How to Survive Being a Math Major
"God does not care about our mathematical difficulties. He integrates empirically."--Albert Einstein
Caveat lector: I can only tell you what worked for me (so far), in a specific environment (Cornell University, 1997-2001). Some of this might be useful to majors in other fields. (Some of the following comes from harangues I've directed at my sister. She majored in English.) Who knows?
It seems to me that there are two kinds of math majors: (a) those who eat, drink and breathe math, and (b) those who don't, but find pleasure in the subject nonetheless. A banal statement? Yes, but it makes a useful distinction. I fall into the latter category, and probably a lot of what I say won't apply to the former. If you're past your freshman or sophomore year, chances are you know all of the below anyway.
Surviving Classes
Nothing ruins a class like a professor who can't teach, especially if the subject doesn't come naturally to you. (If it does, chances are you can compensate without much effort.) The first line of defense is to find out which professors really can't teach (especially if you anticipate that you're not going to be the curve-breaker) and avoid their classes. Listen to fellow students who have taken class A with Professor X, though with a grain of salt; possibly a brilliant student isn't going to notice bad teaching as much (though I know exceptions) and a struggling or unmotivated student might have trouble in the best-taught class. (More banal statements, I know. Bear with me.) Asking other professors strikes me as indiscreet, though if you have a candid advisor you might get some pointers from him/her.
On the flip side, you might also find out which profs are really good (I don't mean easy, I mean effective) and actively try to take their classes and visit their office hours. In math classes, where the interaction generally consists of lecturing and copying down chalkboarded theorems, this might also be a way to get to know profs who might be willing to write you recommendations for grad school.
Your second line of defense, if it turns out that you can't avoid taking a class with a less-than-ideal prof (due to required courses, scheduling issues, etc.), is to find out if the TA is any good. Assuming this is the case, the TA Is Your Friend. In some cases they're better at explaining difficult concepts or problems because they remember what it was like learning them, as opposed to the prof who has had a blinding intuitive grasp of them for the past 30 years. Do Not Be Scared of the TA because s/he is a TA. I made this mistake and it hurt me in several classes. (This applies also to professors and their office hours, if they're good at Explaining Things.) They are not, to my knowledge, there to eat you or make fun of you. They are there to help you.
No help from the TA or the prof? The third line of defense is to talk to fellow students, whether they be higher-classmen or grads who have seen the material before, or others in your class. Getting to know other math majors is a good idea; I didn't do this very well myself, but I've seen what a tightly-knit, friendly and cooperative group of physics majors can accomplish (too many evenings spent at RUPH with my physics-major boyfriend). If you're in a subject where cheating, project/experiment/lab-sabotage, and backstabbing are the norm, you may be less fortunate, but a friend or two never hurts. (I heard horror stories about the genetics lab at Cornell.)
Your fourth line of defense, albeit not an ideal one, is to read the textbook (if any). You should be doing this anyway. The truth is, many of us learn better by working out problems/theorems and having things explained to us, not solely by reading math texts. Just as unfortunately, I can think of several textbooks that are well-written from the viewpoint of someone who already knows the material and is looking for an elegant treatment, but terribly written from the viewpoint of someone who is learning the material for the first time and needs examples, explanations, pictures, not mind-searing elegance. If you're lucky enough to have a decent, readable textbook and the prof follows it somewhat closely, reading relevant sections is never a bad idea. If all else fails, hit the math library (or appropriate library) and look for other texts on the subject. If your prof is wise s/he will have put a few on reserve for exactly this purpose.
Finally, if there's a tutoring service available at your institution, look into it. Shop around until you find a particular tutor (or tutors) you're comfortable with, if you have choices. Perhaps I'm biased, since I tutored both math and writing. These are people who either volunteer or are paid to help you. Frankly, I doubt most of us are in it for the pay. Yes, tutors sometimes ask fellow tutors for help.
None of these will help you if you're not putting in effort (with the exception of those people who put in no effort but are talented enough at the subject that they fly through anyway) or just don't care. Sometimes there are reasons to fall into this category (your uncle is going through cardiac bypass surgery, your effort is going to another class that is much more important to you or your major or whatever). Sometimes there aren't. That's for you to judge.
In the meantime, and these almost shouldn't have to be said, if you're having problems, going to lectures (or sections if available, which they aren't in upper-level math courses at Cornell) and doing the homework in a timely manner are also helpful. (Yes, I've been guilty, on occasion, of failing to do one or the other. But only on occasion.)
Math Homework and Culture Shock
This section is really math-specific.
