Mathematical proof: from mathematics to school mathematics

Abstract

Proof plays a central role in developing, establishing and communicating mathematical knowledge. Nevertheless, it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Adversity Quotient and Resilience in Mathematical Proof Problem-Solving Ability

“…Lesseig, Hine, & Boardman, (2019) who argue "the centrality of proof of mathematics is indisputable." Some opinions also suggested the importance of mathematical proof (Balacheff, 2010;Basturk, 2010;CadwalladerOlsker, 2011;Rocha, 2019). However, Mukmin & Fa'ani (2020) found the fact that in solving mathematical proof problems, there were still so many misconceptions faced by students that resulted in the low mathematical proof ability of students.…”

Discussing Hilbert's 24th problem

“…As a result, their empirical methodology is suggested as a new and fruitful tool to investigate informal notions such as the simplicity of proof. (iv) School mathematics Helena Rocha characterizes, in Mathematical proof: from mathematics to school mathematics [49], 'the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof'. It is observed that, in school mathematics, the understanding of proof differs from the understanding in the mathematical research community and, therefore, also the notion of simplicity of proof may be treated differently in school.…”

Reasoning, Representing, and Generalizing in Geometric Proof Problems among 8th Grade Talented Students

“…Above and beyond the role of argumentation as a formal mathematical requirement, in mathematics education it is instrumental in learning [5] and explanation [6], for formulating arguments entails pondering ideas, organizing knowledge, and comparing different points of view [7]. This study consequently focused on argumentation in mathematics education, understood as the mathematical arguments generated by students and teachers in mathematics classrooms [8].…”