As part of the pedagogy course I’ve been taking, I’m doing a few guest lectures in various courses. I’ve got one coming up in a classical mechanics course (“intermediate”-level, so not Newton’s laws, but stuff the general public doesn’t know much about like Hamiltonians). They’ve been speeding through the core content, so I got to cover a “fun” topic, and after thinking back to my grad school days I chose a topic I think they’ll have a lot of fun with: Chaos theory.

Chaos is one of those things everyone has a vague idea about. People have heard stories where a butterfly flaps its wings and causes a hurricane. Maybe they’ve heard of the rough concept, determinism with strong dependence on the initial conditions, so a tiny change (like that butterfly) can have huge consequences. Maybe they’ve seen pictures of fractals, and got the idea these are somehow related.

Its role in physics is a bit more detailed. It’s one of those concepts that “intermediate classical mechanics” is good for, one that can be much better understood once you’ve been introduced to some of the nineteenth century’s mathematical tools. It felt like a good way to show this class that the things they’ve learned aren’t just useful for dusty old problems, but for understanding something the public thinks is sexy and mysterious.

As luck would have it, the venerable textbook the students are using includes a (2000’s era) chapter on chaos. I read through it, and it struck me that it’s a very different chapter from most of the others. This hit me particularly when I noticed a section describing a famous early study of chaos, and I realized that all the illustrations were based on the actual original journal article.

I had surprisingly mixed feelings about this.

On the one hand, there’s a big fashion right now for something called research-based teaching. That doesn’t mean “using teaching methods that are justified by research” (though you’re supposed to do that too), but rather, “tying your teaching to current scientific research”. This is a fashion that makes sense, because learning about cutting-edge research in an undergraduate classroom feels pretty cool. It lets students feel more connected with the scientific community, it inspires them to get involved, and it gets them more used to what “real research” looks like.

On the other hand, structuring your textbook based on the original research papers feels kind of lazy. There’s a reason we don’t teach Newtonian mechanics the way Newton would have. Pedagogy is supposed to be something we improve at over time: we come up with better examples and better notation, more focused explanations that teach what we want students to learn. If we just summarize a paper, we’re not really providing “added value”: we should hope, at this point, that we can do better.

Thinking about this, I think the distinction boils down to why you’re teaching the material in the first place.

With a lot of research-based teaching, the goal is to show the students how to interact with current literature. You want to show them journal papers, not because the papers are the best way to teach a concept or skill, but because reading those papers is one of the skills you want to teach.

That makes sense for very current topics, but it seems a bit weird for the example I’ve been looking at, an early study of chaos from the 60’s. It’s great if students can read current papers, but they don’t necessarily need to read older ones. (At least, not yet.)

What then, is the textbook trying to teach? Here things get a bit messy. For a relatively old topic, you’d ideally want to teach not just a vague impression of what was discovered, but concrete skills. Here though, those skills are just a bit beyond the students’ reach: chaos is more approachable than you’d think, but still not 100% something the students can work with. Instead they’re learning to appreciate concepts. This can be quite valuable, but it doesn’t give the kind of structure that a concrete skill does. In particular, it makes it hard to know what to emphasize, beyond just summarizing the original article.

In this case, I’ve come up with my own way forward. There are actually concrete skills I’d like to teach. They’re skills that link up with what the textbook is teaching, skills grounded in the concepts it’s trying to convey, and that makes me think I can convey them. It will give some structure to the lesson, a focus on not merely what I’d like the students to think but what I’d like them to *do*.

I won’t go into too much detail: I suspect some of the students may be reading this, and I don’t want to spoil the surprise! But I’m looking forward to class, and to getting to try another pedagogical experiment.

AshleyRPollardMy introduction to Chaos theory came through wargaming, but not WH40K, but rather through articles in the Wargame Developments journal called the Nugget. I was fascinated by the graphs that illustrated how the discontinuity between the expected versus actual outcome came as a result of the maths.

Of course, what a lay member of the public understands tends to simplified to the equivalent of sound bites, if only because most people don’t have the interest or the aptitude to delve deeper into the maths.

I consider myself someone who has taken a peek, but as I never advanced to calculus, my understanding is constrained.

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karlshakI’m curious as to you’re solution to this issue (maybe a follow up after you’ve actually gone through it and there’s no longer the danger of a spoiler). I found chaos hard to teach at the undergraduate level because there you’re, as you say, focusing on things like Hamiltonians and Lagrangians. But the most accessible approach to chaos seems to be via dissipative systems so there’s a bit of a disconnect re. shoehorning the topic into a standard mechanics class. It seems easier to approach in a graduate mechanics class where you can count on students having the mathematical background to do an introduction via Hamiltonian chaos. e.g. via Hamilton-Jacobi theory.

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4gravitonsPost authorSo, oddly enough these guys actually cover Hamilton-Jacobi, and that did end up helping a lot in giving them something to “hang onto” for the lesson. Henon-Heiles is also not “so bad” as a system for understanding chaos, in that you can understand a lot just from the Hamiltonian (or even just from the potential). They can’t do a ton with it analytically but neither could Henon and Heiles anyway, it was mostly numerical. So I think I actually hit a decent sweet spot. It still meant that they couldn’t really calculate much, which would be a problem if this needed to fit into the exam. But in this case it was sort of a bonus topic guest lecture, which avoided that. I did end up using the Henon-Heiles paper, including actually showing them the paper itself, I’m not sure whether they appreciated that aspect but it was kind of fun that one could mostly just go through the paper and find diagrams to answer their questions.

I agree it’s a lot easier to approach with graduate students. Here the line between the two is a bit thinner, because the Bachelor’s program flows right into the Master’s program in the Danish system. But it did feel like they were not quite “mature” enough to get some of what I was trying to convey.

One thing I actually don’t know is how much experience they had with dissipative systems…I know as an undergrad I’d at least seen driven, damped harmonic oscillators before by the time I had intermediate classical, but I don’t actually know if that was true of these guys, I just assumed it was!

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karlshakThat does seem like an excellent sweet spot, though I guess, as you mention, it helped that it was a “bonus” topic. HH is pretty straightforward to describe via the Hamiltonian and equations of motion at least and since the paper isn’t too daunting you even satisfied the desire to expose them to an actual research paper. My problem in that situation would be to try and stop myself from trying to give them an “intuitive feel” for KAM tori and symplectic geometry ! 🙂

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ohwillekeIf you want to look at an extreme Saint John’s College does its level best to teach everything from the original sources of an idea including the sciences. As much as possible teaching, GR, for example, from Einstein’s original publications on the subject. https://www.sjc.edu/

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4gravitonsPost authorI’d heard of them, but I hadn’t realized they even did science working from the original texts. With Einstein for GR that’s already rough (SR is surprisingly tractable), but I shudder to think of what it means for Newtonian physics!

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