To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a…

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a…

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…

Research in psychology and in mathematics education has documented the ubiquity of "intuition traps" -- tasks that elicit non-normative responses from most people. Researchers in cognitive psychology often view these responses negatively, as a sign of irrational behaviour. Others, notably mathematics educators, view them as necessary…

Gestures are associated with powerful forms of understanding; however, their causative role in mathematics reasoning is less clear. We inhibit college students' gestures by restraining their hands, and examine the impact on language, recall, intuition, and mathematical justifications of geometric conjectures. We test four mutually exclusive…

This paper discusses a radically different set of assumptions to improve educational outcomes for disadvantaged students. It is argued that disadvantaged children, when exposed to carefully organized thinking-oriented instruction, can acquire the traditional basic skills in the process of reasoning and solving problems. The paper is presented in…