Early Childhood Mathematics Education Research

Learning Trajectories for Young Children

Julie Sarama, Douglas H. Clements

• Publisher: Routledge
• Year: 2009
• ISBN:
  9781135592493
• Edition: 1
• Topic: Education
• Subtopic: Education General

This important new book synthesizes relevant research on the learning of mathematics from birth into the primary grades from the full range of these complementary perspectives. At the core of early math experts Julie Sarama and Douglas Clements's theoretical and empirical frameworks are learning trajectories—detailed descriptions of children's thinking as they learn to achieve specific goals in a mathematical domain, alongside a related set of instructional tasks designed to engender those mental processes and move children through a developmental progression of levels of thinking. Rooted in basic issues of thinking, learning, and teaching, this groundbreaking body of research illuminates foundational topics on the learning of mathematics with practical and theoretical implications for all ages. Those implications are especially important in addressing equity concerns, as understanding the level of thinking of the class and the individuals within it, is key in serving the needs of all children.

Part I
Introduction

1
Early Childhood Mathematics Learning

“It seems probable that little is gained by using any of the child’s time for arithmetic before grade 2, though there are many arithmetic facts that he [sic] can learn in grade 1.”

(Thorndike, 1922, p. 198)

“Children have their own preschool arithmetic, which only myopic psychologists could ignore.”

(Vygotsky, 1935/1978, p. 84)

For over a century, views of young children’s mathematics have differed widely. The recent turn of the century has seen a dramatic increase in attention to the mathematics education of young children. Our goal is to synthesize relevant research on the learning of mathematics from birth into the primary grades from multiple perspectives. This reveals a field that is fascinating in its new findings, that promises practical guidelines and suggestions for teaching and advances in theory. As just one example, Jean Piaget’s genetic epistemology, as the study of the origins of knowledge itself, has never been more deeply developed or empirically tested than in recent research on early mathematics learning.

In this chapter, we begin with a brief overview of mathematics in early childhood and young children’s learning of mathematics. In the several chapters that follow, we discuss children’s learning of mathematical ideas and skills that are important for young children’s learning (Clements & Conference Working Group, 2004; NCTM, 2006; NMP, 2008), because it is most fruitful for teachers and children to focus on the big ideas of mathematics (Bowman, Donovan, & Burns, 2001; Clements, 2004; Fuson, 2004; Griffin, Case, & Capodilupo, 1995; Tibbals, 2000; Weiss, 2002).

This organization based on content should not be taken as a de-emphasis on other critical components of high-quality mathematics education. For example, processes are discussed within every chapter, because processes are just as important as “facts” and concepts in understanding mathematics. Further, Chapter 13 focuses on specific processes. “As important as mathematical content are general mathematical processes such as problem solving, reasoning and proof, communication, connections, and representation; specific mathematical processes such as organizing information, patterning, and composing, and habits of mind such as curiosity, imagination, inventiveness, persistence, willingness to experiment, and sensitivity to patterns. All should be involved in a high-quality early childhood mathematics program” (Clements & Conference Working Group, 2004, p. 57). This structure of goals is consistent both with recommendations of the National Council of Teachers of Mathematics (NCTM, 2000) and with research on young children’s development of a network of logical and mathematical relations (Kamii, Miyakawa, & Kato, 2004). Finally, it is consistent with the conclusions of another review, that: “The overriding premise of our work is that throughout the grades from pre-K through 8 all students can and should be mathematically proficient” (Kilpatrick, Swafford, & Findell, 2001, p. 10), including conceptual understanding, procedural fluency, strategic competence, adaptive reasoning (capacity for logical thought, reflection, explanation, and justification), and a productive disposition.

That last thread—productive disposition—must also be highlighted. The “habits of mind” named previously, including curiosity, imagination, inventiveness, risk-taking, creativity, and persistence—are components of the essential productive disposition. Children need to view mathematics as sensible, useful, and worthwhile and view themselves as capable of thinking mathematically. All these should be involved in a high-quality early childhood mathematics program. (The companion book includes chapters on learning and teaching contexts, including early childhood school settings and education, equity issues, affect, and so forth that speak to developing children’s productive disposition.)

