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3 Cognitive Foundations for Early Mathematics Learning
Pages 59-94

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From page 59...
... . 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.
From page 60...
... . 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.
From page 61...
... Moreover, although there is some disagreement in the field about the interpretation 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' 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 moving (Antell and Keating, 1983; Starkey and Cooper, 1980; Strauss and Curtis, 1981; Van Loosbroek and Smitsman, 1990; Wynn, Bloom, and Chiang, 2002)
From page 62...
... 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   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")
From page 63...
... . 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.
From page 64...
... Infants dishabituated to the novel set size, suggesting that they considered different sets of two (or three) as similar.
From page 65...
... 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)
From page 66...
... 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)
From page 67...
... "magic experiment" 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.
From page 68...
... 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.
From page 69...
... compared the ability of preschool children to carry out calculations involving numerosities of up to six with objects (called nonverbal) and without objects (called verbal)
From page 70...
... . Summary Toddlers and preschoolers continue to build on the two representational 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)
From page 71...
... 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.
From page 72...
... As is the case for the development of number knowledge, recent research has shown strong starting points for spatial thinking. In contrast to Piaget's 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.
From page 73...
... . Several recent studies have shown that preschool children are able to mentally rotate shapes in the picture plane.
From page 74...
... Given simple maps, 4-year-olds and a ­ majority of 3-year-olds can locate a hidden object in a sandbox (­Huttenlocher, Newcombe, and Vasilyeva, 1999) , children ages 3 to 5½   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)
From page 75...
... . 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)
From page 76...
... 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.
From page 77...
... Children's learning of specific spatial terms also helps highlight spatial categories. These spatial terms include shape words (e.g., circle, square, triangle, rectangle)
From page 78...
... . Thus, it is possible that spatial activities, spatial language, or both promote the development of spatial skills, such as block building and mental rotation.
From page 79...
... Language input and spatial activities appear to be highly influential in the development of spatial categories and spatial skills during the preschool years. DEVELOPMENT OF MEASUREMENT Measurement is a fundamental aspect of mathematics, which "bridges two main areas of school mathematics -- geometry and number" through the attachment of number to spatial dimensions (National Council of Teachers of Mathematics, 2000)
From page 80...
... 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.
From page 81...
... 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.
From page 82...
... . 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 accomplished using spatial coding (Huttenlocher et al., 1999; Sandberg and Huttenlocher, 2001; Stea et al., 2004)
From page 83...
... The third component is cognitive flexibility, which allows for shifting between different 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.
From page 84...
... 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 approximate 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)
From page 85...
... . Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten.
From page 86...
... . Sex Differences in the Relation of Early Puzzle Play and Mental Rotation Skill.
From page 87...
... . The contribution of executive functions to mathematic skills in preschool children.
From page 88...
... Journal of Experimental Child Psychology, 77, 20-29. Gelman, R
From page 89...
... . Development of spatial recognition in preschool children: On Piaget and Inhelder's hypothesis of topological space.
From page 90...
... . Mental rotation in human infants: A sex difference.
From page 91...
... . A sex difference in mental rotation in young infants.
From page 92...
... . Learning about what fits: Preschool children's reasoning about effects of object size.
From page 93...
... British Journal of Developmental Psychology, 25, 103-108. Xu, F., and Spelke, E.S.


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