First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started reading up, through Wikipedia and various other sources, on pure, proof-based mathematics. I was not a math major, but worked what mathematics I could into my studies, taking three courses (vector calculus, ODEs, and linear algebra) beyond the two basic calculus courses and elementary statistics that were required or recommended for my major. I obtained a book on real and complex analysis (which turned out to be terrible for a beginner) and began trying problems. I got some help along the way from my very kind Calculus I professor, and once I got to my second school I was able to sit in on a real analysis course. It was much too fast-paced for me to wrap my head around the proofs as they were presented, but I took thorough notes and enjoyed myself. Two semesters later I was able to take Introduction to Abstract Mathematics (I could already do proofs, but this helped to solidify what I knew and I did learn some new things), and the next semester (after an offer from a very kind professor) I was able to take a reading course in which I combined real analysis (for real this time) and introductory abstract algebra. Since then I have worked through the remainder of the abstract algebra book (covering basic ring theory, field theory, and some more advanced topics in group theory), linear algebra (proof-based this time), and I am currently studying point-set topology. I completed my B.S. last August, and sadly am no longer in an academic setting, so my study of mathematics is now purely self-study.

My pursuing a field different from mathematics should not be taken to mean that my passion for mathematics is any less than that of those who pursue it professionally; I simply feel called to do something different with my life, while pursuing math on the side as much as I can. I find mathematics (especially certain parts of it) to be unspeakably beautiful. I strongly relate to Bertrand Russell’s quote: “Rightly viewed, mathematics possesses not only truth, but supreme beauty.”

Since I’m asking a question, there’s obviously something that I want to know, but don’t. However, here is what I DO know, and do not need to be told: Self-study is very far from ideal. I should try to connect with a professor at a local school to work with. There are different ways in which to study the material, which lead to vastly different amounts of time input and levels of rigor. I have my own ideas on this, and have read others in other questions here, so I do not want that to be a focus of this question. I feel alright with the learning and studying and understanding; what I really need help with is the doing. I am primarily interested to hear people’s thoughts on two things, as indicated in the title: motivation and methods. So, at long last, my questions:

$\bf{Motivation}$: Motivation is a highly personal thing, but what applies to one may apply to others. Do you ever have a difficult time working up the motivation to do math? I am beginning to convince myself that mathematics is difficult, period. I have heard some good stories, but would like to hear more about how difficult it is, and what kind of effort it really takes and that I should really expect, so that perhaps I may not be so easily discouraged when I have a hard time with it. I’m afraid I’ve been subject to a rather vicious cycle: I think I have such a difficult time doing math because I don’t spend enough time doing it; conversely, because it’s difficult, I have a very difficult time convincing myself to sit down, free myself from distractions, and work at it, even though I know very well that that’s what I need to do. Can anyone relate? How did you overcome this difficulty? How do you ‘get in the zone’ where you can do some serious work and really accomplish something?

I spend a lot of time reading up on subjects far beyond my current reach, not in terms of understanding, but in terms of doing. For example, last night I was reading over some notes on the relationships between the mapping class group and Teichmuller space of a surface. It’s an absolutely fascinating and beautiful subject, and I honestly (you don’t have to believe me) understand the concepts. However, I know that in reality I am years away from being able to do such math. Has anyone else gotten so caught up in the higher, more aesthetically pleasing concepts that they had a difficult time trudging through the basics, even though they are necessary for getting to the really beautiful things in the long term?

$\bf{Methods}$: What I’m most interested in is methods for the work that needs to be done after reading/working through a chapter. Without guidance, and without any course syllabus available, how do I wisely select problems to work on? If I can foresee the path a proof will take with a few seconds of thought, should I take the (precious, limited) time to write it down? Are computational problems truly helpful? Should I specifically choose problems that I can’t see a quick solution to? If I want to be on a quasi-school-like schedule and cover perhaps one section per week, I can only do so many problems, but I have a very difficult time choosing, say, five problems when more than that interest me. What are some good criteria for choosing problems to work on? Here’s another aspect to the doing to be done after the reading: how long should I work on one problem before moving on? I know I will not get everything, but how should I balance not letting one problem eat all my time, and not putting in enough effort to make progress with developing thinking and proof-writing skills?

I realize that I have asked many questions. While direct answers are most appreciated, you may also use what I’ve asked just to get a sense of the sort of advice I am looking for.

I’m really baring my soul here, so please take it easy on me. I know this is a great community of mathematicians who have undoubtedly gone through their own struggles and mental turmoil, and I feel confident that I will receive some great advice. I apologize for my extreme long-windedness. Any and all answers are greatly appreciated, as I truly want to turn things around and start doing this the right way, that I may more fully experience the beauty of mathematics. Thank you.