Perceptual Action of Novices and Experts in Operating Visual Representations of a Mathematical Concept

Abstract

In this paper we explore the perceptual actions that allow one to perceive pictures as representing mathematical concepts. The research is based on the cultural–historical approach. Following V.V. Davydov’s ideas on theoretical and, particularly, mathematical thinking, we consider a mathematical concept as being based on a historically determined method of action. Using the eye-tracking system we analyzed the difference between school students, university students, and expert mathematicians in their perception of special pictures (so called ‘external visual representations of the theoretical concept’) when performing a set of tasks, namely choosing a point with given coordinates from a set of points. A standard expert-novice research analysis of dwell time in relevant and irrelevant areas of interest was used. We also compared the gaze paths, the number of visual fixations, and the time each group required to perform the tasks. The directions of the saccades were also analyzed, and our data showed that the vertical and horizontal saccades along the axes prevailed over saccades along other directions, a fact that may be considered as a trace of the ‘Cartesian plane’ concept. The data showed that experts performed the tasks faster and with fewer fixations and they also were able to use additional knowledge flexibly in organizing their perceptive actions. Our results show the fundamental interlacing of conceptual structures and visual processes, in which the latter are organized in accordance with prior knowledge. The specificity of the experts’ Cartesian plane perception corresponds to the late stages of the historical development of this concept. We consider this fact as an empirical confirmation of the relevance of the term “theoretical perception”.