TITLE: Beyond the Exercises?
QUESTION [6 upvotes]: I've entertained and become bored with quite a few interests, though mathematics has more or less been my central passion throughout my high school/middle school life. I've only recently started into formal mathematics and self-teaching, with undegraduate books on number theory, abstract algebra, and the like (I'm currently into Dummit and Foote and Hoffman and Kunze). It's been an exhilarating adventure all right, and it's even seemed to abate some of my minor depression, but I'm starting to wonder if the experience is truly for me.
My regimen is, read the section of the text, do all of the exercises/proofs at the end, capture a few of the more interesting ones on my diary/blog, and never move on without having understood a solution (mostly feasible through this site and Crazy Project). Sometimes I do some independent ventures, but it's almost as if the studying has sapped both my creativity, energy, and incentive to do so (why distract myself with slow, irrelevant, amateur breakthroughs when I can learn so much faster with the texts?).
But this has become somewhat tedious and threatening to me. I vigorously self-studied Mandarin over the summer, and have completely forgotten everything of it already; all of my effort seemed to amount to very little. Is this the same thing? Again I do enjoy my studying, but a great part of my motivation is "Break through the crust to get to the oil," and it's disheartening to imagine my labor will continue to go unrewarded.
I've tried to seek out some sort of mentorship or companionship to ease some of this solidarity-stress, but there's been little luck. My teacher is fine and competent about his job, but seems like a dead-end for this sort of thing. I've sampled the Math Olympiad at my school, but the students and proctors seem wholly interested in streetfighter-mathematics like geometry and functional analysis, rather than my taste for aggregative and proof-based math.
Simply put, is there any way to rekindle my passion or otherwise augment my enjoyment of mathematics given my situation?

REPLY [4 votes]: I would say, for the most part, doing every exercise is a waste of time and can easily kill interest. After reading a section you probably know if your understanding of the material is good, so choose exercises based on making sure you understood the material and on how much fun they look. Doing every problem is just way too tedious(at least for me).
You should also consider trying to find papers (that are relatively accessible) and work on understanding the paper. This is more focused and typically studying some "special topic" can be more interesting and solidify your knowledge in ways doing unmotivated exercises after unmotivated exercise can't do. Plus studying papers leaves more room to ask questions, go down more paths, and discover your own interests. Finding "more specialized" textbooks might be something else to keep your interest. Currently you are studying Dummit and Foote, maybe you should pick up a book on a more specific algebra topic (say geometric group theory for example) and read/study them concurrently. Use the specialized book to give more motivation and connections to the more general book.
I know its difficult especially when self studying, but try to write a paper on some topic (not necessarily research level). Think of some questions, or maybe there is some problem that was mentioned as an aside in some book but has captured your attention and write something on it. For example in my first "real" math class we had a final paper on a some number theory or algebra topic/problem. I did mine on the Burnside problem (but only proved the weak Burnside problem in the paper), I learned a ton and was interested the whole time (although sometimes the research was overwhelming).
If your close to a university it probably wouldn't hurt to attend some math seminars, even if you are pretty sure you won't understand most of it.
Basically, doing just exercises can, in my experience, do more harm than good to mathematical skill, retention of knowledge, and interest.