And now, finally, I give you my own response to Paul Lockhart’s A Mathematician’s Lament. Sorry it’s a bit long.
I enjoy math very much, which makes me an exception among math teachers in general. Many math teachers begrudgingly teach their subject. So many statements in Lockhart’s essay are true. I echo his sentiment as he complains about dispassionate math teachers, saying, “But shouldn’t they [math teachers] at least understand what mathematics is, be good at it, and enjoy doing it?” He encourages math teachers to be mathematicians, just like art or music teachers. And he encourages math teachers to allow their students to be mathematicians too. Students, he says, should have opportunities to play, which is what mathematicians actually do.
Admittedly, I don’t feel like I’m teaching students this “real” kind of mathematics most of the time (the creative, artistic, and fundamental aspects of what mathematicians do). I do my best, but I often do little more than lecture and attempt to keep kids awake with goofy antics. So I understand Lockhart’s point of view on math education. It needs more “art.”
That being said, I feel a bit torn between battling camps when it comes to math education. There are two (often contradictory) goals in the mind of the math educator:
(1) The math educator wants students to like math, think creatively/rationally/logically, and to understand the context in which mathematical ideas are created. This is Lockhart’s position.
(2) Second, the math educator wants students to become good at math and qualified for higher-level math, science, and engineering—ultimately preparing them for college and careers. The math educator recognizes that students going on to college and careers will be expected to have specific mathematical knowledge.
So I think that we, as educators and as a society, must decide if we want to “get students to LIKE math and enjoy it as a creative act” (the first goal) or “make students good at mathematical procedures” (the second goal). That’s the hang up. And the debate is not as easy to resolve as Lockhart would like to make us think.
Points on which I disagree with Lockhart
In my first paragraph I agreed with Lockhart. But now let me take the opposite position (number 2 above).
You see, math is plagued by something that art rarely has to deal with: it is very, very useful in the real world. Some mathematician has described mathematics in terms of this analogy: Mathematics is like a large store, in which there are many shelves housing mathematical tools. The scientists and engineers come into this store and take useful tools off the shelves. Mathematicians, in most cases, did not create these mathematical tools for scientific use. They created them because they were interesting in their own right and beautiful. In fact, most mathematicians would agree with Poincare, “The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.” That being said, math educators still have to equip students with those “mathematical tools” the sciences find so useful. Math just happens to be both beautiful and useful.
Let me give an example:
If you’re a student who has been in my Calculus class, you can DO all sorts of things after a year of being in my class. You understand how to find the area between curves, minimize cost functions, maximize the volume of a container. You know how to apply the quotient rule and the chain rule, and you can do u-substitution with your eyes-closed. And you’re prepared for college courses in math and science. Now imagine you go on to take a Calculus-based Physics course in college (as any math or science major will undoubtedly do). Without having had a formal treatment of Calculus in high school, the task of a college professor teaching Calculus-based Physics would be impossible. The college Physics professor will simply expect that you can do u-substitution with your eyes-closed. In the Physics-class scenario I just laid out, formalism in your high-school math education was a good and necessary approach.
In that same scenario, how would one of Lockhart’s students perform? How does his version of education prepare students who will be math majors in college? If I structure my class like an art class, having my kids work on self-directed projects, standing at easels, playing with mathematics—they’ll gain an understanding of what mathematicians do and they may even learn to love math (good things!). But when they get to that college Physics class and the professor simply expects them to be able to integrate a function, will they be prepared? What if they didn’t discover how to integrate while dabbling at their easel? So in this case, Lockhart’s “artistic” pedagogy fails.
I had a professor who was well-liked. I still see him sometimes and I still think he’s wonderful. His style of teaching was exactly as Lockhart prescribes. His classes were fun and interesting. But he compromised the material in order to do that. I took a course in Modern Geometry and I can’t tell you anything about projective geometry (a central topic) or a host of other important ideas in modern geometry. We did lots of unrelated puzzles and problems, barely ever cracking the book. As I now consider graduate work, I’m a little sad I wasn’t better prepared for graduate-level mathematics.
Can you now see why goal (2) is just as important as goal (1)? For college and career-bound students, I have an obligation to teach certain mathematical skills. For them, my curriculum needs to be systematic, formal, structured, methodical, and well-planned.
Attempting to balance
Clearly a balance has to be struck. If I teach a structured curriculum, my students will be well prepared for their academic future. They might not see the beauty of math; and many won’t get the joy of discovery which is so important to the mathematical experience. However, if I teach only as Lockhart recommends, I risk students not getting the skills they will need.
It’s not an easy battle. Formalism in education is valuable in meeting objectives and preparing students for times when they will actually need certain mathematical skills. But I also agree with Lockhart that we could use a little more space in our curriculum for creativity.
I’d love to just have a math-art class. Sounds like fun. But until then, I (and all the other math teachers) have to do my best to balance a structured curriculum with mathematical play. I’m trying hard. And I hope this blog will, among all my other efforts, spur you on to mathematical play.