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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 enoughâchildren 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 childrenâs mathematics at home, and Chapters 5 and 6 discuss childrenâs 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 preschoolersâ 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 â¢ What is the nature of early universal starting points? These are gen- erally thought to provide an important foundation for subsequent 59
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. â¢ What are some of the important developmental changes in math- ematical understandings in these domains that occur during the preschool years? â¢ What is the relation of mathematical development to more general aspects of development needed for learning mathematics, such as the ability to regulate oneâs 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 knowledgeâknowledge 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 infantsâ 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 Piagetâs (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.
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 61 Beginning in the 1960s and 1970s, researchers began to actively ex- amine early numerical competencies, which led to a revised understanding of childrenâs numerical competence. This research identified a great deal of competence in preschool children, including counting and matching strategies that children use on Piagetâs 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. Infantsâ Sensitivity 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 infantsâ ability to discriminate small set sizes is limited by number rather than by ratio. Huttenlocher, Jordan, and Levine (1994) suggest that infantsâ 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
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, infantsâ 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. 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 â 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 âthis,â âthis,â âthisâ rather than âa set of three thingsâ) (Carey, 2004).
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 63 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). Infantsâ Sensitivity 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, A Â bbott, 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, infantsâ 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, ÂAbbott, and Lutz, 2004; Cordes and Brannon, 2008; Xu, 2003; Xu and Spelke, 2000). Do Infants Have 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
64 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 childrenâs 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 infantsâ 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, H Â iraki, 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.
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 65 Infant Sensitivity to Changes in Set Size Several studies report that infants track the results of numerically relevant transformationsâadding 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 âimpossibleâ result (e.g., 1 + 1 = 1) than at the âcorrectâ 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 correctâthat 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 Wynnâs 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 â 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 infantsâ sensitivity to numerical order r Â elations (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) habituÂated 10-, 12-, 14-, and 16-month-old infants to sequences that were nonequivalent. In the âless thanâ condition, the first display in the pair was always less than the second (e.g., infants were shown two objects
66 MATHEMATICS LEARNING IN EARLY CHILDHOOD followed by three objects). The reverse order was shown for the âgreater thanâ 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 numberâthe object file system. They also expect the appropriate result from small num- ber addition and subtraction transformations (e.g., 1 + 1 = 2 and 2 â 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 infantsâ 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
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 67 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. Childrenâs 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, Gelmanâs (1972) âmagic experi- mentâ 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 âwinner.â 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 featuresâthe number of items in the set and the nature of the transformationâwere needed to exactly represent the results of a calculation. Similarly, Klein and Bisanz (2000) suggest that young childrenâs 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
68 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., â2 and 1 make what?â). 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 childrenâs 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 childâs task was to indicate the total number of disks that were hidden by laying out the same number of disks (âMake yours like mineâ). 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. Toddlersâ 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 preschoolersâ 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 3Â½; black dots to heterogeneous sets of objects at age 4).
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 69 The abstractness of preschoolersâ 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 Mixâs (1999b) finding that the heterogeneity of sets decreases childrenâs 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 (âEllen has 2 marbles and her father gives her 1 more. How many marbles does she have altogether?â) and in the form of number combinations (e.g., âHow much is 2 and 1?â). Children ages 4 to 5Â½ 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
