Problem solving in mathematics and beyond

One doesn’t discover new lands without consenting to lose sight, for a very long time, of the shore.

-André Gide

One of the things that I often faced in my clinical practice and with the students that I mentored was this confusion about acting. “I don’t know what to do, so what should I do? Well nothing, I’ll wait around until I figure out what to do.” No you should put together a bad plan and you should implement it because even if you fail in the implementation you’ll gather information and then you can rectify the plan.

-Jordan Peterson

Math

Math problems are some of the hardest problems humans solve. Besides the obvious things like practicing and learning more theory, are there concrete techniques that can make us better problem solvers?

If you open a book on math problem solving, it will probably talk about heuristics. Heuristics are relevant because they sit in between generating candidate steps automatically (which requires no special teaching or memory aid) and generating them uniformly at random or exhaustively (which is useless). Heuristics may be general (see below) or branch-specific (e.g. major counting techniques, inequalities cheat sheet). They also may be more for information gathering or more for directly taking a step towards a solution.

General heuristics:

• perform change of variables
• How to Solve It
• Schoenfeld, A.H. “Teaching problem-solving skills.”
• Ch. 1 of Larson, L. C. Problem-solving through problems.
• if you’re working from definitions try leveraging theory or vice versa
• name and conquer

Claude Shannon suggests “try to restate [the problem] in just as many different forms as you can”. T. Tao even says “The human brain has got many different modes of thinking. So we have visual modes, we have symbolic modes, we have modes where we are trying to fight some sort of adversary. And by changing the language of your problem, you are activating different areas of your brain.”

Comedian Lee Mack on generating ideas:

I was suddenly in a position where I could perform as many sketches as I could think of in front of millions of people. But I didn’t have enough. So I went off with my old college mate Neil Webster, and we locked ourselves away in a cottage for three days. After a day of achieving nothing I decided that I’d had enough of staring at the fireplace not knowing what to write about, so out of frustration I picked up a magazine and told him to pick a random page number. There were loads of different little articles on the page, so I asked him to narrow it down to a corner. He said top left. It was an article about fishing. I told him that we would both sit and write a sketch about fishing. So we did. […] Then we did it again, a new page, a new random corner. We found that about every one in four mini-sketches had just about a funny enough premise, or key joke, that it was worth exploring a bit more.

Similar techniques are described here. However, in math if we’re solving a particular problem we want to generate ideas in a more constrained manner. Two possible methods are computer tools or bootstrapping by riffing off of your own discoveries while making attempts towards a solution.

Computer tools for generating ideas:

• https://github.com/andrew222651/mathtext/tree/master/info_retrieval
• AI

With regards to bootstrapping, Richard Rusczyk’s top tip is “Do something … At some point you have to stop staring and start trying stuff”. Rusczyk elsewhere: “Notice that we didn’t just sit and stare at the problem and wait for it to solve itself. We have to add lines and variables so we can build equations. Don’t expect to just memorize formulas and bash geometry problems.”

• You can’t write piano music all in your head, you have to press some keys to help prompt ideas. in math, paper is the piano: write stuff down. this can be thought of as freeing mental RAM.
• When deciding whether to try something or not, factor in not just how likely it is to solve the problem in one step but also the fact that the process of trying it may help you generate further ideas

Beyond

The “Build-Measure-Learn” loop in startup strategy is a lot like math problem solving in that it says you don’t have to start with the complete solution, and data generated by exploring the implications of an idea can help find further ideas. See also https://longform.asmartbear.com/posts/extreme-questions/ for some heuristics

Does problem solving training in one domain improve problem solving ability in another domain? Cognitive psychology finds that cross-domain transfer is not automatic. Trinchero, R., “Chess Training and Mathematical Problem-Solving” (2016) says

The results suggest that chess practice can enhance problem-solving abilities in children, but only if chess training conveys problem-solving heuristics to pupils.