aaron maxwell

Solving Math and Physics Problems You Don't Know How to Solve

Years ago, while tutoring a student in physics, I wrote some notes for her about solving class exercises when you don't know how to proceed: how you can start with what you know, and use the exercise as a stepping stone to expand your skill set, to be the kind of person who can solve that problem easily.

After writing it down, I realized that the notes contained a general approach for learning to solve theoretical problems of nearly any technical topic - math, hard science, or engineering. It's not for problems you already understand; those aren't really problems except in an academic sense. It is for situations in which you currently lack some of the skills or knowledge needed, and must find a way to proceed.

Recently while clearing out some old papers, I found a photocopy of those notes the student made for me. I've edited and reproduced them here for anyone who might benefit.

How To Do It

When encountering a problem you don't immediately know how to solve, make a list of the things you know that might be relevant - either from your own experience, or information that is given that you trust to be true.

Make another list of what it is that you don't know, and want to figure out. Usually it will be a shorter list than the first. Include not only the ultimate goal, but everything else that might be relevant that you don't know yet.

Remember (or look up, or write down) some formulas and concepts that you feel might be relevant. You may want to copy them into another list.

With your skills and insight, attempt to build a bridge - some kind of direct relationship - between what you know and what you want to know. Find several relationships between different pieces if you can. Write equations describing them if you can. Draw diagrams or pictures representing these relationships if you can.

If you get stuck, or encounter a new unknown quantity, identify the things you need to know to get unstuck, and repeat this process.

When you have challenges:

  • Review the list of things you know, making sure you understand what they mean.
  • Think about groups of similar concepts, and how they might relate. Examples: power, current, voltage; speed, velocity, acceleration.
  • Think about ways you can test some of your assumptions. For example, you can use an equation to make a prediction that is easy to check. Or you may be able to use units.

Once you do have the understanding needed, the solution becomes easy to see and grasp, and you won't need anyone to tell you how to solve it. This procedure can help you get to that level of understanding.

Thanks to Sunny Lee, the aforementioned student, for digging up the notes and giving them to me.