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Vanderbilt University Peabody College of Education and Human Development

Comparing notes

topics: Math and Science Education, Psychology and Human Development

Ashley Crownover

Teaching students to compare different solutions in order to facilitate learning has become a standard component of the reform mathematics movement. Psychologists have also taken an interest in comparison, seeking to understand how people learn and process information. Recent work by LSI investigator Bethany Rittle-Johnson (assistant professor of
psychology, Peabody College) combines these two approaches, applying cognitive principles identified in psychology research to the teaching of mathematics in the classroom.

Along with collaborator Jon Star at Michigan State University, Rittle-Johnson is in the second year of a three-year grant funded by the U.S. Department of Education's Institute of Education Sciences ($1,014,175 from June 2005 to May 2008). "I've historically been interested in how kids learn math," she says, "and in doing that work, I became increasingly aware of math education and the possibility that some of my work could actually have an impact" on the teaching of mathematics.

In results to be published in the Journal of Educational Psychology (article in process), the researchers reported that a study conducted with seventh-grade students learning algebra produced empirical support for comparison. Students who compared examples of two different ways to solve the same equation learned more than students who studied the same examples one at a time. They became more accurate and more flexible in their equation solving, suggesting that previous cognitive studies done in a laboratory environment are to some degree generalizable to the classroom. A recent study with fifth-grade students learning about estimation also suggests a benefit for comparison.

"People traditionally say, look, we know all these principles from cognitive science, why don't teachers use them?" Rittle-Johnson explains. "But the principles are mostly based on lab studies with college students." These new findings suggest that "prior research on comparison as a basic learning mechanism may be generalizable to a new domain (mathematics), a new age group (school-aged children), and a new setting (the classroom)."

Study results also corroborate that comparison helps students become more flexible in their problem solving. "You need to know problem-solving procedures," Rittle-Johnson says, "but it is especially important to be flexible with them. You need to recognize that they work in a variety of settings, and comparison helps that process. [With comparison] you get a broader range of when a procedure is going to work, and are thus better able to adapt."

This kind of flexibility in problem solving can in turn lead to increased transfer of knowledge, a topic psychologists are very interested in. "Most of the cognitive science literature on comparison is about transfer," Rittle-Johnson says. "If you learn in this context, can you apply it to a new instance? The hope is that if we can get students to compare, we can help them notice what's important so that they're able to transfer [this awareness of deep structure] to new problems."

The atypical collaboration between cognitive psychology and mathematics education has been made possible in part by Vanderbilt's institutional commitment to interdisciplinary work. "The classic model in psychology is to do it alone," Rittle-Johnson says, "but if you're going to get cognitive scientists to do this kind of work, you've got to have a model." The researchers hope their efforts will help move the field forward in this way, as well as eventually produce sufficient evidence to encourage curriculum developers, textbook publishers, and others to incorporate cognitive principles about comparison into their work.