Motivation and methods for self-study

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.

-

I have been self-studying this entire year to prepare myself for graduate school, so I can speak from some limited personal experience.

Motivation is definitely an issue. Earlier in the year, before I chose to specifically focus on studying for my upcoming classes, I simply picked subjects that interested me. With this, however, it is very important to pick a book that is within your range, or else you will become discouraged. For introductions, picking books that have been labeled as "undergraduate" (Springer publishing often does this) is a good choice, and once you've seen a bit of the material, you can then move on to "graduate" versions.

Further, long-winded books that seem to "talk" too much may be actually good for self-study, because often times they give the motivation for the subject that you usually can only gain in a class. This was a problem I had when first tried to learn topology -- I picked a terrible book and had no context and no idea why I was to care about this object called a topological space.

As for picking problems, I, too, have a hard time working on the problems in the text book while self-studying. I find it easier to actually try to prove one of the main theorems in the text on my own, or with some starting help. Of course, it is still important to do as many problems as possible, so I often settle for the easy ones on a first go-ahead.

Is computation important? It depends what you mean. I think to some degree, yes. It is important to get a feel for the specific examples of the objects or ideas you are learning in the abstract. This is the thing that has helped me the most in self-studying. Always relate what you are learning to what you have learned before. It should always be possible, since mathematics is usually concerned with abstraction.

Lastly, while it may seem counterintuitive, learning more than one subject at a time might be beneficial while self-studying. It is easy to get "burned out" on one particular area, perhaps because you've reached a particularly dense proof that took you twenty minutes to get through, and maybe you are now fed-up. Instead of quitting for the day, I've found that my brain is still open to ideas from a less related area. Right now, I go from studying topology to algebra to measure theory almost every day.

EDIT: Although it may be obvious, getting "in the zone" has a lot to do with physical location. While you may be able to think about some interesting problems throughout the day, I've found I'm most satisfied with progress when I become completely focused, in an almost trance-like state. There is a section of my local library that has a quiet cubicle-type area. With no visual or auditory distractions and with a place that your brain associates with only doing mathematics, it is easy to reach this state. Hopefully you can find a similar place. Good luck.

EDIT #2: While this may not be the case for you, it is worth mentioning. In class, I never take notes. I find I learn best by listening, watching, and trying to comprehend what is going on. However, when self-studying, specifically reading a text book, I often need to slow myself down so I can absorb everything. Here, I "take notes," usually by writing out each and every proof. It is time consuming, but no more so than it would be for your professor to do the same on the chalkboard. And if I've just gotten through a hard proof/concept, I often get up and walk and make sure I can recreate the proof in my head.

-
Thanks for the good answer. I actually worked on algebra and linear simultaneously, but didn't like it in the end because I would end up departing one of them for a few weeks for the other, doing several related sections at a time. I would become disconnected from the other subject. However, I may try it again and even switch it up every day, as you have done. I do get burnt out sometimes, and that usually results in a 1 or 2 week period in which I don't do ANY math, so switching to another subject would certainly be more productive. –  Alex Petzke Aug 12 '12 at 3:05
My choice of locations is rather limited where I'm at right now, but hopefully I can find something. Starting two weeks from now I'll be alone most of the time and things will be peaceful. Then the challenge will just be turning off the music and computer! –  Alex Petzke Aug 12 '12 at 3:08
Yes, getting burned out is not hard to do. Actually, I switch learning subjects about every hour or so when I'm in a long stretch of independent studying (with perhaps a few minutes break in between to allow the material to digest). Occasionally this helps me see connections I would have otherwise missed. Oh, and I definitely understand getting away from music and the computer. I'm ashamed to say it, but it's nearly impossible for me to do serious studying in my tiny apartment! Of course, this is not to say I do not love math--a shared sentiment, I'm sure. Again, good luck. –  Three Aug 12 '12 at 5:41
Yes, certainly a shared sentiment. It's great to hear that someone can relate. Thanks for the help! –  Alex Petzke Aug 12 '12 at 14:38

How do you ‘get in the zone’ where you can do some serious work and really accomplish something?

