## The Three Stages of the Problem-Solving Cycle

Essentially every problem-solving heuristic in mathematics goes back to George Polya’s How to Solve It; my approach is no exception. However, this cyclic description might help to keep the process cognitively present.

A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us. During two decades of observing myself and others in the teaching and learning process, I’ve noticed that the most neglected phase is often the first one—understanding the problem.

## 3-Stage Problem-Solving Cycle

The Three Stages Explained

• Understanding the Problem: The most important part of solving any problem is understanding it. Ask yourself or guide others to ask themselves the following questions:
1. What am I looking for?
2. What is the unknown?
3. Do I understand every word and concept in the problem?
4. Am I familiar with the units in which measurements are given?
5. Is there information that seems missing?
6. Is there information that seems superfluous?
7. Is the source of information bona fide? (Think about those instances when a friend gives you a puzzle to solve and you suspect there’s something wrong with the way the puzzle is posed.)
• Strategizing: Now that we think we understand the problem, we choose a strategy or a set of strategies to try to solve the problem. Ten general strategies[1] are:
1. Logical reasoning
2. Pattern recognition
3. Working backwards
4. Adopting a different point of view
5. Considering extreme cases
6. Solving a simpler analogous problem
7. Organizing data
8. Making a visual representation
9. Accounting for all possibilities
10. Intelligent guessing and testing

I have produced videos explaining each one of these strategies individually using problems we have solved at the Chapel Hill Math Circle.

• Implementing: We now implement our strategy or set of strategies. As we progress, we check our reasoning and computations (if any). Many novice problem-solvers make the mistake of “doing something” before understanding (or at least thinking they understand) the problem. For instance, if you ask them “What are you looking for?”, they might not be able to answer. Certainly, it is possible to have an incorrect understanding of the problem, but that is different from not even realizing that we have to understand the problem before we attempt to solve it!

The Cycle

As we implement our strategies, we might not be able to solve the problem, but we might refine our understanding of the problem. As we refine our understanding of the problem, we can refine our strategy. As we refine our strategy and implement a new approach, we get closer to solving the problem, and so on. Of course, even after several iterations of this cycle spanning across hours, days, or even years, one may still not be able to solve a particular problem. That’s part of the enchanting beauty of mathematics.

I invite you to observe your own thinking—and that of your students—as you move along the problem-solving cycle!

[1] Problem-Solving Strategies in Mathematics, Posamentier and Krulik, 2015.

About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.