Paul Lockhart has taught math at Brown and UC Santa Cruz, and for the last 15 years at St. Ann's School in Brooklyn. To read his book A Mathematician's Lament is to realize that we need to question all our underlying assumptions about math instruction.
Lockhart writes:
In place of discovery and exploration, we have rules and regulations. We never hear a student saying, "I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal." Instead we have... the "negative exponent rule" as a fait accompli with no mention of the aesthetics behind this choice, or even that it is a choice.
At Harlem Village Academies, we aspire to teach our kids to grapple with math as inquiry every day, to wonder and explore, to struggle intellectually and learn deeply, not simply to memorize formulas.
When I read the following excerpt to my own kids, they cracked up - and totally agreed:
No mathematician in the world would bother making these senseless distinctions: 2 ½ is a "mixed number," while 5/2 is an "improper fraction." They're equal, for crying out loud. They are the exact same numbers, and have the exact same properties. Who uses such words outside of fourth grade?
Learning to teach math well is much harder when reaching for a higher level of learning:
Of course it is far easier to test someone's knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning.... In practice, the curriculum is not even so much a sequence of topics, or ideas, as it is a sequence of notations.
The essence of mathematics -- and the teaching and learning of mathematics -- is learning how to devise solutions to problems, not memorizing solutions devised by someone else.
Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work. Specific techniques and methods will arise naturally out of this process, as they did historically: not isolated from, but organically connected to, and an outgrowth of, their problem-background.
Lockhart writes about unity and harmony and the beauty of math. He is passionate and some would say radical. I would say right on.