When I wrote about what I learned at the NCTM Annual Meeting back in April, I observed that geometric transformations and mathematical models were common themes at that conference. Transformations seem fairly abstract, even if we math teachers believe that they provide an important connection between geometry and algebra and an important set of unifying patterns that should help students understand some phenomena more deeply. (“Yawn,” says one of my less polite students. Or am I imagining it?) Mathematical models have the potential of being more interesting, but they can easily fall into the trap of being formulaic and empty of much mathematical content. So how do we focus on both of these essential topics while maintaining both student interest and mathematical depth?
Pixar, that’s how. Or that’s one way, at least.
The Museum of Science in Boston is currently hosting an interactive exhibit called “The Science behind Pixar.” This science turns out to be primarily a combination of transformations and mathematical models. Animation, of course, used to be the result of thousands of cels painstakingly drawn by hand by dozens (hundreds?) of artists. But today the use of computer technology lets it be far more elaborate and realistic. Surprisingly, the museum allows photography for this exhibit, so here are some of my photos along with some of my comments relating them to math teaching.
Let’s start with the basics. In several different courses — Geometry, Algebra II, and Precalculus — we teach the most basic isometries: translation, reflection, and rotation. All three of these transformations are important in designing and implementing an animated scene, as shown in the following image:
The question is how to use these transformations to vary a set of parameters and then to see the effects of doing so in real time. The Pixar exhibit includes a wide variety of workstations with physical sliders that viewers can use to control transformations. For example:
Facial expressions are rather complicated, but what about something fairly simple, like a blade of grass?
Well, it’s not so simple after all. It turns out that what we need is a parabola. But not the dry quadratic equation that we usually teach, or the artificial fake-world applications that use quadratics. No, we need a mathematical model that can then be varied by using several different parameters. The Pixar exhibit lets the visitor vary eight different parameters (a process that would be much too complicated in a high-school geometry or algebra class) and immediately view the effects on a scene:
And there’s even a plug for the not-always-loved trigonometry (paired nicely with dilations):
Finally we can return to the school of fish (from Finding Nemo, of course). Emergent behavior arises from a surprisingly small number of parameters: