I'm currently in my final year of a physics major. I have always been interested in math, and decided to take some courses in the math department. I have taken a semester of Mathematical analysis based on Apostol textbook, finishing the first 9 chapters. Recently, I also took a semester of abstract algebra based on Keith Nicholson's Introduction to Abstract algebra, covering the part of group theory and the chapter about p-groups and the Sylow theorems.
However, I didn't do very well on the tests on both these 2 courses. I'm not a guy that really like to attend lectures, since in my opinion this is somehow a passive and inefficient way to learn things. I instead try to prove every theorem on the text, and even if I get stuck, I would still spent hours to come up the proof on my own. After I'm confident with my proof, I looked up the text and compare with my own proof. As for the homework part, I did it in a similar way when reading the textbook's theory part, proving every exercises assigned without looking at any solution, and TA had marked high scores on my homework. After that, I did almost all the exercises on both books. Could anyone give me some advice on how to improve my study method on math? Thnks.
As a former professor of both physics and mathematics, I may have some insight on this question.
My first recommendation is the obvious: If you're not going to class and not doing well on exams, then go to class!! I am extremely skeptical of anyone but towering mathematical geniuses (some of whom I know personally) to learn mathematics without the benefit of classes.
[One of my tricks to get students to come to class is that I say I will solve one problem in class that is not in the textbook (which I wrote!), that will be essentially verbatim on an exam.]
Your core problem seems to be that you think classes are "passive." Hah! Not at all! Ask questions. Anticipate steps in derivations. Think during a lecture.
You're likely to spend more time thrashing around, frankly wasting time, as you seem to be doing by "deriving theorems on your own." And you won't truly know what is and what is not important. Not every theorem is equally important, or equally worth your time and effort. As a student you simply and positively cannot know that.
Next, given that you have a background in physics, you might profit from applying the math to physics problems where you have background and insight. I know I learned how to perform differentiation and integration in high-school mathematics, but I truly understood calculus when I applied it to physics, both Newtonian mechanics and electricity and magnetism.