70 MATHEMATICS LEARNING IN EARLY CHILDHOOD for children from low socioeconomic backgrounds. Children from low socioeconomic backgrounds appear to have more difficulty accessing the numerical meaning of the number words (Jordan et al., 2006), which may be related to their exposure to cultural learning tools (e.g., number symbols, number words) (see Chapter 4 for further discussion). Large Set Sizes To investigate how preschoolers carry out approximate calculations with large numbers, 5-year-olds were presented with comparison and addi- tion problems shown on a computer screen (Barth et al., 2005, 2006). On comparison problems, they were shown a set of blue dots (set sizes ranged from 10 to 58) that were then covered up. Next, they were shown a set of red dots and were asked whether there are more blue dots or red dots. On addition problems, they were shown a set of blue dots that were then cov- ered up. They were then shown another set of blue dots that moved behind the same occluder. Finally, they were shown a set of red dots and were asked whether there were more blue dots or red dots. Subsequent experiments showed that children performed as well when the red dots were presented as a sequence of auditory tones as when they were presented visually. In each condition, performance was above chance and equivalent on compari- son and addition problems, decreasing as the ratio of the red to blue dots approached 1. The ratio dependence of performance indicates that children are using the analogue magnitude system. This system differs from the exact representations of larger numbers that are built up by working with objects arranged in groups of tens and ones (see Chapter 5). Summary Toddlers and preschoolers continue to build on the two representa- tional systems identified for infants: the object file system, which is limited to sets of three or less and provides a representation for each element in a set but no summary representation of set size, and the analogue magnitude system, which provides an approximate summary representation of set size but no representation of the individual elements in a set and no way to differentiate between adjacent set sizes, such as 10 and 11 (Carey, 2004; Feigenson, Dehaene, and Spelke, 2004; Spelke and Kinzler, 2007). Existing research also shows that childrenâs early numerical knowledge is highly context-dependent, often depending on the presence of objects or fingers to represent sets. Although their numerical abilities are limited, young children have considerably more numerical competence than was inferred from Piagetâs research. They are even building early informal knowledge in many other mathematics areas besides representation of the counting
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 71 numbers (see sections below). However, the path from informal to formal knowledge is not necessarily a smooth one. Impressive growth of numerical competence from age 2 to age 6 is stim- ulated by childrenâs learning of important cultural numerical tools: spoken number words, written number symbols, and cultural solution methods, like counting and matching. As shown by Wynnâs (1990, 1992b) research, the acquisition of the understanding of the cardinal meanings of number words is a protracted process. In a longitudinal study, Wynn found that it takes about a year for a child to move from succeeding in giving a set of âoneâ when requested to do so to being able to give the appropriate num- ber for all numbers in his or her count list. The acquisition of such symbolic knowledge is important in promoting the abstractness of number concepts, that is, the concept of cardinality (that all sets of a given numerosity form an equivalence class). It is also important in promoting the exactness of number representations and the understanding of numerical relations, as only children who have acquired this knowledge understand that adding one item to a set means moving to the very next number in the count list (Sarnecka and Carey, 2008). The research concerning these cultural learn- ing achievements is summarized in Chapter 5 in identifying foundational and achievable goals for teaching and learning. It is discussed in Chapter 4 as a major source of socioeconomic differences, connected to differential exposure to talk about number at home and at preschool. DEVELOPMENT OF SPATIAL THINKING AND GEOMETRY Spatial thinking, like numerical thinking, is a fundamental component of mathematics that has its roots in foundational skills that emerge early in life. Spatial thinking is critical to a variety of mathematical topics, including geometry, measurement, and part-whole relations (e.g., Ansari et al., 2003; Fennema and Sherman, 1977, 1978; Guay and McDaniel, 1977; Lean and Clements, 1981; Skolnick, Langbort, and Day, 1982; see Chapter 6, this volume). Spatial thinking has been found to be a significant predictor of achievement in mathematics and science, even controlling for overall verbal and mathematical skill (e.g., Clements and Sarama, 2007; Hedges and Chung, in preparation; Lean and Clements, 1981; Shea, Lubinski, and Benbow, 2001; Stewart, Leeson, and Wright, 1997; Wheatley, 1990). One reason that spatial thinking is predictive of mathematics and science achievement is because it provides a way to conceptualize relationships in a problem prior to solving it (Clements and Sarama, 2007). The mental functions encompassed by spatial thinking include catego- rizing shapes and objects and encoding the categorical and metric relations among shapes and objects. Spatial thinking is also crucial in representing object transformations and the outcomes of these transformations (e.g.,
72 MATHEMATICS LEARNING IN EARLY CHILDHOOD rotation, translation, magnification, and folding) as well as perspective changes that occur as one moves to new locations. Spatial thinking is in- volved in navigating in the environment to reach goal locations and to find oneâs way back to oneâs starting point. Use of spatial symbolic systems, including language, maps, graphs, and diagrams, and spatial tools, such as measuring devices, extend and refine the ability to think spatially. As is the case for the development of number knowledge, recent re- search has shown strong starting points for spatial thinking. In contrast to Piagetâs view, which is in opposition to the gradual unfolding of spatial skills over the course of development, recent evidence shows that infants are able to code spatial information about objects, shapes, distances, locations, and spatial relations. This early emergence of spatial skills is consistent with an evolutionary perspective that emphasizes the adaptive importance of navigation for all mobile species (e.g., Newcombe and Huttenlocher, 2000, 2006; Wang and Spelke, 2002). That said, humans are unique in that their spatial skills are extended through symbolic systems, such as spatial language, measurement units, maps, graphs, and diagrams. Thus, it is not surprising that the trajectory of childrenâs spatial development depends heavily on their spatially relevant experiences, including those involving spatial language and spatial activities, such as block building, puzzle play, and experience with certain video games. Starting Points in Infancy Even young infants are able to segment their complex visual envi- ronments into objects that have stable shapes, using such principles as cohesion, boundedness, and rigidity (e.g., see Spelke, 1990). Infants also perceive the similarities between three-dimensional objects and photographs of these objects (DeLoache, Strauss, and Maynard, 1979). In habituation studies, infants show sensitivity to shape similarities across exemplars (e.g., Bomba and Siqueland, 1983). In addition, they are able to recognize invari- ant aspects of a shape shown from different angles of view (e.g., Slater and Morison, 1985). Infants are also capable of forming categories of spatial relationsâa claim that is widely supported; however, different views exist regarding the developmental sequence for childrenâs understanding of space categories (Quinn, 1994, 2004; van Hiele, 1986). As stated by Bruner, Goodnow, and Austin (1956), categorization entails treating instances that are discrim- inable as the same. Using this criterion, Quinn showed that 3-month-old infants are sensitive to the categories above versus below (e.g., Quinn, 1994) and left versus right (e.g., Quinn, 2004). Both of these categories involve the relationship of an object and a single referent object (e.g., a horizontal or vertical bar). However, infants are not able to code the rela-
