Shadows on proof

In this paper we refer to an experiment in which students of the age range 14-17 have to proof a statement on natural numbers, writing all their thoughts while they are working on this task. We perform a kind of ‘genetic decomposition’ of the statement and single out some parameters, on which we base the analysis of the students’ protocols. The main schemes found in students’ proofs are the authoritarian, the empirical, the ritual and the symbolic. We study the relations of these proof schemes with the context chosen by the students to prove. Some students’ behaviours allow singling out elements suggesting the influence of the algebraic or arithmetic contexts on proving this type of statement: we call it algebraic or arithmetical shadow effect.