Despite its central place in the mathematics curriculum the notion of mathematical proof has failed to permeate the curriculum at all scholastic levels. While the concept of mathematical proof can serve as a vehicle for inculcating mathematical thinking, studies have revealed that students experience serious difficulties with proving that include (a) not knowing how to begin the proving process, (b) the proclivity to use empirical verifications for tasks that call for axiomatic methods of proving, and (c) resorting to rote memorization of uncoordinated fragments of proof facts. While several studies have been conducted with the aim of addressing students' fragile grasp of mathematical proof the majority of such studies have been based on activities that involve students reflecting and expressing their level of convincement in arguments supplied by the researchers, thereby compromising the voice of the informants. Further, research focus has been on the front instead of the back of mathematics. Hence, there is a dearth in research studies into students' thinking processes around mathematical proof that are grounded in students' own proof attempts. Therefore current investigations should aim at identifying critical elements of students' knowledge of the notion of proof that are informed by students' actual individual proof construction attempts.

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.