What Gaze Data Reveal About Coordinating Multiple Mathematical Representations

What Gaze Data Reveal About Coordinating Multiple Mathematical Representations Theodore Wills (twills@temple.edu) College of Education, 1301 W. Cecil B. Moore Avenue Philadelphia, PA 19122 USA Thomas F. Shipley (thomas.shipley@temple.edu) Department of Psychology, 1701 North 13th Street Philadelphia, PA 19122 USA Briana L. Chang (briana.chang@temple.edu) College of Education, 1301 W. Cecil B. Moore Avenue Philadelphia, PA 19122 USA Jennifer G. Cromley (jcromley@temple.edu) College of Education, 1301 W. Cecil B. Moore Avenue Philadelphia, PA 19122 USA Julie L. Booth (julie.booth@temple.edu) College of Education, 1301 W. Cecil B. Moore Avenue Philadelphia, PA 19122 USA Abstract 40 High school students were given a battery of paper and pencil tests, which collectively assessed a variety of spatial abilities, graph and table competencies, conceptual mastery of calculus, and achievement in common topics from typical precalculus and calculus courses. In addition, students completed a computer- presented measure of Coordinating Multiple Representations (CMR), in which they had to assess whether two mathematical representations (e.g. an equation and a graph) depicted the same underlying mathematical function. Gaze data were captured during this measure, using a Tobii T60 eye tracker. Findings suggest that good or poor performance on several paper measures is associated with distinct and specific gaze behaviors. Better achievement scores are associated with fewer fixations near the centerline of the graph, and with fewer point-plotting and function scanning behaviors. These findings are discussed in terms of differing approaches or strategies for engaging in CMR. Keywords: Eye tracking; coordinating multiple representations, graphs. Introduction Students of mathematics are often required to engage with a variety of representations. They may be asked to understand information presented in the form of equations, graphs, tables, and text. The ability to pass flexibly between two or more such representations, a skill called Coordinating Multiple Representations (CMR), is seen as a hallmark of well developed competence, and is associated with better math achievement (Gagatsis & Shiakalli, 2004). However, while there is more than one way to successfully execute CMR, there is little existing research that delineates the sorts of granular strategies that are associated with superior math competencies. This study focuses on CMR strategies associated with graphical representations. Researchers in graph comprehension have delineated a distinction between locally and globally represented information, and the different behaviors or strategies that are necessary to successfully extract these types of information (e.g., Shah, Freedman, & Vekiri, 2005.). Leinhardt et al. (1990) draw a distinction between lower-order techniques in graph interpretation, like point reading, and those associated with higher-order information, like evaluating intervals of increase or decrease. More recently, theorists have argued for meaningful, hierarchical distinctions between reading the data (e.g., reading the value of a single point), reading between the data (e.g., find differences between points, or compute sums or aggregations of points), and reading beyond the data (e.g., infer information that is not explicitly presented; Friel et al., 2001). Differentiation of graphical coordination skills is supported by other research that outlines the relative challenges of different CMR tasks. For example, it is common practice for students to manually graph data provided in tabular format (Kozma, 2000; Roth & Bowen, 2003), but this type of coordination is more challenging with non-linear functions (Demana, Schoen, & Waits, 1993). Likewise, younger students often focus on lower- order information, such as specific points, rather than understanding the function as a continuous entity with a global structure (Elia, Panaoura, Eracleous, & Gagatsis, 2007; Elia et al., 2008; Even, 1998; Monoyiou & Gagatsis,