Intuition and proof in the solution of conjecturing problems

The cognitive relationship between intuition and proof is complex and often students struggle when they need to find mathematical justifications to explain what appears as self-evident. In this paper, we address this complexity in the specific case of open geometrical problems that ask for a conjecture and its proof. We analyze four meaningful cases that, though not exhaustive with respect to all possibilities, can highlight different aspects of the possible tensions between intuition and proof. The main aim of this paper is to show that when a conjecture is constructed based on intuition or perceptive evidence, the proving process can involve a cognitive gap, sometimes difficult for the students to cope with. Research concerning cognitive unity shows that proof is more accessible to students when an argumentation is constructed to justify a conjecture. A proof can be constructed connecting the statements previously used in the argumentation, into a logical chain. However, if the conjecture is based on an intuition or perceptive evidence, cognitive unity cannot be fully realized, and difficulties arise.