I spent a good portion of the day trying to figure out what I wanted to do in my Theoretical Mechanics class tomorrow. We’ve recently begun the chapter on central potentials and I wasn’t sure how far to try to get. I started writing down the big ideas that I thought were important:
- The effective potential idea gets us all the way down to a single variable (from 6 with the x’s, y’s, and z’s of the two particles involved in a central potential interaction)
- Kepler’s second law (which really applies for any central potential). This is the one about the orbit sweeping out an equal area for every equal time period.
- Kepler->Newton (showing how a 1/r^2 force leads to ellipses): more on this below
- Finding relationships for the eccentricity and ellipse axes in terms of the energy and angular momentum of the orbit.
I knew I couldn’t do all that in one class period, but I thought I could probably get through the first two and a half or so.
But that’s when I wondered what the “I can . . .” statement of the day would be and it hit me that I shouldn’t just be plowing through material. Instead I want the “I can . . .” statement to be driving the work, motivating the students to engage and to figure out what other resources they’re going to need to figure out how to prove that “they can.” I realized that repackaging the first two points made it a nice consistent idea that we could really play with in class. The r- and -equations each bring something to the table when understanding orbits and I think if we do them well, we’ll be really set up for the rest on the next day.
It’s funny that I fell off my “standards-decide-the-day” wagon and didn’t even notice. I spent so much of my day today trying to find cool connections between Kepler and Newton that I lost track of the utility of really focussing each class day on a articulatable idea.
One other idea came out of my brainstorming today. The proof that inverse-square-law orbits are ellipses is a beast. The integrals involved are nasty, often utilizing all kinds of “I would have never thought of that” tricks. So I really tried thinking today of other ways to get that point across. I thought about having the students code up some numerical simulations to see that all the orbits seem to be ellipses, but I knew that it would fall short of the goal of proving that all possibilities are ellipses. I talked with a friend in the math(ematics) department about this and we explored a number of interesting pedagogical/curricular issues:
- Do students need to do a derivation like this?
- yes: it’s famous and really interesting, tying together Brahe, Kepler, and Newton
- no: ugh, you’ll never assess it, so why bother
- My friend uses the phrase “it’s good for your soul” to motivate students to try to struggle with a tough idea, even if there’s a good chance it won’t be formally assessed.
- I make it clear what will be assessed and try hard to not have anything else in the class. This has led to cutting material (like catenaries) even if it’s “interesting” if I’m not planning on assessing it. My friend suggests that I’m limiting myself and my students’ learning by doing that.
So what do you think? Here are some starters for you:
- I’m in this class and I want to see if Newton was a genius or if I could have figured this out.
- I’m in this class and I’ll take your word for it that Newton was able to do that nasty integral by hand.
- I like how you try to make each class a stand-alone idea, but I don’t think “articulatable” is a word (and neither does Chrome)
- I don’t understand why you put so much focus on the arbitrary class time breaks. The material is huge and doesn’t have to fit cleanly into those breaks. I say just plow through.
- I say “it’s good for your soul” a lot too. Here’s an example . . .
- I don’t like the “it’s good for your soul” motivation because . . .
- If the students’ numeric results are always ellipses, good enough!
- Do you know if the “equal area in equal time” only works in the reduced mass system?
- I’m the friend from the math(ematics) department and you’ve misquoted me like heck. What I really said was . . .