Last year, I taught conic sections right before spring break. We took a week off, came back and spent two weeks on polar coordinates, and then dove into parametric equations. At the end of parametrics, we wrote parametric equations for conics.
This did not go well.
In hindsight, I can see that it was destined to not go well. They hadn’t seen a conic section in four weeks, one of which was spring break. They were struggling with an entirely new idea, and then I asked them to remember details of things from a while ago. In an ideal world, would students be able to do this? Yes. But spring break makes people forget things. It’s true. I’m sure there’s a study on it somewhere, but I probably read it before spring break, so now I’ve forgotten about it.
I made a note to myself somewhere: “Next year, don’t break up conic/polar/parametric so much.” Our textbook actually puts all of these ideas in one chapter. While that may be overkill – there are some major differences between these ideas – I didn’t tie them together very well, then suddenly expected kids to see connections at the end without weaving them all the way through.
This year, things were different. We pushed sequences and series to before spring break (still not sure if I like that plan), which freed up a large chunk of continuous time after spring break for polar, conic, and parametric. So we did things in that order: a few weeks on polar, a week on conics, and now parametrics.
It’s going so much better! They just learned about hyperbolas last week, so when I asked them yesterday to write parametric equations for a hyperbola, they were so on top of it. And because conics and parametrics are on the same test, they don’t feel like they should have been able to forget about it yet and see this as review.
(It’d be great if students saw the connections between everything, and didn’t see each unit as stand-alone or something to forget about until the final. It’s a long-term goal of mine to make progress on that front, and we’re getting places. But for now, I’ll work with where they are.)
You’d think that teaching related concepts right next to each other would be obvious, wouldn’t you? But I probably didn’t make the decision to do parametric equations of conics until I’d already taught conic sections, and it was a little late by then. But that’s the wonderful thing about re-teaching a class: I’ve found some of the major problems and can fix them. And next year, I can work on more, and maybe even some of the medium sized problems, too.
For instance, now that conics and parametrics are tied together, I’d like to loop polar in to this. I think Sam Shah starts polar by looking at polar equations of conics, and that’s a definite possibility. I’d probably have to move polar to after conics to do that, but it’s feasible. Things to think about it…