全体論的視座からの正負の数の加減の単元構成に関する研究 : 教授学的状況論と代数的思考のサイクルの視点から

抄録

The purpose of this study is to clarify the structure of the teaching unit of addition and subtraction with positive and negative numbers from the holistic perspective both theoretically and practically. We think that the difficulties on teaching and learning algebra result mostly from some fundamental factors such as the gap between elementary mathematics and secondary mathematics, the epistemological character of mathematical expression, and the paradigm behind the teaching of algebra. In particular, it seems to us that the mechanistic and atomistic teaching more or less deprives students of their significance to do mathematics. Therefore, we need to discuss the teaching unit in terms of the alternative educational paradigm and the didactical theory that realizes it as well as students' cognitive development on mathematical expression. In the first half of this paper we overlooked the holistic perspective of education, discussed the theories of the didactical situations and the algebraic cycle of thinking, and set up the provisional framework for designing and analyzing the teaching unit. And in the second half we analyzed the teaching and learning activities of the unit "addition and subtraction with positive and negative numbers" and discussed the structure and characteristics of the teaching unit based on the framework. The results are as follows. 1. The teaching unit of addition and subtraction with positive and negative numbers can be designed as the three stages which consist of the situation for action, the situation for formulation or communication, and the situation for validation. These can be also explained as the process of optimizing equilibration in terms of students' knowing on the one hand, and conceived as the phenomena in which the methods for thinking in the previous stage are successively transformed into the objects for thinking in the next stage on the other hand. 2. If we expect that students realize the significance of algebra during their continuous activities, the classroom lessons should be designed as they can constitute the tight links between their ideas and the milieu in the situation for action, produce the addition and subtraction simultaneously in the situation for formulation, and realize that transformation of the expression makes the link clear in the situation for validation. 3. We can convey the following educational ideas in the teaching unit; The algebraic nature as mathematical language, the humanistic view of mathematical learning, and the constructivist attitude of learning in the classroom lessons.