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Cognitive Foundations for Early Mathematics Learning

Over the past two decades, a quiet revolution in developmental psychology and related fields has demonstrated that children have skills and concepts relevant to mathematics learning that are present early in life, and that most children enter school with a wealth of knowledge and cognitive skills that can provide a foundation for mathematics learning. At the same time, these foundational skills are not enough—children need rich mathematical interactions, both at home and at school in order to be well prepared for the challenges they will meet in elementary school and beyond. (Chapter 4 discusses supporting children’s mathematics at home, and Chapters 5 and 6 discuss children’s mathematical development and related instructional practices.) The knowledge and interest that children show about number and shape and other mathematics topics provide an important opportunity for parents and preschool teachers to help them develop their understanding of mathematics (e.g., Gelman, 1980; Saxe, Guberman, and Gearhart, 1987; Seo and Ginsburg, 2004).

In this chapter we review research on the mathematical development of infants and young children to characterize both the resources that most children bring to school and the limitations of preschoolers’ understanding of mathematics. Because this literature is vast, it is not possible to do it justice in a single chapter. However, we attempt to provide an overview of key issues and research findings relevant to early childhood education settings. These include

  • What is the nature of early universal starting points? These are generally thought to provide an important foundation for subsequent



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3 Cognitive Foundations for Early Mathematics Learning Over the past two decades, a quiet revolution in developmental psy- chology and related fields has demonstrated that children have skills and concepts relevant to mathematics learning that are present early in life, and that most children enter school with a wealth of knowledge and cog- nitive skills that can provide a foundation for mathematics learning. At the same time, these foundational skills are not enough—children need rich mathematical interactions, both at home and at school in order to be well prepared for the challenges they will meet in elementary school and beyond. (Chapter 4 discusses supporting children’s mathematics at home, and Chapters 5 and 6 discuss children’s mathematical development and related instructional practices.) The knowledge and interest that children show about number and shape and other mathematics topics provide an important opportunity for parents and preschool teachers to help them develop their understanding of mathematics (e.g., Gelman, 1980; Saxe, Guberman, and Gearhart, 1987; Seo and Ginsburg, 2004). In this chapter we review research on the mathematical development of infants and young children to characterize both the resources that most children bring to school and the limitations of preschoolers’ understand- ing of mathematics. Because this literature is vast, it is not possible to do it justice in a single chapter. However, we attempt to provide an overview of key issues and research findings relevant to early childhood education settings. These include • What is the nature of early universal starting points? These are gen- erally thought to provide an important foundation for subsequent 5

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60 MATHEMATICS LEARNING IN EARLY CHILDHOOD mathematical development (e.g., Barth et al., 2005; Butterworth, 2005; Dehaene, 1997; but see Holloway and Ansari, 2008, and Rips, Bloomfield, and Asmuth, 2008, for contrasting views). We examine two domains that are foundational to mathematics in early child- hood: (1) number, including operations, and (2) spatial thinking, geometry, and measurement. • What are some of the important developmental changes in math- ematical understandings in these domains that occur during the preschool years? • What is the relation of mathematical development to more general aspects of development needed for learning mathematics, such as the ability to regulate one’s behavior and attention? EVIDENCE FOR EARLY UNDERSTANDING OF NUMBER Preverbal Number Knowledge Delineating the starting points of knowledge in important domains is a major goal in developmental psychology. These starting points are of theo- retical importance, as they constrain models of development. They are also of practical importance, as a basic tenet of instruction is that teaching that makes contact with the knowledge children have already acquired is likely to be most effective (e.g., Clements et al., 1999). Thus, it is not surprising that infant researchers have been actively mapping out the beginnings of preverbal number knowledge—knowledge that appears to be shared by humans from differing cultural backgrounds as well as with other species, and thus part of their evolutionary endowment (e.g., Boysen and Berntson, 1989; Brannon and Terrace, 1998, 2000; Brannon et al., 2001; Cantlon and Brannon, 2006; Dehaene, 1997; Dehaene, Dehaene-Lambertz, and Cohen, 1998, Meck and Church, 1983). A large body of research has examined a set of numerical skills, including infants’ ability to discriminate between different set sizes, their ability to recognize numerical relationships, and their ability to understand addition and subtraction transformations. The study of numerical knowledge in infants represents a major departure from previously held views, which were heavily influenced by Piaget’s (1941/1965) number conservation findings and stage theory. These older findings showed that children do not conserve number in the face of spatial transformations until school age, and they led many to believe that before this age children lack the ability to form concepts of number (see Mix, Huttenlocher, and Levine, 2002, for a review). Although Piaget recognized that children acquire some mathematically relevant skills at earlier ages, success on the conservation task was widely regarded as the sine qua non of numerical understanding.

