Math undergraduate coursework deficiencies

graphic7

I'm currently working on a B.S. in Mathematics at a southern, liberal arts university in the US. I feel as if I've missed out on several classes that any undergraduate who wishes to go onto a much more prestigous university should have (MIT, Stanford, etc.). In this particular area, the only university that offers graduate-level courses is roughly about an hour and a half away, and I'm not financially-stable enough to make the commute. I chose to attend this particular university, because it was all that was available to me at the time -- I earned a high-school diploma about the time I turned 15, and had a lot of difficulty getting into an undergrad program for some reason, so I got into what I could. I'm currently a junior and feel as if I have missed out on several, vital courses. By the time that I graudate, I'll have taken every Mathematics course the university offers. Here's a run-down of the courses:

[completed] Basic Calculus I, II, and III sequence (using Larson's text, which is a basic, plug-and-chug text with very little rigorous treatment -- better than Stewart, IMO. Had very incompetent instructor -- more on this later.)

[completed] Introduction to Abstract Mathematics ('Transition to Advanced Mathematics' by Smith, Eggen, and St. Andre. Typical proofs course taught by a reasonably competent instructor, but I feel that we could've done more if it weren't for the class -- half of which were re-taking it for a second time, because they failed the previous one. I was the only student that completed the course with an A. The next grade was 20 points away).

[completed] Linear Algebra ('Linear Algebra and its Applications' by Lay. Typical one-semester introduction course. Again, taught by a relatively competent instructor, but we could've done much more in the course if the same group of students who keep failing weren't in there)

[completed] Probability and Statistics ('Probably and Statistical Inference' by Hogg and Tanis. A Calculus-based statistical course, taught by the same incompetent instructor that taught the Calculus sequence. I basically came out of the course hating statistics, and having very little understanding. Emphasis was placed on plug-and-chug, while this text was more theoratical. Bad combination, IMO.)

[completed] Ordinary Differential Equations ('Introduction to Ordinary Differential Equations' by Saperstone. Taught by the infamous, incompetent instructor. Emphasis was placed on plug-and-chug, but I think most one-semester ODE courses would suffer from this, as well.)

[completed] Numerical Methods ('Numerical Methods' by Kincaid and Cheney. Same situation as the previous courses -- taught by the same incompetent instructor with a classroom full of students that were struggling to graduate)

[taking Fall of '06] Abstract Algebra (not sure what the text is, yet, but the same failing group is re-taking course, because they failed it the previous year it was offered. Taught by the competent instructor)

[taking Spring of '07] Advanced Calculus (unsure of what text we're using, but the competent instructor I've mentioned previously has said I'll be only person in the course, and that she'll try to 'enhance' the curriculum a bit for me)

In addition to the listed courses, there is also a Special Topics course, where the subject of the course fluctuates from year to year. Sometimes the subject is decent (last year we had 'Complex Variables), but this year it was dedicated to "Problem Solving" (seemed like a way to improve the exit exam scores, IMO). We also have a 'Directed Individual Study' type of course, that I plan on taking with hopefully the competent instructor. It may give me a chance to do some research before I enter graduate school.

This incompetent instructor I've mentioned several times has the policy of "the homework is the test." Exercises that are encountered on the homework are used as a 'problem pool' on exams and homework. This policy may be acceptable in some courses where the homework problems are extremely complicated or lenghty, but that's not the case here -- basic plug-and-chug.

I think I'm missing serveral courses that more 'focused', undergraduate programs offer. Namely, Complex Variables, Real Analysis (two semesters would be nice), and possibly some graduate courses. Unfortunately, my university had no graduate school, so graduate courses aren't available to me, unless I go elsewhere. I'm currently a junior, so I'll be graduating Fall of 07', and I'm just wondering what I should do by then if I want to get into a top graduate school (if I have to go into a Tier-II school, this would be acceptable, but I'd absolutely love to get into Stanford or some equivalent).

