As some of you may know, I am majoring in math and economics. Michelle has already written a lovely post about Course 14 (economics); I want to also talk a little about my experience being Course 18 and how it has differed from my experience doing math in high school.
There are four types of math major at MIT—pure math, general math, applied math, and math with computer science. I do pure math, which will be the subject of this blog post. It’s important to note that there are significant differences among the different tracks—applied math has a whole different set of requirements, of which I have taken very few, and general math has super flexible requirements. Math with computer science, or 18C, requires several computer science classes and math classes that double as computer science classes; I have also taken very few of those. My perspective is that of someone who has been firmly on the pure math track for a few years and who hasn't taken many of the more applied or CS-related classes.
So let me now talk about my pure math classes. The requirements are: one differential equations class, a real analysis class, two algebra classes, one topology class, one additional analysis class (manifolds, functional analysis, or Fourier analysis), one or two seminars (depending on whether you took the “communication-intensive” version of real analysis or another “communication-intensive” class), and at least two additional math classes of your choosing. Here's a page describing the communciation-intensive math classes--basically, they're math classes with a heavy writing/presentation component, so that math majors learn how to talk and write (in LaTeX) about their work.
I recently realized that I am almost done with my major requirements (as in, I just dropped the one remaining class that would fill them out. It has been very abstract and mentally demanding but also satisfying). My college math classes require a lot more mental legwork than my high school ones did, in the sense that I’ve had to do much more struggling and wrestling with many new abstract concepts in my head for long periods of time, and it has been difficult! But worth it!!
Differences from high school:
I think I came into college with the expectation that my classes would be noticeably harder and more time-consuming than they were in high school, and for the most part that turned out to be true, though they ramped up in difficulty gradually rather than all at once. However, I also used to have the sense that I could learn anything I wanted as fast as I wanted, and I have definitely changed my mind on this front and come to terms with my limits in the past 2.5 years of school. One thing I want to note is that if you don’t think high-school math is particularly exciting or do math competitions, but you like to think about abstract concepts, please do keep an open mind about college math classes!
My math classes in high school demanded a lot of memorization, and the problem-solving often turned out to be pretty algorithmic. Apply the concept you learned in class to a problem like the one from class, but with different numbers. After a certain amount of “training,” a sizeable chunk of competition math was like this for me, too. Specifically, I got better at math competitions by taking a lot of old practice tests, so that many of the problems I encountered were variations on old problems I had seen before. I guess I performed moderately well at math competitions, and they were part of what made me want to come to MIT, but I don’t think they were particularly good at showing me what being a math major would be like.
Some things haven’t changed—as in high school, it helps to do practice problems, so you’re familiar with all the possible concepts and theorems you might be able to apply on an exam. It’s just that now there isn’t often enough time to practice enough, depending on what the rest of your schedule looks like. Also, there's less emphasis on memorizing material for exams, and homework is often weighted equally with exams. As a whole, homework is served in larger chunks than it is in high school, so it's important to learn to manage your time. There are also fewer examples in class and fewer problems that are near identical to those in the textbook. Finally, a big change is that everything is proof-based (as I write this, I struggle to remember what it meant for math to not be proof-based). You’re basically given an ever-expanding toolbox of definitions and lemmas and theorems and have to tinker until you can assemble them into solutions to the problems you’re asked to solve—it is a creative activity with strict rules and very little grounding in reality and applications.
A great benefit of college math classes (besides GIRs) is that they’re full of people who actively like to think about math, and hopefully, if you’re a math major, you think math is pretty cool, too. It’s nice doing math with no expectation that it will be applicable in any way. I recently had to teach a section of a textbook for my math seminar, and it was all about applying the theory in the previous sections of the book to a physics problem (the displacement of a cantilever beam at rest with only the force of gravity acting on it, if you’re interested!!!). Even though I was tasked with presenting this part of the book, it was definitely not the section of the book I would consider the most interesting. I personally thought the fact that the material was applicable to a physics problem was much less exciting than the proofs in the previous sections. I am pretty sure no one is taking that seminar to learn about the ways in which math is applicable to physics.
The main trend I have noticed is that math in college is much more intellectually stimulating and abstract than what I encountered in high school. By the way, if you're at all peeved by the way math is taught in your high school, or curious about what math is like when it's separated from its applications, I urge you to read Lockhart’s Lament, which was required reading for the communication-intensive real analysis class (18.100C, now renamed 18.100Q) I took my freshman spring. It criticizes math education in grade school and argues that math is an artistic and creative pursuit that is not taught as such—but should be. He laments the fact that students find math boring and suggests that the mainstream pedagogy is at fault.
