An illustration of the explanatory and discovery functions of proof

Abstract

This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof.

Comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom

“…In the course of devising the strategies, the learners might develop new problem-solving skills. This shows that the inductive proof that learners develop through empirical activities and the deductive proof that they later develop might enhance learners' abilities not only to verify the theorem, but also to discover new knowledge and new ways of problem-solving (De Villiers, 2012;Ding et al, 2009). This also means that the deductive geometric proof development tasks set up in the textbook are in the doing mathematics category, hence they are of high cognitive demand (Stein et al, 2009).…”

“…This also means that the deductive geometric proof development tasks set up in the textbook are in the doing mathematics category, hence they are of high cognitive demand (Stein et al, 2009). As such, the tasks that are set up in the textbook have the potential to engage the learners in making connections among different features of geometric content (Ronda & Adler, 2016), to link formal and informal geometry (Bowie, 2013), and to make logical and clear explanations (deductive proving), hence promoting other functions of proof like explaining and justifying mathematical concepts (De Villiers, 2012).…”

Mathematical proof: from mathematics to school mathematics

“…De Villiers [29] states that, traditionally, justifying or becoming convinced about the validity of a conjecture is the main function attributed to proof; Knuth [13,14] even believes this is the only role that most of the teachers attribute to it. In recent decades, this narrow view of proof has been criticized by authors such as Reid [30].…”

On High School Students’ Communication Skill in Proof-Based Learning

“…The proof is an important science that must be taught in the classroom because it has the function of explaining, discovering, systematic means, creative thinking, communication tools, and verification tools [15][16][17][18]. The method of proof is needed in learning mathematics to show or prove the truth in mathematics in the form of properties or theorems [19].…”