mult.ifario.us

From one of Steve Yegge's rants:

Geometry, trigonometry, differentiation, integration, conic sections, differential equations, and their multidimensional and multivariate versions — these all have important applications. It's just that you don't need to know them right this second. So it probably wasn't a great idea to make you spend years and years doing proofs and exercises with them, was it? If you're going to spend that much time studying math, it ought to be on topics that will remain relevant to you for life.

Sigh.

Unfortunately, there is no canonical ordering of the pedagogical cart and horse in mathematics. Should concepts be rigorously defined and developed at the expense of motivation, or should rigor wait until intuition has been nurtured? My own belief is that the two need to proceed in lock-step so that a student neither becomes bored nor grows careless, but that rigor must be preserved. Rigorous thinking should be the asset that a student carries away from a mathematics course, rather than any particular piece of knowledge or technique. The disclaimer is that I used to be an academic mathematician — completed a Ph.D., spoke at conferences, took an academic job, published some papers, etc. — but I make no apologies for my prejudices about teaching and learning mathematics, as they are the product of knowledge and experience rather than ignorance or malice. (I still have some results that I should write up and publish, but I have a secret plan to spend some of my golden years doing math, something like what R. M. Foster did with his semi-recreational interest in tri-valent symmetric graphs.)

The role of mathematics as queen and servant of science complicates the basic economics of teaching mathematics in a university setting. The various non-honors flavors of calculus are often referred to as "service courses" in that those courses are taught by the mathematics department as a service to other departments. Enrollment drives staffing, be that full-time faculty, adjunct faculty, or graduate students, so it behooves a mathematics department to hang onto calculus courses in order to preserve headcount. This creates tension with other departments who'd like to have the lowest possible barrier to entry and spend time on the smallest body of material. For example, I was on the faculty of the mathematics department at UIC and thus privy to the sort of politics that only people with job security can engage in, and the economics department stirred up a hornet's nest by attempting to hijack calculus for their own marginal gain. The presence of Math 160 "Finite Mathematics for Poets^H^H^H^H^HBusiness" and Math 165 "Calculus for Poets^H^H^H^H^HBusiness" on the list of courses makes me think that the mathematicians lost the battle and won(?) the war in that the watered-down curriculum is present but being taught in the math department.

Watered-down, reduced-rigor service courses are also hard on mathematicians. It's like asking Alain Ducasse to serve Pillsbury "croissants" and convince diners to like it. (Do not poke Alain Ducasse or a mathematician in the belly; they probably will not laugh and might even pop-n-fresh you right in the nose.)

Back to the subject at hand, i.e., the bland preparation of Calculus that's served in schools across the country and obviously left an absence of taste in Steve's mouth. In spite of the fact that the word literally means a method of computing, the meaning and relevance of calculus are not bound-up in the rules and mechanical tricks needed to work contrived physics problems. Calculus is about developing an intuition for functions, about being able to reason clearly about behavior given higher-order observations (e.g., rates of change), about approximation, and about when an estimate is an approximation or not. To further extend the food metaphor, high school (and many college) instructors aren't to blame for the Hamburger Helper ("just add graphing calculator") calculus curriculum, since there's a very high probability that they don't know that Salisbury "steak" isn't really any contiguous part of any one cow — it's the same way they were taught.

More on this later.