This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.
Algebra is used in several different domains in mathematics, but this article has a focus on the algebra that is taught and learned in secondary school (Grade 12 and 13). After having elaborated and presented a theoretical framework for her analysis of proofs, Pedemonte presents some data that has been collected from prospective primary school teachers. These students were attending a course at the University, and their solutions to two open problems were analyzed according to the theoretical framework (the solutions of 7 students’ solutions to each of the two problems were analyzed).