A soft question but I believe important to help get maturity in maths.
Not until I got admitted to a graduate applied math program that I started to learn math. Before that, I was a life science undergraduate that only know some entry level calculus. The more I learn math, the more I started to find it is more than the formulas , but also perspectives that would help to shape a good mathematician. In Terence Tao's real analysis book, he said that a mathematician don't quite think concepts in a way of "object", that they more prefer to ask what a concept can be applied to, rather than what a concept is consisted of. I think that is inspiring and indeed helped me really started to think as a mathematician.
My question is, when you are observing other people or yourself do math, what do you find makes a good mathematician ? What kind of point of views you need to have in order to solve a particular problems in fields like linear algebra, topology, real analysis, statistics, etc? Or, while learning maths, what kind of things I can pay attention to, or I can ask myself about, that would help me to become more to think like a mathematician?
(A bit more about what I used to think before I started learn math: I was pretty good at physics, which is more like modeling by reduction work to me: apply relative few principles and you can start analyze stuff. When it got to maths problem, it doesn't work in that way. I found from equations to equations, though it is "equivalent" between two equations, but the information conveyed can be much different.
Also a lot of times problems are solved by a constructive way, which seems not nature to come up at first glance, but often it indeed make sense once you have a higher view. For example, if you have the idea that the rank and determinant is actually describing "how big" a matrix is, you can easily start to crack down the dimension problems in linear algebra one way or another. But there are more subtlety in writing the inversion of matrix in adj(.)/det(.) that it is not obvious for building up an intuition of adj operator, but it is as well powerful in different application. Nevertheless, not many textbooks would tell you why we should care about adj.)