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

• In Event: 65.063 - Theory-Driven Design of Online Learning Systems, Data Structures, and Analytic Techniques

Abstract

Objective
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 not uncovered a solution. While there are several theories of fraction learning with traditions of research, none of this research employs methods large or nuanced enough to demonstrate either a theory’s potential at scale or how a particular method works. The work reported here uses learning analytics methods with fine-grained log data from an online fraction game to unpack how splitting impacts learning.

Theory
The collection and analysis of game-play data allowed us to look at students’ learning activity at a finer level of granularity than previously possible. Without this level of granularity, we may have found results that supported one of what one might call the “three bears” predictions: 1) a high amount of exploring splitting is good, 2) a low amount of exploring splitting is good, or 3) a medium amount of splitting is “just right.” Prior research on exploration of learning spaces supports all three of these possibilities. For example, cognitive load theory might predict that exploring aspects of the game level extensively that did not lead fairly directly to the solution would be unproductive for learning (Sweller, 2010). Alternately, prior research in the area of preparation for future learning might suggest a prediction that this type of exploring could increase efficiency in learning later in the game and not impair transfer (e.g., Kapur & Bielaczyc, 2012; Schwartz & Martin, 2004).

Methods and Data sources
We measured level of splitting in two ways: maximum node depth measured the amount of repeated splitting and unique board states measured the amount of different fractions created. We found that both of these measures were important.

Results and Significance
The picture we discovered is more interesting and complex than any of those options. In Study 1, both medium and high amounts of fractions created and of repeated splitting were associated with high success on the game level. In Study 2, several patterns were associated with significant gains on the transfer test. First, a medium amount of fractions created and of repeated splitting was conducive to learning, but only if students were doing this activity primarily in the correct one-third fraction space. Second, a high amount of fractions created and of repeated splitting supported learning regardless of the fraction space players were in. Third, creating a high amount of fractions and multiple repeated splits primarily in the one-third fraction space on the prelevel did not support learning if the student followed the same pattern on the postlevel, but it did support learning if the student switched to the medium, thirds strategy (a medium amount of fractions created and of repeated splitting primarily in the thirds space) on the postlevel.

Authors

• Taylor Martin, O'Reilly Media

• Carmen Petrick Smith