I just read this lovely book in one sitting and then wrote a review about it on a website I just discovered, called Goodreads, which follows here slightly edited. (Note that all links go to the Goodreads site.)
A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart
This brand new book, an expanded essay about why math is taught all wrong in schools, is delightfully short, but a great inspiration while I'm studying for my last CSET Math exam in Trig & Calculus. When I started studying for the CSET a college classmate who has has a long career as a chemist was helping me get my mind around some of the new math concepts. He told me that someone had told him that math was all about definitions. Paul Lockhart couldn't disagree more. It's about solving wonderful, fascinating problems, he says.
The author's thesis is that math is an art - the art of solving problems - and we are teaching the grunt work of math, but not the enjoyment of the art.
He starts with a "Lamentation" about how terrible school math is, as if we were teaching kids to read music notes on paper, without ever letting them listen to, play or compose music. Or an art teacher who teaches color theory and paintbrush techniques so that high school art students can do paint-by-numbers pictures.
He understands that is may be necessary for students to understand the things being taught in school, but he'd prefer it if they figured things out by themselves. The role of the teacher would be to give them the space to discover these things.
But then he concludes with "Exultation" explaining (with a few examples) about how delightful math is. It got me all inspired, since I'm in the process of becoming a math teacher!
I find that math is like a game that needs solving, which fits in to his idea of math as art. The way I have taught before (English and German in Danish high schools) is to get the students figure things out as much as possible. I've rarely had much of a lesson plan, other than the requirements of topics that had to be covered (which was quite free in Denmark.) I don't know possible it will be to teach math (rather than train formulas and definitions) when the students have tests to be taken. But I'm inspired now!