The cognitive science and educational psychology literatures provide ample evidence that instructional materials and activities that judiciously combine multiple representations of learning content (MRs) can have significant learning benefits. Much of this literature has focused on learning with a combination of text and figures; only some of it has focused on learning with multiple graphical representations. In order to benefit from multiple representations, students must connect key information across the different representations. Students typically must be supported in doing so (Ainsworth, 2006).
This project studies the use of MRs in the domain of fractions, a very challenging area of mathematics for middle-school students in which graphical representations are used extensively (e.g., pie charts, number lines, fraction strips, set models, etc.) The research focuses on three general (and open) questions that instructional designers face when creating a curriculum that involves the use of MRs: First, when multiple representations of learning materials are used, how frequently should learners switch between representations? Second, what kinds of activities are most effective in helping students make connections between different representations? Third, what fraction of the students? time should be devoted to making connections between representations, relative to activities centered on a single representation?
Researchers from Carnegie Mellon University and the University of Freiburg (Germany) investigate these questions in the context of an established educational technology: intelligent tutoring systems. These types of software tutors have been shown to improve students' mathematics learning in a number of scientific studies. A set of authoring tools created in a lab at Carnegie Mellon make the development of these tutors more cost effective and more accessible to education researchers than it used to be.
During a three-year grant period, the project will (1) create web-based intelligent tutors as supplemental activities for fractions learning; these tutors support activities in which students work with interactive graphical representations of fractions, and make connections between the representations, and (2) conduct controlled experiments in Pittsburgh middle schools to investigate the three research questions outlined above.
The proposed research will result in principles for learning with MRs. It will produce new knowledge about how fraction representations can best be used to support robust learning. The proposed research has the potential to produce more effective fractions instruction in the lower and middle grades, and thereby facilitate later mathematics learning. The proposed software tutors will be made freely available on the Mathtutor website (http://webmathtutor.org).
The notion that "a picture is worth a thousand words" may never be more true than in education. Take fractions, for example. You may remember how in elementary school, you or your children learned to understand fractions using pictures of circles, number lines, and rectangles broken into colorful pieces. Although many elementary school math curricula use these graphical representations of fractions, there is great variety in how they do so. This variety reflects the fact that we do not know, in a scientific way, how these representations can best be used in fractions instruction. Is using multiple representations better than sticking to just one, say, the number line? How often should we alternate between representations? How can we help students make connections between representations, as a way of getting at the fundamental concepts behind them? Fractions are not the only domain in which multiple graphical representations are used extensively in instruction. Chemistry is another example and there are many others. What was said about fractions is also true of these other domains: there is little scientific knowledge about whether and how to use multiple graphical representations to best support learning. More solid scientific knowledge is highly desirable, as it might lead to better, more effective curricula. Our research investigated, through a series of classroom studies, how 4th and 5th grade fractions instruction can best employ multiple graphical representations. The research makes three key contributions: First, we created a web-based intelligent tutoring system for fractions learning, called the Fractions Tutor, that can serve as supplementary fractions instruction for students nationwide. Second, we established a set of general research-based instructional principles for how best to support learning with multiple graphical representations. Third, we showed that an intelligent tutoring system that embodies these principles (namely, the Fractions Tutor) can help students learn effectively, over and above their regular fractions instruction in school. Like other intelligent tutoring systems, the web-based Fractions Tutor helps students learn through complex problem-solving activities. The Fractions Tutor covers 10 fundamental fractions topics and represents over 10 hours of supplemental instruction for typical students. It uses graphical representations (circle, rectangle, and number line) in all tutored activities. It aligns with the Common Core and other standards. Using the Fraction Tutor as platform, we conducted five research studies, four of them in actual classrooms, in which over 3,000 research participants (students in grades 4-6) used the Fractions Tutor. We found that the use of multiple graphical representations can be effective in fractions learning, provided that: the software prompts students to explain how each representation relates to fundamental fractions concepts; the activity sequence frequently alternates between different fractions topics and between different fraction representations (as opposed to doing them in "blocks"); students carry out activities that involve only a single graphical representation and activities that involve multiple graphical representation; in activities with multiple representations, there is support for sense making and there is support for fluency; by sense making we mean figuring out how different representations relate to each other; by fluency we mean practice so that students learn to quickly and accurately see how different representations relate to each other (e.g., whether they depict equiivalent fractions) without much effortful reasoning in the sense-making activities, students use an example of a solved problem with a more familiar representation to guide the work with the unfamiliar representation. In our fourth and final classroom study, students in grades 4 and 5 worked for 10 hours with the Fractions Tutor. To measure their learning, we tested their fractions knowledge both before and after. A week after working with the Fractions Tutor, students’ test scores had improved by 32% relative to the test they took before working with the Fractions Tutor. On test items that measured students’ deep understanding of fractions concepts, students scored 48% higher a week later, compared to their earlier scores. Thus, the Fractions Tutor was very effective in helping students learn fractions concepts, even on top of their regular classroom instruction. The work confirms a widespread educational practice that did not previously have scientific support, namely, the use of multiple graphical representations within a single domain. It also adds a significant amount of knowledge as to how multiple graphical representations can best be used. The research may have a positive impact on educational practice in two ways: First, the Fractions Tutor provides effective fractions supplementary instruction for the elementary grades and is available to anyone (https://fractions.cs.cmu.edu). Since a deep understanding of fractions is a key facilitating factor for later mathematics learning, the research may even influence later K-12 mathematics learning. Second, the principles for instruction with multiple representations provide guidance to instructional designers in a range of domains, not just fractions. As such, the research has the potential to lead to better, more principled instruction in a range of domains.
