by Jonathan Kujawa
“Meaningless! Meaningless!”
says the Teacher.
“Utterly meaningless!
Everything is meaningless.”
— Ecclesiastes 1:2
This spring I taught one of my favorite classes. Officially, it goes by the name of Discrete Math and covers various standard topics: sets, functions, basic logic, modular arithmetic, counting problems, graph theory, etc. Unofficially, it is our department’s boot camp for people who want to take advanced math classes.
Up through high school math, and even in most freshman and sophomore math classes, the goal of a math class is for students to be able to use standard tools to solve standard questions. We are teaching students to distinguish between a screw and a nail, and to know to twist one with a screwdriver and hit the other with a hammer. Most problems and techniques are chosen more for teachability and testability, than for being interesting or useful. What matters is The Answer. It is right or wrong, and it’s right because someone said so. We shouldn’t blame the teachers for this state of affairs. They suffer as much as anyone under our educational system.
But math is meant to be a creative, human activity. In Discrete Math our students are deprogrammed of their twelve years of bad math education. I teach it using a slow version of the Socratic method, sometimes called Inquiry-Based Learning. Students are given a well-chosen series of problems to solve, but many are ambiguous or open-ended. They collaborate, share ideas, present written and oral solutions, and they critique each other’s work. For students used to being force-fed, Clockwork Orange style, it’s liberating, exciting, exhilarating, and terrifying all at once.
The first challenge is to get the students to deeply internalize the fact that words have meaning. If we’re going to talk about even numbers, then we’d better all agree on which numbers are even. Is 104 even? How about -2, or 0, or π?
Worse, they discover there are no God-given definitions in mathematics. Students are always a bit shocked to realize it was people just like them who came up with the math they take for granted. Even the equals sign was invented by someone, after all. Definitions can be contentious. Read more »