From proof image to formal proof ‒ a transformation. (English)

Rezat, Sebastian (ed.) et al., Transformation ‒ a fundamental idea of mathematics education. New York, NY: Springer (ISBN 978-1-4614-3488-7/hbk; 978-1-4614-3489-4/ebook). 269-289 (2014).

Summary: We propose the notion of proof image as an intermediate stage in a learner’s production of a proof. A proof image consists of the cognitive structure in the learner’s mind that is associated with the given proof. It consists of previous constructs that the learner has selected for potential use in the proof to be constructed and of the links between these previous constructs, links that the learner expects to play a role in the proof. In this chapter, we focus on the transition of a proof from proof image to formal proof. We do this within the theoretical framework of Abstraction in Context, leaning on Davydov’s notion of abstracting, according to which abstraction proceeds from an unrefined and vague form to a final coherent construct. We exemplify this transition by means of the story of K, who constructs a proof for a theorem in analysis from his proof image. We discuss in more detail the notion of proof image by means of the story of L, another learner, this one being related to bifurcations in dynamical systems.