The purpose of this study is to clarify what everyday concepts of fractions appear in classes and how children form the meanings of fractions in a classroom, using the Vygotskian theories about everyday concepts and mathematical concepts, and the ZPD. Firstly, everyday concepts and mathematical concepts are defined according to the Vygotskian theories, and then a child's concept formation process is described using these theories and an idea called sublated concepts (Yoshida, 2000). Secondly, the new aspect (partition-quantity fraction) is introduced in addition to the other four aspects of the fraction: partition fraction, quantity fraction, ratio fraction and quotient fraction, which have standardly been used in Japan. In particular, the partition-quantity fraction [A] is closely related to children's everyday concepts. In conclusion, the following are pointed out: 1) Children's everyday concept of fractions became the driving force for forming the fraction concept. 2) Children formed the meanings of the partition-quantity fraction [A] in a classroom, while focusing on the partitioning activities. Therefore, in this situation, they were not conscious of the quantity of the fraction. Then children formed the meanings of the partition-quantity fraction [C] in a classroom, while focusing on the quantity of the fraction. 3) The teacher played the following three roles: firstly, setting up a suitable situation for children to form the meanings of fractions; secondly, understanding well the children's everyday concepts and helping children create the appropriate ZPDs; and finally, using the appropriate learning materials.
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