As a part of my undergraduate education, I took a class called Mathematical Problem Solving. The professor presented us with a list of problems from which we chose at least five to solve and demonstrate to the class over the course of the semester. As a part of our solution, we had to choose and identify a problem-solving strategy (from George Polya’s list of strategies) that we would then apply to the problem. This was both a good introduction to a wide range of problem-solving strategies and a powerful example of how different students see and solve problems in different ways.
When I started teaching, I sometimes tried to incorporate the strategies into my instruction. The challenge was providing a variety of problems that spanned both the standards and the strategies, giving students the opportunity to practice both in a meaningful way. To help teachers address this challenge, Didax now offers Problem Solving Practice Cards for grades 3 through 5 that provide both a problem and a suggested problem-solving strategy.
Strategies Build Procedural Fluency and Conceptual Understanding
By pairing problems from five key strands with a variety of problem-solving strategies, these cards provide grade-level practice that builds both procedural fluency and conceptual understanding. When students have an effective, efficient strategy for solving a problem, they become fluent thinkers and doers of math. The exposure to rich problems helps students build conceptual understanding as they explore the essential concepts in each strand. Both of these—procedural fluency and conceptual understanding—are key elements in building rigor in mathematics. (For some additional discussion on rigor and other ideas for developing procedural fluency, take a look at this earlier blog post.)
Problem Solving Practice Cards make it easy for students to identify the problem and a strategy they could use to solve it. The sample cards below show the problem and the suggested strategy, along with the Going Further task.
For some students, this kind of problem-solving thinking is natural and maybe even “easy.” These students see the relationships in the math and make connections quickly. Other students may ignore the suggested strategy and resort to brute force and algorithms to try to find a solution. Still others will need a little more support to find a solution, or may find it easier with a different strategy altogether. What all these students have in common is a need to go further and extend the problem. Each problem-solving card includes a Going Further question or task that will help students extend the mathematical concept to a new problem.
As a teacher, it seems like the really good ideas take time to put into practice. These new Problem Solving Practice Cards reduce the time from idea to implementation, and provide problems that will help students build concepts and grow mathematically. Happy problem solving!