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 addition 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 covered 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 comparison 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 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). Existing research also shows that children’s 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 Piaget’s research. They are even building early informal knowledge in many other mathematics areas besides representation of the counting



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