A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form (Bellevue Literary Press, 2009) is a a book by Dr. Paul Lockhart, a mathematician who taught at Brown and U.C. Santa Cruz and then decided to devote himself to teaching math to K-12 students at a private school in Brooklyn. What he's lamenting is the present state of mathematics education.
The argument is radical. Lockhart advocates doing away with the formal step-by-step progressive method of teaching kids mathematics. Starting kids with arithmetic, then moving up through algebra, geometry, trigonometry, and eventually getting to calculus (ideally). He says most people don't really need math (at least beyond an elementary checkbook balancing or calculating square feet of carpeting level) and instead we should just teach kids to play around and "discover" math.
Lockhart gives a number of examples of things I assume he does with his students where they're asked to discover proofs for simple concepts in geometry or number theory. The students are given a statement or a figure (e.g. a triangle drawn inside a box) and then asked a question about it (e.g. how much of the area of the box does the triangle take up) and then the students just play around to see if they can figure it out.
The claim is that the students will then discover how math works, see the beauty of it, and basically fall in love with it as the author has. Hopefully I'm not misrepresenting Lockhart's claims since I'm about to criticize them a bit.
Here's my perspective as the department chair of the science and math department at a community college and a homeschooling parent of two children.
Lockhart is working with a special group of children at St. Anne's in Brooklyn. I checked out their website and it's an exclusive school that admits really bright kids. While they may work in St. Anne's under the tutelage of an innovative Ph.D. mathematician, I wonder how well these ideas would work in a typically inner city public school with a typical elementary school math teacher.
Also, while Lockhart claims most adults don't need or use formal algebra, geometry, or trigonometry; some people do go on to major in science or math in college. I can't imagine that those who go through Lockhart's suggested program would be prepared to enter college and go into a typical Calculus I class.
While it's admirable to want to pass on the beauty of mathematics to kids, I think it needs to be done in conjuction with a more formal progression through mathematical concepts. At least that's the bet I made with my kids who, even though homeschooled, use a formal math curriculum (Saxon). We also try to play around a bit so they see how useful and interesting math really is!
One thing I do see at our community college is that public school math education fails miserably. All incoming students at our institituion take a placement exam in math and English. Many students place into remedial math courses (even those students who just graduated with a NYS high school diploma a few months earlier). Non-science and math students have to take a course called College Algebra to graduate - when I look at the topics covered it reminds me of what I had in 9th grade algebra some 30+ years ago. Many of our high school graduates instead place in Pre-Algebra, a 6th grade-level course covering topics like fractions and decimals.
I just don't think Lockhart's ideas are the answer for the failure of the public school system in math education. I don't think I'll discuss my ideas either, since they'll offend most people! Maybe some other time.