Clustering eye-movement data uncovers students’ strategies for coordinating equations and diagrams of vector fields

Abstract

In mathematics education, students are repeatedly confronted with the tasks of interpreting and relating different representations. In particular, switching between equations and diagrams plays a major role in learning mathematical procedures and solving mathematical problems. In this article, we investigate a rather unexplored topic with precisely such requirements—that is, vector fields. In our study, we first presented a series of multiple-choice tasks to 147 introductory university students at the beginning of their studies and recorded students’ eye movements while they matched vector field diagrams and equations. Thereafter, students had to solve a similar coordination task on paper and justify their reasoning. Two cluster analyses were performed including (i) transition and fixation data on diagrams and options (Model 1), and (ii) additionally the number of horizontal and vertical saccades on the diagram (Model 2). In both models, two clusters emerge—with Model 1 distinguishing behaviors related to representational mapping and Model 2 additionally differentiating students according to representation-specific demands. Model 2 leads to a better distinction between the groups in terms of different performance indicators (test score, response confidence, and spatial ability) which also transfers to another task format. We conclude that vertical and horizontal saccades reflect executive actions of perception when approaching vector field coordination tasks. Thus, we recommend targeted interventions for mathematics lessons; these lessons must focus on a visual handling of the vector field diagram. Further, we infer that students’ difficulties can be attributed to covariational reasoning, thereby indicating the need for further investigations. From a methodological perspective, we reflect on the triangulation of eye-tracking and verbal data in (multiple-choice) assessment scenarios.

A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus

“…Between any two arrows, there exists a field vector rather than space being empty [13,14] Right click on any spot reveals the arrow starting at that spot. When the box Scan field is activated, the smooth continuation of the field is experienced when inserting additional arrows (vector path) by clicking, holding, and moving the mouse 3 A vector field has components F x and F y ; Both components can depend on input variables (x and y) [13-15, 39, 40] Clear separation of F x and F y in the control panel; User must enter x and y explicitly; Decomposition of vectors can be shown 4 Vector decomposition [7,13,15,39,40] Highlighting F x and F y within a user-defined rectangle and along a vector path using different colours 5 Partial vector derivatives [7,11].…”