Teaching Fractions: Sense-Making Before Fluency
Using data collected from fourth- and fifth-grade students with fractions tutor cognitive learning software, researchers at Carnegie Mellon and the University of Wisconsin determined that an instructional model that emphasizes making sense of a fractions concept using graphical representation before demonstrating fluency in using graphical representations produces significantly enhanced learning gains. Research further showed that both representational understanding and fluency are needed for students to benefit from the tutoring system.
How it worked:
Data was collected from 599 4th- and 5th-grade students from five elementary schools in the United States using fractions tutor cognitive learning software. Students worked with the Fractions Tutor for about 10 hours on 80 problems during their regular mathematics class.
The Fractions Tutor uses multiple interactive graphical representations (circles, rectangles, and number lines) that are typically used in instructional materials for fractions learning. The Fractions Tutor covers a comprehensive set of topics ranging from identifying fractions from graphical representations, to equivalent fractions and fraction addition.
We contrasted two experimental factors:
- Support for representational understanding — students used a worked example with a familiar representation as a guide to make sense of an isomorphic problem with a less familiar representation.
- Support for fluency understanding — students had to visually estimate whether different types of graphical representations showed the same fraction.
The data the instructor used:
Students in the experiment received a pretest on the day before they started to work with the Fractions Tutor. The day after students finished working with the Fractions Tutor, they received an immediate posttest. One week after the immediate posttest, students were given a delayed posttest. All three tests were equivalent (i.e., they contained the same items with different numbers). Students worked with the Fractions Tutor for about 10 hours total and had to complete each tutor problem. All interactions with the Fractions Tutor were logged.
Results were based on the analysis of pretest, immediate posttests and delayed posttest from 428 students and confirmed the hypothesis that a combination of instructional support for representational understanding and representational fluency is most effective. The interaction between support for understanding and fluency was significant, F(2, 351) = 3.97, p < .05, ηp² =.03, such that students who received both types of support performed best.
The design of the tutor interfaces and of the interactions students engaged in during problem solving were based on a number of small-scale user studies, a knowledge component model developed based on Cognitive Task Analysis of the learning domain, and a series of in vivo experiments which were previously analyzed in DataShop.
The Fractions Tutor used multiple interactive graphical representations (circles, rectangles, and number lines) that are typically used in instructional materials for fractions learning. The Fractions Tutor covered a comprehensive set of topics ranging from identifying fractions from creating graphical representations to identifying improper fractions from graphical representations. The Fraction Tutor comprises about 10 hours of supplemental instructional material. Students solved tutor problems by interacting both with fractions symbols and with the graphical representations. As is common with Cognitive Tutors, students received error feedback and hints on all steps. Each step was logged in the DataShop for later analysis. In addition, each tutor problem included conceptually oriented prompts to help students relate the graphical representations to the symbolic notation of fractions.
The larger impact
Future research should investigate whether our findings are specific to the domain of fractions learning, and to the acquisition of representational understanding and representational fluency in making connections between multiple graphical representations.
Graphical representations are universally used as instructional tools to emphasize and illustrate conceptually relevant aspects of the domain content. Furthermore, in any given domain, students need to develop representational fluency in using graphical representations to solve problems, and they need to effortlessly translate between different kinds of representations. But representational understanding and representational fluency are not limited to learning with graphical representations: representational understanding and representational fluency also play a role in using symbolic and textual representations. For example, should students acquire representational fluency in applying a formula to solve physics problems before understanding the conceptual aspects the formula describes, or should they first conceptually understand the phenomenon of interest and then learn to apply a formula to solve problems related to that phenomenon? This is a crucial question for instructional design and one that remains open.