Teaching Multivar to Outdoorsmen
Unexpected bonus teaching at a school in rural Vermont with a significant outdoor component: teaching basic concepts from multivariable differential calculus is actually rather easy. When students learn that level curves and gradient vectors are perpendicular to each other it makes perfect sense to them: if you’re going for maximum challenge you go perpendicular to a trail. Once they associate the x- and y-axes with east/west and north/south, respectively, partial derivatives as measures of angle inclination become pretty obvious. There are natural connections between surfaces and heights and mapping altitude with lat/long using a GPS-enabled device. For functions of three variables it’s easy for them to associate each point in space with meteorological data. Everything involving visualizing surface becomes easier when there’s physical association and memory along with it. In fact, if it wasn’t so cold out there I’d have class on back trails behind campus.
Having class right next to the fiber and visual arts studios is also helpful. You can do a lot of demonstrations with marker-on-fiberglass planes and fabric models of surfaces that wouldn’t be as three-dimensional even with the latest version of Mathematica. (That, and the projector in my classroom doesn’t do greens very well and it makes projecting complicated/shaded things rather difficult.)