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[ptarmigan]

I apologize in advance for asking a question that you can't know the answer to, but I'm interested in any kind of comments you can give me.

I'm 34.  It has taken me a lot of years to finish my undergrad degree (in math) because I've worked full time (at a real, career-type job) while doing so and have largely treated school as a hobby.  I'm getting my degree from a 4-year college (not a university; there are no graduate programs).  It has become clearer and clearer to me over the past couple of years that I want to study math forever.  I have always liked it "well enough" but once I got past the calculus sequence into the proof-based stuff I was hooked. 

So, I've applied to eight grad schools, and we'll see what happens.  This post isn't about whether I can or will get in, but about the choice to go, assuming that I do get in somewhere with funding.

I guess what I'm wondering is, exactly how hard IS grad school?  Besides the calc sequence (except diff eq, which I somehow never managed to fit in), I've taken a proofs course (it's a gateway course at my school), discrete math, one semester of advanced calc, two semesters of non-Euclidean geometry, linear algebra, and prob/stats.  I'll have the second semester of advanced calc and abstract algebra next semester and then graduate.  I have A's in all of my math courses other than Calc 3, where I got a B.  If my school had plusses and minuses, nearly all of them would be A+'s.  My average in advanced calc is about 99%. 

However, I don't feel at all like a math genius.  I can only read my advanced calc textbook(s) with a pencil and paper and a fair amount of effort.  Even though I've done extremely well at advanced calc, and enjoyed it thoroughly, I have found the material difficult.  I took the Putnam last weekend and I'll be lucky if I got a couple of points.

In my second geometry course, I had to write a paper.  It turned out that writing papers in math is absolutely thrilling and I enjoyed every minute of it.  It wasn't original research, of course, but...wow.  I bored everyone around me for weeks talking about it and worked on it for 12 hours at a time on more than one occasion.

Is enthusiasm and being reasonably good at math enough to succeed at grad school, with work?  What is the population of other math graduate students like?  Are they math geniuses?  (I'm talking about students at schools in the middle or bottom half of the NRC rankings, not people at MIT or Stanford, etc.)  Am I going to totally flunk out?

[ptarmigan]

It's probably obnoxious to follow-up on one's own post, but I think I understated my difficulties in math above.  I have to work to understand ANY undergraduate math textbook, not just the advanced calc one.  I get frustrated a lot.  I actually cried (briefly) in my discrete math class during an exam when I had trouble with a recursively defined sequence.  I make algebra errors constantly.  I could go on...

[greyscale]

There are people better-qualified than me who will probably comment, but I hope I can be a little bit helpful at least. I did my undergrad in math and went on to grad school in computational biology. I was an undergrad at a very intense school where a solid grasp of abstract math (analysis, algebra, topology, etc) always seemed just out of my grasp, though in retrospect I probably could have learned it solidly if the pace and content of the classes had been less intense. Compared to my classmates, I was not cut out for math grad school, nor was I interested. So I really know nothing about grad school at the sorts of universities you're considering -- and you certainly are much better prepared than I would have been. I didn't have enough patience to work carefully through the books like you have. I think that almost all serious math students hit the point where they need to grind their way through each chapter with a pen and paper to really understand it, by the way. That's not a weakness.

But if you're interested, you might consider statistics (or applied math). It's not that it's easier or less competitive, per se, but there are so many research problems in statistics (either on its own or in an applied context like biostatistics) that I suspect you could find a good topic and make a solid contribution. There's not the same sense of it being full of geniuses. And serious statistics has plenty of proofs, too, so it could be just as satisfying to you.

[mathgrad]

These are really great questions to be asking yourself as you prepare for this very serious but very rewarding undertaking. 

How hard is grad school in math?  Well, just as it should be, it is very hard.  Enthusiasm can get you pretty far, as long as it can be channeled into regular, hard work.  I can't say that talent doesn't matter, but certainly most of my cohorts who are 'super geniuses' put in a lot of time (50-60-80 hours per week) in studying things they love.

In terms of your background, I think you are about where most students who come into lower ranged programs stand.   You've just begun to understand the basic parts of algebra and analysis.  Such students usually struggle at first, but if they adapt to the jump in difficulty and commit themselves to hard, consistent work, they can do well.

Will you take courses in the spring term?  What will you study?  If there is any way possible, you should really think about doing a graduate level algebra or graduate introductory analysis. course.  Perhaps with some faculty member you like a lot. 

I did this in your situation and it helped me in many ways.  First of all, it felt like a less scary introduction to graduate level math.  Second, it helped me a lot when I was actually in gradschool, having seen a lot of the material before.

Best of luck.  You remind me of myself when I was going through the same kind of indecision nonsense.

[ptarmigan]

Will you take courses in the spring term?  What will you study?  If there is any way possible, you should really think about doing a graduate level algebra or graduate introductory analysis. course.  Perhaps with some faculty member you like a lot.

Thanks, I appreciate your feedback.  I wish I could take a graduate-level course in the Spring, but that won't work, for two reasons:

(1) I'm already taking as many classes as I can handle (given my job) with Advanced Calc II, Abstract Algebra, and a 1-hour seminar, and

(2) My school doesn't have graduate-level courses.

[kedves]

I'm not in math.  But I wonder what you would like to do with the degree.  You say that you would like to study math forever and you like writing papers in it.  Do you also want to teach?  What sorts of courses would you like to teach?

[cgfunmathguy]

I'd like to follow up on a couple of comments. First off, I went to grad school to get my master's when I was 37 so I could start a THIRD career.