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 73 tionship between an object and a diagonal bar, showing that certain kinds of spatial relationship are privileged over others. Somewhat later, at 6 to 7 months, they are sensitive to the category of between relationships (Quinn et al., 1999). This spatial category is more complex than above/below or left/right, as it involves the relation of an object to two referent objects (e.g., two bars). At around this same age, infants form other, rather subtle spatial concepts. For example, they are sensitive to the functional differ- ence between a container and a cylindrical object that does not have a bottom, even though these objects are highly similar visually (Aguiar and B Â aillargeon, 1998; Baillargeon, 1995). Infants and toddlers also have impressive abilities to locate objects in space using both landmarks and geometric cues. Infants as young as 5 months are also able to use enclosed spaces that define a shape (e.g., walls of a sandbox) to code the location of objects (Newcombe, Huttenlocher, and Learmonth, 1999). By 12 months, children code distance and direction and use this information to search for objects hidden in displays (Bushnell et al., 1995). By 16 to 17 months, they are able to use the rectangular shape of an enclosure as well as landmark cues (both adjacent to the hiding loca- tion and at a distance from it) to search for objects (Hermer and Spelke, 1994, 1996; Huttenlocher, Newcombe, and Sandberg, 1994; Learmonth, Newcombe, and Huttenlocher, 2001). Mental Transformation of Shapes Mental rotation (the ability to visualize and manipulate the movement of two-dimensional and three-dimensional objects) and spatial visualization (holding a shape in mind and finding the shape in more complex figures, combining shapes, or matching orientations) are fundamental spatial skills essential for mathematics learning (Linn and Peterson, 1985). Several re- cent studies have shown that preschool children are able to mentally rotate shapes in the picture plane. In one study, Marmor (1975) showed that children as young as age 5 years are able to mentally rotate visual images in the picture plane to determine whether one image is the same as another. Similarly, Levine et al. (1999) showed that children as young as age 4Â½ are able to perform above chance on mental transformations involving rotation and translation. In tasks requiring spatial visualization (e.g., holding an image, such as a block letter, in mind for later comparison to a standard block letter), children between ages 4 and 5 perform poorly unless the visualized image is in the same orientation as the comparison object, whereas children be- tween ages 6 and 10 were not adversely affected by differences in orienta- tion (Smothergill et al., 1975). Furthermore, spatial ability in manipulating orientation at age 7, but not at ages 3 to 5 (Rod-and-Frame Test, Preschool
74 MATHEMATICS LEARNING IN EARLY CHILDHOOD Embedded Figures Test), predicted spatial visualization abilities much later, at age 18 (Ozer, 1987). This developmental shift in spatial visualization ability is most likely to reflect differences in mental rotation ability and perspective-taking. Thus, when children are better able to mentally ma- nipulate images held in mind (e.g., imagining the letter âFâ and mentally rotating it clockwise or counterclockwise), they will be more accurate at determining how these images will appear from various viewpoints. Similarly, when a child is asked to imagine what an object would look like from another personâs perspective, this task is more easily accomplished when the child can mentally imagine the scene and move either themselves or the objects in order to match another personâs perspective of the scene. For example, a child is sitting at a desk that has a toy car to the left of a pencil on top of the desk. A teacher is sitting on the other side of the desk, opposite the child, and asks the child to arrange the toy car and the pencil so that they would match what the teacher sees. The task becomes easier if the child can imagine the desk with the two objects and mentally âwalkâ to the other side of the desk to figure out the answer (pencil on the left, toy car on the right) or can imagine the objects and mentally rotate them so that they are in the 180 degree position. As mental rotation and perspective-taking ability increase over time, such factors as changes in orientation become less problematic in tasks in which one must match something displayed in a different orientation than the visualized object. The fact that early spatial visualization measures during preschool were not correlated with later spatial visualization may suggest that the foundations for spatial abilities, such as mental rotation and perspective-taking, are molded in these formative years and are highly susceptible to change, more so than during later elementary education. This has important implications for findings that display gender differences in spatial performance on such tasks as mental rotation by age 4Â½ (Levine et al., 1999) and socioeconomic differences by second grade (Levine et al., 2005). That is, these differences in spatial ability may largely be the result of experiential differences during early childhood, and the preschool pe- riod may be an especially important time to begin addressing these issues through educational programs that foster spatial learning. The early emergence of mental rotation ability may be related to preschoolersâ success with map use. Given simple maps, 4-year-olds and a Â majority of 3-year-olds can locate a hidden object in a sandbox (ÂHuttenlocher, Newcombe, and Vasilyeva, 1999), children ages 3 to 5Â½ â Recent evidence shows a sex difference in mental rotation for 4- and 5-month-old infants that is not attributable to experimental factors (see Moore and Johnson, 2008; Quinn and Liben, 2008). Implications from these studies suggest there may be an advantage in early spatial learning for boys.
