Defining in Classroom Activities

This paper discusses some aspects concerning the defining process in geometrical context, in the reference frame of the theory of figural concepts. The discussion will consider two different, but not antithetical, points of view. On the one hand, the problem of definitions will be considered in the general context of geometrical reasoning; on the other hand, the problem of definition will be considered an educational problem and consequently, analysed in the context of school activities. An introductory discussion focuses on definitions from the point of view of both Mathematics and education. The core of the paper concerns the analysis of some examples taken from a teaching experiment at the 6th grade level. The interaction between figural and conceptual aspects of geometrical reasoning emerges from the dynamic of collective discussions: the contributions of different voices in the discussion allows conflicts to appear and draw toward a harmony between figural and conceptual components. A basic role is played by the intervention of the teacher in guiding the discussion and mediating the defining process.