 238-260. Ceci, S.J., and Papierno, P.B. (2005). The rhetoric and reality of gap closing: When the “have- nots” gain but the “haves” gain even more. American Psychologist, 60(2), 149-160. Chau, M., and Douglas-Hall, A., (2007, September). Low-income Children in the United States: National and State Trend Data, 16-2006. Mailman School of Public Health at Columbia University, National Center for Children in Poverty. Available: http://www. nccp.org/publications/pdf/text_761.pdf [accessed August 2008]. Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., Sarama, J., and DiBiase, A. (2004). Engaging Young Children in Mathemat- ics: Findings of the 2000 National Conference on Standards for Preschool and Kinder- garten Mathematics Education. Mahwah, NJ: Erlbaum.

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1 MATHEMATICS LEARNING IN EARLY CHILDHOOD Copley, J.V. (2004). The early childhood collaborative: A professional development model to communicate and implement the standards. In D.H. Clements, J. Sarama, and A-M. DiBiase (Eds.), Engaging Young Children in Mathematics (pp. 401-414). Mahwah, NJ: Erlbaum. Duncan, G.J., Dowsett, C.J., Claessens, A., Magnuson, K., Huston, A.C., Klebanov, P., Pagani, L.S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., and Duckworth, K. (2007). School readiness and later achievement. Deelopmental Psychology, , 1428-1446. Fuller, B. (2007). Standardized Childhood: The Political and Cultural Struggle oer Early Education. Palo Alto, CA: Stanford University Press. Gelman, R. (1980). What young children know about numbers. Educational Psychologist, 15, 54-68. Ginsburg, H.P., and Russell, R.L. (1981). Social class and racial influences on early math- ematical thinking. Monographs of the Society for Research in Child Deelopment, 6(6), 1-68. Ginsburg, H.P., Klein, A., and Starkey, P. (1998). The development of children’s mathemati- cal thinking: Connecting research with practice. In I. Sigel and A. Renninger (Eds.), Handbook of Child Psychology, Volume : Child Psychology and Practice (5th ed., pp. 401-476). New York: Wiley. Ginsburg, H.P., Goldberg Kaplan, R., Cannon, J., Cordero, M.L., Eisenband, J.G., Galanter, M., and Morgenlander, M. (2006). Helping early childhood educators to teach math- ematics. In M. Zaslow and I. Martinez-Beck (Eds.), Critical Issues in Early Childhood Professional Deelopment (pp. 171-202). Baltimore: Paul H. Brookes. Hughes, M. (1986). Children and Number. Oxford: Blackwell. Jacobson Chernoff, J., McPhee, C., and Park, J. (2007). Preschool: First Findings from the Third Follow-Up of the Early Childhood Longitudinal Study, Birth Cohort (ECLS-B). Washington, DC: U.S. Department of Education, Institute of Education Sciences, Na- tional Center for Education Statistics. Jordan, N.C., Huttenlocher, J., and Levine, S.C. (1994). Assessing early arithmetic abilities: Effects of verbal and nonverbal response types on the calculation performance of middle- and low-income children. Learning and Indiidual Differences, 6, 413-432. Kagan, S.L., Kauerz, K., and Tarrant, K. (2008). The Early Care and Education Teaching Workforce at the Fulcrum: An Agenda for Reform. New York: Teachers College Press. Lee, J.S., and Ginsburg, H.P. (2007a). Preschool teachers’ beliefs about appropriate early literacy and mathematics education for low- and middle-socioeconomic status children. Early Education and Deelopment, 1(1), 111-143. Lee, J.S., and Ginsburg, H.P. (2007b). What is appropriate mathematics education for four- year-olds?: Pre-kindergarten teachers’ beliefs. Journal of Early Childhood Research, 5(1), 2-31. National Association for the Education of Young Children and National Council of Teachers of Mathematics. (2002). Early Childhood Mathematics: Promoting Good Beginnings. A joint position statement. Available: http://www.naeyc.org/about/positions/pdf/psmath. pdf [accessed February 2008]. National Association of Child Care Resource and Referral Agencies. (2007). Parents and the High Price of Child Care. Arlington, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. National Research Council. (1999). How People Learn: Brain, Mind, Experience, and School. Committee on Developments in the Science of Learning. J.D. Bransford, A.L. Brown, and R.R. Cocking (Eds.). Commission on Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

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