Learning Fractions by Splitting: Using Learning Analytics to Illuminate the Development of Mathematical Understanding

Abstract

The struggle with fraction learning in kindergarten through Grade 12 in the United States is a persistent problem and one of the major stumbling blocks to succeeding in higher mathematics. Research into this problem has identified several areas where students commonly struggle with fractions. While there are many theories of fraction learning, none of the research on these theories employs samples large enough to test theories at scale or nuanced enough to demonstrate how learning unfolds over time during instructional activities based on these theories. The work reported here uses learning analytics methods with fine-grained log data from an online fraction game to unpack how splitting (i.e. partitioning a whole into equal-sized parts) impacts learning. Study 1 demonstrated that playing the game significantly improved students’ fraction understanding. In addition, a cluster analysis suggested that exploring splitting was beneficial. Study 2 replicated the learning results, and a cluster analysis showed that compared to early game play, later game play showed more optimal splitting strategies. In addition, in looking at the types of transitions that were possible between a student’s early cluster categorization and later cluster categorization, we found that some types of transitions were more beneficial for learning than others.

Notes

¹ For a general reference on cluster analysis, options in any given type of analysis, and interpretation of results, please see Witten, Frank, and Hall (2011).

² Please see the discussion of the effect size in “Study Limitations.”

FIGURE 12 Test score by cluster change, Study 2.

³ Even splits are more advanced than 2ⁿ splits, as they mix splits with factors of 2 and other factors (in our case factors of 3).