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.

**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).

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