One of the vocab quizzes from the previous post talked about how the student preferred math to English because math produced “objective answers.” When I shared that quiz with some other folks, it produced an interesting exchange.
Stanford student: Good read, definitely spot on with the objectivity of math.
Me: Not at all about the objectivity of math. Writing a good math proof on a hard topic is, for instance, just as subjective as writing a good English paper. The difference is that expectations are much higher in English because you’ve been using it so much longer. Lower school is mostly arithmetic most places. If you didn’t start learning English until you started learning algebra, you wouldn’t be able to do much in it either. Except that, I suppose, the brain is hardwired for more language sophistication than math sophistication for most people.
And, to be fair, they are applied to two different kinds of problems. Essentially, math manipulates objects and language manipulates people. Which is harder?
Stan: OK, fair point as I’d consider my proofs for my logic class to be graded somewhat like essays. But what he’s faced is pretty objective. Manipulating objects = far easier than manipulating people in my opinion. That whole free will thing…
Me: Exactly. Which is why you think math is more straightforward and less subjective than English.
2nd student (currently at SJS): Hmmm…I like that quote about “math manipulates objects while language manipulates people”…that’s a really good way to put it and makes clear why English in its very essence requires subjectivity. However, if you’re just concerned about the manipulation of objects in math, doesn’t that make a proof very objective (either it proves or it doesn’t)? I’m not very facile with proofs, but I’ve always been under the impression that the subjectivity in proofs comes from a sense of aesthetics, which would involve the manipulation of people more than objects…
Also, is it fair to say that the brain is more hardwired for language than math? In many ways, math seems to me significantly more logical than language (even though there shouldn’t be gaps in an analytical paper, the creative process certainly doesn’t flow the way two-column proof does); doesn’t the success of your lab course in math show
that? Using fairly succinct labs with good (but sparse) directions, it’s very possible for students to recreate the achievements of great mathematicians, but I very much doubt similar system could encourage student to generate work of the same quality of Shakespeare…
Me: In the manipulation of objects, ways that are shorter and hence more efficient are probably preferred by most people to ways that are less efficient. In spite of the fact that until Diff Eq your exposure to math has generally been to problems that have exact answers, most things don’t (either inherently or because they are too complicated at this stage of our ability)–think back to the finite-difference stuff at Exxon, for instance. A math procedure that is more robust and handles more cases effectively is to be preferred to one that can only handle a few things, so no, math is not nearly as objective as it’s presented in school, which is another reason to be concerned about what we teach in math and how. Further, when something is proven is quite subjective and depends on the perspective of the prover. Did we not go over the section on proof in the little red book on mathematics last year?
I don’t know whether the brain is in fact more hardwired for language than math, but it’s a reasonable guess that it is. The labs (thank you for the shout-out) guide you to recreate the achievements of some great mathematicians, but not necessarily in the same way and certainly not at the same speed they did. It took Newton years to develop what you go through in math 3 in a few months.
And your comparison to Shakespeare is inapt. In recreating Newton (as it were), you are recreating his basic results, using neither his form of expression nor his language. I assume when you talk about students’ recreating Shakespeare, you’re referring to both his language and his form of expression, not simply his plot structures–which I could indeed get students to discover in an English course in less time than it took you to master differentiation last year, I suspect.
