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 approach utilized by the director turned into as follows: each pupil needed to show the theorems consecutively. No lecture took place at all. nobody became allowed to inform his answers to every person else and every scholar acquired the following theorem to show, impartial of the opposite college students, as soon as he had proved the previous one correctly, and as long as he had understood the reasoning. This changed into a completely new interest for me, and one that I grasped with outstanding enthusiasm and a zeal for knowledge. Already, with the first theorem, i was a long way ahead of the others, and whilst my friends had been still struggling with the eleventh or 12th, I had already proved the hundredth. there has been simplest one younger fellow, now a medicine pupil, who could come close to me. even as this technique is excellent, strengthening, as it does, the powers of deduction and inspiring self sustaining wondering and opposition among college students, generally speakme, it can likely no longer be adapted. For as lots as i can see its advantages, one ought to admit that it isolates a positive power, and one does not obtain an overview of the complete concern, that may simplest be done with the aid of an amazing lecture. as soon as one has acquired a amazing variety of material thru [...] for college kids, this technique is manageable only if it deals with small fields of effortlessly, comprehensible information, in particular geometric theorems, which do now not require new insights and thoughts.

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?