Mathematics in Early Childhood

There are at least eight reasons for the recent surge of attention to mathematics in early childhood. First, increasing numbers of children attend early care and education programs. In 1999, 70 percent of four-year-olds and 93 percent of five-year-olds were enrolled in preprimary education, up from 62 and 90 percent, respectively, in 1991 (U.S. Department of Education, 2000, p. 7). Several states are instituting universal pre-K,¹ with about 1 million children enrolled in 1999, and that number is increasing (Hinkle, 2000). In 2001, about two-thirds of all four-year-olds were enrolled in universal pre-K, with that ratio increasing (Loeb, Bridges, Bassok, Fuller, & Rumberger, in press; Magnuson, Meyers, Rathbun, & West, 2004). Various government agencies, federal and state, provide financial support for pre-K programs designed to facilitate academic achievement, particularly programs for low-income children.

Second, there is an increased recognition of the importance of mathematics (Doig, McCrae, & Rowe, 2003; Kilpatrick et al., 2001). In a global economy with the vast majority of jobs requiring more sophisticated skills than in the past, American educators and business leaders have expressed strong concern about students’ mathematics achievement.

Third, the mathematics achievement of American students compares unfavorably with the achievement of students from several other nations, even as early as first grade and kindergarten (Stigler, Lee, & Stevenson, 1990). Some cross-national differences in informal mathematics knowledge appear as early as three to five years of age (Starkey et al., 1999; Yuzawa, Bart, Kinne, Sukemune, & Kataoka, 1999). Children in East Asia and Europe learn more advanced math than most children in the U.S. are taught (Geary, 2006). (Similar contrasts appear between East Asian and other Western countries as well: Aunio, Ee, Lim, Hautamäki, & Van Luit, 2004.)

Fourth, the knowledge gap is most pronounced in the performance of U.S. children living in economically deprived urban communities (Geary, Bow-Thomas, Fan, & Siegler, 1993; Griffin, Case, & Siegler, 1994; V. E. Lee & Burkam, 2002; Saxe, Guberman, & Gearhart, 1987; Siegler, 1993). That is, differences are not just between nations, but also between socioeconomic groups within countries. A stark example occurred in our own research project when a child stepped up to research assistant grinning and showing her “Happy Birthday” crown. The assistant asked how old she was. The girl stared without responding. “Can you show me on your fingers?” The girl slowly shook her head “no.” Cross-cultural differences raise concerns of equity regarding children’s pre-K experiences and elementary schools’ readiness to adapt instruction to children at different levels of mathematical development. Many government-funded programs serve low-income children, who often experience difficulties in mathematics and are at increased risk of school failure (Bowman et al., 2001; Natriello, McDill, & Pallas, 1990). These children need to build the informal knowledge that provides the basis for later learning of mathematics. Thus, equity demands that we establish guidelines for quality early mathematics education for all children.

Fifth, researchers have changed from a position that very young children have little knowledge of, or capacity to learn mathematics (e.g., Piaget, Inhelder, & Szeminska, 1960; Piaget & Szeminska, 1952; Thorndike, 1922) to theories that posit competencies that are either innate or develop in the first years of life (Baroody, Lai, & Mix, 2006; Clements, Sarama, & DiBiase, 2004; Doig et al., 2003; Gelman & Gallistel, 1978; Perry & Dockett, 2002). We will discuss these issues in more depth; here, it suffices to say that it is clear that young children can engage with substantive mathematical ideas.

Sixth, early knowledge strongly affects later success in mathematics (Denton & West, 2002). Specific quantitative and numerical knowledge in the years before first grade has been found to be a stronger predictor of later mathematics achievement than tests of intelligence or memory abilities (Krajewski, 2005). What children know early affects them for many years thereafter (Horne, 2005; NMP, 2008). Mathematics knowledge on school entry is a stronger predictor than any of a host of social-emotional skills. The most powerful preschool avenue for boosting fifth grade achievement appears to be improving the basic academic skills of low-achieving children prior to kindergarten entry (Claessens, Duncan, & Engel, 2007).

Seventh, research indicates that knowledge gaps appeared in large part due to the lack of connection between children’s informal and intuitive knowledge (Ginsburg & Russell, 1981; Hiebert, 1986) and school mathematics. This is especially detrimental when this informal knowledge is poorly developed (Baroody, 1987; Griffin et al., 1994). High-quality experiences in early mathematics can ameliorate such problems (Doig et al., 2003; Thomson, Rowe, Underwood, & Peck, 2005).

Eighth, traditional approaches to early childhood, such as “developmentally appropriate practice” (DAP) have not been shown to increase children’s learning (Van Horn, Karlin, Ramey, Aldridge, & Snyder, 2005). We need ways to keep the probable benefits of DAP, such as socioemotional growth (Van Horn et al., 2005), and yet infuse the young child’s day with interesting, equally appropriate, opportunities to engage in mathematical thinking (cf. Peisner-Feinberg et al., 2001).

