A common trick by math teachers is to give students problems that look different on the surface but require the same solution.
Asking students to do side-by-side comparisons of math equations is a potent tool in learning, says a recent study in the Journal of Educational Psychology. These comparisons can promote deep relational learning and the development of theory-level explanations.
What types of comparisons are most effective in promoting learning in math? Do students learn more when they compare equivalent problems, different problems with the same solution or different solutions to the same problem?
Two researchers, one from Vanderbilt University and the other from Harvard University, conducted a study of 162 7th- and 8th-grade students who worked in pairs to compare math equations. Based on their study, the authors report that comparing different solutions to the same problem is the most conducive to learning.
“All comparisons are not equally effective,” the researchers write. “In the case of equation solving, comparing solution methods was more effective for supporting conceptual knowledge and procedural flexibility than comparing equivalent equations or comparing problem types.”
In this study, the researchers examined the effectiveness of learning algebra by:
- comparing equivalent problems solved with the same solution method
- comparing different problem types solved with the same solution method, or
- by comparing different solution methods to the same problem.
Participants (81 boys, 81 girls) were from 9 pre-algebra classes from a rural public school, a suburban public school and an urban private school. Students were from 5 advanced and 4 regular math classes taught by 5 different teachers. The students ranged in age from 11.9-15.1 years with a mean age of 13.1 years and were mostly Caucasian (5% African American, 5% Asian/Indian, and 1% Hispanic). Approximately 14% of students received free or reduced lunch.
Pairs of students within a classroom were randomly assigned to compare solution methods, problem types or equivalent equations. In the compare equivalent condition, equations of the same type were paired, such as 2(x + 3)=8 and 5(y + 4)=10. In the problem types condition, different problem types were paired, such as 2(x + 3)=8 and 6(h +1)=3(h + 1).
Students studied 12 example pairs with a partner and answered explanation prompts. Half of the examples demonstrated a conventional solution and the other half a shortcut solution.
Use of shortcut
The packets included one guided practice problem on which to use a particular shortcut method to solve the equation and 4 independent practice problems on which students could use either the conventional or shortcut method. (In the methods condition, students were asked to solve 2 practice problems each in two different ways, whereas 4 different equations were presented in the packets for the other conditions.)
Each student had his or her own packet. Students were instructed to describe each solution method to their partners and answer the accompanying questions first verbally and then in writing. Students were asked to solve the problems on their own, then compare answers with their partner and have their answers checked by an adult.
The researchers evaluated the impact of the different conditions on 3 components of math competence:
- procedural knowledge (ability to execute action sequences to solve problems),
- procedural flexibility (incorporates know- ledge of multiple ways to solve problems and when to use them), and
- conceptual knowledge (an integrated and functional grasp of mathematical ideas).
Students who compared equivalent equations used the shortcut steps less often than students in either of the 2 other groups, indicating less flexible use of solutions.
Students who compared solution methods had higher accuracy on conceptual knowledge items in the posttest than students in the other 2 groups.
Comparing solutions is superior
“In summary, comparing solution methods generally led to greater conceptual knowledge and procedural flexibility than comparing equivalent or different problem types,” the authors write. The study took place in students’ math classes over 5 consecutive classroom periods. On Day 1, students completed the pretest and on Day 2 an instructor gave a brief scripted introduction to students, worked through a solution with the class using a conventional solution method and modeled appropriate work with a partner. The pairs of students worked on their packets on days 2, 3, 4 after a brief whole-class lesson introducing a problem feature.
On Day 5, students took the posttest and 2 weeks later the retention test. The same assessment was used for pretest, posttest and the retention test. It was modified from an assessment used by the 2 authors in previous research in this topic area.
To be effective, comparison requires careful support, the researchers caution. Scaffolding for effective comparison was embedded in the instructional materials. Worked examples, carefully crafted explanation prompts and peer collaboration seemed to support explanation during partner work in the classroom. Teacher-led whole-class discussion would further enhance these benefits.
“At the same time, we caution that a poorly planned or implemented comparison is unlikely to facilitate learning,” they write.
“Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving,” by Bethany Rittle-Johnson and Jon Star, Journal of Educational Psychology, Volume 101, Number 3, 2009, pps. 529-544.