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 75 can find a Âhidden toy in an open field (Stea et al., 2004) and children ages 5 and 6 can navigate the hallways of an unfamiliar school (Sandberg and H Â uttenlocher, 2001). In order to succeed on these tasks, children must recognize the correspondence between the map and the actual space of a similar shape, scale distance (which we discuss further in the section on measurement), and perform mental rotation of the map with respect to ac- tual space. Successful use of maps among preschoolers has occurred when the maps were oriented with respect to the space and mental rotation was limited to the vertical plane (in order to match ground-based perception of the space). Increasing the complexity of mental rotations required to realign spaces causes maps to become increasingly difficult for preschool children and is most likely to explain some of the difficulty children show in interpreting maps even into the elementary school years (Liben and Downs, 1989; Liben and Yekel, 1996; Piaget and Inhelder, 1967; Uttal, 1996; Wallace and Veek, 1995). Although the level of sophistication in mental transformation matures dramatically throughout childhood, the initial abil- ity to mentally transform objects in space at the preschool age allows for productive interactions with spatial representations, such as maps. Learning Spatial Terms: Relation to Spatial Mathematical Skills As summarized above, infants form spatial categories from an early age (e.g., Quinn, 1994, 2004). These visual categories may lay the foundation for the later learning of the spatial terms that label these categories (e.g., Mandler, 1992). However, it is also possible that linguistic input guides the learning of spatial concepts, highlighting certain preverbal spatial concepts and not others, perhaps leading to the formation of new spatial concepts. An example of how language can shape a preexisting nonverbal concept is provided by recent evidence showing that English-speaking infants form categories for tight/loose fit, a relation that is labeled in Korean but not in English (e.g., Casasola and Cohen, 2002; Hespos and Spelke, 2004; McDonough, Choi, and Mandler, 2003). By 29 months of age, English- speaking infants still categorized tight-fit containment relations when these were contrasted with loose-fit containment, but they no longer categorized loose-fit containment. By adulthood, English speakers do not pay atten- tion to fit, categorizing tight and loose fit as âinâ (McDonough, Choi, and Mandler, 2003). Thus, in this case exposure to English seems to play a selective function, highlighting some preexisting categories (in versus on) while downplaying others (tight fit/loose fit). Exposure to spatial language during spatial experiences also appears to be particularly useful in âthe learning and retention [of spatial concepts by] . . . inviting children to store the information and its labelâ (Gentner, 2003, pp. 207-208). Gentner found that children who heard specific spatial
76 MATHEMATICS LEARNING IN EARLY CHILDHOOD labels during a laboratory experiment that involved hiding objects (âIâm putting this on/in/under the boxâ) were better able to find the objects than children who heard a general reference to location (âIâm putting this hereâ). Moreover, this was true even two days later, without further exposure to the spatial language (Loewenstein and Gentner, 2005). Similarly, Szechter and Liben (2004) observed parents and children in the lab as they read a childrenâs book with spatial-graphic content. These researchers found a re- lation between the frequency with which parents drew childrenâs attention to spatial-graphic content during book reading (e.g., âthe rooster is really tiny nowâ) and childrenâs performance on spatial-graphic comprehension tasks. Cannon, Levine, and Huttenlocher (2007) have also examined the parentsâ use of spatial language during puzzle play in a longitudinal study in which parent-child dyads were observed during naturalistic interactions every four months from age 26 to 46 months. Their findings show that puzzle play is correlated with childrenâs mental rotation skill at 54 months for boys and girls. However, for girls but not boys, amount of parent spatial language during puzzle play (controlling for overall language input) is also a significant predictor of mental rotation skill at 54 months. This finding may be related to gender differences in the way in which spatial information is coded (e.g., Kail, Carter, and Pellegrino, 1979; Lourenco, Huttenlocher, and Fabian, under review). Understanding of Geometric Shape and Shape Composition Various proposals have influenced views on childrenâs shape categories. Piaget and Inhelder (1967) proposed a developmental sequence in which children first discriminate objects on the basis of topological features (e.g., a closed shape, which has an internal space defined by the closed boundary, versus an open shape, which has no defined internal or external boundaries) and only later on the basis of Euclidean features, such as rectilinear versus curvilinear. Still later, according to this theory, children are able to discrimi- nate among rectilinear shapes (e.g., squares and diamonds). However, this sequence has been called into question on the basis of evidence that young children are able to represent the projective (e.g., curvilinear or rectilinear) as well as the Euclidean aspects of shape (e.g., Clements and Battista, 1992; Ginsburg et al., 2006; Kato, 1986; Lovell, 1959). A different stage framework, proposed by van Hiele (1986), posits that children first identify shapes at the visual level on the basis of their appear- ance, then represent shapes at the âdescriptiveâ level on the basis of their properties, and finally progress to more formal kinds of geometric thinking that are based on logical reasoning abilities. Consistent with van Hieleâs first stage, preschoolersâ early shape categories are centered on prototypes
