Abstract
We conducted a design experiment for clarifying the transition process from geometry in primary mathematics to geometric proof in secondary school for the seventh grade's teaching unit "plane geometry" which was designed as three subunits of (1) the discovery game of geometric figures, (2) the jintori game, and (3) the discovery game of geometric constructions and the geometric proof. In this paper we examined the students' thought processes when they could develop the geometric constructions through the discovery game and explore the geometric proof on the superposition of two triangles which are placed at the arbitrary positions, and aimed at abstracting the factors of students' development towards geometric proofs. The factors of the transition towards logical proofs that we abstracted from our analysis of the design experiment are the following points. 1. Cognitive aspects are ・to find the geometric constructions by using and combining the figures and their properties, ・to confirm and limit the extent where the law works using inductive reasoning and empirical explanation, ・to make the proposition so that inductive reasoning can be changed to deductive reasoning, and ・to reinterpret the procedures of geometric constructions as the conditions for justification. 2. Aspects of the recognition of figure and shape are ・to find out figures and use them for geometric constructions and inferences, ・to conceive shapes included in the figure in terms of various relations and correspondences, ・to see figures as the variables which can be transformed dynamically, and ・to see figures as the representations of relations through showing reasonings diagrammatically. 3. Social aspects are ・to learn based on conjectures, refutations, and consensus among the participants, ・to make conjectures while assuming the criticism by the other students, ・to develop conjectures by examining the criticism and the counter-example by the other students, and ・to reinterpret and express the others' explanations more explicitly. 4. Aspects of the teaching unit structure are ・to envision the fundamental situation and the unit structure as a series of the situation for action, the situation for formulation and the situation for validation in the theory of didactical situation, ・to pose a proof problem so that the learned procedures of geometric constructions can be reconstructed as the conditions for justification, and ・to make geometric transformations and geometric constructions interact with each other at the final stage of the unit.
