At an orientation for new and prospective math majors yesterday, I was asked to speak for about eight minutes giving tips on how to succeed in the major. Here is roughly what I said:
“For those of you who recently became math majors, welcome to the major. For those of you who are prospective majors, welcome to this meeting. I have been asked to speak for eight minutes giving tips on how to succeed in the major.
How to succeed in eight minutes: holy cow, I don’t know!
As you are no doubt aware by now, the upper division courses that form the core of the math major are very different from lower division courses. In lower division courses you are mainly concerned with computing things. How do I compute this integral? How do I compute this other thing? How do I get an A on the final exam? On the other hand, upper division courses are more about understanding concepts. Why is this definition the way it is? Why is this theorem true? Why is this hypothesis necessary? What is it good for?
This can all be very confusing, and my main piece of advice is to keep trying and don’t get discouraged or intimidated.
I’ll tell you a story from my own undergradute days. When I was taking complex analysis (like Berkeley’s Math 185), I was very confused. We wrote a complex number as
and I was OK with that. But then we talked about integrals and wrote
At this point I got lost, because I didn’t know what , , or meant! I somehow managed to struggle through the exercises, but I didn’t know what the heck was going on. Meanwhile, my classmates nodded sagely during the lectures, making me feel like a complete idiot. At the end of the course, I nervously went to the professor’s door where the grades were posted. It turned out that I had the highest score on the final and got an A in the class. I almost went to the professor to complain about my grade and say that he must have mixed up my name with someone else’s, because there was no way I could have done so well!
About two years later, when I was learning differential forms (another confusing and difficult subject), suddenly a light bulb went off, and I finally understood what we had done in complex analysis.
Anyway, I should also say a bit about what is not in the math major. From reading undergraduate texts, one could get the misleading impression that everything is all figured out, except for a couple of famous unsolved problems; you just have to memorize everything and you’re done. Nothing could be farther from the truth. Mathematical knowledge is currently exploding. The explosion in science and technology, which you can see in everyday life for example in the computer revolution, has a counterpart in mathematics, which however is less publicly visible because no one can understand it. Most of the problems that have been solved and theorems that have been proved have been done in the last 20 or 30 years. Now most of the math in lower division courses was figured out by the 19th century, with the exception of some of math 55 (discrete math). Most of the math in the upper division courses was done by the early part of the 20th century, so you are now learning about the automobile, but not yet about microprocessors.
When you do research at the frontier of knowledge, it is a very different experience. When you see an undergraduate homework problem, you know that it has a short solution, and you can probably figure it out in at most a few hours. In research, you often don’t even know what the question is! And when you do formulate a question, it might be impossibly hard. So the problem is to find a direction in which you can make progress. In addition, in teaching there is a host of additional challenges; and in applied math you are faced with real world problems which are not formulated as mathematical equations and for which it may be difficult to use mathematical techniques.
I encourage you to look at the blue sheet which lists various opportunities to gain research experience. In particular, the REU’s (Research Experiences for Undergraduates) are a great opportunity do some research over the summer. The problems you work on are sometimes a bit lame (so that they can be approached in a short time without much background), but give you crucial practice in working on problems when there is no answer at the back of the book — because no one knows the answer!
Since I am just a pure math advisor, I don’t know as much about the opportunities in applied math or teaching. When you have questions about these kinds of opportunities, or about where the math you are learning is going, you should ask the professors in your classes or your faculty advisor, who will be happy to talk to you about these things. And of course you can also ask me.
Well, I guess that was eight minutes, so thank you.”