How Do Children Acquire Number Concepts ECI 314 Early Childhood Mathematics

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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 understand. 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 one drop). 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 & Piaget, 1...
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