Beyond the Exercises?

Question

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?

Answer 1

Take a look at coursera, there you can enroll in formal classes over the 'net. Stay around here, if you spot a question you should be able to answer, try your hand at it. The motivation of teaching somebody/helping somebody out is powerful.

But it could also be that your enthusiasm ran out. If so, don't torture yourself by going on.

Answer 2

I can't retain information without a goal. Do you have one? Is there a particular "big theorem" you're interested in understanding? Or is there a particular topic in Dummit and Foote or Hoffman and Kunze that you want to know more about?

If not, peruse a library (or the internet) for books whose introductions or tables of contents excite you. Perhaps you'll find you don't know enough ring theory to fully understand it. That's fine, you can always go back to Dummit and Foote.

I admire your desire to fully learn foundations and your diligence in completing exercises -- certainly the pendulum swings too far away from those things for many people. But you might find it more exciting to try something new, even if you can't fully digest it one go.

(Also: I have no idea what you mean by "streetfighter-mathematics like geometry and functional analysis" or why you contrast those subjects with "proof-based math." I invite you to read some literature in either field. I think you'll find them quite proof-based.)

Answer 3

The key to maintaining interest in and enthusiasm for any undertaking you enjoy and in which you'd like to invest time and energy is to "keep it fun" and to "find community" with others who share your interest. Trust me, I know how important this can be. I tend to be avid about everything I undertake, and have learned the hard way that if I don't make a point of "lightening up", I can burn out easily: all-or-nothing burns anyone out. And like you, it is very easy for me to lose interest in anything unrelated to what I'm currently obsessed with! ;-) So you might have to force yourself to take breaks (at least that's what I have to do!)

Sure, be serious about and persistent in rigorous study, but make room for having fun with it. Remember recess time from years ago? They work: and they're meant to help, not hurt. Sometimes we need breaks. And taking breaks doesn't slow your progress down; indeed, if you're like me, I do a lot of processing when not explicitly studying, processing what I've studied: and that's been crucial in my learning and mastery of math: giving time for things to be digested and settle firmly in my mind. And try playing with the abundant software that's readily available: e.g., try exploring what you're learning in Dummit & Foote by using, say, GAP?

Create challenges for yourself; set goals; reward yourself along the way! Don't waste time on "busy work" (like computational exercises that you understand thoroughly).

As for community: Personally, I find Math.SE to really help, in this respect. Answering questions, dropping in on "chat", etc: it really helps alleviate feeling alone and isolated. Math.SE has MANY self-studyiers who ask and answer questions, visit the chat room, etc.

What about applying to various "summer camps" sponsored by universities?

Start a blog, and post on others' math-related blogs. That's a good way to reach out and connect with like-minded folks.

Answer 4

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.