New research looking at comparison and the strategy of learning concepts before learning procedures as a way to help middle schoolers learn new math concepts has been co-authored by Bethany Rittle-Johnson, assistant professor of psychology and human development at Peabody.
“We found that comparing different ways to solve a problem helped middle-school students become more flexible problem solvers and better understand the concepts behind the methods,” Rittle-Johnson said.
Rittle-Johnson and her colleague and co-author, Jon Star, assistant professor at the Harvard Graduate School of Education, also found that comparing different solution methods was more effective than comparing different problems solved using the same solution. Overall, students should not just learn one way to solve a math problem; rather, they should learn multiple ways and be encouraged to compare the benefits and drawbacks of each, she said.
The findings are summarized in two studies, one recently published in the Journal of Experimental Psychology and the other in the Journal of Educational Psychology.
“In a past study, we found that seventh graders who compared two different ways to solve equations were both more accurate and more flexible in their equation solving. In our recent studies, we found similar benefits for fifth graders learning about estimation,” Rittle-Johnson said.
Rittle-Johnson has also co-authored a study finding that students benefit more from being taught the concepts behind math problems rather than the exact procedures to solve the problems. The findings offer teachers new insights on how best to shape math instruction to have the greatest impact on student learning.
The research by Rittle-Johnson and Percival Matthews, a Peabody doctoral candidate, was published recently in the Journal of Experimental Child Psychology.
“Teaching children the basic concept behind math problems was more useful than teaching children a procedure for solving the problems. These children gave better explanations and learned more,” Rittle-Johnson said.
“This adds to a growing body of research illustrating the importance of teaching children concepts as well as having them practice solving problems.”
In math class, teachers typically demonstrate a procedure for solving a problem and then have children practice solving related problems, often with minimal explanation for why things work.
“With conceptual instruction, teachers explain a problem’s underlying structure. That type of instruction enables kids to solve the problems without having been taught specific procedures and also to understand more about how problems work,” Matthews said. “When you just show them how to do the problem they can solve it, but not necessarily understand what it is about. With conceptual instruction, they are able to come up with the procedure on their own.”
The study also examined whether having the students explain their solution to problems helped improve their learning. To test this, the researchers used the conceptual teaching approach with all students, and had one group explain their solution while the other did not. They found no discernable difference in performance between the two groups. While self-explanation has been found to be beneficial in previous studies, Rittle-Johnson and Matthews found that when the students were given a limited time to solve the problem, the benefit disappeared. This led them to suggest that part of the benefit of self-explanation may come from the extra time a student spends thinking about that particular problem.
“Self-explanation took more time, which left less time for practice solving the problems,” Matthews said. “When time is unlimited, self-explanation gives students more time to repair faulty mental models. We found conceptual explanation may do the same thing and make self-explanation less useful.”
Rittle-Johnson is an investigator in the Vanderbilt Kennedy Center for Research on Human Development and in the Peabody Learning Sciences Institute. Research for both studies was funded by the U.S. Department of Education.