For these reasons, there has been much recent interest in, and attention to, the learning and teaching of mathematics to the young.

Young Children and Mathematics Learning

Given the opportunity, young children possess an informal knowledge of mathematics that is surprisingly broad, complex, and sophisticated (Baroody, 2004; B. A. Clarke, Clarke, & Cheeseman, 2006; Clements, Swaminathan, Hannibal, & Sarama, 1999; Fuson, 2004; Geary, 1994; Ginsburg, 1977; Kilpatrick et al., 2001; NCES, 2000; Piaget & Inhelder, 1967; Piaget et al., 1960; Steffe, 2004; Thomson et al., 2005). For example, preschoolers engage in substantial amounts of foundational free play.² They explore patterns, shapes, and spatial relations, compare magnitudes, and count objects. Importantly, this is true for children regardless of income level and gender (Seo & Ginsburg, 2004). They engage in mathematical thinking and reasoning in many contexts, especially if they have sufficient knowledge about the materials they are using (e.g., toys), if the task is understandable and motivating, and if the context is familiar and comfortable (Alexander, White, & Daugherty, 1997).

What do children know about math before they come to school? More than a century ago, G. Stanley Hall (1891) included specific mathematics skills in his survey of the “content of children’s minds upon entering school.” About 40 years later, Buckingham and MacLatchy (1930) similarly surveyed the number abilities of entering first graders. Since then, several reports have described what children know as they enter school, providing valuable data for writers of standards, goals, and curricula, as well as for teachers. A brief review of previous surveys of mathematics knowledge at school entry indicates that children generally are acquiring mathematical knowledge at earlier ages, but that children have never entered schools as tabulae rasae. The following is a brief discussion; some results are summarized in Table 1.1 to provide another view of the pattern of these findings.

Number

Of children entering first grade in Berlin, Hall (1891) reported that the percentages of children that “had the idea of” two, three, and four were 74, 74, and 73, respectively. Except for a handful of questions on shapes, these were the only questions asked concerning mathematics. Buckingham and MacLatchy (1930) reported that entering first graders possessed “remarkable” ability with counting, reproducing and naming numbers, and number combinations. At least 90 percent could verbally count to 10, 60 percent to 20. The majority of the 1,356 children enumerated objects to 20. Success in reproducing numbers declined with increasing number size, yet over 75 percent were successful with each of the numbers five, six, seven, eight, and 10 at least once; 70 percent identified a set of 10 at least once. Similarly, Brownell (1941) concluded that about 90 percent of first grade entrants could count verbally and count objects to 10 (his review of previous research revealed that about 10 percent could verbally count to 100 and between half and two-thirds could count to 20); about 60 percent could identify groups up to 10 represented concretely without pattern; over 50 percent could reproduce sets up to 10 given verbal directions; about 80 percent could identify the number which was “more”; about 75 percent could reproduce a set of four or five given a model set, whereas about 50 percent were successful with a set of seven; and about 75 percent dealt successfully with number combinations with small numbers of visible objects and from 38 to 50 percent with verbal presentations. More recently, Callahan and Clements (1984), in a survey of 4,722 first graders entering urban schools in the years 1976 to 1980, found that a relatively high percentage of children stopped verbal counting in the 10–19, 20–29 and 30–39 intervals, but beyond 50 most counted until the interviewer stopped them at 100.

A recent study found that by the end of the school year, a large percentage (88 percent) of kindergarten children understood the concept of relative size (e.g., can count beyond 10 and understand and can use nonstandard units of length to compare objects). By the spring of first grade, most children (96 percent) mastered ordinality and sequence (the understanding of the relative position of objects); and about three-quarters (76 percent) demonstrated proficiency in adding and subtracting basic whole units. Moreover, by the spring of first grade, about one-quarter (27 percent) demonstrated proficiency in multiplying and dividing simple whole units (Denton & West, 2002).

