What my mistakes reveal about my thinking.

 

As I work through introductory number theory, I have started noticing that my mistakes are not random. They cluster around a very specific behavior in my mind. I tend to switch viewpoints too quickly. Instead of staying inside one definition or one structure long enough, I jump to a more general interpretation before the foundation is stable.

This shows up clearly in modular arithmetic. For example, when I first learned that two residue classes $[i]$ and $[j]$ in $\mathbb{Z} / n\mathbb{Z}$ are equal if and only if $i \equiv j \pmod{n}$, I understood the definition but immediately tried to generalize it. I started reasoning about the classes almost as if they were single numbers, not sets, and occasionally I would try to compare them by looking directly at representatives instead of the congruence relation. The definition had not yet settled into my mind as an object.

Another example: when working with congruence equations, I sometimes tried to cancel terms without checking if the cancellation was valid modulo $n$. This is not a computational mistake. It is a conceptual one. I was treating modular arithmetic as if it behaved exactly like the integers, forgetting that cancellation only works cleanly when the modulus and the value being cancelled are coprime. Once I wrote this out carefully, the issue became obvious:

If

$$ax \equiv ay \pmod{n},$$

I can cancel $a$ only when $\gcd (a,n)=1.\mathrm{<br>}\mathrm{<br>}$ 

Without that condition, I risk losing solutions or introducing ones that were never valid.

These are the types of mistakes that keep repeating. Not because I misunderstand the math, but because I switch to a higher level of generality faster than the definitions can support.

The interesting part is that these errors are actually a good diagnostic tool. They show me exactly where my mental model is incomplete. When I rush into abstraction, the gaps in the foundation reveal themselves as soon as I try to use a property that does not exist.

The cure has been simple but effective: slow the step from “definition” to “application.” When I write out the definitions explicitly, the mistakes disappear. When I rely on intuition that is not fully formed, they reappear.

So this post is really about the role mistakes play in shaping my mathematical mindset. They tell me which concepts are stable in my head and which ones are still blurry. And as I keep going through Silverman, the goal is not to eliminate mistakes, but to use them to sharpen the structures I am building.