LINK: The Department Chair, whose musings on Europe are all entertaining, links to a paper on the problems with math education. I sympathize (quite a bit) with the argument of the paper: I lost my interest in math with Algebra I, had it briefly revived with Geometry, then lost it altogether in Algebra II, which had, so far as I could tell, nothing to do with anything. Calculus was a wash: I enjoyed--very much--the concept of calculus, thinking in terms of differentials and integrals--but the actual work not very much. I was not an engineer, I was not going to be an engineer; I had zero interest in figuring out the volume of oddly-shaped solids or comparing two random functions with no relation to anything. Not surprisingly, I dropped the class for Public Speaking (my teacher, bless her heart, told me I was the only one in the class who appreciated what was really beautiful about calculus; I still suspect this was a way of her telling me I was not very good at it).

There's a claim in the paper I find interesting and relates to other fields as well:

By concentrating on the what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the 'truth' but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity--to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs--you deny them mathematics itself.

The idea that the explanation matters as much as the truth of a statement is appealing on both a substantive and a pedagogical level. As substance it indicates that possessing the truth is not enough--that truth is inseparable from the means of arriving at it, that to some extent the meaning of truth is determined by the means by which it is discovered. A premium is placed on failure and creativity--what one is trying to do is make connections, and the best and most useful connections are made each person taking up the task for themselves. (There also appears to be a claim, backed up by the author's frequent mentions of the history of math, that the truth is seldom known ahead of time, but almost always discovered--there's a danger in treating a priori truths as ones that must simply be believed or accepted, even if they're a priori)

As pedagogy, this relates to a significant problem in the teaching of political theory. Is the point of, say, an intro course to introduce students to the thought of significant figures in the history of political theory, or to introduce them to particular ways of answering political problems? On the one hand, we expect students to know the importance of fortuna and virtu in Machiavelli, the appeal to heaven in Locke, the censor of the general will in Rousseau--certainly, without the ability to explain these things, the students won't really have understood what is going on in the text. But the text itself is a means of answering a question: why does Machiavelli think historical examples are useful? Why does Locke mention natural law? Why does Rousseau focus so much on institutions? Ideally, one teaches both--the answer is determined, in part, by the method, and we need to ask whether both the method and the answers are good. But there's often a tendency to think in terms of sequences (e.g. early modern, late modern, liberalism), or else of disparate topics that don't necessarily link up (ancients and moderns, but rarely the medievals who help make sense of the oddities of modern political thought).

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