A Study on the Developmental Process from Empirical to Theoretical Conception : Focusing on the Process of Geometric Construction and its Justification

Abstract

This study aims at clarifying the process by which 7th grade students develop their empirical conceptions into theoretical ones. In particular, this paper explores how they develop their geometric conceptions by analyzing two kinds of classroom teaching practices concerning geometric construction. In the first classroom teaching practice study, it is suggested that students can interactionally develop their geometric concept of a kite as well as their way of constructing it. Initially, students conceived a kite as its shape and its collection of properties. They then began to use it as a tool for their geometric constructions, and gradually started to understand the relationship between the procedure of construction and the properties of the geometric figure. Finally, they could not only use the tool explicitly and mentally, but also become aware of the relationships between the properties of the kite itself. That is, their conception of the geometric figure sustained their construction activity and at the same time the activity promoted their understanding of the properties of the geometric figure. However, although we also intended in our teaching plan that students could eventually prove their construction procedure, they were unable to do so. As a result of this, we observed classroom lessons at the university-attached school and analyzed some factors for students to succeed in finding this proof. The results and implications for teaching about geometric construction are as follows. Firstly, students naturally use geometric figures as their cognitive tools in construction. Therefore we think that in the classroom lessons teachers should encourage students to become conscious of and reflect on their own cognitive activities. Secondly, in order to succeed in justifying the construction procedure, students need to not only differentiate it from the product, but also to use this procedure as a condition of proof. Thirdly, if we admit the existence of empirical proof, students have the potential to find some proof and justify it by activating their own image schema.