Michael Paul Goldenberg <firstname.lastname@example.org> wrote:
>I think that at least one point here is that mathematics is an open >enterprise, constantly evolving, never to be completed. While that may be >a commonplace in the mathematics community, it's hardly well known to the >general public. Few of us are taught that mathematics is a growing body >of knowledge that represents human ingenuity and inventiveness. Until I >met graduate students in mathematics during my college years, I didn't know >there WAS any mathematics past calculus.
Why is it important to teach that mathematics is a growing body of knowledge? I mean, it's certainly true, and it is better to be aware of the wide world than not -- but how much attention should be paid to this in an 8th grade math class?
Oh sure, one wants the NSF to get funding, etc., so people need to be wordly enough to recognize the existence of researchers. Then, too, as a mathematician, I would like to see more people develop the world-view of mathematicians and apply the general principles of abstraction and reason to every aspect of their lives (though even many mathematicians do a poor job of this -- for example, some think their internal proofs of their own political prejudices somehow carry as much weight as the proofs they produce in more appropriate contexts). But what does this have to do with changing the curriculum to reflect all the new math developed since WWII?
The bulk of the mathematics students learn in K-12 dates from the 18th century or earlier. It is dressed up in 20th century notation, and takes some of its emphases from the late 19th/early 20th century -- and is frequently applied to modern contexts -- but there is little "modern" mathematics in it. Perhaps a few topics in the BC calculus saw their first rigorous proofs in the 19th century -- not that the BC calculus students are given complete proofs.
One could do quite well (at least as far as mathematics content is concerned) with textbooks well over a century old. US Math Olympiad students know, for instance, that Hall and Knight's "Algebra" (it may be called "Higher Algebra" -- I don't have it in front of me), an English textbook dating from the 1880s, aimed at preparing students for University entrance exams, is an INCREDIBLE encyclopedia of useful concepts and tricks.
Clearly one wants to give students a sense of modern applications. But the essential core of mathematics is actually pretty ancient. (That doesn't need to be so horrible, does it? Isn't there a thrill in being part of an ancient tradition, imparting the distilled wisdom of generations long past to the next generation? Is there no romance at all in seeing a child wrestle with a concept the Greeks wrestled with over two millenia ago?)