Its the same old passive learning (reading, listening to lecture, barely engaging with the content) vs active learning (deconstructing theorems, pondering about each hypothesis and why each is necessary, connecting the various results together, trying to formulate possible
relations between objects even if they're not explained to you explicitly, looking at mathematical objects from different perspectives)
I'll give you a couple of examples (I'm a 2nd year undergrad):
My Geometry B exam comprised a part of Linear Algebra Addendums (Jordan's Standard Form for matrices), a part on Projective Geometry, and a part on Theory of Hypersurfaces viewed in Projective, Affine and Euclidean Spaces. I had a professor for the first 2 and a professor for the latter.
The first professor did his theory of Jordan through pure Linear Algebra, which was painful to me because I couldn't quite understand where did things spawn from, but I got an intuition that everything could be viewed as a Module over a Principle Ideal Domain and then verified it and I was like "it all makes sense now" (the theorems were all easier to understand too but I had to study it also the linear algebra way and it wasn't so bad after the realization). What I did was I had an intuition, tried to verify it (through the use of books and AI, too), and effectively looked at a mathematical theory formulated through Linear Algebra from a purely Algebraic point of view.
Another example: studying projective geometry, I came across the notion of "projecting cone" (or whatever it is called in English) and couldn't figure out its use in any way...
Then, reflecting about conics and quadrics, I realized that any non degenerate hypersurface that exists in a projective space can be projected from a subspace that does not intersect the space the hypersurface exists in, to form a degenerate hypersurface, and that subspace is the singular point of the hypersurface and vice versa, given a degenerate hypersurface you can section it with subspaces that do not intersect the vertex (to keep the rank constant) and eventually obtain a non degenerate hypersurface (again, excuse me if the terminology is wrong but I'm mostly directly translating from Italian).
This represents an instance of looking at mathematical objects that seemingly have nothing to do with each other and finding a connection between the two.
The perk of doing such things is that once you do reach the realization, you are never forgetting the knowledge you attained, but you HAVE to have the intuitions ON YOUR OWN and only use the appropriate tools to verify. For this, you need to sit down, breathe deeply, and ponder about the implications and all that. You probably got it by now.
Bonus points if you go to follow lectures, the professor asks a question and you take your time to find an answer.
Once my Algebra B (Ring and Field and Module theory) professor asked a question about an invertible element of a field such as Z/mZ[x]. I went further and tried to prove what an invertible element might look like, and produced a 5 pages mini-pseudo-paper proving the result. It was something that one could consider "research", it was fun as fuck, I dumped like 50 hours in a week and that's what got me trying to aim for a PhD to do math research.
Conclusion: use your intuition, explain to yourself the notions in a simplified matter that retains all the info (eg Cayley-Hamilton's theorem asserts that this does that, instead of the usual Let this be etc etc therefore etc etc
, I hope it is clear), be curious and investigate connections and what happens when hypothesis are taken away in the formulations of theorems, and have fun. If you have any questions ask away no problem. Remember this thing, mathematics is one. People divide it into courses for convenience, but it's all connected.