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 77 and the similarity of perceptual surface qualities of a shape are used to determine category inclusion. For example, preschoolers do not accept an inverted triangle as a triangle or nonisosceles triangles as triangles (e.g., Clements et al., 1999). Moreover, they tend to regard squares as a distinct category and not as a special kind of rectangle with four sides that are equal in length (Clements et al., 1999). Preschoolers sometimes overextend shape labels to nonexemplars. For example, they sometimes extend the label ârectangleâ to right trapezoids as well as to nonrectangular parallelograms that have two sides that are much longer than the other two (Hannibal and Clements, 2008). By the elementary school years, childrenâs shape categories incorporate deeper knowledge of rules and theories that are definitional (Burger and Shaughnessy, 1986; Satlow and Newcombe, 1998). The timing of the shift from relying on characteristic perceptual features to relying on defining fea- tures varies depending on the shape. For example, Satlow and Newcombe (1998) report that this shift occurs between ages 3 and 5 for circles and rectangles, prior to second grade for triangles, and during second grade for pentagons. During the preschool years, the main change in shape categories is an increasing tendency to accept atypical exemplars of shapes as members of the categoryâthat is, to extend shape categories beyond prototypical examples (e.g., Burger and Shaughnessy, 1986; Usiskin, 1987). The ability to broaden shape categories to include nonprotoypical examples depends on exposure to a variety of exemplars rather than to just prototypical examples such as equilateral and isosceles triangles (e.g., Clements et al., 1999). Neither Piagetâs nor van Heileâs stage theories recognize preschool- ersâ ability to represent and categorize shapes. Childrenâs learning of specific spatial terms also helps highlight spatial categories. These spatial terms include shape words (e.g., circle, square, tri- angle, rectangle), as well as words describing spatial features (e.g., curved, straight, line, side, corner, angle), spatial dimensions (e.g., big, little, tall, short, wide, narrow), and spatial relationships (e.g., in front of, behind, next to, between, over, under). Between ages 2 and 4, children learn terms for novel shapes more readily than other features, such as novel color or texture words (Heibeck and Markman, 1987; OâHanlon and Roberson, 2006). Fuson and Murray (1978) found that over 60 percent of 3-year-old children could name a circle, a square, and a triangle. By age 5, 85 percent of children could name a circle, 78 percent a square, and 80 percent a triangle. In addition, 44 percent could correctly name a rectangle (Klein, Starkey, and Wakesley, 1999). In a shape word comprehension study, results were similar. Clements et al. (1999) report that over 90 percent of children, ranging in age from 3 years 5 months to 4 years 4 months, could correctly point out a circle, and by age 6 years, 99 percent of children could do so. Only
78 MATHEMATICS LEARNING IN EARLY CHILDHOOD a few children in the younger group incorrectly chose an ellipse or another curved shape. For a square these numbers were also high yet somewhat lower: 82 percent of children in the younger group responded correctly, and 91 percent of 6-year-olds did so. Some children in the younger group incorrectly identified nonsquare rhombi as squares. Accuracy for triangles and rectangles was significantly lower (60 and 50 percent, respectively, for children ranging in age from 4 to 6). Children also learn spatial words for shape dimensions (e.g., big, small, tall, short, wide, narrow) and words for the relationships of shapes (e.g., in, on, under, in front of, behind, between). For example, Clark (1972) reports that for each pair of dimensional adjectives, children learned the unmarked term before the marked term, that is, they learned big before little. Note that asking how big something is does not presuppose its being big or little, whereas asking how little something is carries the presupposition that one is asking about little things. The same is true for other pairs such as tall/short. The learning of these terms, like other words, is highly related to their fre- quency of occurrence in child-directed speech (e.g., Levine et al., 2008). Children who hear greater amounts of spatial language have been found to perform at higher levels on a variety of nonverbal spatial tasks, including the WPPSI-3 Block Design subtest and a mental rotation task (Levine et al., 2008). This correlation may rest on the association of spatial language and spatial activities. Furthermore, spatial language may serve to focus childrenâs attention on spatial relationships and lead to deeper pro- cessing of this information (e.g., forming categories of shapes and spatial relations). However, parentsâ spatial language to 3- to 5-year-old children has been found to occur more frequently during such activities as block and puzzle play than during other activities, such as book reading (Levine et al., 2008; Shallcross et al., 2008). Furthermore, higher amounts of parent spa- tial language occur during guided block play in which there is a goal than during free play with blocks (Shallcross et al., 2008). Thus, it is possible that spatial activities, spatial language, or both promote the development of spatial skills, such as block building and mental rotation. Summary As for number, there are strong starting points during infancy for learn- ing about space, including shapes, locations, distances, and spatial rela- tions. These early starting points, however, like those for number, undergo major developments during the preschool years and beyond. Moreover, developmental rates and the competencies achieved are highly dependent on access to spatial activities, spatial language, and learning opportunities at home and at school. Children are equipped to comprehend and reason about shape at an