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61 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING Beginning in the 1960s and 1970s, researchers began to actively ex- amine early numerical competencies, which led to a revised understanding of children’s numerical competence. This research identified a great deal of competence in preschool children, including counting and matching strategies that children use on Piaget’s conservation of number task (see the discussion in Chapter 5). As we detail, infant and toddler studies have largely focused on the natural numbers (also called counting numbers). However, they have also examined representations of fractional amounts and proportional relations as well as geometric relationships, shape categories, and measurement. Moreover, although there is some disagreement in the field about the in- terpretation of the findings of infant and toddler studies as a whole, these findings are generally viewed as showing strong starting points for the learning of verbal and symbolic mathematical skills. Infants’ Sensitiity to Small Set Size Infant studies typically use habituation paradigms to examine whether infants can discriminate between small sets of objects, either static or mov- ing (Antell and Keating, 1983; Starkey and Cooper, 1980; Strauss and Curtis, 1981; Van Loosbroek and Smitsman, 1990; Wynn, Bloom, and Chiang, 2002). In a typical habituation study, infants are repeatedly shown sets containing the same number of objects (e.g., 2) until they become bored and their looking time decreases to a specified criterion. The infant is then shown a different set size of objects or the same set size, and looking times are recorded. Longer looking times indicate that the infant recognizes that the new display is different from an earlier display. Results show that infants (ranging in age from 1 day old to several months old) can discrimi- nate a set of two objects from a set of three objects, yet they are unable to discriminate four objects from six objects, even though the same 3:2 ratio is involved. These findings indicate that infants’ ability to discriminate small set sizes is limited by number rather than by ratio. Huttenlocher, Jordan, and Levine (1994) suggest that infants’ ability to discriminate small sets (2 versus 3) could be based on an approximate rather than on an exact sense of number. Several studies suggest that the early quantitative sensitivity displayed by infants for small set sizes is actually based on their sensitivity to amount (surface area or contour length) which covaries with numerosity, rather than on number per se (Clearfield and Mix, 1999, 2001). That is, unless these variables are carefully controlled, the more items there are, the greater the amount of stuff there is. In studies that independently vary number and amount, Clearfield and Mix (1999, 2001) found that infants ages 6 to 8 months detected a change in amount (contour length or area) but not a

