I like much of Lockhart's attitude. Math is fun, and giving students an experience of that fun will do more for math education than better standards or testing ever will. But, as educational pedagogy, the idea that students should "work on what interests them" or take a problem-solving approach to learning is not new, nor even untried.
Lockhart seems to think math is the only subject which is so gratuitously misrepresented, made boring, and reduced to pointless trivialities. Obviously he does not talk to any of his colleagues. The same lament could me made of history, or literature, or geography. Talk to mrissa about having to take six years of "The Explorers" because the multi-year curriculum was rewritten every year. Or go watch "Dead Poets Society".
Nor, for all his enthusiasm for an artistic approach, does he show much appreciation for how artists and musicians are actually trained. There's lots of boring pedagogical stuff there too. It's not as bad as the analogies he starts the book with--- imagine years of paint-by-numbers before getting to paint, or years of music theory before playing any music. But, musicians do get drilled on music theory and scales. Artists do have technical exercises and art history. And yes, sometimes they are pretty damn divorced from anything the students are interested in doing at the moment.
Finally, he ignores that there is a math-based skillset that really is useful. You'll have a hard time taking even high-school physics without a well-developed mathematical toolkit. There's a certain minimum amount to tackle chemistry too. If we expect adults to have a reasonable grasp on science (which also ought to include appreciation for how science actually works) the only way to do the science is to have the necessary math. If we expect adults to be "numerate" then that boring practical stuff has to be introduced somewhere along the line.
Does everybody have to take trigonometry? Certainly not. I don't care about the half-angle formulas either. It would be far better for students to have experienced real mathematical discovery. (Lockhart has some choice words about how this process is *most* stifled by high school geometry.) But while Lockhart thinks he's talking about a radical rework of math education, he's really advocating a tremendous change in the system as a whole. What can be learned from similar impulses in other areas?
For somewhat similar responses, see an excerpt and some criticism from Bill Kerr.