Problem-solving in math is not just about identifying the keyword in a word problem, the word that will tell you whether to apply the quadratic formula or take a derivative. Problem-solving is often a lot more playful than that. Seriously!
I spend time working on pure math research but also on helping students with their current math homework problems. Honestly, I don’t remember the answers to everything. What do I do, then?
- Draw a picture. Using the info I’ve got, draw as complete a picture as I can, or a couple. What’s missing from the picture? What could be a bit more general? This works for, “How high does the apple get if it is thrown at 30 degrees and velocity 1.2 m/s?” as well as for, “What is the probability that max(X,Y) is less than .5 if X and Y are the coordinates of a point chosen randomly from the unit square?” In the first problem, my picture would include some representation of gravity and I’d start thinking of critical points or the line of symmetry for a parabola. The second picture might remind me that I need to use area for my calculation.
- Look for similar problems in my past. Obviously this only works directly if you’ve done a similar problem! However, you can use this idea more generally: the probability problem I mentioned above is, once you draw the picture, just a ratio of areas. I’ve computed a lot of areas before. The apple-throwing problem could be like a physics problem, like a precalc problem about the vertex of a parabola, like a critical point problem in calculus… There are a lot of analogies we can make and they often hold a clue to the solution.
- Look for similar problems in someone else’s past. Ask a friend. Read the book for examples. Search the internet for examples. I’m not saying you should offload your work on someone else: I’m saying that judicious use of your other resources is smart. Maybe you won’t find the exact same problem but you’ll find something similar, and then you’ll have the insight to solve your own problem. This often happens in real research: just this week I was reading a friend’s old research paper and saw something that looked familiar, and I was able to say, “Hey, those polynomials are just what I need right now for my own project!” A few weeks ago at a conference I talked to a famous guy in my field and asked how he’d had a new insight, and he said, I was talking to an old friend who is in physics about my problem, and he said it looked just like a system of particles that was really explained in 1989. Yep — that was enough!
- Just follow my computational nose, or hands, or whatever. Seriously, it’s like doing a puzzle: just start writing things down and doing things that are true (adding the same thing to both sides of an equation, taking square roots of everything in sight, multiplying by x, differentiating all the things). Follow the rules of logic, but not in a logical way, and see what happens — just like trying that puzzle piece in every wavy free segment until it fits. Sometimes if you just write you will discover things. Rely on your hands, not your brain, for a few minutes.
Play with it. Juggle the pieces of your problem. Lie on the floor and stare at the ceiling. Get a change of scenery. Eat some chocolate. Try to explain it to someone else. Write stuff down.