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62 MATHEMATICS LEARNING IN EARLY CHILDHOOD change in number. Thus, if they were habituated to a set of two items, they did not dishabituate to a set of three items if that set was equivalent to the original set in area or contour length. However, recent findings indicate that infants are sensitive to both continuous quantity and to number (Cordes and Brannon, 2008, in press; Kwon et al., 2009). Furthermore, Cordes and Brannon (2008) report that, although 6-month-old infants are sensitive to a two-fold change in number, they are sensitive to a three-fold change only in cumulative area across elements, suggesting that early sensitivity to set size may be more finely tuned than early sensitivity to continuous quantity. Other studies that provide support for early number sensitivity include a study showing that 6-month-old infants can discriminate between small sets of visually pre- sented events (puppet jumps) (e.g., Wynn, 1996). This result is not subject to the alternative explanation of discrimination based on amount rather than number, like the findings involving sets of objects. However, it is possible that even though the rate and duration of the events have been controlled in these studies, infants’ discrimination is based on nonnumerical cues, such as rhythm (e.g., Demany, McKenzie, and Vurpillot, 1977; Mix et al., 2002). Indeed, in one study in which the rate of motion was not a reliable cue to numerosity, 6-month-olds did not discriminate old and new numerosities (Clearfield, 2004). A set size limitation also is seen in the behavior of 10- to 14-month-olds on search tasks (Feigenson and Carey, 2003, 2005; Feigenson, Carey, and Hauser, 2002). For example, in one study 12-month-olds saw crackers placed inside two containers. The toddlers chose the larger hidden quantity for 1 versus 2 and 2 versus 3 crackers, but they failed to do so on 3 versus 4, 2 versus 4, and 3 versus 6 crackers (Feigenson, Carey, and Hauser, 2002). The authors suggest that this failure is due to the set size limitation of the object file system.1 When cracker size was varied, the toddlers based their search on the total cracker amount rather than on number. Similarly, 12- to 14-month-olds searched longer in a box in which two balls had been hidden after they saw the experimenter remove one ball, than they did in a box in which one ball had been hidden and the experimenter removed one ball (in actuality there were no more balls in either box, as the experimenter sur- reptitiously removed the remaining ball). They also succeeded on 3 versus 2 balls but failed on 4 versus 2 balls. That is, they did not search longer in a box in which four balls were hidden and they saw two removed than in a box in which they had seen two hidden and two were removed. The failure 1 The object file system refers to the representation of an object in a set that consists of small numbers, the objects are in a 1-to-1 correspondence with each mental symbol, and there is no summary representation of set size (e.g., three items are represented as “this,” “this,” “this” rather than “a set of three things”) (Carey, 2004).

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6 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING on 2 versus 4, which has the same ratio as the 2 versus 1 problem, suggests that they were using the object file system rather than the analog magnitude system, which is second system that represents large set sizes (4 or more) approximately. Furthermore, in this study, the toddlers based their search on the number of objects they saw hidden rather than on the total object volume. Thus, at least by 12 months of age, it appears that children can represent the number of objects in sets up to three (Feigenson and Carey, 2003). A subsequent study shows that this set size limit can be extended to four if spatiotemporal cues allow the toddlers to represent the sets as two sets of two (Feigenson and Halberda, 2004). Infants’ Sensitiity to Large Set Size Recent studies have shown that infants can approximate the number of items in large sets of visual objects (e.g., Brannon, 2002; Brannon, Abbott, and Lutz, 2004; Xu, 2003; Xu and Spelke, 2000; Xu, Spelke, and Goddard, 2005), events (puppet jumps) (Wood and Spelke, 2005), and auditory sets (Lipton and Spelke, 2003) that are well beyond the range of immediate apprehension of numerosity (subitizing range). Consistent with the accumulator model, which refers to a nonverbal counting mechanism that provides approximate numerical representations in the form of analog magnitudes, infants’ discrimination of large sets is limited by the ratio of the two sets being compared rather than by set size. Thus, at 6 months of age, when infants are habituated to an array of dots, they dishabituate to a new set as long as the ratio between two sets is at least 2:1. By 10 months of age, infants are able to discriminate visual and auditory sets that differ by a 2:3 ratio but not by a 4:5 ratio (Lipton and Spelke, 2003, 2004; Xu and Arriaga, 2007). Importantly, these studies controlled for many continuous variables, suggesting that the discriminations were based on number rather than amount (e.g., Brannon, Abbott, and Lutz, 2004; Cordes and Brannon, 2008; Xu, 2003; Xu and Spelke, 2000). Do Infants Hae a Concept of Number? Infants may be able to discriminate between sets of different sizes but have no notion that all sets that have the same numerosity form a category or equivalence class (the mathematical term for such a category). This no- tion is referred to as the cardinality concept (e.g., the knowledge that three flowers, three jumps, three sounds, and three thoughts are equivalent in number). Number covers such matters as the list of counting numbers (e.g., 1, 2, 3, . . .) and its use in describing how many things are in collections. It also covers the ordinal position (e.g., first, second, third, . . .), the idea of cardinal value (e.g., how many are there?), and the various operations on

