How to solve (almost) any math problem

academics High School math
By Evan T.

Math is all about problems -- questions for which you don’t currently know the answer -- and problems can be really frustrating. That feeling of being stuck, for me, goes from a scattered confusion to a mind-numbing blankness. It’s really easy to shut down and give up, so the first step to solving any math problem is to persist! Don’t let the frustration get the best of you. Just be patient, take a deep breath, clear your mind, then get a new sheet of paper and start again. I know that’s all much easier said than done, but with practice you’ll get better and better at working through the struggle. Plus, often the harder you work, the more satisfied you’ll feel when you find an answer.

Besides persistence, I find it useful to have a guidebook for getting unstuck. Once I realize I’m stuck, I’ll just pull stuff from this list and hope something works:

Rewrite the problem

Try to rephrase the problem in your own words. Could you explain the problem to a classmate that’s never seen the problem before? Can you come up with equivalent problem statements (e.g. if you’re trying to show that x is divisible by 15, you could instead show it’s divisible by 5 and 3).

Write down what you know

Maybe we know A is a 2x2 matrix with rank = 2. This means A must be invertible, so we could write down its inverse. Then we also know its determinant is not 0. This also means the only vector we can multiply by A to get the zero vector is the zero vector itself. Even writing relevant formulas counts here. If you keep writing down things you know, maybe one will help you get closer to your solution.

Make the problem easier

If you still have no idea where to start, make the problem easier! Instead of trying to place L-shaped tiles on an 8x8 chessboard, try a 2x2 or a 4x4 first. Trying to show something for all integers? Try showing it for a few small integers first. Maybe you’ll learn something from the smaller cases that help us solve the larger one. 

Guess and check

Looking for zeros of a polynomial or an N for a limit proof? If you’ve got no ideas on how to come up with a value, you may as well guess some values and check if they work. Even if they don’t work, maybe you’ll learn something about the desired value (e.g. maybe it should be larger/smaller).

Draw a picture

Especially for problems with functions or combinatorics, pictures can help you develop intuition for what the problem’s asking and maybe possible solution approaches. To take this a level up, grab some props and act it out!

Tidy up your work

Get a blank sheet of paper, erase your chalkboard, clear your tablet. Rewrite your work in a more organized way. Sometimes you’ve already got all the pieces you need written down, they just aren’t organized enough to come together.

Look for similar examples

Pattern recognition is key to problem solving. Ask yourself if you’ve seen similar problems before, then take some time to go back and understand how they work. Try to figure out which variables / values in the example relate to which values in your current problem.

Conclusion

The steps in this list won’t necessarily fix all your math problems, but they should at least help you get started. If you’re interested in more problem-solving resources, take a look at George Pólya’s How to Solve It. In it, he talks about a complementary approach that involves making plans and trying them out. 

At the end of the day, problem-solving is really about practice. With more practice and working through struggle you can get better at problem-solving so you just have to stick to it! As a bonus, none of these methods are necessarily math-specific. I use these all the time in computer science and physics, and I’d even argue these are useful for life in general!

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