- I want to study math full time, and
- I'm tired of having a job where people are surprised that I'm smart.
If I do go with the full-time grad school idea, there are three general paths I could take.
This is the most academically appealing path. What I seem to like best in math is stuff that involves a small number of axioms and seeing what comes of them. (This could change; I don't have a lot of maturity in math yet.) I think I would enjoy (as much as one ever does) writing a master's thesis or doctoral dissertation in pure math, once I got far enough along for such a thing to become possible. (My paper on Laguerre planes last semester was as close as I've gotten, and that was thrilling and I found I could easily work on it for hours and hours.)
Having this degree would broaden my career prospects in the regular world to a degree, and enable me to try to get a job teaching at a college or, if I want to continue doing research, at a university. Academic jobs (other than horrible, low-paid adjunct positions) are often difficult to come by, however, even if you want "only" a community college job.
Something in applied math - perhaps Operations Research - would be the most career-applicable path. Applied math interests me somewhat less than math with no conceivable application, but is still very interesting, and, though the thought doesn't thrill me, it seems very possible to imagine writing a paper in an applied math area. As far as academic careers go, a higher applied math degree would be about the same (as best I can tell) as a higher pure math degree. They have professors in both, of course, at universities, and for lower-level teaching-oriented positions I doubt it makes any difference.
Sally joked the other day that I could go to her interim school and get my PhD there, where they have a math education doctoral program that seems very research-oriented. I find this a very intriguing option (in general, not just at that school) because it has such strong highs and lows to it, in my view.
A PhD in math education is not at all helpful for a regular job in industry, as best I can guess, even though you do learn a lot of advanced math in the course of getting one. It pretty much limits you to the education field, if you want your degree to really count for something. It also does not feel as prestigious to me, which I admit is a consideration. ("I have a PhD in math" just sounds so much better to me than "I have a PhD in math education.")
Yet the topic of math education interests me greatly. I feel I would happily read any number of books or papers about it.
Yet I'm not sure the idea of doing original research in math education sounds that great. It doesn't really sound much better to me than the idea of doing original research in say, psychology, and I wouldn't go do that, and not just because my background for it is wrong. I like the research and want it done, but I'm not sure I should be the one doing it.
If I were to get a PhD in math education, the kinds of jobs that would be available to me would be the usual academic jobs (where I could teach either math - using my math education focus to do so more effectively, perhaps, especially if I specialized in post-secondary ed - or math education itself, like to future teachers), or I could probably get some kind of job in the public school system doing something like program or curriculum development, etc. (I'm not sure exactly what jobs exist, but there are surely jobs along those lines.)
A PhD in math education, or really any PhD, would probably make it a bit harder for me to get a high school teaching job, at least in a public school, since I would have the deadly combination of no teaching experience at that level + being required to be paid more than someone with a Bachelor's. But maybe a fancy private school would like to hire me, and I'd certainly have teaching experience in general, since teaching college courses generally happens as part of a program like that. (It's explicitly required at Sally's interim school, and would be part of TAing in any program.)
One question central to all of these considerations is really what kind of career I want to have, or at least, what kind of career I want to try next. I worry that I am not really suited, psychologically, to the general type of work I do now - i.e., working in an office. I seem to be an inveterate slacker in those types of jobs. And while having a more interesting and challenging office job - as I might after getting a higher degree - might help, it might also involve elements that I find even more impossible than what I face now, like project management.
I have never tried teaching, but I think it might be a kind of job that I would be good in. I enjoy teaching people things quite a lot, certainly. It seems like it would be very engaging and, while you can slack off as a teacher, it wouldn't have the same kind of slippery-slacker-slope thing going on, I think. I also fundamentally really like school - almost everything about school - and it would be nice to be a part of that kind of system.
Ultimately, then, I really don't know what I want to do and how I want to go about it, at all. My current plan is to apply to various schools for Fall 2010 and try to decide what to do between applying and getting the results of those applications.