Proof validation in Euclidean geometry: a comparison of novices and experts using eye tracking

Abstract

This dissertation investigates and compares the methods of proof validation utilized by novice and expert mathematicians within the realm of Euclidean geometry. With the use of eye tracking technology, our study presents empirical evidence supporting claims previously studied only through the use of verbal protocols. Our investigation settles a series of contentious results surrounding the practical implementation of the generalized validation strategy called zooming out (Inglis and Alcock, 2012; Weber, Mejia-Ramos, Inglis, and Alcock, 2013). This strategy analyzes the overall structure of a proof as an application of methods or logical chunks. Settling the debate through use of longer and more complicated proofs devoid of blatant errors, we found that validators do not initially skim-read proofs to gain structural insight. We did however confirm the practical implementation of zooming out strategies. The literature identifies within the proof validation process specific differences between novices and experts. We are interested in a holistic understanding of novice and expert validations. We therefore present the direct comparison of entire validation processes that assess the similarity of novice and expert overall validation attempts. We found that the validation processes of novices and experts share a certain degree of similarity. In fact novices tend to be closer to experts than to other novices. And when validations are clustered, the groups are heterogeneous with regard to mathematical maturity. Our investigation expands the proof validation literature by including diagrams in the proof validation process. We found that experts tend to spend more time proportionally on the diagram than novices and that novices spend more time on the text. Furthermore, experts tend to draw more connections within the diagram than novices as indicated by a higher proportion of attentional changes within the diagrams. Experts seem to draw on the power of visualizations within the mathematics itself, spending more time on conceptual understanding and intended connections.