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 79 earlier age and in more complex detail than originally thought. By pre- school, they benefit from learning about a variety of shapes, both typical and atypical, and this knowledge is impacted by their acquisition of spatial language. Language input and spatial activities appear to be highly influ- ential in the development of spatial categories and spatial skills during the preschool years. DEVELOPMENT OF MEASUREMENT Measurement is a fundamental aspect of mathematics, which âbridges two main areas of school mathematicsâgeometry and numberâ through the attachment of number to spatial dimensions (National Council of Teachers of Mathematics, 2000). The development of measurement skills usually starts with directly comparing objects along one dimension. Thus, children generally succeed in measuring length prior to area and volume (Hart, 1984; but see Curry and Outhred, 2005, for early success in measur- ing volume when the task involves successive filling of a container). Certain skills, such as sensitivity to variations in amount, can be thought of as precursors to mature measurement skills and have been observed in infants. The ability to directly compare the lengths of objects is an early emerging skill and initially appears to be perceptually based (Boulton-Lewis, 1987). Infants demonstrate awareness of variations in amount in one di- mension (e.g., noticing height) as early as 4 months (Baillargeon, 1991) and can discriminate between two objects based on height at 6 months (Gao, Levine, and Huttenlocher, 2000). For example, 6-month-old infants and 2-year-old toddlers are able to discriminate the length of dowels when they appear in the presence of a constant, aligned standard but not when there is no standard available with which to compare them (Huttenlocher, Duffy, and Levine, 2002). Subsequent studies show that infants and toddlers are responding to the relative size of the standard and the test objects (Duffy, Huttenlocher, and Levine, 2005a, 2005b). This result is in line with the theory (Bryant, 1974) that relative coding precedes absolute coding. The ability to discriminate lengths in a more precise manner (distinguishing two heights that are fairly close without a present, aligned standard) develops some time between ages 2 and 4. However, even by age 4, childrenâs sensitivity to variations in size is often influenced by the relation between two objects. This early reliance on a standard to assess size may seem to contrast with findings by Piaget and his colleagues showing that young children do not spontaneously use a standard to measure objects (Piaget, Inhelder, and Szeminska, 1960). Piaget and colleagues argue that before school age, chil- drenâs ability to encode metric information is limited because they lack the ability to make transitive inferences that are involved in measurementâthat
80 MATHEMATICS LEARNING IN EARLY CHILDHOOD is, if A = B (the measure) and B = C, then A and B are equivalent. How- ever, unlike Piaget and colleaguesâ task, in which the child was required to spontaneously use a stick to compare the heights of two towers that were not aligned, the experiments showing much earlier skill involve a visually aligned standard. So far we have been discussing the development of the ability to dis- criminate linear extents and not the understanding of equivalence/nonequiv- alence of these extents, or sensitivity to amount transformations (adding or subtracting amounts). Although, as reviewed above, researchers have examined these topics with respect to discrete sets, there is little work on these topics with respect to continuous amounts. However, some evidence indicates that the ability to order continuous amounts is present at least by the preschool years. For example, Brainerd (1973) found that kindergart- ners could arrange three balls of clay according to weight and could arrange three sticks according to length. Understanding Units and Conventional Measurement Early sensitivity to linear extent in relation to a standard is far from the mature ability to measure length. It is not until age 8 that children typically succeed in discriminating between objects of different lengths when there is not a constant aligned standard present. This ability is much closer to conventional measurement than the skill displayed by children up to age 4 (Duffy et al., 2005a). These changes in sensitivity to variation in amount from age 4 to age 8 may be related to exposure to conventional measure- ment at school and the ability to form and maintain images with certain attributes. However, developing a sophisticated conceptual understanding of linear measurement has a surprisingly long developmental time course (e.g., Copeland, 1979; Hiebert, 1981, 1984; Miller, 1984, 1989). Conventional measurement involves several basic operations. First, it is important to realize that the units must be equal in size and must be specified. Second, the chosen unit must be repeated if it is smaller than the object being measured. Finally, the chosen unit must be subdivided when a whole unit does not fully cover the object or the remaining part of an object (Nunes, Light, and Mason, 1993). Young children have difficulty understanding the importance of using an equal size unit. Miller (1984) showed that preschoolers between ages 4 and 5 have difficulty appreciating that the size of pieces (or units) must remain constant in measurement situations. In a well-known example, Miller found that preschool children who are asked to divide candy evenly among children consider it fair to break the last piece in half if they run out of pieces. In other words, as long as everyone gets a piece, they are not concerned that the pieces are unequal in size. In a study in which 5- and 6-year-olds were asked to make rulers by writing in the numbers, Nunes
