More lazy email(ish) excerpts. This one from a conversation about polar coordinates, and my Calculus II students’ struggles with them last semester.
I don’t fully understand the usefulness of polar coordinates as traditionally presented. I feel like a great deal of the focus is on the pretty pictures we get when trig functions interact with polar coordinates. Cardioids and all that.
When I taught Calculus 2 this past semester, it was clear that my students were struggling to sort out differences between cartesian coordinates and polar coordinates. They knew how to convert between polar and cartesian coordinates, but they didn’t seem to know why one would do that, nor did they seem to see polar coordinates as a self-contained system. Polar coordinates were always (in their minds) in relation to cartesian coordinates.
My students struggled to think about an angle as an independent variable that could change (and correspondingly a radius as a dependent variable that could change).
They couldn’t view a function defined in polar coordinates as a dynamic relationship. They could identify points one at a time. They could make their graphing calculators display polar graphs. But they couldn’t think about the process of tracing out a polar graph. This seemed to be true even for students who could talk dynamically about cartesian graphs (increasing, decreasing, approaching an asymptote-this was terminology my students could apply to cartesian graphs, but not to polar ones).
The precise conceptual nature of the relationship between polar coordinates and cartesian parametric equations is unclear to me. My students saw some relationships that they couldn’t quite articulate. I’m interested in exploring this territory a bit.
For the record, I was flying blind through this material last semester. It was the first time I had taught polar coordinates in any serious way. I have never taught precalculus. I had never taught Calculus 2 before. So I kept bumping into obstacles that I hadn’t imagined would be there.