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Unformatted text preview: is not a good number to choose to illustrate the
logico-mathematical nature of number concepts because two is a perceptual
number. Small numbers up to four or five are perceptual numbers, as will be
explained shortly. However, two can also be a logico-mathematical number for
an adult, who has constructed logico-mathematical numbers. We chose the
number two because, with two counters, we could illustrate other relationships
such as “similar,” “different,” and “the same in weight.”
Children go on to construct logico-mathematical knowledge by putting
previously made relationships into relationships. For example, by coordinating
the relationships of “same” and “different,” children become able to deduce diat
there are more animals in the world than dogs. Likewise, by putting four twos
into relationships, they become able to deduce that 2 + 2 + 2 + 2 = 8, that 4x2
= 8, and that if 4x = 8, x must be 2.
Piaget thus recognized external and internal sources of knowledge. The
source of physical and social knowledge is partly external to the individual. The
source of logico-mathematical knowledge, by contrast, is internal. This state Theoretical Foundation 6 ment will be clarified shortly when we discuss the two kinds of abstraction dis
tinguished by Piaget. Let us first review the conservation-of-number task, which
will clarify the differences among the three kinds of knowledge. The Conservation-of-Number Task
Conservation of number refers to our ability to deduce, through logical rea
soning, that the quantity of a collection remains the same when its spatial ar
rangement and empirical appearance are changed. The procedure described
below (Inhelder, Sinclair, & Bovet, 1974) may appear rather standardized.
However, each interview must be adapted to the particular child, especially with
regard to the latter s understanding of the terms used in quantification.
Materials: About 40 counters, 20 red ones and 20 blue ones
Procedure A. Equality
The interviewer makes a row of 8 blue counters and asks the child to
put out the same amount of red ones (“as many as,” “the same much,”
The interviewer records the child’s response. If necessary, the red
and blue counters are put in one-to-one correspondence, and the child
is asked whether the two rows have the same amount.
Note: At least 7 counters must be used because small numbers
up to 4 or 5 are perceptual numbers. Small collections such as “oo”
and “ooo” are called perceptual numbers because they can be dis
tinguished at a glance. When 7 objects are presented, however, it
is impossible to distinguish “ooooooo” from “oooooooo” with cer
tainty by merely looking at them. Small numbers greater than 4
or 5 are called “elementary numbers.”
The interviewer says, “Watch carefully what I’m going to do” and modi
fies the spatial arrangement in front of the child’s watchful eyes by spac
ing out the counters in one row and/or pushing them close together in
the other row (see Figure 1.1). The following questions are then asked:
“Are there as many blue ones as red ones [running his or her finger
along each row], or are there more here [indicating one row] or more
here [indicating the other row]?” and “How do you know?”
If the child has given a correct conservation answer with a logical ex
planation, the interviewer says, “But another boy [or girl] said there How Do Children Acquire Number Concepts? 7 Figure 1.1. The arrangement of the counters when the question is asked
about conservation. are more in this row [indicating the longer row] because this row is
longer. What do you think? Are you right, or is the other child right?”
If, on the other hand, the child gave an answer of nonconservation,
the interviewer reminds him or her of...
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This document was uploaded on 03/13/2014.
- Spring '14
- The Land