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How Do Children Acquire Number Concepts ECI 314 Early Childhood Mathematics

There are likewise more animals than dogs in every

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Unformatted text preview: nserve continuous and discontinuous quantities. Children of indigenous peoples such as the Aborigines in Australia (Dasen, 1974; De Lemos, 1969) and the Atayal in Taiwan (Kohlberg, 1968) have also been found to conserve without any instruction. The ages of attainment vary from one group to another, but the fact of this attainment remains certain. Researchers who have studied deaf children (Furth, 1966), blind children (Hatwell, 1966), and children and adolescents with severe mental retardation (Inhelder, 1943/1968) have also reported that these children attain the conservation of continuous and discontinuous quantities. Logicomathematical knowledge is thus universal because there is nothing arbitrary in it: 2 and 2 make 4 in every country. The words “one, two, three . . . ” are differ­ ent from “uno, dos, tres . . .,” but the numerical ideas underlying these words are universal. There are likewise more animals than dogs in every culture. If B is larger than A, and C is larger than B, adults and older children in every part of the world can also deduce that C is larger than A. On the basis of Piaget’s research and theory, and its cross-cultural verifica­ tion, I hypothesized in 1980 (Kamii, 1985) that if children construct their own number concepts, they should be able to construct numerical relationships out of these numbers that are in their heads. They should be able to invent arith­ metic for themselves because all numbers are created by the repeated addition of “one.” The idea of 5, for example, is (1 + 1 + 1 + 1 + 1), and 5 + 3 is therefore (1 + 1 +1 + 1 + 1)+(1 + 1 + 1). This hypothesis was amply verified in 1980-81 and 1981-82 by Georgia DeClark’s first graders (Kamii, 1985). It continues to be verified every year in many other classrooms in the United States and abroad. THE IMPORTANCE OF A SCIENTIFIC EXPLANATORY THEORY For centuries, education has been an art based on opinions called “philoso­ phies.” While education is still an art, it entered a scientific era when it embraced behaviorism, associationism, and psychometric tests. Associationism is less rig­ orous than behaviorism, but both grew out of empiricism, according to which knowledge is acquired by internalization from the environment. Both scientifi­ cally proved the importance of reinforcement, and both have been verified all over the world. These scientific theories grew out of empiricist common sense 16 Theoretical Foundation and reinforced the commonsense belief that drill and reinforcement enhance the internalization of knowledge. Behaviorism and Piaget’s constructivism are both scientific theories that have been verified all over the world. The question that must be answered is: How can two scientific theories be so contradictory and both be true? The answer to this question is that behaviorism and Piaget’s constructivism are related in the way illustrated in Figure 1.5a. This figure shows that Piaget’s theory can explain everything behaviorism can explain, but that the converse is not true. As a biologist, Piaget (1967/1971) pointed out that all animals adapt to reward and punishment, and that higher animals like dogs can anticipate the appearance of meat when they hear a bell, for example. Piaget also explained what behaviorists call “extinction” by saying that when the meat stops appear­ ing, the dog stops anticipating its appearance. While Piaget’s theory can thus explain everything behaviorism can explain, behaviorism cannot explain children’s acquisition of knowledge in a broader, deeper sense. Only Piaget’s theory can explain scientifically why children all over the world become able to conserve quantities without a single lesson in conser­ vation. Once children become able to conserve quantities solidly, no amount of reinforcement can extinguish their logic....
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