On Fri, 05 Jul 2013 17:41:04 -0600, Robert Hansen <email@example.com> wrote:
> Some, well, mainly Lou, have questioned my hypothesis that there has > been a decline. Although, I think he means a decline in student ability, > not a decline in the quality of mathematics education. I recall being > exposed to so many derivations in my classes and producing many of them > myself. Those are mostly gone today.
I was graduated from high school in 1962; derivations, and "producing them myself" were gone by then---if they'd ever been there in the first place. They were in the textbooks, but the "teachers" ignored them. The closest I ever came to derivation in trigonometry was working trig identities using the standard identities---which we never bothered to derive. Only two or three of the fifteen us were any good at that---and the "teacher" handle us by calling on us to present the solutions to the hard ones. In so doing, he got us to think about those hard ones while not having to think about them himself.
I see no reason to suppose that my high school was much different from most others. Certainly the students in my college calculus courses, in 1962-3 and 1963-4, gave no indication of having anything in their backgrounds much deeper than what I had.
It wasn't until I took my first calculus course as a freshman in college that I encountered real concern about proofs and derivations. I was fortunate in that my professors there made proofs and derivations about half of their calculus sequence---all four semesters of it. And they examined us on those proofs and derivations, not only in the calculus sequence, but in the junior and senior comprehensive exams all math majors had to pass in order to be graduated. (Many of my colleagues during that first calculus sequence would not have agreed that any of this proof-and-derivation business was "fortunate"; a substantial fraction of them show themselves to be completely at sea at that business.)
So, yes. I do question your hypothesis that there has been a decline, and I do mean a decline in the quality of mathematics education.
- --Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver