Keith Devlin recently published in his
MAA column an essay by Paul Lockhart
titled A
Mathematician's Lament, a critique of American primary school
mathematics education. It's a fascinating read, and the quickest 25 pages
I've read lately.

Marvin Minsky has written an
essay, What
makes Mathematics hard to learn? In it he suggests some ideas for
teaching math effectively, and relates a couple of poignant anecdotes about
math education, my favorite of which is this one:

...that child [who had been struggling with learning to
multiply] had a larger-scale complaint: "Last year I had to learn the
addition table and it was really boring. This year I have to learn another,
harder one, and I figure if I learn it then next year there will be another
one and there'll never be any end to this stupid nonsense." This child
imagined 'math' to be a continuous string of mechanical tasks— an
unending prospect of practice and drill. It was hard to convince him that
there would not be any more tables in subsequent years.

These got me thinking about my mathematics education. Perhaps the single
most lasting skill I've picked up from doing math is learning how to prove
things: not those two-column geometry proofs that everyone is tired of, but
really knowing how to put together a line of argument that is both easy to
follow and watertight. And there is not just one way to do that. Like
writing, there is both art and science in it: choosing terminology and
metaphors carefully; designing the structure and scaffolding of an argument;
putting in enough details, but not too many; being economical with words.

Moreover, from mathematics I picked up an inclination to ask why things are
true. In math— unlike in any natural science— you can always get
down to the bottom of things by asking "Why?" until you get back to the
axioms. Frankly, that's an amazing idea.

Yet, I saw very little of the "art of proof" and the "mathematician's
skepticism" until I started going to
a math circle in high school.
Most of my public school math classes focused on learning recipes rather than
talking about the *why*. When we learned, say, the quadratic formula,
we did the derivation (based on completing the square) in class. But that
derivation was soon squirreled away so we could do "applications", which is a
fancy name for plugging numbers in ad nauseum. Personally, I find this
tragic: the trouble is that in mathematics, the process is far
more important and interesting than the result. More pragmatically, a student
who remembers the idea of completing the square can derive the quadratic
formula (and more) anytime, but students are given little incentive to think
about, or remember, that beautiful idea. Most students will forget the
quadratic formula once they have stopped using it for a few years, and if
their experience with mathematics is that it is just a bunch of facts, they
are more than likely never going to get that quadratic formula back.

The result of these methods, as Lockhart notes, is that pretty much no one
today knows what mathematics is (not even math teachers). And students are
being led to think that they dislike math when they actually just dislike
whatever it is that they're being taught in math classes. Doing math is a
deeply creative and enlightening process, but those aspects of mathematics
are very rarely seen by the general public.