“Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.”
So says Paul Lockhart, and I couldn’t agree more. It’s great having cooperative students who will correctly follow directions in solving problems — or should I say exercises — but following directions is a cheap virtue. As Lockhart observes, you don’t develop your mental faculties that way. On March 28, 2008, I wrote a brief laudatory piece about Lockhart’s fascinating essay, which he has now turned into an irritating book, also called Mathematician’s Lament. That’s too bad, as he has a lot of valid things to say. But most readers will be unable to see what’s good because it’s surrounding by so much that’s annoying. In particular, Lockhart seems to take an extreme view in favor of throwing out all curriculum and all direct instruction, replacing everything with student-directed problem solving. I say “seems to take” because in fact that’s not actually his position; it’s just that he gets so carried away with his radical POV that everything else gets lost. So, if you read this book, you need to star the following sentence in particular:
If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side.
Keep that in mind. It’s just that everything Lockhart discusses is in fact all the way on the other side. Consider, for example, this provocative chapter title: “High School Geometry: Instrument of the Devil.” Certainly some students do like geometry, though Lockhart claims that those students would like it even more if it were taught his way. And surely most adults remember their high school geometry class with something less than fondness. The big complaint about high school geometry — and here I agree with Lockhart — is that the central themes of proof and definition are presented so woodenly. Writing proofs about claims that are obvious feels arbitrary and useless, and yet that’s what most of the early months of geometry are filled with. And the two-column format is arbitrary and restrictive, a peculiar American custom that no real mathematician would ever use. As Lockhart observes, “A proof should be an epiphany from the gods, not a coded message from the Pentagon.” But it’s rare experience in high school geometry for students to spend a long time struggling with a non-obvious problem, then to come up with a non-obvious conjecture, and finally to write a convincing proof that shows how the conjecture connects with other knowledge. That’s how it should be done.
A similar issue occurs with definitions:
Definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or a structural property. Historically this comes out of working on a problem, not as a prelude to it.
All of this, of course, is driven by one’s concept of what math really is. Lockhart is a pure mathematician, viewing problem-solving and puzzle-solving as rewarding for their own sake, and I agree with him there. But his contempt for applied mathematics will do nothing but turn off most of his readers. It’s important for students to understand that applications come after the math is developed and hardly ever motivate the discovery of new mathematics, but it’s also important for them to work with those applications. Some students will be motivated by that, and everyone will learn something that their future teachers will expect. Nevertheless, Lockhart’s characterization of what math really is is spot on:
Math is not about a collection of “truths” (however useful or interesting they may be). Math is about reason and understanding.
Unfortunately this characterization flies in the face of so much of what is expected of math teachers and math students. MCAS and SATs and science teachers inadvertently encourage the “collection of truths” misconception, even though they of course also want reason and understanding.
Finally, I need to mention the subtitle of Dan Meyer’s blog, dy/dan. The subtitle is simply less helpful. This characterization may seem like an odd one, especially when it’s the subtitle of a blog that’s well worth reading. But Meyer’s resolution to be less helpful is an important one. Like most math teachers, my unthinking inclination is usually to try to be helpful, to answer questions, to point students in the right direction. But Lockhart’s response to a certain question from a student is to observe that “the right thing for me to do as your math teacher would be nothing.” In other words, to be less helpful. That, in the long run, is what will actually be helpful to the student. I just wish that Lockhart had limited himself to a more tempered criticism and had been clearer about taking a balanced approach; he will turn off too many readers who would have a lot to gain from his wisdom if they could only pay attention to what’s good rather than what’s irritating in this provocative book.