Designing Practice

Here’s a handout I gave my students this week (link):

I’ve written about variation theory before. The idea is that, from each problem to the next, only one thing changes. This does three things. First, it scaffolds learning for students who are having a hard time. There is less to focus on because most things stay the same from each problem to the next. Second, it helps students focus on important features of the problem. They can make connections based on cause and effect – changing the problem in this way caused the solution to change in that way. Third, it provides opportunities for more confident students to see shortcuts and make connections they might not make otherwise because of the patterns in the problems.

I don’t want to take too much credit here. This handout is far from perfect. I often design practice activities like this because practice is important, and I can throw something like this together in a few minutes that targets an issue I see students having. In this case, many students seemed to understand the meaning of the inequality sign, but not how to “solve” an inequality as they would solve an equation. For instance, for questions like #2, a lot of students would say something like “x=2.” Which is a true statement! But is not what it means to solve an inequality, in the sense of describing the set of all possible solutions. So we spent some time talking about that distinction, and I threw this together to help students practice and make connections between what it means to solve an equation and what it means to solve an inequality.

But helping my students work through these problems, I noticed a benefit I hadn’t before. The two places that a number of students got stuck were problem #2 and problem #6. And in each case, most of those students had successfully solved the problem above. So instead of needing to explain a problem from scratch, I could explain the problem they were stuck on by connecting it directly to a problem they had already solved and knew how to do — building a bridge from what they already knew to something new. It really changed the character of the conversations I was having with students, and I think it helped them to make connections they wouldn’t have made otherwise.

Practice is important, but how teachers structure practice isn’t often a popular conversation. It’s easy to dismiss things like this as just “worksheets” or “drill and kill” or whatever. But I think this was a really effective tool to address an issue I saw my students having, and I learned something new about how to structure practice in a way that helps students who are having a hard time.