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 81 and Bryant (1996) found that they failed to space the numbers even ap- proximately equally. In another study reported by Nunes and Bryant, children ages 5 to 7 had no trouble answering whether a 7 cm or 6 cm ribbon is longer. However, when asked whether a 2 inch or 2 cm ribbon is longer, 5-year-olds performed at chance. Although 7-year-olds performed above chance, they still performed significantly worse when the units were unequal than when they were equal, even though all the children knew that an inch is longer than a centimeter. Even first through third graders have difficulty understanding the importance of equal size units on rulers. Pettito (1990) gave children in elementary school a choice of rulers with which they could measure a line. She found that the majority of first and second graders were content to use a ruler with units that varied in sizeâin fact, only about half the third graders chose the standard unit. Preschool children also have difficulty understanding that changes in the units of measure change the numerical answer (1 foot = 12 inches), but they do not change the length of the object being measured. Preschool children commonly fail to grasp the fundamental property of a unit, that a whole object can be segmented into parts of various sizes without changing the overall amount of what is being measured. They often count discrete parts of objects as being examples of a whole rather than grouping objects and counting amounts based on meaningful units (e.g., the two halves of a plastic egg each count as eggs versus the combination of the two pieces is one egg) (Shipley and Shepperson, 1990; Sophian and Kailihiwa, 1998). Similarly, Galperin and Georgiev (1969) gave kindergarten children two equal cups of rice and had them empty the cups by putting spoonfuls of rice into piles on a table using either a tablespoon or a teaspoon. When asked which group of piles contained more rice (correct answer is neither), a majority of the children chose the one made with the teaspoon because it contained more piles rather than choosing the group made with the table- spoon, which had fewer but larger piles. Thus, they were influenced by their propensity to count the overall number of piles. In this sense, childrenâs skill at counting can interfere with their understanding of measurement. These findings highlight that part-whole relationships are fundamental to understanding the relationship between units and wholes (see Sophian, 2002, for a review). A mature understanding of units of measure also entails the realiza- tion that the smaller the size of the unit, the larger the number of units the object will encompass. Research by Sophian, Garyantes, and Chang (1997) showed that preschool children have difficulty understanding this inverse relation, but that with instruction they can learn it. Young children do demonstrate some understanding of measurement principles, such as the inverse relation between unit size and the number of units after training or when measurement activities are set in a familiar context (e.g., part of a normal everyday routine or using familiar objects). Sophian (2002) taught
82 MATHEMATICS LEARNING IN EARLY CHILDHOOD preschool children ages 3 and 4 to correctly judge whether more small objects or more large ones would fit in a designated space. In pretest trials, the children incorrectly chose the larger object, but after six demonstration trials of watching the experimenter place objects of the two sizes, one by one, into two identical containers, they performed significantly better on posttest trials. These results identify the difficulties very young children have with understanding units and suggest that preschoolers (ages 2 years, 9 months to 4 years, 7 months) benefit from instructional intervention high- lighting the relation between unit size and number. Thus, young children show some understanding of fundamental mathematical concepts that are relevant to measurement if given the opportunity to explore these concepts in interactive, supportive contexts. Scaling and Proportion Children demonstrate early use of fundamental skills related to mea- surement and proportional reasoning in their use of maps. A critical fac- tor for success in map use is scaling, which is related to measurement and proportional thinking. Scaling refers to the ability to code distance and un- derstand how distance on a map corresponds to distance in the real world (Huttenlocher et al., 1999). Newcombe and Huttenlocher (2000, 2005) review the hierarchical nature of spatial coding, suggesting that various systems of coding spatial location are available, and their use depends on a mix of factors (e.g., cue salience in the external environment, complexity of movements required for action by the viewer). Furthermore, the availability of these systems appears as early as 6 months for both externally referenced and viewer-centered systems, which is much earlier than is predominantly reported in the literature. In relation to map use, children not only need to code locations in space but also to accommodate changes in scale, which requires a form of measurement (e.g., comparing the distance between two locations on a map and the corresponding distance between two loca- tions in the real world). Scaling has been assumed to involve proportional reasoning and therefore to occur much later in development, between ages 10 and 12 (Piaget and Inhelder, 1967). However, evidence of early success using maps by children ages 3 to 6 indicates that scaling, at least in these cases, may represent a precursor to more precise measurement and is ac- complished using spatial coding (Huttenlocher et al., 1999; Sandberg and Huttenlocher, 2001; Stea et al., 2004). REGULATING BEHAVIOR AND ATTENTION Infantsâ and young childrenâs mathematical development also takes place in the context of cognitive and behavior regulation, which when
