Reflections on How to Solve It

Introduction

I am rereading George Polya’s excellent book How To Solve It: A New Aspect of Mathematical Method. I greatly enjoyed my first reading and it has had a profound impact on the way in which I teach. Some of the lessons from the books appear to common sense, but others are deeply impactful. I am hoping for inspiration. I frequently find myself faced with students who have all of the knowledge they need to complete a problem/assignment, but are paralyzed by indecision on starting the problems.

Part 1

It is often the case that I can help students solve a problem with simple questions such as “What is the problem asking?”, “What is given in the problem?”, “What kind of relationships do you know between the given information and your goal?”. Students requiring only this minimum amount of information are my current focus for intervention. Polya suggest developing a set of questions for students to internalize through repeated modeling by the teacher.

The Four Phases

Polya suggest dividing a problem into four phases can help group these questions.

• Understand
• Plan
• Implement
• Reflect

Understand the Problem

Sufficient time should be spent to develop a thorough understanding of the problem. In the beginning this should occur as a discussion with the teacher and the class. The teacher should model appropriate questions, with emphasis on students using these questions independently later, during the lecture. Questions such as “What is the question asking?” and “Do we have sufficient information to solve the problem?” and “Have I solved similar problems before?” are all helpful.

Make a Plan

Creating a plan can be “long and tortuous.”¹ However, this is the most important part of being able to solve the problem, and in many respects more true to the idea of mathematics that the computation. Again, as the instructors we should be modeling questions that help students find the plan on their own, but also ingrains in them appropriate questions for when they are working individually.

Implement the Plan

At this point, students should be implementing their plan from the previous step. This is where the vast majority of my students want to start the problem solving process and thus struggle greatly with any king of rigorous problem. The teacher should still be offering questions, such as “Do you understand why this step works?” without digging to deep “Does this generalize to all similar problems?”

Reflect upon the Solution

This is a critical step that is often overlooked by both teachers and students. After solving a problem, this is a good opportunity for building more mathematical connections and developing problem solving skills. At the most basic level students could check any difficult computations or the logic of their arguments for correctness. At a deeper level, students could develop questions that are similar to the process they just developed to help retain these ideas.

• Math
• How to Solve It
• Pedagogy
• Polya