The purpose of this study is to develop the prescriptive model for designing social interactions in an elementary mathematics class which is effective and applicable to teaching practices at elementary school level. In this paper we verify the effectiveness of this model through a teaching experiment of ‘fractions’ conducted for two fourth-grade classrooms in a school. The teaching experiment had four characteristics: a teaching material focusing on meanings of fractions, recall of the definition of fractions learned in third grade, illustrative representations of tapes, and an applied problem which promotes mathematical generalization. As a result of analysis, we found out the following four results.
Firstly the fundamental process of being conscious, solving by the individuals, solving by small group, being reflective, and then making agreement, in particular the small group activity, contributed to solving the problem. Also, it was suggested that the children’s solving activities progressed from their ‘individual’ solution to ‘quasi-general’.
Secondly the intentional support of illustrative representations by a teacher was quite effective. Namely some children in one classroom were able to solve the problem of fractions for themselves and explain the reason why their answer was correct clearly by using two kinds of illustrative representations of tapes. In addition, the children negotiated that the length of one-third of a tape whose length was two meters was twothirds meter, as the solution of a problem, by differentiating two kinds of meanings of fractions. Furthermore, it was confirmed that the children also explained such reason by translating illustrative representations into symbolic representations and generalized their solution of the original problem when they solved an applied one.
Thirdly three kinds of social interactions such as social interaction with others, social interaction with the self and social interaction with representations, were activated by setting small groups. These kinds of social interactions contributed to developing the children’s deeper understanding of fractions.
Lastly we could have two concrete suggestions for the improvement of teaching fractions by a comparative analysis of children’s activities in two classrooms. It was so crucial for the children to have an additive view of the definition of fractions in order to solve the problem. It was important for the children to realize two kinds of the structure which were embedded in illustrative representations of tapes.
These four findings demonstrate the effectiveness of the prescriptive model of social interactions for teaching practices in elementary school mathematics.