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 83 stimulated and supported can promote mathematical learning. Research suggests that executive function is more strongly associated with success- ful transition to formal schooling than IQ or entry-level reading or matheÂ matics skills (Diamond, 2008). Executive function is defined as having three core components (Diamond, 2008). The first is inhibitory control, which is the ability to stay on task and do what is most necessary, even in the face of an inclination or impulse to do something else. The second is working memory, which is the ability to keep information in mind while still ma- nipulating it or changing it mentally; âworking memory may be thought of as a short-term âworking spaceâ that can temporarily hold information while a participant is involved in other tasksâ (Passolunghi, Vercelloni, and Schadee, 2007, p. 166). In mathematics specifically, this allows for performing mental arithmetic, such as addition or subtraction. The third component is cognitive flexibility, which allows for shifting between dif- ferent tasks, demands, priorities, or perspectives. As Diamond explains, executive function, particularly the inhibitory control component, is very similar to self-regulation but tends to focus more on cognitive tasks and less on social situations. Multiple executive function skills may be valuable in early math learning. These include the ability to stay on task and ignore distractions, the ability to follow the teacherâs directions, the ability to keep two strategies in mind at the same time, the motivation to succeed, the abil- ity to plan and reflect on oneâs actions, and the ability to cooperate (Leong, n.d.; McClelland et al., 2007). The link between mathematics success and executive function may have different underlying causes. Blair and colleagues (2007) review neu- roscience research indicating that, in adults, there may be a relationship between mathematical skills and executive functioning at the neural level. Reviewing changes in mathematics curriculum for children, they also found that, increasingly, automatized knowledge is emphasized less and tasks that require executive function skills (pattern-solving, relational reasoning, and geometry concepts) are emphasized more. This is an area that will continue to shed light on the relationship between executive function and mathemati- cal development as more research is conducted. Some studies have explicitly found a link between executive function and early math skills. In a study of 170 Head Start children, Blair and Razza (2007) found that multiple aspects of self-regulation (including inhib- itory control, effortful control, and false belief understanding, along with fluid intelligence) all made independent contributions to childrenâs early math knowledge. Similarly, McClelland and colleagues (2007) adminis- tered the Heads-to-Toes task to more than 300 preschool-age children. The â Different studies use different terms for concepts encompassed by executive function, such as self-regulation and behavioral regulation.
84 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 âstrengthening 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 skillsâ (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 childrenâs 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 infantsâ 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. Childrenâs 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 childrenâs under- standing of measurement begins with length, which is perceptually based,
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING 85 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 childrenâs math- ematical development. The next chapter explores variation in childrenâs 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 childrenâs mathematical development. REFERENCES AND BIBLIOGRAPHY Aguiar, A., and Baillargeon, R. (1998). Eight-and-a-half-month-old infantsâ reasoning about containment events. Child Development, 69, 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, 85, 50-62. Antell, S.E., and Keating, L.E. (1983). Perception of numerical invariance by neonates. Child Development, 54, 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, 38, 13-42. Baillargeon, R. (1995). Physical reasoning in infancy. In M.S. Gazzaniga (Ed.), The Cognitive 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, 98, 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 Development, 78, 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, 35, 294-328. Boulton-Lewis, G.M. (1987). Recent cognitive theories applied to sequential length measuring knowledge in young children. British Journal of Educational Psychology, 57, 330-342. Boysen, S.T. and Berntson, G.G. (1989). Numerical competence in a chimpanzee (Pan troglo- dytes). Journal of Comparative Psychology, 103, 23-31.
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