Neeum Zawan <m.n### [at] ieeeorg> wrote:
>  Reminds me of a bunch of math graduate students I've known. Fairly slow
> at doing integrals, etc. But a physics student or a good engineering
> student is much faster at doing them. It's because the math students are
> not concerned with calculations (for the most part) - but the others are.

From my PDE's textbook:

"...Our efforts will be largely devoted to proving mathematically the existence
of solutions. ... This may seem like wasted or misguided effort, but in fact
mathematicians are like theologians: we regard existence as the prime attribute
of what we study.  But unlike theologians, we need not always rely upon faith
alone."

>  In any case, my advisor in grad school had a no-calculators policy for
> exams. And this was for engineering courses, where you have to calculate
> stuff with real numbers, and not just symbolically. I always felt that
> if I were a professor, I'd do likewise for exams. I wonder how long
> before such professors are shunned by their own colleagues.

The professors I thought the most highly of were the ones who basically said,
"Open book, open notes, calculators allowed--but it won't help."  I really do
think estimation is an undervalued skill though.  The ability to stand there,
figure out orders of magnitude in your head, and come up with some quick
approximations can save a lot of time blindly pursuing totally inappropriate
methods.  That sounds obvious, but it's really remarkable how completely some
people lack basic problem-solving and analysis skills.

>  I often wonder. A lot of people (including myself sometimes) feel that,
> say, mathematics education is degrading over time. In my undergrad
> institution, a few decades ago, the lowest math class was introductory
> calculus. As the years went by, they needed to put a remedial precalc
> course that was strictly not for credit. Then as time went by, they
> converted that to a proper course with credits, and put a remedial
> algebra course for no credits. Now that course is offered for credit.

Maybe some people need a more remedial course, so they add a remedial course,
just to get everyone up to speed.  Suddenly, the bottom rung of the ladder isn't
the bottom anymore, so all the people who were struggling in precalc move up a
step.  And if you're not falling off the bottom, why worry?  People just expand
to fit the container, in a sense.

>  Likewise, some of the things they often teach in the first year of
> graduate school here in mathematics is often taught in the 4th year of
> undergrad in universities in other countries (I just saw one where it's
> taught in the 3rd year of undergrad).

"it" being what?  Just curious...

>  Another explanation is that as a percentage (and of course, in absolute
> numbers) of the population, more and more students are getting educated
> and graduating high school, and the education system is having trouble
> keeping up. Put another way, the percentage of people age 18 in this
> country who have a "solid" background in mathematics may have actually
> stayed the same (not gone worse) - it's just that the percentage
> graduating and moving on to university is higher (and not just because
> standards have gone down).

This actually seems plausible.  Still, they choose their program, so to allow
more people through, the standards must be relaxed.  As I said above, relaxed
standards in turn reduce effort put forth by students, compounding the problem.
Just a guess, of course.

>  I think what matters is maintaining a fairly good average (not just of
> students, but the whole population).

Let's hear it for mediocrity!

 - Ricky

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