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).

Recent studies have shown that infants can approximate the number of items in large sets of visual objects (e.g., Brannon, 2002; Brannon, Abbott, 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, infants’ 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, Abbott, and Lutz, 2004; Cordes and Brannon, 2008; Xu, 2003; Xu and Spelke, 2000).

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 notion 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