On Solving Problems

It’s been a while since my last post. That said, I’ve kept the habit of solving “one problem a day,” so in truth I’ve never stopped learning. But since it’s also part of my teaching preparation, and sometimes even material for class, I haven’t been able to share it publicly. Still, as I work through problems each day, I often find myself wondering: Isn’t this something others might want to know? Or: Why is it that I was able to solve this? And perhaps such self-analysis could be useful to you as well.

After all, if you’re reading this, your real concern is likely: How can I score higher on regular tests? or How can I reach the passing mark on entrance exams? To achieve that, at least in mathematics, what truly matters is being able to solve problems on your own and get the correct answer. In fact, you could say that studying mathematics essentially is “solving problems.”

So then, based on my own daily practice, I’d say the process of solving a problem generally follows this flow:

Look at the problem statement.

Clarify what is being asked. Is it “Find this,” “Prove that,” or something else?

Identify what information is given.

Recognize what kind of problem it is. Is it similar to that problem I’ve seen before? Is it this type? Ah, an integer problem. Or a plane vector problem. Once you can recognize the pattern, the solution method usually comes to mind right away. For problems at the University of Tokyo level, I’ll often jot down a quick list: Method A, Method B, Method C…

If step (4) doesn’t work?that is, if I can’t grasp what the problem is really asking?then I move on to what I call the “hard problem strategies.” That means trying concrete examples, considering an easier related problem, or writing down what I know. In short, gradually chipping away at the hard problem.

Proceed with the calculations and steps according to the method chosen.

Check the answer. For instance, if I get something like sin x = 2, something’s wrong. On a multiple-choice test, if my result isn’t among the options, that’s also suspicious.

If it’s a written test, neatly write out the final solution on the answer sheet or in my notebook.
(For my own teaching research)

Reflect on the process?like a post-game review. Which method did I use? Why was I able to think of it? I write down this analysis for myself.

Repeating this flow every day, I’ve noticed in step (9) that there really are “solution parts” or “go-to moves” for certain situations. In other words, in this kind of situation, you usually do this. If I were to summarize these recurring solution patterns?let’s tentatively call them “mathematical schemas”?they might be quite useful to others. I expect there may be around 200 of them. And as a working teacher, I think it might be a good use of my spare moments to write them down.

Recently, I’ve been checking the blog stats, and it seems someone stops by every day. That made me think I should probably write something. Hopefully, I can post from time to time without interfering with my classes, and in a way that’s genuinely helpful to you.

Until next time.
September 9, 2025
Nasuno Kumao

Notice: Above sentences translated by ChatGPT.