December 2015

A Mathematician's Delight
Mathematician's Delight
What does mathematics look like if, instead of burying yourself in equations and formulas and jargon, you begin by simply asking questions? MacLaren’s faculty set out in pursuit of that question in this semester’s Faculty Seminar, a twice-yearly affair in which all the teachers in the school gather regularly after school for conversation (often heated) about an exciting text (often difficult) or an aesthetic endeavor (often frustrating) we can practice as a group. In past years, this has entailed sitting down over food and drink to discuss Dante or The Nichomachean Ethics or quantum physics. It has also entailed drawing flowers and singing medieval music in four-part harmony. The aim is to enliven our minds and deepen our discourse, to step outside our own fields of expertise and be wonder-struck amateurs in someone else’s, to experience the same bafflement and thrill we try to elicit in our students.

This semester our quarry has been math, and it has been a doozy. First we read and discussed Paul Lockhart’s classic of math pedagogy A Mathematician’s Lament. Lockhart’s aim is twofold: first, to reintroduce mathematics as pure art, a field of imaginative play; and second, to skewer all our notions of math as a practical endeavor, a means to an end, simply a tool for engineers and scientists and the writers of standardized tests. The faculty had, let us say, strong feelings about the book, and those feelings did not all concur.

Kara Hrbacek, who led one of the seminars on Lockhart, is a humanities teacher and among the book’s champions. “I loved everything he had to say,” she said. “To take something that is so process-oriented or concrete in its application and move it to the realm of beauty—I just love that. It’s like a breath of fresh air through a window that’s been locked for too long.”

A veteran math teacher, Paul Lilley was already conversant with Lockhart’s work and similarly convinced that math education in America is “a mess.” For Mr. Lilley, the value of the discussion lay in speaking with colleagues in other fields about how we’ve gotten there and how we can fix it. “The mess is an open question,” he said. “That’s why it’s so interesting! It’s different when you hear Andrew Swanson or somebody else who isn’t a math teacher reflecting on that as well. It gives you a much wider perspective.”

To immerse ourselves in these questions more fully, we tried to do some math—real math, in Lockhart’s contentious formulation, the kind of math that students aren’t taught. Led by Kathryn Maslow (who, for the record, teaches both math and humanities), we spent two evenings doing mathematics that was largely free of formulas and definitions and was characterized instead by a more open-ended, investigatory approach.

“My hope was to help everyone on the faculty experience mathematics in a way that was accessible but also pure,” Ms. Maslow said. “There are concepts in mathematics that are confusing and intriguing and get at the heart of the questions that humans ask, [and which] don’t have strict application in, say, science or business.”

This meant exploring set theory, a branch of mathematical logic that deal with collections of well-defined objects, called sets. Ms. Maslow raised the possibility of an infinite set (all even numbers, for example, or all fractions) and caused us to wonder what it even means to talk about the size of an infinite set, and the implications thereof. It meant, too, playing with taxicab geometry, in which we dealt with the ramifications of throwing a seemingly inalterable geometric rule—the idea that a line is the shortest distance between two points—out the window. We felt more like philosophers than mathematicians, more like puzzle-makers than pupils.

And this was Ms. Maslow’s intent: “People who were interested in philosophy could really dive into the philosophical implications, but math people could also dive into the technical aspects. All sorts of people in the room were willing to engage in the implications, and that was really exciting to me.”

Which is not to say that there wasn’t a good deal of head-scratching and furrowing of brows. And this, too, was Ms. Maslow’s intent: “It was definitely difficult for a lot of people, but there were some moments when people came to an understanding, and those were my favorite moments to see. People were struggling through, drawing these little squares, and all of a sudden they would have this realization [of] what they were doing, what they were accomplishing. They would see the reality, the beauty of what they were doing. Those are some the greatest moments you can have: trudging through and then all of a sudden you have this moment where you see something, you make something happen, and you’ve gotten yourself there.”

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