Comparing Methods Of Solving Problems

 

When we teach students the steps needed to solve certain types of problems, students end up memorizing the steps rather than taking time to understand the mathematical principles behind each step.  It's also hard to "tell" students those principles because it often goes in one ear and out the other but researchers at Vanderbilt University has found a way to help students with this. 

They discovered if you ask students to compare different ways of solving the same problem, it encourages higher level thinking while demonstrating a deeper understanding of the concepts.  It has been found that comparison is a more natural way of learning for humans. People use comparison so much in real life such as when they shop, they look to see which deal is a better one, or they compare routes from point A to point B.  

These researchers even took time to determine which problems are the best to use comparison with.  As mentioned earlier, one type of problem is to compare the different ways to solve problems. Another type of problem to compare are those problems which are confessable and finally, comparing correct with incorrect strategies. They discovered when students compared these three types of problems, they acquired greater conceptual knowledge and were more flexible when solving problems.

Furthermore, the researchers looked at the best time during the learning process to use this comparison technique. One thing they discovered was that students could compare methods if they understood at least one method.  After some practice, students showed they were able to compare the similarities and differences between two or more unknown methods but this needed more support from teachers.

When introducing comparison to students, teachers need to provide clear and visible examples to students while showing the solutions at the same time. In addition, students did better at figuring out the similarities and differences between mentors when teachers used well defined vocabulary, terms, gestures and visual clues.  Furthermore, teachers should provide questions to prompt clear explanation of key points and summarization of the major points brought up during the comparison. 

These same researchers are getting ready to explore how using the comparison method impacts student attitude towards mathematics, or for teachers to use it to decide if students need additional support. Fortunately Vanderbilt University has a site with the materials in a downloadable form to show you how to implement this method.  They have activities for linear equations, functions and linear equations, solving systems of linear equations, polynomials and factoring, and solving quadratics.  

I downloaded the first topic on linear equations.  The 27 page module has several activities from "Which is correct" to "Why does it work?" to "How does it work?" to "Which is better?" for multiple scenarios.  Each scenario shows two students doing the same problem and students are asked specific questions they discuss during a think, pair, share activity with a big idea at the end.  It provides the worksheets students need to do the activities.

Furthermore, there is a teachers guide which provides suggestions on when each activity should be done during the lesson, at the beginning, the middle, or the end. There are also questions provided to help guide the discussion students are asked to do.  I like they've provided the materials needed to try this method in class.  Let me know what you think after you've checked it out because I'd love to know.  Have a great day.