followed by three objects). The reverse order was shown for the “greater than” condition. At test, the 14- and 16-month-olds showed more interest in the opposite relation than the one that was shown, suggesting that they represented the less than and greater than relations, whereas 10- and 12-month-olds did not. However, Brannon (2002) presents evidence that infants are sensitive to numerical order relations by 11 months of age.
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). The set size limitation has been interpreted as reflecting one of two core systems for number—the object file system. They also expect the appropriate result from small number addition and subtraction transformations (e.g., 1 + 1 = 2 and 2 − 1 = 1), at least when amount covaries with number. Somewhat later, by 10 months of age, infants discriminate equivalent from nonequivalent sets, and by 14 months of age they discriminate greater than from less than relationships. Because many of these studies did not control for continuous variables that covary with number (i.e., contour length and surface area), the basis of infant discriminations is debated. However, recent studies indicate that infants are sensitive to both number of objects in small sets and to continuous variables, and they may be more sensitive to number than to cumulative surface area. Infant studies also have examined sensitivity to approximate number by using larger sets of items (e.g., 8 versus 16). These studies have found that infants can discriminate between sets with a 2:1 ratio by age 6 months and between those with 2:3 ratios by age 9 months as long as all set sizes involved are greater than or equal to 4, that is, 6-month-olds fail to discriminate 2 versus 4.
We also note that infants’ early knowledge of number is largely implicit and has important limitations that are discussed below. There were no number words involved in any of the studies described above. This means that learning the number words and relating them to sets of objects is a major new kind of learning done by toddlers and preschoolers at home and in care and education centers. This learning powerfully extends numerical knowledge, and children who acquire this knowledge at earlier ages are provided with a distinct advantage.
Just as much of the infant research has a focus on theorizing about and researching the nature of infant representations of number, so, too, does some research on toddlers and preschool children. The goal is to understand