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6 MATHEMATICS LEARNING IN EARLY CHILDHOOD number (e.g., addition and subtraction). The notion of 1-to-1 correspon- dences connects the counting numbers to the cardinal value of sets. Another important aspect of number is the way one writes and says them using the base 10 system (see Chapters 2 and 5 for further discussion). Knowledge of number is foundational to children’s mathematical development and gradually develops over time, so not all aspects of the number are present during the earliest years. Several studies (e.g., Starkey, Spelke, and Gelman, 1990; Strauss and Curtis, 1984) examined whether infants understand that small sets that share their numerosity but contain different kinds of entities form a cat- egory (e.g., two dogs, two chicks, two jumps, two drumbeats). Starkey and colleagues (1990) examined this question by habituating infants to sets of two or three aerial photographs of different household objects. At test, infants were shown novel photographs that alternated between sets of two and sets of three. Infants dishabituated to the novel set size, suggesting that they considered different sets of two (or three) as similar. Whereas these studies might be regarded as suggesting that infants form numerical equiva- lence classes over visual sets containing disparate objects, these studies may have tapped infants’ sensitivity to continuous amount rather than number, as described above (Clearfield and Mix, 1999, 2001). That is, unless careful controls are put in place, sets with two elements will on average be smaller in amount than sets of three elements (e.g., Clearfield and Mix, 1999, 2001; Mix et al., 2002). Findings showing that infants consider two objects and two sounds to form a category would not be subject to this criticism and thus could be considered as strong evidence for abstract number categories. In an impor- tant study, Starkey, Spelke, and Gelman (1983) tested whether infants have such categories. While the results seemed to indicate that 7-month-olds regarded sets of two (or three) objects and drumbeats as similar, several attempts to replicate these important findings have called them into ques- tion (Mix, Levine, and Huttenlocher, 1997; Moore et al., 1987). Thus, whether infants have an abstract concept of number that allows them to group diverse sets that share set size remains an open question. The find- ings, reviewed below, showing that 3-year-olds have difficulty matching visual and auditory sets on the basis of number, and that this skill is related to knowledge of conventional number words, suggest that the ability to form equivalence classes over sets that contain different kinds of elements may depend on the acquisition of conventional number skills. Kobayashi, Hiraki, Mugitani, and Hasegawa (2004) suggest that the methods used may be too abstract to tap this intermodal knowledge and that when the sounds made are connected to objects, for example, the sound of an object landing on a surface, evidence of abstract number categories may be revealed at younger ages, perhaps even in infants.

