From proof image to formal proof-a transformation

The idea to investigate the notions of proof image and formal proof emerged within our research on construction of knowledge and of justification as a specific case of construction of knowledge. Indeed, justification is a central and crucial component of mathematical reasoning. In the process of justifying a mathematical phenomenon, learners frequently need to expand their knowledge and to construct new knowledge. The aim in our previous research was to elucidate the intricate relationships between processes of justification and the emergence of new (to the learner) knowledge constructs. In a previous study (Dreyfus and Kidron 2006), processes of knowledge construction of a solitary learner whom we call L were investigated. The learner was constructing knowledge about bifurcations of dynamic processes. While we were acutely aware that the core of the constructing process is justification, it was only later (Kidron and Dreyfus 2010a) that we paid attention to the question of what justification means to the learner and analyzed the relationship of this meaning of justification for the constructing actions and the patterns of knowledge construction. It is indeed important to elucidate what we mean by justification. Fischbein (1982) identified a gap between mathematical proof and justification in everyday life. Rather than starting from formal mathematics, justification takes into account the learner's point of departure with its intuitive thinking, visual intuitions, and verbal descriptions. For example, the solitary learner in our previous studies wanted to gain more insight into the phenomena causing the second bifurcation point.