When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial.
After reading "The Art Of Problem Solving" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much:
The method used by the director was as follows: each student had to prove the theorems consecutively. No lecture took place at all. No one was allowed to tell his solutions to anybody else and each student received the next theorem to prove, independent of the other students, as soon as he had proved the preceding one correctly, and as long as he had understood the reasoning. This was a completely new activity for me, and one which I grasped with incredible enthusiasm and an eagerness for knowledge. Already, with the first theorem, I was far ahead of the others, and while my peers were still struggling with the eleventh or twelfth, I had already proved the hundredth. There was only one young fellow, now a medicine student, who could come close to me. While this method is very good, strengthening, as it does, the powers of deduction and encouraging autonomous thinking and competition among students, generally speaking, it can probably not be adapted. For as much as I can see its advantages, one must admit that it isolates a certain strength, and one does not obtain an overview of the whole subject, which can only be achieved by a good lecture. Once one has acquired a great variety of material through [...] For students, this method is practicable only if it deals with small fields of easily, understandable knowledge, especially geometric theorems, which do not require new insights and ideas.
I feel that this type of environment is something you don't often see, especially in the US -- perhaps that's why so many of our greats are foreign born. As I understand it, he does go on to say that he wouldn't particularly recommend that method of study for higher mathematics, though.
A similar question was posed to mathoverflow where Tim Gowers (Fields Medal) went on to say that he recommended similar methods to study: link
I'm not quite certain that I understood the context of it all, though. Upon asking a few people whose opinion mattered to me, I was told that it if time were precious to me, it would be a waste going about studying mathematics in that way, so I'd like to get some perspective from you math.stackexchange. How do you go about studying your texts?
Edit: Broken link added fixed.