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65 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING Infant Sensitiity to Changes in Set Size Several studies report that infants track the results of numerically relevant transformations—adding or taking away objects from a set. That is, when an object is added to a set, they expect to see more objects than were previously in the set and when an object is taken away, they expect to see fewer objects than were previously in the set. Wynn (1992a) found that after a set was transformed by the addition or subtraction of an object, 5-month-old infants looked longer at the “impossible” result (e.g., 1 + 1 = 1) than at the “correct” result. However, as for numerical discrimination, subsequent studies suggest that their performance may reflect sensitivity to continuous (cumulative size of objects) amount rather than to numerosity (Feigenson, Carey, and Spelke, 2002). For the problem 1 + 1, infants looked longer at 2, the expected number of objects, when the cumulative size of the two objects was changed than at three, the impossible number of objects, when the cumulative size of the objects was correct—that is, when the cumulative area of the three objects was equivalent to the area that would have resulted from the 1 + 1 addition. Cohen and Marks (2002) suggested an alternative explanation for Wynn’s results. In particular, they suggest that the findings could be at- tributable to a familiarity preference rather than to an ability to carry out numerical transformations. For the problem 1 + 1 = 2, they point out that infants more often see one object, as there was a single object in the first display of every trial and thus, based on familiarity, look more at 1 (the incorrect answer). A similar argument was made for looking more at 2 for the problem 2 − 1. Although their findings support this hypothesis, a more recent study by Kobayashi et al. (2004) provides evidence that infants look longer at 1 + 1 = 3 and 1 + 2 = 3 than at 1 + 1 + 2 and 1 + 2 = 3 when the first addend is a visual object and the second addend consists of a tone(s). This paradigm cannot be explained by the familiarity preference because, for each prob- lem, infants see only one element on the stage. Order Relations A few studies have examined infants’ sensitivity to numerical order relations (more than, less than). One habituation study showed that 10- and 12-month-olds discriminated equivalent sets (e.g., a set of two followed by another set of two) from nonequivalent sets (e.g., a set of two fol- lowed by a set of three) (Cooper, 1984). In another study, Cooper (1984) habituated 10-, 12-, 14-, and 16-month-old infants to sequences that were nonequivalent. In the “less than” condition, the first display in the pair was always less than the second (e.g., infants were shown two objects

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66 MATHEMATICS LEARNING IN EARLY CHILDHOOD followed by three objects). The reverse order was shown for the “greater than” condition. At test, the 14- and 16-month-olds showed more inter- est in the opposite relation than the one that was shown, suggesting that they represented the less than and greater than relations, whereas 10- and 12-month-olds did not. However, Brannon (2002) presents evidence that infants are sensitive to numerical order relations by 11 months of age. Summary The results of infant studies using small set sizes show that, very early in life, infants have a limited ability to discriminate sets of different sizes from each other (e.g., 2 versus 3 but not 4 versus 6). The set size limitation has been interpreted as reflecting one of two core systems for number—the object file system. They also expect the appropriate result from small num- ber addition and subtraction transformations (e.g., 1 + 1 = 2 and 2 − 1 = 1), at least when amount covaries with number. Somewhat later, by 10 months of age, infants discriminate equivalent from nonequivalent sets, and by 14 months of age they discriminate greater than from less than relationships. Because many of these studies did not control for continuous variables that covary with number (i.e., contour length and surface area), the basis of infant discriminations is debated. However, recent studies indicate that infants are sensitive to both number of objects in small sets and to continu- ous variables, and they may be more sensitive to number than to cumulative surface area. Infant studies also have examined sensitivity to approximate number by using larger sets of items (e.g., 8 versus 16). These studies have found that infants can discriminate between sets with a 2:1 ratio by age 6 months and between those with 2:3 ratios by age 9 months as long as all set sizes involved are greater than or equal to 4, that is, 6-month-olds fail to discriminate 2 versus 4. We also note that infants’ early knowledge of number is largely implicit and has important limitations that are discussed below. There were no number words involved in any of the studies described above. This means that learning the number words and relating them to sets of objects is a major new kind of learning done by toddlers and preschoolers at home and in care and education centers. This learning powerfully extends numerical knowledge, and children who acquire this knowledge at earlier ages are provided with a distinct advantage. Mental Number Representations in Preschool Children Just as much of the infant research has a focus on theorizing about and researching the nature of infant representations of number, so, too, does some research on toddlers and preschool children. The goal is to understand

