Towards a Comprehensive Conception of Mathematical Proof

There is overwhelming evidence that students face serious challenges in
learning mathematical proof. Studies have found that students possess a
superficial understanding of mathematical proof. With the aim of
contributing to efforts intended to develop a comprehensive conception of
mathematical proof, literature search was conducted to identify areas where
research could be directed in order to increase proof understanding among
students. To accomplish this goal, literature on modes of reasoning involved
in proof construction, ideas on the classification of activities that constitute a
proof path, and categories of proof understanding are exemplified using
mathematical content drawn from Real Analysis. These exemplifications
were used to illustrate the connections between modes of reasoning and
levels of proof understanding. With regard to students’ fragile grasp of
mathematical proof this critique of literature has revealed that many previous
studies have given prominence to proof validations while there is lack of
crucial interplay between structural and inductive modes of reasoning during
proving by students. Hence, it is suggested in this paper that current research
could also focus on mechanisms that promote an analytic conceptions of
mathematical proof that are comprehensive enough to allow students to
engage in more robust proof constructions.