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Unformatted text preview: 50 counters. The child is given one of the glasses, and the
interviewer takes the other glass. The interviewer then asks the child to “drop a
counter into your glass each time I drop one into my glass.” When about 5
counters have thus been dropped into each glass with one-to-one correspon
dence, the adult says, “Let’s stop now, and you watch what I am going to do.”
The interviewer then drops one counter into his or her glass and says to the child,
“Let’s get going again.” The adult and the child drop about 5 more counters into
each glass with one-to-one correspondence, until the adult says, “Let’s stop.” Theoretical Foundation 14
The following is what has happened so far: The interviewer now asks, “Do you and I have the same amount, or do you have
more, or do I have more?”
Four-year-olds usually reply that the two glasses have the same amount.
When asked, “How do you know that we have the same amount?” 4-year-olds
explain, “Because I can see that we both have the same amount.” (Some 4-yearolds reply that they have more, and when asked how they know that they have
more, their usual answer is “Because.”)
The interviewer goes on to ask, “Do you remember how we dropped the
counters?’ and 4-year-olds usually give all the empirical facts correctly, including
the fact that only the adult put an extra counter into his or her glass at one point.
In other words, 4-year-olds remember all the empirical facts correctly and base
their judgment of equality on the empirical appearance of the two quantities.
By age 5 or 6, however, most middle-class children deduce logically that
the adult has one more. When asked, “How do you know that I have one more?”
the children invoke exactly the same empirical facts as the 4-year-olds.
One-to-one correspondence is made empirically in this task, but children
who have not constructed number concepts (logico-mathematical knowledge)
can get only empirical knowledge from this correspondence. When they have
synthesized hierarchical inclusion and order, on the other hand, it becomes
obvious to them that there is one more counter in the adult’s glass.
If the child says that the adult’s glass has one more counter, the interviewer
goes on to ask the next question: “Suppose we continued to drop counters in
the same way (with one-to-one correspondence) until supper time. Would you
and I have the same number, or would you have more, or would I have more?”
The 5- and 6-year-olds divide themselves into two groups at this point. The moreadvanced children say, “You would always have one more, no matter how long
we went on.” By contrast, the less-advanced children give more concrete an
swers, such as “You don’t have enough counters to keep going until supper time,”
or “I can’t tell because we haven’t done it yet.”
These responses indicate that the 5- and 6-year-olds may have constructed
small numbers, but not large ones. As Piaget (Piaget & Szeminska, 1964) pointed
out, number concepts seem to be constructed progressively, up to about 7 first,
then to about 15, and later to about 30. The Universality of Logico-Mathematical Knowledge
Cross-cultural research has documented that children all over the world
become able to conserve number (discontinuous quantities) as well as con- How Do Children Acquire Number Concepts? 15 tinuous quantities such as amounts of water and clay. The conservation of con
tinuous quantities is discussed in some detail in Chapter 3. Studies in Aden
(Hyde, 1959), Algeria (Bovet, 1974), Iran (Mohseni, 1966), Martinique (Piaget,
1966), Nigeria (Price-Williams, 1961), Montreal and Rwanda (LaurendeauBendavid, 1977), Scotland and Ghana (Adjei, 1977), and Thailand (Opper,
1977) are among the investigations giving unequivocal support to the state
ment that children all over the world become able to co...
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