I really enjoyed teaching this past week, and I thought I should get down what happened to cheer me up for those weeks to come that might not go as well.
The calculus of variations stuff that I did this week (and planned last weekend) went really well. In class on Monday, we really went after the notion of proving that the shortest distance between two points is a straight line. We decided to put some time into showing that a two segment line with a kink is necessarily longer than the single straight line. It took us a while to figure out the best way to parametrize it, but it was really cool that they were clear that, while we did prove that simple thing, it isn’t the same as showing any other curve is worse (or longer, in this case). One student, though, articulated a path to go from showing a single kink is bad to saying that maybe any (non-straight) curve can be thought of as having lots of kinks. We decided at the end of class to make that “proof” be the standard for the day.
Then, on Wednesday, we attacked the problem using the Euler derivation (which they had read about and watched some vids on ahead of time). Some of the cool things we stumbled onto were:
- Euler shows that the best curve has no second derivative (literally no curve!). But it doesn’t specify which straight line does the trick. For that you have to add in the boundary conditions (where the two points that you care about are). I really liked that because we have to do the same thing for the Brachistochrone problem.
- The result is that . Yes you can make that happen with no curvature (y”(x)=0), but you can also do it by getting the slope (y'(x)) to be huge. I asked what they thought of that, and one of the students pointed out that if you go looking for a minimum, you can sometimes find a maximum. They realized that having a single kink curve where the kink is up at infinity would do the trick. Awesome.
- I told them that I wanted the standard to be “I can derive the Euler equation” but I told them they could try to change my mind. We had a great discussion about that at the end of class and they seemed happy to leave it with the derivation. We talked about how there are certain steps that are vital (like the integration by parts step that allows you to cancel like I talked about in my last post). I’m excited to see those student screencasts this week.
In my other class (Modern Physics), the students requested a day on General Relativity this semester (this was during our mind mapping on the first couple of days). I struggled with how to go about that, but, in the end, I punted and asked the students to come to class ready to throw out anything they were able to learn about GR through popular media. We did that right at the beginning of class and got a great list on the board (space/time curvature, black holes, equivalence principle, etc). I thought the day went pretty well after that, but there was one logical sequence that we circled around that I thought was cool: It takes infinite energy to push something to go the speed of light -> light travels at the speed of light -> light doesn’t represent infinite energy -> light has no mass -> light can’t feel a force -> light must travel in straight lines -> light is seen to bend around stars -> straight isn’t what we thought. That seemed to help with the weirdness of curved space/time, a little anyways.
In Modern we’ve now shifted from relativity to what I call “light is a particle.” I decided to start with blackbody radiation, but I was torn about teaching the statistical mechanics necessary to understand the derivation of the equation that determines the spectral intensity of a hot body. So, again, I punted and decided to focus on why we study this in the first place.
We talked about the Rayleigh Jeans approach to understanding blackbody radiation. I told them that there were two major issues to work out: 1) how many standing waves can exist in a cavity in a particular frequency range, and 2) what is the average energy of a particular mode of energy if that mode is in thermodynamic equilibrium with the cavity? This took some work, of course, but I told them that they represented two major milestones of 19th century physics. I then reminded them how poorly the Rayleigh Jeans prediction is and how it leads to the ultraviolet catastrophe.
Ok, the table was set. Then I told them what Planck did, and I asked if they’d have been brave enough to do the same. Specifically, I pointed out (again, they’d already read all this) how Planck left (1) above alone, and showed that a minor tweak to (2) did the trick. That minor tweak was just to assume that the oscillators in the wall of the cavity could only have quantized energies. Only I didn’t say that. I said quantized amplitudes. I’ve noticed that gets them more engaged. Especially after I grabbed a board and asked what they’d think if I told them that I could turn that board into a pendulum that would only oscillated at a fixed set of amplitudes. I was asking if they could live with that if it meant it explained away the ultraviolet catastrophe.
Then I decided maybe an analogy would help. I asked them to consider having an equation that accurately predicted the traffic on our local road, such that they could consult only that when crossing, and that they’d never go astray. Then I asked what they’d do if that same theory/equation suggested that chickens could talk. Wow, what a fun conversation! Planck was brave indeed.
Friday marked the first set of oral exams this semester in either class. I thought they went quite well, and I loved how they set the scene for a collaborative learning session. Probably my favorite moment was when I asked a students how he’d rate both his performance and, separately, his true knowledge of the standard he’d been assigned. He responded that he agreed with his classmates that his performance was a little rough (we ended up settling on giving him a 2 on my rubric), but that, after the conversation, he thinks he really understands it now. Cool.
It was a fun week, and I’m really having fun with both of these classes. I hope the good times keep rolling.