Abstract:

Many times, in our undergraduate mathematical experience, we are confronted with students that don't connect the subjects that they have learned with those that they are learning and that they will eventually teach. Also, a big number of students, especially non Mathematic's ones, show difficulties when a proof construction is requested. So, it is usual to hear, from some students, that they like Mathematics but hate proofs. To invert this situation must be part of the teacher's role as early as the elementary grades, continued at all learning levels.
In fact, it is very important to provide opportunities for the students to establish links between the multiple, mathematical and/or other, subjects. On the one hand, the teaching practice approaching several languages and knowledge areas motivates the students. On the other hand, allows them to integrate the various subjects as a whole. In particular, the significant and diversified learning experiences that a future mathematics teacher is exposed to will certainly influence positively its mission and help “... to develop their competencies in using certain didactic notions for analyzing a mathematical topic.” (Gómez and González, 2009).
Moreover, despite the fact that doing Mathematics implies making proofs, since we need them to understand, establish and communicate mathematical knowledge, the Mathematics curricula and the teaching practice usually associate the notion of proof with upper levels of learning. Consequently, “... students are deprived of experience with an essential element of mathematical thinking and sense making until late in their mathematical education. Furthermore, when they do experience proof, it seems alien rather than a natural extension of things they have learned.” (Stylianides, A. J. and Stylianides, G. J., 2009).
We start with a few references to the Pythagorean Theorem that attest the widely known popularity of this result. Based on recent educational research, namely of the previously cited authors, we highlight the role of proof in mathematical learning and the teacher’s role in the enhancement of proving activities. We also focus on these relevant and related notions: conceptualization of proof, teacher in action, didactic analysis. Finally, inspired on Loomis' book entitled The Pythagorean Proposition, we propose some proofs as possible ways to enhance mathematical proof making and, in particular, the activity of proving in Mathematics.

Keywords:

loomis book, pythagorean theorem, proof making, proof conceptualization, teacher in.