Number and Mathematical Reasoning

NUMBER AND MATHEMATICAL REASONING

Related points emerge when we scrutinize Piaget’s other claims, such as the idea that children younger than 6 years do not conserve number, and so might say, for example, that a row of four buttons, all spread out, has more buttons in it than a row of four bunched closely together. This certainly sounds like they have failed to grasp the con-cept of numbers, but experiments have shown that even very young children do have some numerical ability. In one study, 6-month-olds were shown a series of slides that displayed different sets of objects. The items shown varied from one slide to the next, but each slide contained exactly three objects. One slide, for example, might show a comb, a fork, and a sponge; another might show a bottle, a brush, and a toy drum; and so on. Each slide also differed in the spatial arrangement of the items. They might be set up with two on top and one below, or in a vertical column, or with one above and two below.

With all these variables, would the infants be able to detect the one property that the slides shared—the fact that all contained three items? To find out, the experimenters used the habituation technique. They presented these sets of three until the infants became bored and stopped looking. Then they presented a series of new slides in which some of the slides showed two items, while others continued to show three. The infants spent more time looking at the slides that displayed two items rather than three. Evidently, the infants were able to step back from all the particulars of the various slides and detect the one property that all the slides had in common. In this regard, at least, the infants appear to have grasped the concept of “threeness” (Starkey, Spelke, & Gelman, 1983, 1990).

Toddlers, too, have more mathematical skill than Piaget realized and, even at this early age, grasp some aspects of what counting is all about. Thus, when asked to count, one 2-year-old counted “1, 2, 6,” and another said “1, 13, 19.” But what is important is that they used these series consistently and realized that each of these number tags has to be applied to just one object in the set to be counted. They also realized that the tags must always be used in the same order and that the last number applied is the number of items in the set. Thus, the child who counted “1, 13, 19” confidently asserted that there were 13 items when she counted a two-item set, and 19 items when she counted a three-item set. This child is obviously not using the adult’s terms but does seem to have mastered some of the key ideas on which counting rests. (For more on the young child’s grasp of mathematics, see Barth, Kanwisher, & Spelke, 2003; Brannon, 2003; Cordes & Brannon, 2008; Gallistel & Gelman, 2000; Gelman, 2006; Lipton & Spelke, 2006; McCrink & Wynn, 2004; N. S. Newcombe, 2002.)

Once again, though, we need to ask: If preschool children have a basic grasp of counting skills, why do they fail Piaget’s tests—for example, his test for conservation of number? In part, the problem may lie in how the children were questioned in Piaget’s studies. In these procedures, the child is typically questioned twice. First, the two rows of items are presented in an evenly spaced manner, so that both rows are the same length. When asked, “Which row has more, or do they both have the same?” the child quickly answers, “The same!” Now the experimenter changes the length of one of the rows—perhaps spreading the items out a bit more or pushing them more closely together—and asks again, “Which row has more, or do they both have the same?”

Why is the same question being asked again? From the point of view of the child, this may imply that the experimenter did not like his first answer and so, as adults often do, is providing him the opportunity to try again. This would obviously suggest to the child that his first answer must have been wrong, and so he changes it.

Of course, this misinterpretation is possible only because the child is not totally sure of his answer, and so he is easily swayed by what seems to be a hint from the experi-menter. In other words, Piaget was correct in noting the limits of the preschool child’s knowledge: The child’s grasp of numerical concepts is tentative enough so that even a slight miscue can draw him off track. But this does not mean the child has no under-standing of numbers or counting, and, in fact, if we question the child carefully, provide no misleading hints, and simplify the task just a little (by using smaller numbers of items), preschool children reliably succeed in the conservation task (Siegal, 1997).