It really helps to develop a strong intuitive understanding of the material you’re learning, although sometimes it gets difficult because there are no good analogies to real life. Sometimes you get stuck rereading a definition over and over and over, trying to flip back to earlier definitions that the current definition refers to in an effort to regain understanding of all the concepts that the concept at hand relies on, but to no avail, because there’s some other relevant concept from a math class you took two years ago that you need to revisit, so you look it up on Wikipedia, but by the time you understand it again, you forget what you were originally looking at, so you’re left scrambling to retrace the string of things you referred to, and at this point you still haven’t even started to solve the actual problem…
Sometimes. But most of the time it’s not that bad. Sometimes it’s just tedious, and sometimes you have to write out long expressions with a lot of symbols in order to rigorously explain something that’s much easier to explain intuitively. Lockhart's Lament is nice, but at some point you also have to buckle down and slosh around in tedium and make sure that all your symbols are written correctly.
Getting used to math in college and dealing with impostor's syndrome:
I took 18.022 my freshman fall and found that it helped ease me into the sort of proofs and thinking that the rest of my math classes required. Perhaps if I had jumped immediately into 18.100 I would have been overwhelmed; I know for sure that I wouldn’t have been able to handle 18.701. There are people who can, and seeing other freshmen sail through more advanced classes definitely freaked me out at times and stoked my first touches of impostor’s syndrome. Now, though, I don’t think I can adequately stress the importance of not going too fast. Because math classes often rely on definitions and material from prerequisites, these prerequisites are often super useful, unless you've actually learned the exact material from the class you want to skip. Of course, there are exceptions, but I often find it a lot more difficult to grasp mathematical concepts that I've forgotten or skipped than to pick up other new material, like a programming language.
I took 18.701 sophomore fall, and I attempted to take 18.702 last spring, but I ended up dropping it because I was overwhelmed by all the other stuff I was trying to do/learn. I was taking 18.125 concurrently. This semester, I once again made the poor choice of taking three math classes simultaneously, but I ended up dropping one of them last week because I simply could not handle it. I did not have enough time or energy to wrap my head around all the material, and I’m finally (finally) coming to recognize the importance of learning things well and deeply rather than learning them as fast as possible.
Bear in mind that this sequence just happens to be what I ended up with, and it is not a recommendation that everybody should take those classes in this order. I am hesitant to give general advice about what classes to take and when because it depends so much on your personal background and how much time you have to devote to the class!
A relevant article about impostor's syndrome and the feeling of "racing" to learn math is “The Wrong Way to Treat Child Geniuses” by Jordan Ellenberg, a former “child prodigy” who is now a math professor at UW-Madison. This one might be particularly relevant to people who didn’t grow up being praised for being “good at math” or winning awards at math competitions. (This one’s also particularly hard to access without a WSJ subscription, sigh…) I’ve reproduced a relevant paragraph below:
One of the most painful aspects of teaching mathematics is seeing my students damaged by the cult of the genius. That cult tells students that it's not worth doing math unless you're the best at math--because those special few are the only ones whose contributions really count. We don't treat any other subject that way. I've never heard a student say, "I like 'Hamlet,' but I don't really belong in AP English--that child who sits in the front row knows half the plays by heart, and he started reading Shakespeare when he was 7!" Basketball players don't quit just because one of their teammates outshines them. But I see promising young mathematicians quit every year because someone in their range of vision is "ahead" of them.
I think that MIT students, especially freshmen, are prone to psyching themselves out comparing themselves to all the people around them who have won the IMO or who were doing calculus in middle school—but Ellenberg points out that this type of thinking sounds absurd when applied to other fields and skills--and there is no reason that it should apply to math more than any other field.
One of the first math majors I met at MIT had never done math competitions in high school and hadn't had much exposure to higher-level math (i.e. calculus and beyond) coming into MIT, but he loved the math classes he was taking in college, so he began to register for more and more of them until he was thinking seriously about pursuing it as a profession. He was initially intimidated by his peers but enjoyed math so much that it was not a chore at all for him to devote significant amounts of time to mastering his coursework. He later became involved in research and is now pursuing his PhD in math at MIT!
In summary, if you think math is cool, please consider continuing to study it in college, but bear in mind that college classes aren't exactly like high school classes. And if you don't think math is cool, maybe it hast to do with the way math is taught in your school. Or maybe not--not everyone is destined to be a math major! And if you're intimidated and convinced that you'll never be "good enough at math", because other people seem to be so far "ahead," well that's almost certainly not the case--it's much more important that you enjoy the subject and don't try to jump ahead so quickly that you lose enjoyment of the subject in an attempt to "catch up." If anyone has specific questions about classes or anything like that, I would also be happy to try to help you individually. I know I have been pretty absent from the blogs--life update coming soon--but I'm getting back in the swing of things.
Sending you all strength and luck for pi day and all subsequent college and major choosing!