Learning with Multiple Graphical Representations in a Complex, Real-World Domain: Intelligent Software Tutors for Fractions

Principal Investigator: 
Project Overview
Background & Purpose: 

Although there is much evidence that instructional materials and activities that judiciously combine multiple external representations of learning content (MERs) can have significant learning benefits, there are many open questions that instructional designers face when creating a realistic curriculum that involves the use of MERs. We investigate three open research questions about learning with MERs within a rich, complex, real-world domain. We investigate these questions as we apply a proven educational technology: Cognitive Tutors, to the domain of fractions learning, a very challenging area of mathematics for middle-school students.


The participants in the planned experiments are 5th and 6th-grade students. The experiments will be carried out in the context of students’ regular classroom instruction at local middle or elementary schools.

Research Design: 

The project team will conduct comparative, experimental research to investigate how MERs can best be used to cause robust student learning (i.e., the research will generate causal results). In three planned experiments, we will investigate three open issues related to learning with multiple representations, by comparing experimental conditions in which students learn with multiple versions of intelligent software tutors for fractions learning. We will develop these tutors with dedicated authoring tools (CTAT). The open issues relate to the frequency of switching between representations, the most effective activities for helping students make connections between different representations, and the relative amount of time that should be devoted to making connections between representations, compared to activities centered on a single representation. In each experiment, we will compare learning outcomes with different versions of the intelligent tutor that differ only by the feature at issue in the given experiment.

This project collects original data using assessments of learning and achievement tests and detailed logs of student-tutor interactions (records of online usage).

To assess learning outcomes, we will create a test instrument that assesses students’ reproduction and transfer of conceptual and procedural knowledge. The test will include standardized test items, gathered from various readily-available sources (primarily state tests), as well as items adapted from the fractions literature, and new items created with the help of a retired middle-school math teacher.

To analyze the data from the planned experiments, one-factorial MANCOVAs will be used to investigate the between subjects effects of the experimental variations at the post-test and delayed post-test, using data from the pre-test as a covariate. In addition, we will analyze log data of students’ interactions with the fraction tutors to study how different tutor versions may affect their learning behavior.


In a classroom study with 132 sixth-grade students in a US middle school, conducted prior to receiving the REESE grant, we investigated the effect of multiple graphical representations (v. a single graphical representation) crossed with support for self-explanation (v. no support) in an intelligent tutoring system for fraction conversion and fraction addition. Students learned more with multiple graphical representations of fractions than with a single representation, but only when prompted to self-explain how the graphics relate to the symbolic fractions representations.

Publications & Presentations: 

Prior to receiving the current REESE grant, the project produced the following publications:

Rau, M. A. (2008). Flexible knowledge of fractions with multiple graphical representations in intelligent tutoring systems. Unpublished Diploma Thesis. Albert-Ludwigs-Universität, Freiburg im Breisgau, Germany.

Rau, M., Aleven, V., & Rummel, N. (2009). Intelligent tutoring systems with multiple representations and self-explanation prompts support learning of fractions. In V. Dimitrova, R. Mizoguchi, B. du Boulay, & A. Graesser (Eds.), Proceedings of the 14th International Conference on Artificial Intelligence in Education, AIED 2009 (pp. 441-448). Amsterdam: IOS Press.

This paper won the Best Student Paper Award AIED 2009 Conference.

Other Products: 

We will build intelligent web-based software tutors for fractions learning that support the use of MERs, to a much greater degree than current curricula, in ways guided by the results of the experimental studies. The software tutors will be made available on the web to teachers and students who would like to use them.