Mathematics Problem Solving Strategies

Problem solving is the key activity in any mathematics course.  Through the act of problem solving, you hone your critical thinking skills as well as your organizational skills.  Most people will agree that solving problems (any problem) always begins with an organized plan or approach.  

There are many mathematics problem solving techniques that have been developed over the years. The most commonly employed is the approach developed by George Polya in the 1940s.  His 4-step approach is a simple plan that can be followed to get organized and begin working on the solution to a problem.  

Polya’s 4-Step Problem-Solving Process

1. Understand the Problem
2. Devising a Plan
3. Carry Out the Plan
4. Looking Back

Step 1: Understand the Problem

It may seem obvious, but gaining an understanding of what is being asked is a critical first step to solving a problem.  In order to gauge your understanding of the problem, ask yourself questions like the following: 

• What are you being asked to do in the problem?
• Is there sufficient information that you can use to solve the problem?
• If you were asked, could you restate the problem in your own words?
• Could you create a picture or diagram that could assist you in solving the problem?

Step 2: Devising a Plan

You’ve probably heard the phrase – If you don’t plan, you plan to fail.  Coming up with an approach to solving a problem is the next important step in this process.  It is important to know that there is no one way to solve a problem.  In fact, there are multiple pathways to a solution (depending on the type of problem).  Any plan to solve a mathematical problem will involve many of the following activities: 

• List what you know so far.   
• List the item(s) that you’ve been asked to find.
• Draw a picture or diagram if you feel it will be beneficial. 
• Determine which information you’ve provided is necessary to find the solution and which information is not. 
• Is this problem similar to one you’ve seen or solved before? Recall how that problem was solved.
• Identify formula(s) needed to achieve the solution to the problem. 
• Be creative, make a guess, sometimes a solution pathway will occur to you spontaneously.

Step 3: Carry out the Plan

Now that you’ve made your plan, it is time to put it to work.  As you work through your plan, do the steps make logical sense?  Be ready to potentially alter your plan as you go along.  Do you have confidence in your plan?  If not, consider revising your plan or reaching out to your professor for guidance.  

Step 4: Looking Back

When you solve a problem, it is important to take a moment to reflect upon what you have done.  Consider what worked and what didn’t work. Reflection can be an important process to help you devise strategies that you can use for future problem solving. 

• Can you check your solution?
• Does the solution make sense to you?
• Can you see another pathway to the solution?
• Can you use this solution or approach to help you solve a different problem?

Conclusion

While the strategies discussed on this page can be useful, ultimately it is up to you to devise the plan that works best for you.  No matter what process you choose, taking the time to understand a problem, devise a plan, carry it out, and reflect upon it will form a framework upon which you will improve your own problem solving and organization skills.  

Reference

Polya, G. (1957). How to solve it. A new aspect of mathematical methods. (2nd ed.). Doubleday Anchor Books.