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Comparison Adds Up

New research advocates for having students compare problems and solutions when learning math

July 21, 2021
Math teacher in front of white board

From deciding on the best brand of paper towels to weighing one treatment option to another, we use comparison to solve problems all the time. But can this common learning strategy be translated to the classroom?

In their new paper, Harvard Graduate School of Education professor Jon Star and Vanderbilt professors Bethany Rittle-Johnson and Kelly Durkin advocate for the use of comparison in mathematics. “We want students to engage in careful thinking, reasoning, and sense-making in math class,” says Star. “Teachers already try to do this in many ways. We are proposing another very powerful way for teachers to do this, which is for teachers to get students to compare and contrast multiple ways that math problems can be solved.”

What to compare?

It may be helpful, the authors say, for math teachers to provide students with two options for solving a problem. These options can be two correct strategies, a correct strategy with an incorrect strategy, or even two confusable problem types. Allowing students to compare problems and solutions lets them notice, discuss, and reason about meaningful similarities and differences to deepen their understanding of the math that underlies the problems.

A few examples include:

  • Asking students to compare two different ways for solving the same problem and then to discuss which way might be better for the problem, and why.
  • Comparing a problem solved correctly with a second strategy that has a common error can help students better understand the “why” behind the mistake.

Instructional support and scaffolding for comparison

Trying to understand a single method for solving a math problem can sometimes be challenging, which suggests that comparing two methods could be even more overwhelming for students. To take full advantage of the benefits of comparison, teachers need to think carefully about how they are managing and directing students’ attention so that comparison is possible.

A place to start is with the choice of what two problems or methods to compare. “When teachers are thinking about what problems to put next to each other to compare, they should look for problems and methods that are mostly similar but a little different, and where the differences are what teachers hope students notice,” Star says. “When this is true, the differences will really pop out and will stick with students.”

A few additional strategies that can help teachers implement comparison effectively in the classroom include:

  • Displaying both solutions at the same time, side by side. Comparison is easiest when both problems are visible at the same time, such as side by side on the board or on the same piece of paper.
  • Aligning the visual display of the problems. Where and how the to-be-compared problems are placed matters. The problems should be aligned so that it is easy to see similarities and to notice differences.
  • Underlining or color-coding important information to highlight similarities and differences. This helps focus student attention and makes the important details stand out.
  • Pointing. Teachers and students can use pointing and gesturing as a way of guiding student attention to key similarities and differences.
  • Asking clear, prompting questions to encourage student explanations of key points. Once students notice similarities and differences, teacher questioning can prompt students to make meaning of the compared example and to support deep thinking.

Read more about the research behind comparison here.

Find additional resources for teaching using comparison here.  

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Key Takeaways
  • Comparison plays a key role in how we learn — and can be leveraged in math class. 
  • To avoid information overload, compare no more than two problems. These problems should be mostly similar and a little bit different. The difference should be what you’d like students to focus on.
  • Present the problems as clearly as possible to students. Make sure it’s easy for students to move back and forth between the examples and be consistent in how you label and discuss the examples.
  • Scaffold students with prompts, questions, gestures, and other cues to direct their attention to key steps and details. Ask higher-order thinking questions as well to ensure students are tapping into underlying, broad concepts.
  • Provide them with time to reflect and process their learning to help ensure they can apply these strategies in other situations.
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K-12 Learning and Teaching STEM Learning