If you're a math major, chances are you've survived or are surviving calculus (differential and integral calculus, the beginnings of linear algebra and differential equations, and vector calculus at Cornell, and probably elsewhere, with variations) without too much trouble.
If you go on to do applied/computational math, then you're probably going to see more of the "same" (using "same" in a loose, non-isomorphic sense). I haven't pursued this route (two years of comp sci were enough), but it seems that learning Mathematica, Maple, or Mathlab (or related programs), as well as computer programming in general, is either helpful or necessary.
If you decide, instead, to try theoretical math, you may run afoul of what my advisor, Prof. Marshall Cohen, called math culture shock. (I did.) This is when you are confronted with lectures that are entirely theoretical and your homework consists entirely of proofs, and your brain, which has been conditioned to compute things, not prove them in a manner that the average math prof or TA considers rigorous (or elegant), implodes.
Well, perhaps it's not that bad, but there's a leap between the first two years of calculus and upper-level theoretical courses. When I took applied algebra, I'd already been through honors intro analysis and so the notion of "rigorous proof" (however clumsily executed on my part) wasn't foreign. It was foreign to any number of engineering-oriented folks who were taking the course, though the proofs in applied algebra were nowhere near as strenuous as those in honors analysis.
Realize, if you are or might be in this position, that there is a world of difference between evaluating random integrals and proving, oh, the existence of the Lebesgue integral for a class of functions. Math profs and TAs are brutally effective at finding unjustified assumptions and leaps of unlogic in your proofs. They aren't usually doing this to be mean. In math, where proofs often involve abstract notions and not experiments or tests in the real world, notions of "right" have to be downright anal. Get used to it. Ask people to point out just why your cute three-line argument doesn't work. After a while, it starts to sink in. Give yourself time, get help if needed, and be kind to yourself.
Maintaining Your Sanity
Assuming you don't eat, drink and breathe math alone, you are likely interested in subjects other than math. Insofar as your schedule permits, indulge yourself. Take electives. (If you're a double or triple major, perhaps this doesn't quite apply.) My idea of an elective, for example, was a language or history course, not more math, or physics, or computer science. Chances are that's different from your idea of an elective (physics may strike you as a reasonable "leisure course"). Hobbies might also perform the same function. In any case, my brain works better if I can do something that's not math and that's "just for fun" once in a while. Perhaps yours does too. (We will ignore my boyfriend, who took 3 physics courses and 1 math course one semester, and actually lived through the experience.) I don't mean to imply these other courses are "easy," or to be blown off. I adore history. But it let me rest the math-brain and use the history-brain for a bit.
Have a social life, or as much of one as you are comfortable with. In my case, that's not very. At the same time, realize (especially if you like socialization) that you're probably going to have to compromise in order to get your work done. Some people like to schedule out every hour of their day. Others save their funtime until after they've gotten a decent amount of work done. I don't recommend the opposite, saving work until after a decent amount of free time, since I've seen it fail spectacularly. (But people keep trying.) It also helps to have friends so there's a support network (both for you and for them) in case things get rough.
Exercise. Eat. Sleep. Maybe this is hypocritical. I would have gotten a lot less exercise if going to Cornell didn't involve so much walking to classes. Sometimes taking a walk helps you regain perspective on life, something I need often (just because I'm stuck on half my problem set doesn't mean it's the end of the world, it means I'm having a lousy week). Whenever my parents called, they never asked what my grades were. They asked if I was eating well, getting enough sleep, and in good health. Insofar as it is possible, be good to your body. It's the only one you have.
It's okay not to do well sometimes. This is situational. If you want to get a doctorate in math, C's in honors classes won't help. If life has blown up in your face, maybe you won't get that problem set completely done. It happens. (It happened to me. I don't claim to have been a model math major.) If you're in your first analysis class and you're discovering that you're a lot better at algebra (or statistics, or geometry, or...) than analysis, don't beat yourself up about it. I can't swim (but I float and splash nicely), figure out chemistry, or dance, either. You can work harder for the moment, or accept a lower grade with good humor, or avoid the subject in the future. There are few people in this world who are good at everything all the time, if there are any at all.
Where to Go Next
What can you do with a bachelor's in math?
You should have some idea of this from far more reliable sources, but what the hey. You could become an actuary or go into consulting or some area of industry. You could go on to graduate school, get a doctorate, and happily ensconce yourself in the ivory tower. You could go on to teach math at whatever level. You could write sf/f with math in (I don't recommend this as a primary source of income, but it can be done). Or you could be confused.
The hope is, of course, that you do have some interest in math remaining after ~4 years of undergraduate education and that said interest hasn't been beaten out of you. Go forth and spread the math!
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