Paul Lockhart's Measurement starts with a triangle. If you connect each of a triangle's vertices to the midpoints of the opposite sides, they all seem to meet right in the middle--but do they? How do you prove it? From this exercise, Lockhart spins out an explanation of the basic premises of most grade school level (and perhaps beyond grade school level, what do I know) mathematics. "Measurement" is the key term here, that in doing mathematics we are measuring things according to other things, breaking down polygons into constituent shapes, measuring the paths of vectors in a space of any number of dimensions, fitting triangles into circles and slicing up cones. It's mostly stuff you might remember from high school, but which, unless you went on to a career in STEM, you have probably mostly forgotten. But it's presented in a way that reinvigorates the material, not as a set of dreary problems, but as a kind of elegant unfolding and shifting, toggling back and forth between different modes to reveal fundamental relationships.
I take that back: there are lots of problems. They just aren't the kind that I remember, which start with "Imagine a ladder ten feet high," or "Two trains leave Pittsburgh at 10:03 a.m." Instead, they sound like these: "How can we measure the length of a helix?" Every curve can be straightened without metric distortion. Is the same true for surfaces?" "How many corners are on a four-dimensional cube?" Some of these problems are explained, but some you're supposed to take a break and explore on your own. I didn't do this, but I appreciated the invitation all the same. I liked that the problems kept the number to a minimum. Lockhart emphasizes that what this mathematics about is not application, the kind of "real-world" motivation that is vogue in education today, but the imaginary "pure" mathematics that is never exactly reproduced in the messiness of the real world. In this way, Lockhart emphasizes the beauty and the elegance of mathematics. It's a subtle argument not just for a way of looking at math, but a way of teaching math, one that forsakes ladders and trains for the two simple section titles: "Size and Shape" and "Time and Space."
I'll probably remember very little of the specifics here, except maybe for the explanation of conic sections as lines in projective space, something I'd never heard about before, and the disturbing fact that there's no way to measure the perimeter of an ellipse algebraically. But I really did appreciate the opportunity to rethink some of the mathematics I learned, and see it in a new way. Lockhart is right when he says that mathematics can be elegant and beautiful. My dad, a professor of physics, unlocked an understanding in me when he told me, when I was a kid, that mathematics is a kind of language. I suppose that, as Lockhart writes, it's a kind of art as well.
