A post at Language Log explores the use of mathematics in linguistics. It closes with

Anyhow, my conclusion is that anyone interested in the rational investigation of language ought to learn at least a certain minimum amount of mathematics.

Unfortunately, the current mathematical curriculum (at least in American colleges and universities) is not very helpful in accomplishing this — and in this respect everyone else is just as badly served as linguists are — because it mostly teaches thing that people don’t really need to know, like calculus, while leaving out almost all of the things that they will really be able to use. (In this respect, the role of college calculus seems to me rather like the role of Latin and Greek in 19th-century education: it’s almost entirely useless to most of the students who are forced to learn it, and its main function is as a social and intellectual gatekeeper, passing through just those students who are willing and able to learn to perform a prescribed set of complex and meaningless rituals.)

Before getting into economics and after getting out of physics, I took calculus and found it very useful and interesting for its own sake. I do see that the way calculus is taught in the US is geared toward engineers and physicists, but I have a hard time thinking of what mathematics would substitute for calculus in the undergraduate curriculum if the goal was to teach students something useful. It can’t be analysis or topology. I took abstract algebra as an undergraduate and found it esoteric and boring. Discrete mathematics? OK maybe statistics, but don’t you need integration for that? Help me out here, if you had the choice, what would you replace calculus with? And remember the goal is to teach something useful.

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May 29, 2009 at 5:38 am

ToddLinear Algebra

May 29, 2009 at 10:50 am

tayloran intro to proofs course or a problem solving course

May 29, 2009 at 12:51 pm

tylerhI would replace calculus with.. Calculus.

There is no substitute.

You can’t do anything beyond cookbook statistics without calculus. For example, the transformation between marginal and cumulative distributions is mere witchcraft without the concepts of integration/differentiation. The meaning of the p-value for most statistical tests are intuitively obvious for those who understand calculus, but just an arbitrary table of numbers otherwise.

Some mentioned linear algebra. There certainly needs to be more and better linear algebra instruction. But one of the key applications of linear algebra is: differential equations. Also, good luck explaining the derivations of linear regression from quadratic error function without calculus.

Indeed, *any* discussion of optimization is pretty limited without a knowledge of calculus. Any discussion of (local) optimality or any gradient method requires derivatives.

Admittedly, calculus could be taught a lot better. For example most classes spend a month or more on weird substitutions that no one ever uses because one can look them up in widely available reference books. The examples should be drawn from wider examples than physics. I, for one, would spend more time on asymptotic behaviors because this is such a great basis for “order of magnitude” reasoning — that is, rigorously isolating the key parts of process so that simple relations emerge from complicated-seeming situations.

But the idea that calculus is “Greek” is a deeply misguided viewpoint. It’s half-right, in that mathematical thinking is a language that takes years of daily practice to master. But the reason that calculus is the “gateway” class is because it is the foundation for almost everything scientific that has happened since the Enlightenment. It’s no accident that the founders or Calculus, Leibniz and Newton, are widely regarded as two of the greatest thinkers of the Enlightenment.

May 30, 2009 at 10:22 pm

jeffYou have said it much better than I could, thanks.

May 29, 2009 at 9:28 pm

mikePhilosophy.

And by that, I mean studying the nature of mathematics. In real life you don’t need to know how to push the symbols around; what you really need to know is what your tools are about and what they can be used for.

Calculus is about scale and difference.

Algebra is about possibilities.

Probability is about information.

And Mathematics as a whole are about relationships (as opposed to things).

It’s surprising how often I talk to people who don’t grasp just those fundamentals. Normal people need to learn the what and the why, and can leave the how to the mathematicians and their computers.

June 1, 2009 at 1:24 am

aramLinear algebra and (discrete) probability.

Regression, i.e. minimizing ||Ax-b||, can be thought of geometrically, as projecting a vector onto a subspace. I think this gives more insight than just taking derivatives and having the answer fall out as if by magic.

And more generally, the problem with intro calc is that it teaches a lot about how to calculate things, but not so much how to understand things. I agree that talking more about asymptotics would be an improvement, but I would rather students know about things like Bayes’ rule.

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