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Unformatted text preview: n réfléchissante, which
has been translated as “reflective” or “reflecting” abstraction. Piaget also oc
casionally used the term constructive abstraction, which seems easier to
Having made the theoretical distinction between empirical and construc
tive abstraction, Piaget went on to say that, in the psychological reality of the
child, one cannot take place without the other. For example, we could not con
struct the relationship “different” if all the objects in the world were identical.
Similarly, the relationship “two” would be impossible to construct if children
thought that objects behave like drops of water (which can combine to become
Conversely, we could not construct physical knowledge, such as the knowl
edge of “red,” if we did not have the category “color” (as opposed to every other
property such as weight) and the category “red” (as opposed to every other color).
A logico-mathematical framework (built by constructive abstraction) is thus
necessary for empirical abstraction because children could not “read” facts from
external reality if each fact were an isolated bit of knowledge, with no relation
ship to the knowledge already built and organized. This is why we said earlier
that the source of physical knowledge is only partly in objects and that the source
of social knowledge is only partly in conventions made by people.
While constructive abstraction cannot take place independently of empiri
cal abstraction up to about 6 years of age, it becomes possible later. For example,
once the child has constructed number (by constructive abstraction), he or she
can operate on numbers and do 5 + 5 + 5 + 5 and 4x5 without empirical ab
straction from objects.
The distinction between the two kinds of abstraction may seem unimpor
tant while children are dealing with small numbers up to 10 or 20. When large
numbers such as 999 and 1,000 are involved, however, it becomes clear that
numbers cannot be learned by empirical abstraction from sets of objects. Num 10 Theoretical Foundation bers are learned by constructive abstraction as the child constructs relationships.
Because these relationships are created by the mind, it is possible for us to under
stand numbers such as 1,000,001 even if we have never seen or counted
1,000,001 objects. The Synthesis of Hierarchical Inclusion and Order
Piaget went on to explain that the development of number concepts re
sults from the synthesis of two kinds of relationships: hierarchical inclusion and
order. These are explained below.
Hierarchical Inclusion. If we ask 4-year-olds to count 8 objects arranged in a
row, they often count them correctly and announce that there are “eight.” If we
then ask them to “show me eight,” they often point to the eighth object, saying
“That one” (see Figure 1.3a). This behavior indicates that, for this child, the words
one, two, three, and so on, are names for individual elements in a series, like
“Monday, Tuesday, Wednesday,” and so forth. For this child, the word eight
stands for the last object in the series and not for the entire group.
To quantify a collection of objects numerically, the child has to put them
into a relationship of hierarchical inclusion. This relationship, shown in Figure
1.3b, means that the child mentally includes “one” in “two,” “two” in “three,”
“three” in “four,” and so on. When presented with 8 objects, the child can quan
tify the collection numerically only if he or she can put them mentally into this
hierarchical relationship. Figure 1.3. (a) The absence and (b) the presence of hierarchical
inclusion In a child's mind. How Do Children Acquire Number Concepts? 11 Four-year-olds’ reaction to the class-inclusion task helps us understand how
difficult it is for young children to make a hierarchical structure (Inhelder &
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