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 naturalistic 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 development (Piaget, 1941/1965). However, Gelman’s (1972) “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. After children learned to choose the winner, the experimenter altered 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 nonverbal; 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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