UC Berkeley

Symbolic fractions are notoriously difficult to learn, and this difficulty has been characterized in terms of cognitive, mathematical, and educational challenges. In this dissertation I present evidence from several eye-tracking studies that illuminate participants’ cognitive approaches to proportional reasoning, and analyze them from the perspectives of psychological research on relational reasoning, as well as modern education research. In my first two studies, I had participants perform fraction comparison tasks while I measured their eye movements to assess how they approached various types of problems. In the first of these studies, I compared the performance and eye gaze patterns of 5th-graders, who were just beginning to learn fractions, with those of college students. I sought to test the hypothesis, based on relational complexity theory, that children have difficulty learning fractions in part because they have difficulty integrating relationships among mental representations. This effect was present in the data, however, there were additional performance decrements unrelated to relational complexity that are better explained by the development of inhibition and cognitive flexibility. Further, I found that children who did not comprehend fraction concepts, as evidenced by their performance, still exhibited similar eye movements to those who performed well, suggesting that they encoded the relevant numerical relations even though they were not able to interpret them correctly. These findings underscore the cognitive and mathematical complexities inherent in proportional reasoning. In the second fraction comparison study, I investigated the extent to which adults applied various relational integration skills to proportional reasoning problems, and whether doing so impacted their performance. I found that they performed better on trials that could be solved more easily by componential than magnitude processing. Specifically, when there was a readily-available multiplicative factor between the two fractions, they made fewer within-fraction saccades consistent with magnitude calculation – and when they made fewer of these saccades, they performed more efficiently. This work highlights the ways in which relational thinking can support proportional reasoning. In the dissertation, I place the results of these first two studies in the context of prior research on fraction understanding, and point to possible implications for pedagogy. In a third study, I collaborated on an investigation of children’s learning trajectories during the acquisition of fraction knowledge, comparing two curricular approaches. We found that individual prior knowledge, classroom environment, and the curriculum with which the students engaged all influenced their acquisition of fractions knowledge. As is evident in this dissertation, experimental psychology and educational research provide different, and equally productive, lenses with which to explore the learning of fractions. By integrating across these theories and methods of analysis, we can more fully characterize and promote the development of proportional reasoning.