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6 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING how and when young children represent small and larger numbers. To do this, special tasks are used that involve hidden objects, so that children must use mental representations to solve the task. Sometimes objects are shown initially and are then hidden, and sometimes objects are never shown and numbers are given in words. These tasks are quite different from situations in which young children ordinarily learn about numbers in the home or in care and educational centers, and they can do tasks in home and naturalis- tic settings considerably earlier than they can solve these laboratory tasks (e.g., Mix, 2002). In home and in care and educational settings, numbers are presented with objects (things, fingers), and children and adults may see, or count, or match, or move the objects. The objects do not disappear, and they are not hidden. Children’s learning under these ordinary conditions is described in Chapter 5. Here we continue to focus on theoretical issues of representations of numbers. Small Set Sizes Like infants, 2- to 3-year-olds show more advanced knowledge of number than would be predicted by previous views. As noted previously, conservation of number was considered to be a hallmark of number de- velopment (Piaget, 1941/1965). However, Gelman’s (1972) “magic experi- ment” showed that much younger children could conserve number if the spatial transformation was less salient and much smaller set sizes were used. In this study, 3- to 6-year-olds were told that either a set of two mice or a set of three mice was the “winner.” The two sets were then covered and moved around. After children learned to choose the winner, the experimenter al- tered the winner set, either by changing the spatial arrangement of the mice or by adding or subtracting a mouse. Even the 3-year-olds were correct in recognizing that the rearrangement maintained the status of the winner, whereas the addition and subtraction transformations did not. Huttenlocher, Jordan, and Levine (1994) examined the emergence of exact number representation in toddlers. They posited that mental models representing critical mathematical features—the number of items in the set and the nature of the transformation—were needed to exactly represent the results of a calculation. Similarly, Klein and Bisanz (2000) suggest that young children’s success in solving nonverbal calculations depends on their ability to hold and manipulate quantitative representations in working memory as well as on their understanding of number transformations. Huttenlocher, Jordan, and Levine (1994) gave children ages 2 to 4 a numerosity matching task and a calculation task with objects (called non- verbal; see Box 3-1). On the matching task, children were shown a set of disks that was subsequently hidden under a box. They were then asked to lay out the same number of disks. On the calculation task, children were

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6 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 3-1 Clarifying Experimental Misnomers Researchers have used tasks in which two conditions vary in two important ways, such as in Huttenlocher, Newcombe, and Sandberg (1994). In one condi- tion, children are first shown objects, and then the objects are hidden. Number words are not used in this condition. In the other condition, children never see objects but must imagine or generate them (e.g., by raising a certain number of fingers). Here the numbers involved are conveyed by using number words, either as a story problem or just as words (e.g., “2 and 1 make what?”). In their reports, researchers call the first condition nonverbal and the second condition verbal. But these labels are a bit misleading, because they sound as if nonverbal and verbal are describing the children’s solution methods. In this report we use language that mentions both aspects that were varied: with objects (called nonverbal) and without objects (called verbal). shown a set of disks that was subsequently covered. Following this, items were either added or taken away from the original set. The child’s task was to indicate the total number of disks that were hidden by laying out the same number of disks (“Make yours like mine”). On both the matching and transformation tasks, performance increased gradually with age. Children were first successful with problems involving low numerosities, such as 1 and 2, gradually extending their success to problems involving higher numerosities. Importantly, when children re- sponded incorrectly, their responses were not random, but rather were ap- proximately correct. Approximately correct responses were seen in children as young as age 2, the youngest age group included in the study. On the basis of these findings, Huttenlocher, Jordan, and Levine (1994) argue that representations of small set sizes begin as approximate representations and become more exact as children develop the ability to create a mental model. Exactness develops further and extends to larger set sizes when children map their nonverbal number representations onto number words. Toddlers’ performance on numerosity matching tasks indicates that, as they get older, they get better at representing quantity abstractly. This achievement appears to be related to the acquisition of number words (Mix, 2008). Mix showed that preschoolers’ ability to discriminate numerosi- ties is highly dependent on the similarity of the objects in the sets. Thus, 3-year-olds could match the numerosities of sets consisting of pictures of black dots to highly similar black disks. Between ages 3 and 5, children were able to match the numerosities of increasingly dissimilar sets (e.g., black dots to pasta shells and black dots to sequential black disks at age 3½; black dots to heterogeneous sets of objects at age 4).

