many-body systems; composite objects

Every time I teach mechanics (and this is something like the 21st year I have taught it at the undergraduate level) I learn something new. This week we are talking about many-body systems; I had two epiphanies (both trivial, but still): The first is that the description of the object in terms of a center-of-mass vector and then many difference vectors away from the center of mass (one per "atom") is purely a coordinate transform. Indeed, it is just generalized coordinate system that is related to the Newtonian coordinates by a holonomic transformation. Awesome! So when the Lagrangian separates into external and internal terms, this is just a result of the appropriateness of that transformation.

The second is that the definition of the many-body system is completely arbitrary. It should be chosen not on the grounds of being bound or solid or connected but rather on the grounds of whether choosing it that way simplifies the problem solution. Both of these realizations are simple and obvious, but it took a lot of teaching for me to get them fully. I am reminded as I realize these things that the physics concepts we expect first-year undergraduates to manipulate and be comfortable with are in fact pretty damned hard.

the answer "that question is ridiculous" must be accepted

My friends who work in education of the young (the 'fuzz included) like to quote studies that show students answering without comment or concern questions like "Farmer Jake has 13 sheep and walks them 21 miles. How old is Farmer Jake?" There are many mixed-up reasons for this problem; some relate to rote learning; some relate to the artificial dichotomy set up between reading and math; some relate to the decontextualized ways we teach math; some relate to the testing environment that saturates schools; and so on. I feel all these things!

Imagine we want to see students using their common sense and their judgement with every question they consider and answer. I think that would be good. How do we foster this kind of thinking and exercise of common sense? I think we have to let the students call "bullshit".

Here's an example: "Johnny has twelve toy cars. He gives eight to Frances. How many does he have left?" Obviously we should accept the answer "four". But we should also accept the answer "No way! Who would give more than half of his toy cars to someone else?" If we don't accept that answer, we are saying to the students "calculate without thinking". That might be okay for quantum physicists (though I disagree), but it isn't okay for the rest of us.

don't make students hate math!

In my previous post I implied, inadvertently, that we should only teach useful math. That was not my point. My point was that we should not teach math if the effect of that teaching is to cause most students to hate it. Of course if we could teach it such that the effect was to cause most students to love it, I would be all for teaching it!

don't teach math!

In a nice conversation about writing for education, Adam Gidwitz (the author of A Tale Dark and Grimm) pointed me to The Mathematician's Lament. The book makes (much more clearly than I) a point I have been making informally for years: If you want students to know and love math, you definitely should not teach it in school! (Same for literature.) The mathematics requirements in school empty the subject of its meaning and point, and are useless to boot. How many non-scientists use the quadratic formula, ever? Discovering the formula would be fun, using it is a drag (and exceedingly rare).

complex numbers

I showed the students complex numbers today, in the context of solving the damped harmonic oscillator. I took my time, and treated it as a cultural romp rather than a physics problem, so I enjoyed myself. What I was surprised to learn is that almost every student in the room knew that the square root of negative one is i. What is up with high school math that every student in the room learned what an imaginary number is, while not a single one learned how to program a computer? Which is easier, which is more relevant to them, and which is more natural given their interests and materials? I would say: Programming, programming, and programming!