_{2}, by Natalie Angier, from today's New York Times. The article describes physics classes taught by Faye Cascio to ninth-graders in a magnet school, who are answring questions like the following:

You fall into a swiftly moving river and are in need of a flotational device. You see a life preserver bobbing three meters downstream of you and another one the same distance behind. Which preserver should you swim toward?

If you assume that the river is moving at a constant speed, you can look at this in a frame of reference that moves with the river; in that frame each flotation device is the same distance away, so it doesn't matter which one you swim towards. (Incidentally, I wonder if it is easier or harder to teach this

*without*invoking the word "relativity"; I suspect it's harder, because if you mention to the students that they're doing relativity, they may be aware that this is more modern physics and therefore expect it to be more difficult. But Galilean relativity really is straightforward. This particular problem is of course doable in a frame of reference anchored to the shore, but the algebra's a bit uglier.

3) A plane flying into a headwind will have a lower speed, relative to the ground, than it would if it were flying through still air, while a plane traveling with the benefit of a brisk tailwind will have a comparatively greater ground speed. But what about a plane flying through a 90-degree crosswind, a breeze that is buffeting its body side-on? Will its ground speed be higher, lower or no different than it would be in unruffled skies?

The ground speed is slightly larger; it's the length of the hypotenuse of a right triangle with legs equal to the ground speed in still air and the speed of the wind. (But notice that the derivative of the plane's ground speed with respect to the wind speed is zero at zero wind speed. High school freshmen don't know this, though.)

The students can articulate their reasoning because, for one thing, they have no choice.

This is something I've been trying to get through my own students' head on their homework, by taking into account their ability to explain the solutions when grading; for the most part it seems to work, although since it's on homework (and not in class) there are always some students who don't bother explaining their work and figure that the points they'll lose are made up for by the time they don't have to spend on the work.

Nearly all scientists and educators agree that somehow, at some point during their pedagogical odyssey, most Americans get the wrong idea about what science is, and what it is not.

Carl Sagan once said that scientific education in this country was structured so that it would actually attract the people he thought would be

*worst*at being scientists -- people who like routine and who want to just plug and chug their way through the calculations. I agree with this, and I'd add that mathematical education is much the same way. What one does in math classes through at least midway through college bears so little resemblance to mathematical research that the mathematical community probably loses a lot of potentially very good mathematicians, simply because their classes turn them off. (However, the process

*might*produce people who can competently

*use*mathematics, which is something worth keeping in mind. We must remember that part of what we do is to teach the rest of the world how to use our tools.)

## 4 comments:

My immediate answer to the “airplane in a crosswind” problem was the exact opposite of yours: I thought the wind should be orthogonal to the ground track of the plane, so you have to fly into the wind a bit, and actually end up with a lower ground speed.

Harald,

you have to fly into the wind a bit if you want to maintain the same direction. But the problem didn't specify that interpretation. (It didn't specify my interpretation, either.)

Actually I think the phrase 'a breeze that is buffeting its body side-on' suggests the wind is perpendicular to the aircraft body, not its ground track, so the problem does at least imply Isabel's interpretation.

If the wind is actually perpendicular to the body, then how do the wings generate any lift?

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