Proof and proving in the classroom: Dynamic Geometry Systems as tools of semiotic mediation

Abstract

The objective of this paper is to discuss the didactic potential offered by the use of a Dynamic Geometry System (DGS) in introducing students to theoretical thinking and specifically to the practice of proof. Starting from a discussion about what constitutes the general objective in developing students' sense of proof, the notion of Theorem is introduced as a system consisting of a statement, a proof, and a theory within which such a proof make sense. The paper discusses how the use of Dynamic Geometry Systems may support the teacher's intent to achieve her educational goal. The Theory of Semiotic Mediation provides the main theoretical reference for describing and explaining the role of Dynamic Geometry Systems in fostering the development of students' sense of theorems. Examples will be presented drawn from teaching experiments involving 9^th and 10^th grade students, illustrating how semiotic potential may unfold in the solution of specific tasks. The didactical focus consists in exploiting the semiotic potential of a DGS, especially the use of Cabri tools and particular dragging modalities that may function as tools of semiotic mediation.

Keywords:

• semiotic potential
• Dynamic Geometry System (DGS)
• Theory of Semiotic Mediation (TMS)
• dragging
• conditional statement

Notes

1. An earlier version of this paper was presented at Seventh Congress of the European Society for Research in Mathematics Education (CERME7) in Rzeszów, Poland, in February 2011 (Mariotti 2011).

2. An exception is that of mathematical induction, which is explicitly treated, and to which a specific training is devoted. But, mathematical induction is very rarely presented in comparison with other modalities of proving, which are commonly considered natural and spontaneous ways of reasoning.

3. Actually a DGS provides a larger set of tools, including for instance ‘measure of an angle’, ‘rotation of an angle’ and the like. This implies that the whole set of possible constructions do not coincide with that attainable only with ruler and compass (see Stylianides and Stylianides 2005 for a full discussion).