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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity
Clarifying Experimental Misnomers
Researchers have used tasks in which two conditions vary in two important ways, such as in Huttenlocher, Newcombe, and Sandberg (1994). In one condition, children are first shown objects, and then the objects are hidden. Number words are not used in this condition. In the other condition, children never see objects but must imagine or generate them (e.g., by raising a certain number of fingers). Here the numbers involved are conveyed by using number words, either as a story problem or just as words (e.g., “2 and 1 make what?”). In their reports, researchers call the first condition nonverbal and the second condition verbal. But these labels are a bit misleading, because they sound as if nonverbal and verbal are describing the children’s solution methods. In this report we use language that mentions both aspects that were varied: with objects (called nonverbal) and without objects (called verbal).
shown a set of disks that was subsequently covered. Following this, items were either added or taken away from the original set. The child’s task was to indicate the total number of disks that were hidden by laying out the same number of disks (“Make yours like mine”).
On both the matching and transformation tasks, performance increased gradually with age. Children were first successful with problems involving low numerosities, such as 1 and 2, gradually extending their success to problems involving higher numerosities. Importantly, when children responded incorrectly, their responses were not random, but rather were approximately correct. Approximately correct responses were seen in children as young as age 2, the youngest age group included in the study. On the basis of these findings, Huttenlocher, Jordan, and Levine (1994) argue that representations of small set sizes begin as approximate representations and become more exact as children develop the ability to create a mental model. Exactness develops further and extends to larger set sizes when children map their nonverbal number representations onto number words.
Toddlers’ performance on numerosity matching tasks indicates that, as they get older, they get better at representing quantity abstractly. This achievement appears to be related to the acquisition of number words (Mix, 2008). Mix showed that preschoolers’ ability to discriminate numerosities is highly dependent on the similarity of the objects in the sets. Thus, 3-year-olds could match the numerosities of sets consisting of pictures of black dots to highly similar black disks. Between ages 3 and 5, children were able to match the numerosities of increasingly dissimilar sets (e.g., black dots to pasta shells and black dots to sequential black disks at age 3½; black dots to heterogeneous sets of objects at age 4).