To emphasize Kedves' point, you need to have a goal beyond graduate school. Why, besides studying more mathematics, do you want an advanced degree? Mine was to teach at a community college, and I knew that I needed a master's to do that. Define your goal. If you're willing to work toward the goal, you should do fine.

Next, you need to realize that even "supergeniuses" need access to pencil and paper whenever they read mathematics. My thesis advisor, who is one of the five smartest people I know, would read journals at his desk with a stack of blank paper beside the journal and a pencil or pen in hand. It is how you should be reading any writing in mathematics.

Finally, you are expected to have some understanding of the basic abstract concepts walking into graduate school. So, if you're taking a graduate-level numerical analysis course, it would be expected that you understand the high points of an undergrad course in numerical analysis. If you don't know these things, you'll be expected to fill in the gaps by reading on your own (mostly).

Is it difficult? Yes. Is it rewarding? If you love mathematics, yes.

[ptarmigan]

I don't really know what I want to do.  I've worked in industry for a lot of years now; I have a semi-professional type of position.  I wouldn't mind doing something similar [in a rough sense] to what I do now, but as one of the professionals instead.  That would take me in more of an applied direction if I were planning ahead, presumably.

I've always wanted to teach.  I guess I don't really see myself doing heavy research on an ongoing basis, so probably something like a CC professor, or a professor at a 4-year college like the one I am graduating from. 

If I get in somewhere with funding, and go, I'll be taking a large pay cut, but I don't particularly care whether it ultimately pays off or not.  The idea of continuing at my current job just to accumulate wealth sounds boring, and when I think of what I would do with money, I think "buy a house", and then I think, "Great, then I'll be even more trapped."  I'm completely out of debt and have some (small) savings, and the entire point of getting here was to free myself to go do something different if I want.

Working on a PhD in math sounds like a really challenging and interesting thing to do for a few years.  If I ended up back in this exact job with an unused PhD, it wouldn't kill me.  Doing something with it would be even better.

I probably sound kind of aimless and unmotivated, but I just feel like I'd rather pursue whatever I want to do at any given time than try to have a plan for the rest of my life.  Nevertheless, I of course want to pursue the degree in such a way as to give myself options for the future. 

[tinyboss]

Chiming in with the others to say that the only way to read a math textbook is with pencil and paper.  Nobody (I've ever met) can just read them like a novel and get much out of it, at least not the first time through.

I'm your age and just finishing my first semester of grad school in math.  In fact, our stories are nearly identical, except that I finished my undergrad at a place where I could take a few grad courses in my final semesters.  If you get funded at a reasonable school, and if there are no problems with relocation etc., then I think you should do it.  The fact that you started to get excited after calculus, when you took your first proof course, is a good indicator that you'd enjoy further study.

You might consider approaching a prof you like about taking a reading course, i.e. a one-on-one course between you and the prof.  It's a big favor to you if the prof agrees, but if you have built some good relationships then one might be willing.  If you're not able to take any actual graduate courses, a reading course or two will help to mitigate that weakness in your application.  I know you said your schedule is tight, but it'll make a big difference in your admission chances.

[ucprof]

The fact that you like writing papers on the subject and you like doing proofs is a very good sign for your potential in research.  However the courses you are currently taking are the minimum level for good PhD programs in Mathematics.  So expect to have some catching up to do if you get into a serious PhD program.  Also expect that you are not the only one who has to catch up. If you really have a passion for the subject then you are not at all crazy to do the degree.  I suggest to apply to broad based programs that have some faculty in a variety of areas of mathematics, including a reasonable group in computational mathematics.  If you get some background in that while in graduate school it will open up more career options beyond just the academic teaching/research.  Your current college likely does not have enough breadth or depth in Math (given your courses and what you say) for you to really have a good idea what sub area to study - you will need a year or two to sort that out and it is an important decision not to be taken lightly.   I know someone who finished a PhD in Math at approximately the age of 40 after a prior career in industry.  That person is now a full prof (in math) at a very well known R1 and has a great life. 

[ptarmigan]

Thanks, all of you.  I'm finding these replies much more encouraging than I expected.

[daniel_von_flanagan]

However the courses you are currently taking are the minimum level for good PhD programs in Mathematics.  So expect to have some catching up to do if you get into a serious PhD program. 

Second this.  By graduation you will not have really had a suitable selection of courses from which your future ability is easily judged, and you could end up in an institution which does not match your potential.  I don't really know the solution to this, other than maybe to look at programs that have a stand-alone Masters degree (these are rare in math in the US, but not unheard-of). - DvF

[ucprof]

Second this.  By graduation you will not have really had a suitable selection of courses from which your future ability is easily judged, and you could end up in an institution which does not match your potential.  I don't really know the solution to this, other than maybe to look at programs that have a stand-alone Masters degree (these are rare in math in the US, but not unheard-of). - DvF

Good comment by DvF.  Case in point, the person I know who got the PhD around age 40 and is now tenured at an R1 started out at one university and moved after a year or two to another one - the latter being the top place in the US in their subdiscipline.  Getting a PhD at a top place is not a requirement for tenure at an R1, but it sure helps - mainly because people take your file more seriously if you have a letter from the best in the field that says you are the best young person in the field.  There are plenty of examples of people with PhDs from a top 30-40 school who are tenured at an R1, however if you look at the PhD institution of the faculty at the R1s you will see a trend and it is mostly that they have PhDs from a few select places.

[daniel_von_flanagan]

Even if (s)he is not looking for an R1, it helps for almost any job to have the degree from a top school.  Also, this calibration works both ways - you shouldn't go to a school that is too far beyond you. - DvF