ヴィゴツキー理論に基づく「分数概念の素地となる子どもの生活的概念」に関する日米比較調査 : 分数概念の構造と原理の同定

抄録

Vygotsky categorized concepts into two types: everyday concepts and scientific concepts (the latter is called mathematical concepts in this paper). Although Vygotsky pointed out that these concepts are interrelated in concept formations, he did not mention how these concepts themselves are formed. Yoshida (2000, 2004), therefore, posed a concept formation process using the idea of "sublation" through the following three stages: everyday and mathematical concepts 1) conflict with each other, 2) are lifted to higher levels, and 3) are preserved as a unified concept, or sublated concepts. Based on this model of process, this paper aims at clarifying how children's everyday concepts prepare the ground for fraction concepts before they encounter fraction concepts in systematized lessons at school. Following the interview survey for 21 second graders and the questionnaire survey for 39 third graders in Japan in 2001, this paper compares the questionnaire survey for 23 fourth graders in the U.S. in 2004 with the Japanese ones. The main findings in this paper are as follows. Broadly speaking, the questionnaire data on the everyday concepts of fractions from American children are almost identical to those of Japanese children, although there are small differences in detail (e.g. the variety of answers by American children). The use of "everyday concepts of fractions" is readily visible in these data. For instance, responses such as "half is something less than a whole," or "half is something divided evenly into some parts (not in two)" show that the concept of half remaining ambiguous. The "structure of fractions" composes of (A) fractions as quantity (the object of the fractions is quantity), (B) fractions as ratio and (C) fractions as operation (the object of these fractions is the relation between quantity and quantity) , and (D) fractions as number (the object of the fractions is number) (Yoshida, 2002a). This "structure of fractions" was derived theoretically, and is shown in Figure I. Finally, relating the results of the surveys with the "structure of fractions," two fundamental principles (P1 and P2) emerge, which run through the "structure of fractions" (cf. Figure2). P1 is a principle of equality, in which equality of size in fractions is in common to all kinds of fractions. P2 is a principle of comparing and relating two quantities or two numbers, or more specifically, a principle of relating with ONE-whole. For example, you can describe the "quantity" of juice in a cup as "1/3 of a cup" (fraction (A)) relating the amount of juice as a part with the cup as ONE-whole. You can compare and describe the "relation" between Sylvia's and Daniel's oranges as Sylvia's orange is "1/4 of Daniel's" (fraction (B)). While children have to regard the quantity of Daniel's oranges as ONE-whole, some children describe that "Daniel likes oranges more than Sylvia." Furthermore, a "number 1/5" is positioned in a number line relating with ONE-whole, or 1. Although the principles reflect essential aspects of fractions as mathematical concepts, children do not become aware of these in their everyday life as demonstrated in the surveys.