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 matching 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 equivalence 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 indications 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 situations. 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 preschool children to carry out calculations involving numerosities of up to six with objects (called nonverbal) and without objects (called verbal) (the former 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 performance between nonverbal and verbal calculations was particularly marked

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