Teaching Math – Abstract (Not Concrete) Understanding Adds Up
What’s the best way to teach math? It’s a big question, but research at Ohio State University’s Center for Cognitive Science challenges a commonly held (though perhaps informal) notion in instructional design that concrete examples aid the learning and application of mathematics more than abstract proofs and representations. The idea that mastery of abstract quantities and concepts actually provides the learner with a better, i.e., more practical, set of tools for problem solving seems counter-intuitive, but researcher Jennifer Kaminski and her team believe they have proof. Kaminski et al. looked at whether students who received instruction using concrete examples performed differently from those who were encouraged to master the concepts abstractly. What they found was that the group who were instructed in more concrete terms and examples were less able to apply the knowledge to new situations.
“These findings cast doubt on a long-standing belief in education…. The belief in using concrete examples is very deeply ingrained, and hasn’t been questioned or tested.” – Vladimir Sloutsky, co-author
Ohio State’s Research Communications quotes Kaminski as saying:
“Teachers often use real-world examples in math class, the researchers said. In some classrooms, for example, teachers may explain probability by pulling a marble out of a bag of red and blue marbles and determining how likely it will be one color or the other.
But students may learn better if teachers explain the concept as the probability of choosing one of n things from a larger set of m things.”
This research might help explain why so many people find word problems (and the semantic or linguistic use of mathematics) so daunting in mathematics and physics. In Kaminski’s words:
“The issue can also be seen in the story problems that math students are often given. For example, there is the classic problem of two trains that leave different cities heading toward each other at different speeds. Students are asked to figure out when the two trains will meet.
The danger with teaching using this example is that many students only learn how to solve the problem with the trains.
If students are later given a problem using the same mathematical principles, but about rising water levels instead of trains, that knowledge just doesn’t seem to transfer.”
Sloutsky sees a role for word problems, however, just not as an instructional aid:
“It is very difficult to extract mathematical principles from story problems. Story problems could be an incredible instrument for testing what was learned. But they are bad instruments for teaching.”
Kaminski’s and Sloutsky’s study should provide useful insight for those looking at ways to better teach subjects like mathematics, physics, signal analysis, algorithm design, dynamics, logic or economics. It should be noted that Kaminski and Sloutsky worked with Andrew Heckler of Ohio State’s Physics Department on parts of the study.
References.
Concrete Examples Don’t Help Students Learn Math, Study Finds
Students Learn Better When the Numbers Don’t Talk and Dance
Kaminski et al., LEARNING THEORY: The Advantage of Abstract Examples in Learning Math, Science 25 April 2008: 454-455, DOI: 10.1126/science.1154659.