Wax on, wax off

 I have noticed that the way I learn mathematics does not look linear. I do not move smoothly from definitions to lemmas to theorems. Instead, I seem to oscillate between a vague big picture and a very narrow struggle with a single example. For a long time I worried that this meant I lacked discipline or that I was learning incorrectly. Lately, I have started to think this is simply how my understanding actually forms.

It usually begins with a rough sense of what a topic is about. I might know, for example, that primes behave “better” than composite numbers, or that modular arithmetic modulo a prime feels cleaner. At that stage everything is verbal and slightly foggy. I cannot prove much, but I have a sense that something structural is going on. Then I zoom in hard on a single problem and get stuck there for a while.

A good example was the congruence $a^2 \equiv 1 \pmod p$. On the surface it is a trivial equation. I knew the answer almost immediately, but knowing the answer and understanding why it is forced are very different things. I spent an unreasonable amount of time staring at the factorization $(a-1)(a+1)$ and asking why that argument suddenly fails when the modulus is not prime. At that level, everything felt small and frustrating. I was no longer thinking about primes in general, just about why $2 \cdot 4 \equiv 0 \pmod 8$ breaks the logic.

Eventually, something shifts. The local struggle starts to illuminate the bigger picture instead of obscuring it. The issue was not the equation at all, but the presence or absence of zero divisors. Once I saw that, a lot of previously fuzzy statements snapped into focus. Cancellation works mod primes because zero divisors do not exist. Inverses exist for the same reason. Fields are not an abstract upgrade but a guarantee that factorization behaves itself. The small example ended up teaching me more about the structure than any high level summary ever did.

This is the point where I zoom back out. The original vague intuition is still there, but it is now anchored to something concrete. I am no longer just saying that primes are special. I know exactly where the specialness enters and exactly how it shows up in computations. The big picture has changed because the small picture was taken seriously.

Learning this way is slow and it often feels inefficient. I can spend a lot of time on what looks like a toy problem feeling very stupid. But I am starting to see that this back and forth is not a failure of method. It is the method. The big ideas do not descend fully formed. They condense around specific points of resistance, and those points are usually small, almost embarrassing examples.

Writing this down is partly a way of reassuring myself. There is no correct pace for understanding. There is only the cycle of zooming out to orient yourself, zooming in until something breaks, and then zooming out again with a slightly sharper view. If progress feels uneven, that is probably because it is real.

Also, here's a sketch I sketched.