Three and a half years ago I wrote a post about Paul Lockhart’s book, Mathematician’s Lament. Now he has a new book, oddly titled Measurement.
Why, you ask, is that title odd? It’s because the book is only peripherally about measurement. Mostly it’s about exploring a whole host of mathematical ideas, of which measurement plays at best a minor role. In fact, the “real world” plays only a minor role, since the book is about mathematical reality, not about what most people would consider practical applications.
Mathematical ideas…hmmm, many people are surprised to hear that mathematics is about ideas at all. They imagine that it’s about formulas and algorithms. Lockhart firmly minimizes (but doesn’t eliminate) formulas and algorithms, definitely concentrating on Big Ideas. Some of these are in the form of suggestions to the learner, such as Do Not Ignore Symmetry and Try Things — two suggestions that I will attempt to persuade my geometry students to accept in the soon-to-start academic year. Other Big Ideas are in the nature of mathematical content, such as Symmetry (implicit within Lockhart’s advice not to ignore it) and Proportion.
We swing back and forth from the big to the small. Here’s a variety of major and minor quotations (by Lockhart) along with comments (by me, in italics):
- “The best way to solve a problem is to find an ingenious way not to have to solve it at all.” This big idea has profound implications. I’ve emphasized it before; I’ll try to do it more this year.
- Or consider another big-picture thought: “The tangling and untangling of numerical relationships is called algebra.” That’s certainly not how most people think of algebra…but of course it’s correct. I’ll have to see what my students think algebra is.
- “To be like a geometric thing, a real thing has to be the right size; namely, it has to be about our size. It has to be roughly at the scale at which we humans operate. Why? Because we’re the ones who made up the mathematics!” Maybe. Worth thinking about.
- And here’s an intriguing detail: Find the volume of a pyramid by dilation, starting with connecting the center of a cube to its vertices. Never thought of it that before. I’ll try it.
- Perhaps surprisingly, Lockhart has a lot of material about conic sections, relating them to each other in many ways, including projective geometry. He also has a slightly different perspective on dilation than the one I’m used to. To him, it refers to stretching in one direction only (such as a circle to an ellipse). To our textbook and to the Geometer’s Sketchpad, it refers to a proportional projection from a single point. To Wolfram, it has yet a third meaning: “A similarity transformation which transforms each line to a line whose length is a fixed multiple of the length of the original line.” It’s all food for thought: we like to teach how definitions are arbitrary and that there are reasons for picking one rather than another. As Humpty Dumpty said, “The question is, which is to be master — that’s all.”
- Finally, a lovely paragraph we find on our way to establishing Heron’s formula: