This reminds me of a fairly common answer to "what good is a math major?" or "what good is a PhD in math if I don't want to be a professor?", namely that for most values of X, it's easier to teach a mathematician to do X than it is to teach an X-ist mathematics. I'm not sure what good this answer is, though, because it presupposes that having someone who can do both X and math is a Good Thing. That sounds plausible but I want a proof. Or at least some convincing numerical evidence.
According to Zeno, it was somewhat controversial that a two-year college degree in California was made to require more math than a high school diploma in the same state; this reminds me of what I noticed when I was studying for the GRE. The verbal section of the GRE seemed to me to be harder than the verbal section of the SAT -- it had more advanced vocabulary, more complicated reading passages, and so on. But the mathematics section seemed easier! It's as if the test-writers assume that people forgot math in college. (Since you're going to ask: I got a perfect score on the math section and would have been ashamed of anything less.) The average score in my department, depending on the sample, is somewhere between 789 and 794 out of a possible 800. But you'd expect a bunch of entering PhD students in math to be nearly perfect on math they should have learned by the ninth grade. I concluded, upon taking the test, that there was no score I could get on the math section that could possibly make my application look better -- that the test could only hurt me. (Fortunately, it didn't.)
Zeno also writes:
With enough sang froid, a math teacher could probably pose as an English teacher for a much longer time than an English teacher could do the same in a math class. Math doesn't have the wiggle room or the space for discourse that other subjects allow.
It's an interesting experiment. Sokal affair, anyone?
I'm not sure what good this answer is, though, because it presupposes that having someone who can do both X and math is a Good Thing.
ReplyDeleteI spent 2 years in a math PhD program (about a decade ago) before deciding that it was not the lifestyle for me, so I have thought about this question a fair amount.
My conclusion was that math grad school trains you to learn new things very quickly -- more than a few departmental seminars were on topics I'd never even heard of, and I was expected to at least partially follow along.
A better way of thinking about your formulation is that an X-ist may be stuck at about 33% competency when a job requires (say) X, Y, and Z; a mathematician can get up to speed quickly in all three.
The "learning new things quickly" explanation makes sense. I wonder if that's more true of grad school in math than in other fields?
ReplyDeleteNot to get all egotistical here, but maybe mathematics is genuinely more difficult than everything else (partly because it's almost completely independent of real-word intuition). And thus people who have the chops to become mathematicians are genuinely smarter than other people.
ReplyDeleteAs I comment in Zeno's blog, math is a priori, so students could find out in class if the answers were correct or not. In other fields, a familiarity with the literature is required. You could perhaps relate false information to undergrads and get away with it briefly, because unlike math, they would have to do research out in "the real world" in order to check your answers. In a posteriori fields most students simply memorize what their professor tell them in class and take it as the gospel truth. This is relatively harmless, because credentialism should guarantee that the professor relates what is standard in the field.
ReplyDeleteHaving been to a few philosophy seminars, I bet very few mathematicians could get away with pretending to be philosophers: its not so much about what they say (at least to undergrads), but they communicate in such a different manner. In their seminars, they take a list of people who want to ask questions at the end, and the question session often goes longer than the seminar itself. Then in physics (my own subject) - how many mathematicians could teach labs?
ReplyDeleteKea: how many mathematicians could teach labs?
ReplyDeleteVery few, I'd imagine. Math majors tend to take relatively few lab courses in college (I had no labs beyond freshman chem and sophomore physics), so we'd be floundering if called on to conduct a lab course (or to try to fake a lab session). That's why I tend to reserve my barbs for those professors in poli sci, lit, or comp who snarkily tell me math is merely utilitarian and not important in higher ed. Anyone who tells me that is sure to blanch at the notion of a class-swap contest.
Steve also raises a good point about the difference between math and other subjects. Whereas any level of math class can be expected to have at least a handful of students ready to catch a teacher's mistakes, there are plenty of other courses where it would be necessary to check references and track down details; students wouldn't have it at their fingertips and would thus be vulnerable to anything plausible the teacher might say.
While my humanities colleagues bridle at the suggestion that I could fake it in their classes longer than they could fake it in mine, it's an indication of the difference in our subjects rather than any difference in our academic attainments. Math exposes you faster.
I recall reading in the news some years ago a report about a study of testing in high school level math and English. The test covered something like trig, and the English test was an essay on some common book assigned in high school.
ReplyDeleteA bunch of math and English teachers were assigned to grade the tests, but with no key provided by the person who created the test. Each teacher could decide his or her own method of grading.
The result they found was that math teachers were *far more likely to disagree about how the tests would be graded*. Not only did they differ in what grades they assigned, but it was not very uncommon that a student who did well in one teacher's grading would do poorly in another's, because, for example, one teacher might allow partial credit and another would not, or one teacher would insist on seeing every step in a solution, whereas another would not, and some students would suffer quite badly due to this.
By contrast, the English teachers' grades largely correlated to basic competence in writing, and it turns out they pretty much agreed about how to identify that.
I have a theory that reading and writing decently is the most difficult skill that is very widely mastered, because spoken language is the most important and nearly the most fundamental technology that human beings have, with writing following close behind. There is, IIRC, even some evidence that we evolved partly in response to pressures to use language. Even a dull person will occasionally, if you listen carefully, surprise you by saying something perfectly beautifully, speaking pithily, and using exactly the right words to evoke the desired ideas, emotions and, perhaps, images. Such nuggets may be buried among egregious solecisms, mangled syntax, cliches, and unpretty and ungainly expressions, but if you listen I believe you will find them, hinting at the potential most people have to speak and write expertly, if not beautifully.
This being so, I think it mainly suffices for students to read, think about, discuss, and write about literature for them to get better and better at it. The learning comes largely through experience. It's more akin to how one becomes good at chess than how one becomes a good student of math. After the beginning levels, the teacher plays a role more like a coach, a guide, and a sounding board than an imparter of knowledge.
By contrast, with math at every step the teacher reveals the substance of the subject.
I suspect that in English, this situation may reverse again in specialized areas of literary study, where accumulated knowledge of historical developments, traditions, parallels between different genres, etc. become more important. In a class like that, I suspect that it would again become more difficult for the math teacher to fake it.