I am asking for what, if any, the preferred study skills of approaching classes are like in higher math education are like **in general, ** and I am not asking necessarily for personal anecdotes for this question (though they are welcome if it's all you have to share)?
My question is what are the better studying methods for learning higher mathematics(i.e., subjects involving more proofs instead of calculations like Analysis, Topology, Axiomatic Set Theory, Abs. Algebra, etc.)?
Let me explain what I mean by giving you an example. As an undergrad at first, courses like the Calculus sequence, Differential Equations could be approached by learning how to solve problems even if you weren't able to understand the proofs(which were usually skipped over in the textbook by the professor).
But now, that I have taken Linear Algebra, and I am about to take Analysis I usually take a much different approach:
$ $$\cdot$$ $ I start by rewriting all the definitions and theorems and proofs of a chapter and memorizing them. Then I will work through the examples, and ideally get to the exercises and finish off the chapter. And also spend time reflecting on the topics to get a more intuitive understanding of the concepts involved. I skip very little, anything if at all from the books I'm working with even if the topics are skipped in class.
This approach is very effective, especially when I am studying a topic for my own interests, and I am really able to understand things at a level that my peers usually have trouble following. The draw back is that I move at a much slower pace than my peers and I end up struggling in a class towards the end of a semester because I am behind on the syllabus. I end up risking a low grade though all but once I have managed an A- or above. Also am able to work out Jech's Set Theory which others have told me is out of my league but I actually find it the right amount of challenging using this approach.
The method followed by the math department at my school feels very superficial and not as the affective long term. Usually, professors do not focus on the reasoning or intuition behind concepts. Tests are oriented so that we memorize by rote the proofs of the major theorems and regurgitate them on exams.
Let me give more concrete questions:
I asked a math professor once, and he said that real mathematics is usually done where you understand 2 or 3 pages a day of a text on your first reading of it. Is this true for graduate level work for the average student??
Do students who follow my immersive way of studying tend to have an advantage over those who don't when we get to grad school?
In terms of learning theory and doing exercises, how much importance is recommended I place on each?
4. Are there study techniques used in more advanced math courses (like engaging in discourse with peers, focusing more on memorization before attempting to do problem sets, taking notes in a particular way) that are more fruitful than others?