Paul Lockhart’s terrific essay about the state of mathematics education and what should be done: A Mathematician’s Lament (25 page must read PDF):

G.H. Hardy’s excellent description:

A mathematician, like a painter or poet, is a maker
of patterns. If his patterns are more permanent than
theirs, it is because they are made with ideas.

So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the
biologists. Ideas about language and culture? No, not usually. These things are
all far too complicated for most mathematicians’ taste. If there is anything
like a unifying aesthetic principle in mathematics, it is this: simple is
beautiful. Mathematicians enjoy thinking about the simplest possible things,
and the simplest possible things are imaginary.

By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the
subject. It is like saying that Michelangelo created a beautiful sculpture,
without letting me see it.

By concentrating on 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. So no, I’m not complaining about the presence of facts and formulas in
our mathematics classes, I’m complaining about the lack of mathematics in our
mathematics classes.

If teaching is reduced to mere data transmission, if there is no sharing of
excitement and wonder, if teachers themselves are passive recipients of
information and not creators of new ideas, what hope is there for their
students? If adding fractions is to the teacher an arbitrary set of rules, and
not the outcome of a creative process and the result of aesthetic choices and
desires, then of course it will feel that way to the poor students.

Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children. In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

It’s perfectly simple. Students are not aliens. They respond to beauty and
pattern, and are naturally curious like anyone else. Just talk to them! And
more importantly, listen to them!

Read the whole thing. This essay has reinforced some beliefs of mine about software engineering, teaching and parenting.

Slashdot has a decent thread on the piece.