Patterns of reasoning in classroom

In this paper we report on some patterns of reasoning, which emerged during an activity of proving a mathematical statement performed by nine grade and university mathematics students. The statement in question involves drawing figures, working in arithmetic and in algebra. As for secondary students we detected fluency, flexibility and ability of verbalizing their reasoning. In particular, we will focus on the behavior of a student who through drawings succeeded in giving meaning to algebraic manipulation. The solutions of the university students were conditioned by the burden of the formal style used in university course of mathematics.