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6 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING The abstractness of preschoolers’ numerical representations was also assessed in a study (Mix, Huttenlocher, and Levine, 1996) examining their ability to make numerical matches between auditory and visual sets, an ability that Starkey, Spelke, and Gelman (1990) had attributed to infants. The researchers presented 3- and 4-year-olds with a set of two or three claps and were asked to point to the visual array that corresponded to the number of claps. The 3-year-olds performed at chance on this task, but by age 4, the children performed significantly above chance. In contrast, both age groups performed above chance on a control task that involved match- ing sets of disks to pictures of dots. Another study assessed the effect of the heterogeneity of sets on the ability of 3- to 5-year-olds to make numerical matches and order judgments. The results replicate Mix’s (1999b) finding that the heterogeneity of sets decreases children’s ability to make equiva- lence matches. However, heterogeneity versus homogeneity of sets did not affect their ability to make order judgments (i.e., to judge which of two sets is smaller) (Cantlon et al., 2007). Mix (2002) has also examined the emergence of numerical knowledge through a diary study of her son, Spencer. In this study, she found indica- tions of earlier knowledge than the experiments described above might indicate. Spencer was able to go into another room and get exactly two dog biscuits for his two dogs at 21 months of age, long before children succeed on the homogeneous or heterogeneous set matching tasks described above. Indeed, Spencer himself had failed to perform above chance on these laboratory tasks. Thus, it appears that early knowledge of numerical equivalence may arise piecemeal, and first in highly contextualized situa- tions. For Spencer, his earliest numerical equivalence matches occurred in social situations (e.g., biscuits for dogs, sticks for guests). Whether this is a general pattern or whether there are wide individual differences in such behaviors is an open question (also see Mix, Sandhofer, and Baroody, 2005, for a review). Levine, Jordan, and Huttenlocher (1992) compared the ability of pre- school children to carry out calculations involving numerosities of up to six with objects (called nonverbal) and without objects (called verbal) (the for- mer calculations were similar to those described above in the Huttenlocher, Jordan, and Levine, 1994 study). The calculations without objects (called verbal) were given in the form of story problems (“Ellen has 2 marbles and her father gives her 1 more. How many marbles does she have altogether?”) and in the form of number combinations (e.g., “How much is 2 and 1?”). Children ages 4 to 5½ performed significantly higher on the calculation task when they could see objects and transformations than on the calculation tasks when they could not see objects or transformations. This was true for both addition and subtraction calculations. This difference in perfor- mance between nonverbal and verbal calculations was particularly marked

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 MATHEMATICS LEARNING IN EARLY CHILDHOOD Heads-to-Toes task asks children to do the opposite of what the instructor tells them. So, for example, if the instructor asks the children to touch their head, they are to touch their toes. This task measures behavioral regula- tion (a component of self-regulation), in that it requires children to employ inhibitory control, attention, and working memory. The researchers found that behavioral regulation scores significantly predicted emergent math scores. The researchers conclude that “strengthening attention, working memory, and inhibitory control skills prior to kindergarten may be an ef- fective way to ensure that children also have a foundation of early academic skills” (p. 956). Espy and colleagues (2004) specifically studied the roles of working memory and inhibitory control with almost 100 preschoolers. They found that both components of executive function contributed to the children’s mathematical proficiency, with inhibitory control being the most prominent. Passolunghi and colleagues (2007) studied 170 6-year-olds in Italy. They examined the roles of working memory, phonological ability, numerical competence, and IQ in predicting math achievement. They found that working memory skills significantly predicted math learning at the beginning of elementary school (primary school in Italy). SUMMARY This chapter underscores that young children have more mathematics knowledge, in terms of number and spatial thinking, than was previously believed. Very early in life, infants can distinguish between larger set sizes, for example 8 versus 16 items, but their ability to do so is only approxi- mate and is limited by the ratio of the number of items in the sets. The set size limitation is thought to reflect one of the two core systems for number (Feigenson, Dehaene, and Spelke, 2004; Spelke and Kinzler, 2007). Further- more, young infants’ early knowledge of quantity is implicit, in that they do not use number words, which means that learning number words and relating them to objects is one of the major developmental tasks to occur during early childhood. Toddlers and preschool children move from the implicit understand- ing of number seen during infancy to formal number knowledge. Spoken number words, written number symbols, and cultural solution methods are important tools that support this developmental progression. Young children also learn about space, including shapes, locations, distances, and spatial relations, which also go through major development during the early childhood years. Children’s acquisition of spatial language plays an important role in the development of spatial categories and skills. In addition to learning about number and shape, early childhood also includes development of measurement, which is a fundamental aspect of mathematics that connects geometry and number. Young children’s under- standing of measurement begins with length, which is perceptually based,

