The first time I started thinking seriously about how to teach mathematics was when I read “A Mathematician’s Lament,” by Paul Lockhart. He’s an impressive mathematician with real credentials, but what I like best about him is his straightforward, common sense way of talking about things. He’s direct and clear, and funny.
In school I was a good math student because I was a good student, but like many high school kids I know, math suddenly became very difficult for me when it got advanced. I was surprised by that, and discouraged. With hindsight I can see what was happening: I had gone through early math education merely imitating what the teacher had done, not fully understanding the reasons why he or she was doing it. I was good at finding patterns and copying them, but I wasn’t developing number sense or any kind of deep appreciation for what math really is. When things got difficult my understanding was too superficial to carry me through. If I had learned to ask why, rather than how, I would have been better off.
In graduate school I had a statistics professor who was a real purist: appalled that we political science types wanted to run regressions on a computer without really understanding the math behind it, he did away with the syllabus and gave us 8 weeks of intensive calculus instead. It had to be done by hand, and there were no tutors available that semester. So, with nowhere else to turn, I sat in a coffee shop and taught myself calculus for 30 hours a week. I was 27, and that semester I got my math confidence back.
Working in K-12 education is showing me what math education should really be like: why Singapore math is so excellent, and how important it is to develop number sense and math fact fluency in students. Above all I’ve learned that I was wrong to say “I’m not a math person” when I was growing up. Math, after all, is just thinking, and all people are thinking people.
In a way it makes sense that I had to learn how to think about math from an essay. You really should read the entire thing. In the mean time, here are some of Lockhart’s best observations, with quotes from the essay.
- Studying math is exciting.
“[T]he fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.”
2. Math doesn’t just mean practicing a techniques for solving problems. It’s about asking questions and exploring.
“The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid ‘exercises.’ ‘Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.’ What a sad way to learn mathematics: to be a trained chimpanzee.”
“A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?”
3. A good mathematician is a good thinker. Good thinking is a skill no matter what discipline you’re studying. Being good at math helps you get better at everything else.
“But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).”
4. The best math teachers know how to make the lesson exciting and suspenseful.
“I can understand the idea of training students to master certain techniques— I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.”
The Classical Classroom 