I start with an explicit and reasonable mathematical goal in mind. By that I don't mean "do a certain number of problems from this book," I mean "learn the material necessary to prove this interesting result" or "learn the material necessary to understand how to interpret this interesting computation." The keyword is "interesting": if I can't drive myself to work using my curiosity, I admit that I usually can't do it.

This is as far as picking an appropriate thing to do rather than motivating yourself to do it. Finding motivation in general is not a problem limited to doing mathematics, but see, for example, How To Beat Procrastination. (In terms of that post, the strategy I describe above is about increasing value.)

What are some good criteria for choosing problems to work on?

To be honest, I have never successfully forced myself to work independently on problems from a textbook, so I have no advice about how to go about doing this. If you follow the strategy above, you'll instead set yourself as exercises whatever results seem necessary to pursue your goal.

-
Thanks, that helps. I think it may just be a little difficult when my goal is something like 'become reasonably good at the basics of point-set topology so I can delve into deeper areas like algebraic topology someday.' I just have such broad goals at this point that it's difficult to know what's going to matter most in the end. –  Alex Petzke Aug 12 '12 at 2:52
@Alex: okay, so find a topological theorem you find interesting (it does not have to be point-set topology, e.g. it might be the computation of the fundamental group of the circle) and backtrack to learn all of the topology you need to understand the proof of the theorem. –  Qiaochu Yuan Aug 12 '12 at 3:02
Interesting. Your approach really makes me question the linearity of the strategy I've been using (obtain good book on subject, read each chapter, do problems, move on to next book, repeat...). It does sound rather boring when I put it in writing like that. I'm just trying to avoid gaps in my knowledge. –  Alex Petzke Aug 12 '12 at 3:13
@Alex: of course different things work for different people. What I'm saying is that the linear approach doesn't work for me (I get bored). Maybe you can make it work for you. I sympathize with the desire to avoid gaps in your knowledge; I took a lot of classes precisely to avoid such gaps, but that isn't an option available to everyone... in any case, in some sense gaps are inevitable, and the best you can do is put yourself in a position where they are relatively easy to identify and fill in. As far as more general comments about nonlinear learning, see Ravi Vakil's comments at... –  Qiaochu Yuan Aug 12 '12 at 3:21
math.stanford.edu/~vakil/potentialstudents.html (Ctrl+F for "tendrils"). As far as reading goes, my pattern is generally to read quickly whatever I can (introductions to books, blog posts, answers on math.SE) and let everything roll around in my brain for awhile until things start to click, usually once I start working towards an explicit and reasonable mathematical goal. –  Qiaochu Yuan Aug 12 '12 at 3:21

Being a self-studier many decades out of school and geographically remote from a viable math program, I've experienced many of the sentiments you articulate so well.

This response is more a practical solution for the near term. I might offer two suggestions.

First, there is a great, free video lecture series on algebra taught by Benedict Gross at Harvard. To be succinct, he and it are outstanding. He follows "Artin." You can get a used copy. There are two editions. As part of the course, you can see the assigned problems. However, I would get the second edition. In it there are a few, mostly doable problems specific to each section - just the right amount. Then there are some pretty tough ones lumped together at the end of each chapter's problems.

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

Also, here are Artin's assignments for his course at MIT:

http://math.mit.edu/classes/18.701/

Secondly, it pays if you are learning from a book to pick one that is accessible. A particular suggestion is Ireland and Rosen's "A Classical Introduction to Modern Number Theory."

Here are homework problems:

http://www.math.brown.edu/~jhs/MA0156/MA0156Homework.html

In this case, the course was taught by an outstanding author and mathematician in the field, Joe Silverman, so I feel this is very much to my advantage.

-