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5 COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING and an important feature of their learning during this period is that they have difficulty understanding units of measure. Young children can become successful at this when given appropriate instruction. It is also important to note that across early childhood, mathematical development that is situated in an environment that promotes regulation of cognitive activities and behavior can improve mathematical development. More specifically, when young children have an opportunity to practice staying on task, to keep information in mind while manipulating or chang- ing it mentally, and to practice shifting between differing tasks, mathematics learning is improved and in turn improves these regulatory processes. Although we discuss universal starting points for mathematics devel- opment in this chapter, there are, of course, differences in children’s math- ematical development. The next chapter explores variation in children’s mathematical development and learning outcomes and the sources of this variation. We also discuss the role of the family and informal mathematics learning experiences in supporting children’s mathematical development. REFERENCES AND BIBLIOGRAPHY Aguiar, A., and Baillargeon, R. (1998). Eight-and-a-half-month-old infants’ reasoning about containment events. Child Deelopment, 6, 636-653. Ansari, D., Donlan, C., Thomas, M.S.C., Ewing, S.A., Peen, T., and Karmiloff-Smith, A. (2003). What makes counting count: Verbal and visuo-spatial contributions to typi- cal and atypical number development. Journal of Experimental Child Psychology, 5, 50-62. Antell, S.E., and Keating, L.E. (1983). Perception of numerical invariance by neonates. Child Deelopment, 5, 695-701. Baillargeon, R. (1991). Reasoning about the height and location of a hidden object in 4.5 and 6.5 month-old children. Cognition, , 13-42. Baillargeon, R. (1995). Physical reasoning in infancy. In M.S. Gazzaniga (Ed.), The Cognitie Neurosciences (pp. 181-204). Cambridge, MA: Bradford Press. Barth, H., LaMont, K., Lipton, J. and Spelke, E.S. (2005). Abstract number and arith- metic in preschool children. Proceedings of the National Academy of Sciences, 102, 14116-14121. Barth, H., LaMont, K., Lipton, J., Dehaene, S., Kanwisher, N., and Spelke, E. (2006). Non- symbolic arithmetic in adults and young children. Cognition, , 199-222. Blair, C., and Razza, R.P. (2007). Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Child Deelopment, , 647-663. Blair, C., Knipe, H., Cummings, E., Baker, D.P., Gamson, D., Eslinger, P., and Thorne, S.L. (2007). A developmental neuroscience approach to the study of school readiness. In R.C. Pianta, M.J. Cox, and K.L. Snow (Eds.), School Readiness and the Transition to Kinder- garten in the Era of Accountability (pp. 149-174). Baltimore, MD: Paul H. Brookes. Bomba, P.C., and Siqueland, E.R. (1983). The nature and structure of infant form categories. Journal of Experimental Child Psychology, 5, 294-328. Boulton-Lewis, G.M. (1987). Recent cognitive theories applied to sequential length measuring knowledge in young children. British Journal of Educational Psychology, 5, 330-342. Boysen, S.T. and Berntson, G.G. (1989). Numerical competence in a chimpanzee (Pan troglo- dytes). Journal of Comparatie Psychology, 10, 23-31.

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