Persistence in Problem Solving

This is a project that I had my students do when I taught linear algebra at the University of Western Ontario in the summer of 2020. The Persistence Techniques mentioned in the Project Description are at the bottom of the page. You can also find a printable version of these techniques here.

Intended Audience

Students enrolled in any math course. The Persistence Techniques below mention linear algebra concepts, but that's not essential.

Learning Outcomes

After completing this project multiple times over the course of a term, students will be able to...

1. Identify situations in which a persistence technique could be useful.

2. Choose persistence techniques best suited to a given situation.

3. Explain why these persistence techniques are best sutied to that situation.

4. Successfully implement persistence techniques 75% of the time.

Core Concepts

The goal of this project is to build students ability to persist in the face of difficult problems, even when they get stuck.

Project Description

In this assignment, you will reflect on and answer the following questions:

1. What gave you the most difficulty recently? This could be a concept, an example, or a homework problem.

2. If nothing gave you difficulty, choose a problem that you think would have been difficult for your peers, and describe how you would have assisted them to be persistent. What persistence techniques would help your peers most for this problem, and why?

3. What persistence techniques did you use? How did you use them? Which were most effective, and why? Which were least effective, and why?

4. Were you successful in overcoming this difficulty?

   • If not, how do you plan to?

Persistence Techniques

The following is a list of techniques you can use to help you keep on pushing through when a math problem gets challenging. I and others have found these techniques extremely helpful. There are lots of persistence techniques out there, and not every technique works well for everyone and in every situation. Moreover, just because a technique isn’t listed here doesn’t mean it isn’t helpful! In fact, I encourage you to try to find ways, both here and elsewhere, that help you!

Record your thoughts

You can write these down, you can make a voice/video recording, etc.

• Try to articulate what’s stumping you.

• What ideas have you had? Why didn’t they work?

Try some examples

• Replace variables with values. For instance, if the problem talks about a matrix A, but doesn’t say what the matrix is, try seeing what happens when you replace A with a specific matrix. Then try it again with a different matrix, and again with yet another different matrix. 

  • Can you spot a pattern?

• Change some of the values to something simpler. For instance, if the problem has to do with vectors in R4, try it with vectors in R3. 

  • Can you solve this new problem? Can you modify what you did with this new problem to solve the original problem?

Learn from your past experiences

• Have you seen similar problems in the past?

  • These could be example problems, other homework problems, exam problems, even problems you saw worked out in a YouTube video!

• Are there parts of the problem at hand that are similar to what you’ve seen in the past?

• Compare those past experiences to the problem at hand. 

  • What aspects are the same? What aspects are different?

  • Will the methods you used to solve the old problems work here? If not, can you modify them so they will?

Look for relevant facts

• Is there a fact (or facts) you learned in lecture or the book that can be useful here?

  • This could be a theorem, equation, definition, etc.

• Look for keywords from the problem to help direct you to the right section of the book or the right lecture video. For instance, if the problem mentions eigenvalues, there’s a good chance that you can find something useful in the eigenvalue sections of the book.

• Does the fact you’ve found apply in this situation? 

  • If not, why not? Can you rethink your approach to the problem at hand so that it does?

  • If so, will it help you get closer to the goal of the problem?