Basically, I'd like to open up a dialogue in this thread with several people that may have been or seen people in the situation I'm presently in. It's really discouraging looking at several of my peers at other universities, and seeing what they're taking. On a slightly more positive note, Spivak should be arriving within the next day or so via priority mail.

0rthodontist

I was at an inferior school for my first year of college where the situation was like you had. Fortunately I transferred to UMass. Much was my dismay when I found out that several of my two-semester courses at the inferior school would only substitute for a one semester course here, or even counted for nothing but a credit! If you want my advice, I'd say transfer to a better school and take an extra year to finish your undergrad if you can, especially if you want to go to grad school.

ircdan

I guess if I was in your situation I would try to do several directed independent studies with the professor you mentioned, and if he says no, then ask another professor. You might also want to do independent studies with more than one professor, since it's unlikely you'll be able to do two the same semester with the same professor I'd think. (I don't know what else you are taking though so disregard that if you are already overloaded with courses)

I would ask him/her if you could study material usually taught in courses like:

Topology
Complex Analysis
Calculus(maybe a continuation of your advanced calculus I class)
Linear Algebra(with more proofs, use a more advanced book)
Algebra(just more Algebra, maybe a continuation of your current course)

So that would be 5 directed independent studies, again unlikely you can do 2 with the same professor the same semester I think.

I am not in your situation but I am a math major, and I do commute about 3 hours a day so it's prevented me from taking certain graduate courses I wanted to take, but I have taken alot more math courses than you have and I have 2 semesters left, counting this one.

About the research, I don't know. I've had a few opportunities to do research and I chose to take mathematics courses instead. I think the courses you take now will benefit you more in the future(I don't know if this is true, but I think it is). All the math you learn now you need to know eventually and you will need to know it forever if your going to graduate school. On the other hand, I think research might look better on applications to graduate school. So it's a guess a matter of what's more important to you, getting into a good school(and a better school is better at getting students to do good research I'd think), or learning more mathematics in general, which of course again, all the math you learn now you will see it again later in graduate school, and it's stuff you need to know.

graphic7

Right, transferring to a better undergraduate program is probably a better option, unfortunately, I'm not sure what that would mean for the financial aid I'm receiving. I also have about 80 hours of credit, and I've been under the impression that undergrad programs do not like accepting transfer students with a significant amount of credit hours already.

graphic7

One of the professors, in particular (the competent one), seems to be willing to work with me. Unfortunately, I haven't take her up on the chance on a number of occasions. I've been "burnt out" for the better part of a year, because I had take so many courses with the incompetent one. But now, as graduation is getting closer and closer, and after trying (unsuccessfully) to go into industry, I've decided that being a professional mathematician is for me. Anyway, this particular professor (from what I gather) is planning on modifying her usual teaching style/curriculum for this Advanced Calculus course in the Spring, because I'll be the only student in there. Also, from what I've heard, references are a very important part of the admission process for various graduate programs, and I have no doubt that I'd be able to get a superb one from her.

ircdan

That's great to hear! Yea I've heard that references are important also. I'm applying to graduate schools myself really soon(early october probably). Anyways another thing to consider is the GRE, in particular, the mathematics GRE. It's roughly 50% calculus, 25% algebra, 25% everything else. Anyways doing really well on this can help alot. If anything it will maybe show the admissions committee you can do math(if you do well), it's a timed test and you don't have much time, it works out to about 2.5 minutes per question or so. Anyways taking more courses will help you do better on this. Also remember that as long as you get in somewhere you can keep studying mathematics. Just work hard wherever you end up going:) Goodluck!

Edit: Oh cool the only student in the course, pretty neat!

leright

Also, to add to ircdan's list, I would recommend studying some PDE.

0rthodontist

Well, I transferred in and got 63 credits (APs + one year) and I probably had more than 63 that I didn't get credit for, I don't remember. I know there were some people in my transfer orientation session who had 70+ coming from a community college. IF your financial aid is OK, it's probably worth a try.