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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity
change in number. Thus, if they were habituated to a set of two items, they did not dishabituate to a set of three items if that set was equivalent to the original set in area or contour length.
However, recent findings indicate that infants are sensitive to both continuous quantity and to number (Cordes and Brannon, 2008, in press; Kwon et al., 2009). Furthermore, Cordes and Brannon (2008) report that, although 6-month-old infants are sensitive to a two-fold change in number, they are sensitive to a three-fold change only in cumulative area across elements, suggesting that early sensitivity to set size may be more finely tuned than early sensitivity to continuous quantity. Other studies that provide support for early number sensitivity include a study showing that 6-month-old infants can discriminate between small sets of visually presented events (puppet jumps) (e.g., Wynn, 1996). This result is not subject to the alternative explanation of discrimination based on amount rather than number, like the findings involving sets of objects. However, it is possible that even though the rate and duration of the events have been controlled in these studies, infants’ discrimination is based on nonnumerical cues, such as rhythm (e.g., Demany, McKenzie, and Vurpillot, 1977; Mix et al., 2002). Indeed, in one study in which the rate of motion was not a reliable cue to numerosity, 6-month-olds did not discriminate old and new numerosities (Clearfield, 2004).
A set size limitation also is seen in the behavior of 10- to 14-month-olds on search tasks (Feigenson and Carey, 2003, 2005; Feigenson, Carey, and Hauser, 2002). For example, in one study 12-month-olds saw crackers placed inside two containers. The toddlers chose the larger hidden quantity for 1 versus 2 and 2 versus 3 crackers, but they failed to do so on 3 versus 4, 2 versus 4, and 3 versus 6 crackers (Feigenson, Carey, and Hauser, 2002). The authors suggest that this failure is due to the set size limitation of the object file system.1 When cracker size was varied, the toddlers based their search on the total cracker amount rather than on number. Similarly, 12- to 14-month-olds searched longer in a box in which two balls had been hidden after they saw the experimenter remove one ball, than they did in a box in which one ball had been hidden and the experimenter removed one ball (in actuality there were no more balls in either box, as the experimenter surreptitiously removed the remaining ball). They also succeeded on 3 versus 2 balls but failed on 4 versus 2 balls. That is, they did not search longer in a box in which four balls were hidden and they saw two removed than in a box in which they had seen two hidden and two were removed. The failure
The object file system refers to the representation of an object in a set that consists of small numbers, the objects are in a 1-to-1 correspondence with each mental symbol, and there is no summary representation of set size (e.g., three items are represented as “this,” “this,” “this” rather than “a set of three things”) (Carey, 2004).