Proof image

The emergence of a proof image is often an important stage in a learner’s construction of a proof. In this paper, we introduce, characterize, and exemplify the notion of proof image. We also investigate how proof images emerge. Our approach starts from the learner’s efforts to construct a justification without (or before) attempting any formal argument, and it focuses on the process by which a complete but not necessarily communicable image of that justification becomes available to the learner and provides explanation with certainty. We consider the interplay between the learner’s intuitive and logical thinking and, using the theoretical framework of Abstraction in Context, we trace the construction of knowledge that results from and enables progress of this interplay. The existence and identification of proof images and the nature of the processes by which they emerge constitute the theoretical contribution of this paper. Its practical value lies in the empirical analyses of these processes and in the potential to apply them to the design of tasks that support students in constructing their own proofs images and proofs. We believe that such processes are likely to considerably enrich students’ mathematical experience.