Instruction-based clinical eye-tracking study on the visual interpretation of divergence: How do students look at vector field plots?

Abstract

Relating mathematical concepts to graphical representations is a challenging task for students. In this paper, we introduce two visual strategies to qualitatively interpret the divergence of graphical vector field representations. One strategy is based on the graphical interpretation of partial derivatives, while the other is based on the flux concept. We test the effectiveness of both strategies in an instruction-based eye-tracking study with N ¼ 41 physics majors. We found that students' performance improved when both strategies were introduced (74% correct) instead of only one strategy (64% correct), and students performed best when they were free to choose between the two strategies (88% correct). This finding supports the idea of introducing multiple representations of a physical concept to foster student understanding. Relevant eye-tracking measures demonstrate that both strategies imply different visual processing of the vector field plots, therefore reflecting conceptual differences between the strategies. Advanced analysis methods further reveal significant differences in eye movements between the best and worst performing students. For instance, the best students performed predominantly horizontal and vertical saccades, indicating correct interpretation of partial derivatives. They also focused on smaller regions when they balanced positive and negative flux. This mixedmethod research leads to new insights into student visual processing of vector field representations, highlights the advantages and limitations of eye-tracking methodologies in this context, and discusses implications for teaching and for future research. The introduction of saccadic direction analysis expands traditional methods, and shows the potential to discover new insights into student understanding and learning difficulties.

Visual cues improve students’ understanding of divergence and curl: Evidence from eye movements during reading and problem solving

“…1). Prior research has shown that students' performance regarding this problem was only marginal above the chance of guessing correct-even after explicit instruction [6]. Students mainly struggled with the correct interpretation of partial vector derivatives [6,10].…”

“…This interdependence is known as the representational dilemma [2]. One example which has recently been studied in this context consists of the visual interpretation of twodimensional vector field plots with respect to divergence or curl [3][4][5][6]. Vector fields are mathematical structures that assign a vector to every point in space.…”

“…Prior research has shown that students' performance regarding this problem was only marginal above the chance of guessing correct-even after explicit instruction [6]. Students mainly struggled with the correct interpretation of partial vector derivatives [6,10]. Based on previous work [6], we created an instruction with visual cues that emphasize important conceptual features in the vector field diagram which are crucial to connect Eq.…”

Student understanding of graph slope and area under a curve: A replication study comparing first-year physics and economics students

“…To examine students' response processes during graphical task solving, eye tracking has been proven to be a powerful tool to complement the research with a data resource on students' visual attention [15]. Previous eyetracking research on students' understanding of kinematic graphs showed a strong link between the visuospatial abilities of a student and the understanding of kinematic graphs, which is linked to the tendency of low-spatial students to interpret graphs literally [16].…”

Role of diagrams in problem solving: An evaluation of eye-tracking parameters as a measure of visual attention

“…Eye tracking also gives insight into the role of visual attention during problem solving in mathematics and science [8,9]. However, there is a limited number of PER studies that have used eye tracking [10][11][12][13][14][15][16][17][18][19][20][21][22]. In addition to information on spatial and temporal characteristics of visual attention, eye-tracking data may provide information on cognitive load.…”