Saturday, March 17, 2007

So Advanced, It's Simple

Keeping it simple can be a bad bargain when it means failing to explain, failing to convince, or failing to support us as we move up to the next level of complexity. A handful of examples along these lines:

1. Kinematics

The curriculum for Newton's Laws traditionally begins with 1-D kinematics, and traditionally emphasizes the reduction of 2-D problems to 1-D problems via component analysis. It is an important feature of the world that phenomena can be sliced up and viewed one component at a time. But just because you can work one component at a time, that doesn't mean you always have to.

Over the years I've come to feel that trying to build a 2-D understanding on a 1-D foundation is a losing strategy. It's sort of like trying to build a rectangle out of line segments: it works fine, as long as you have an eternity to spend on it. In my book manuscript, I've taken the approach of presenting kinematics in two dimensions from the very beginning. We look at all kinds of curvilinear motions and draw lots of pictures, as we learn what the position, velocity, and acceleration vectors mean. The vectors themselves are allowed to spend a little time in the limelight, instead of being chopped up and used for components. Circular motion becomes relatively easy and loses its ad hoc character.

This approach does require that we learn some "vector calculus," but I have found that the essential facts are relatively few and surprisingly easy even for nonmajors to grasp - perhaps because working with vectors and trajectories in space allows you to bring your visual imagination to bear in ways that sketching a_x-t graphs does not.

2. Walking

To zeroth order, you could posit a human being walking in a straight line at a constant speed. But this is not so much walking as it is floating - and it doesn't feel right. The next order of complexity seems crucial for understanding the process. It seems to me that over the cycle of each stride, your center of mass speeds up during the "launch" phase, glides ahead for a moment, and then slows down during the "heel strike" phase. A graph of the acceleration component over time leads to a graph of the net force exerted by the floor on your feet; this allows you to visualize the friction-force vector flipping back and forth, maintaining the net force at zero in a time-averaged sense. But now it's a "dynamic zero" that we're seeing, not a "floating zero" of the sort that we would associate with a pebble drifting through space.

This is certainly a more complicated picture than the simple proposition of "walking in a straight line at a constant speed." But here we may have a case in which the zeroth-order approximation is so far from the reality and the felt experience that it's actually confusing.

3. Steady state flows

The book I'm using right now for introductory physics confines its treatment of heat conduction to the steady state. The idea behind restricting to the steady state seems to be one of shielding students from the complexity of the full dynamics of diffusive heat transfer. In a way this makes sense, but the trouble is that it doesn't acknowledge the fact that the steady state viewed in isolation is highly counterintuitive. If there's construction on the highway and two lanes constrict down to a single lane, have you ever in your life observed the traffic speed to double at the constriction point? Of course not; but in the steady state, this is what would happen. (Once I did actually take part in a steady-state traffic jam, though it wasn't on the interstate. It was at the Blue Water Bridge in Port Huron, Michigan: one lane bridge, seven lane toll plaza; roughly two minutes to advance each car length on the bridge, roughly fifteen minutes to advance each car length on the plaza. Multiplying the number of lanes by seven divided the speed by about seven!)

Frustrated by the troubles my students were having with conduction problems and the steady state, I sat down one night last week and did some numerical work on the temperature diffusion equation, funneling the output into a handful of little movies. One of them is embedded below; for a clearer version, click here; some explanation is attached to the YouTube video. Feel free to send me an e-mail and I'll send you .mov files and/or Mathematica code. (This was some quickie programming the night before class....)

We discussed this movie and the accompanying T(x,t) animation for quite some time. Far from being intimidated or confused by the complexity of the diffusion-equation behavior they were seeing, the class seemed to gain a much clearer idea of just what I had been asking them to do by focusing on the steady state in the first place.

Given the physicist's overwhelming desire to simplify (which I described earlier in Is Physics a Liberal Art?), it's worthwhile to keep in mind Einstein's famous corrective: "Everything should be made as simple as possible, but not simpler."


Andrew said...

Re 1: I've had the same experience, in teaching both math and physics. In principle, one dimension is simpler, but it always turns out to be more confusing for students than two.

I think that we each have two computing engines available for tackling problems: a grammatical engine and a spatial engine. As long as we are doing algebra, we're using the former; two dimensional pictures and diagrams, of course, use the latter.

Somehow, when students look at 1D kinematics problems, their spatial engine isn't even turning over. They're looking at these problems strictly algebraically.

But at the same time they're itching to use geometrical intuition. When first shown a graph of the height of a projectile (moving in 1D) versus time, they almost universally interpret it as a 2D trajectory.

The same is true in mathematics, where it seems that people don't really understand the definition of the limit and of continuity until they see them in 2D.

Andrew said...

Re 3: This is also an important point. We make simplifying assumptions, we study special cases, we concentrate on problems we can actually solve. But there's a tendency not to tell students we are doing this. Sometimes they have correct intuitions which conflict with the unrealistic model, which either sows doubt or reinforces the idea that physics class has nothing to do with real things.

If we show them how we're making simplifying assumptions, it gives them some context, some sense of where they are placed in the scope of real problems. And it gives them an idea of how simplifying assumptions are made - which is probably the most important thing anyway.