In a Netherlands study, entering first graders scored an average of 75 percent on the researchers’ assessment (Heuvel-Panhuizen, 1996). Most children had mastered relational concepts and were quite familiar with numerals (written symbols, “1,” “2,” “3” … ) to 10 (97 percent; 81 percent for “14”). On a pictured board game, the great majority could identify numbers that come after or before another number. An even higher percentage could color two (99 percent) to nine (84 percent) pictured marbles. Performance on addition and subtraction problems was variable, from 80 percent on a countable board game problem, to 39 percent on a noncountable, open-ended subtraction problem, with performance on countable problems solved by about 79 percent, and others solved by about 50 percent. An interesting addition to this study was that several categories of experts in education were asked to predict these scores. They expected mastery of the relational concepts, but their predictions of number abilities were substantially lower. For example, they thought about half the children would know numbers to 10, compared to the actual average of 97 percent. They predicted 25 percent for numbers before and after, compared to the actual 59 percent to 86 percent. Their estimates for addition and subtraction were proportionately less accurate, often a mere tenth of the actual averages. The children’s scores—and substantially lower estimates of them by experts—were substantiated by replications in Germany and Switzerland (although the estimations tended to be somewhat higher, proportionately).

Many studies have been conducted of kindergartners (for a review, see Kraner, 1977). In 1925, children 4.6 to 6 years showed 100 percent performance for naming the number of items in collections of one and two, from 85 percent to 95 percent on three, 15 percent to 65 percent for four, and 0 percent to 12 percent for five (Douglass, 1925). When not correct, most children’s estimates were within one to two for numbers up to 10. These skills improved with age. Bjonerud (1960) reported the average performance of children (from a public and university demonstration school) for both rote and rational counting was 19, although there was little facility with number sequences other than the sequence of numbers by one (about 25 percent could count by tens). All were able to recognize sets of less than four immediately; some were able to recognize sets up to nine, mostly by counting. Almost 90 percent solved simple addition word problems and approximately 75 percent solved subtraction problems. Rea and Reys (Rea & Reys, 1971; Reys & Rea, 1970) assessed the mathematical competencies of 727 entering metropolitan kindergartens. Skills in counting, recognizing, and comparing small groups were generally well developed, more so than ordinal concepts. Over 50 percent counted beyond 14 and over 75 percent beyond 10 (rote and rational). Between 70 and 95 percent successfully identified the number of objects in groups containing from one to eight objects. The task of forming groups of three, seven, and 13 was performed successfully by 82, 55, and 34 percent respectively. When three numbers in sequence were provided, approximately 90 percent of the children could supply the next number; however, providing only one number cue and asking for the number before or after greatly decreased the percentage of correct responses. Kraner (1977) concluded that the average kindergartner possessed mathematics knowledge on a par with that of the average first grader of earlier years.

A few studies have surveyed the prekindergarten child’s mathematical knowledge. McLaughlin (1935) investigated the number abilities of children from three to five years of age. The average verbal and object counting (without checking the cardinal principle) was to 4.5 and 4.4 for three-year-olds and 17.6 and 14.5 for four-year-olds (which included children up to 60 months). There was no substantial ability to count backwards. Only some three-year-olds could match groups of two or three whereas most four-year-olds were beginning to count as a means of matching, forming, naming, or combining number groups (percentages were not provided). In his investigation of the acquisition age (defined as an 80 percent success rate) of number concepts of three- to six-year-olds from five middle-class locations, Kraner (1977) reported the following ages of concept attainment: verbal counting to three, 3-6; to four, 4-0; to nine, 4-6; and to 18, 5-0; object counting to two, 3-6; to eight, 4-6; and to 20, 5-6; recognizing and comprehending cardinal numbers to five, 5-0; to 10, from 5-6 to 6-0; and identifying more than, 5-6, less than, 6-6, and one more or less, above 6-6. Clements (1984b) identified the following abilities of middle-class entering four-year-olds, in average accuracy: object counting to 10, 59 percent; choosing more, 51 percent; after, before, between, 30 percent; counting on and back, 19 percent; equalizing, 20 percent one-to-one correspondence, 58 percent; identity conservation, 72 percent; equivalence conservation, 62 percent; verbal arithmetic problems, 49 percent; concrete arithmetic problems, 46 percent.

An Australian study showed that most entering preschoolers could count eight objects and between 25 and 41 percent could tell what one more or one less would be (Thomson et al., 2005). Less competence was reported for Scandinavian prekindergarteners, who did not master any of the assessed counting skills (Van de Rijt & Van Luit, 1999). Prescho...

Table of contents

• Cover Page
• Half Title Page
• Title Page
• Copyright Page
• Contents
• Preface
• Appreciation to the Funding Agencies
• Part I: Introduction
• Part II: Number and Quantitative Thinking
• Part III: Geometry and Spatial Thinking
• Part IV: Geometric Measurement
• Part V: Other Content Domains and Processes
• Notes
• References
• Index