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Following long chains of reasoning

October 1, 2010
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One of the most seminal quotes I’ve ever heard about what students should learn in a high school science class comes from Bruce Sherwood, author of the amazing Matter and Interactions curriculum. This is by far the best calculus based physics curriculum out there, and here’s a great email from Bruce on what he expects students to know coming into this course.

One of our reasons for creating Matter & Interactions was to have a college course that didn’t simply repeat the high school physics course. In the preface we even say, “It is assumed that you have had an introductory physics course in high school.” Yet it turns out that among the mainly engineering students we have at NCSU, about 20% of them have never had any physics before. However, there is no correlation between prior physics and performance on tests. You can either see this as good news or bad news. A possible interpretation is that M&I starts in a very different place and goes in a very different direction, so it’s new to everyone, and those without prior physics aren’t at a big disadvantage, fortunately.

What I see as extremely important is that students 1) have had the experience of working though a fairly long chain of reasoning, with many steps, and 2) that they ascribe meanings to symbols. In contrast, many of our students have in their prior courses (not just physics!) encountered only problems that involved at most one step in reasoning. A close corollary is that their expectation is that science in general and physics in particular consists of the instructor telling you what formula to use, and telling you what numbers to put into the formula, and then you evaluate it, and that supposedly means you have learned something, when in fact you have learned nothing.

Concerning the meaning of symbols, our students have for the most part adequate skills in algebra and basic calculus manipulations, but they ascribe no meaning to these manipulations. It is all syntax and no semantics. For example, dy/dx isn’t a ratio of small (infinitesimal) quantities, it is a prescription telling you to look for an x, and if you find a number written above and to the right of the x, you’re supposed to move it in front of the x and decrement by 1. And dS/dE is completely meaningless, because there’s no x in the denominator and you couldn’t find any x in the S function so there’s no way to carry out the prescription.

Another example: Problem asks students to find the final kinetic energy K. They use the energy principle and find K is 100 J. They then set this equal to (1/2)mv^2 and solve for v. Next they put v in the formula K = (1/2)mv^2 and find K is 100 J. We think these extra steps come from the fact that apparently K doesn’t have any real meaning; rather K is what you get when you multiply a half times m times the square of v. Until you’ve done that, you don’t really have K.

Drives you crazy, and makes you deeply lament what has been done to these bright but educationally mistreated kids.

Notice that I’ve said nothing about specific physics content. Give me a student who has had practice really thinking, and making logical deductions, and I can work with that student and contribute to the student’s further intellectual development. Of course it would be a plus for the student to have already had experience with the major concepts: motion, force, Newton’s second law, energy, momentum, charge, current, field. But the larger issues are the crucial ones. A good high school physics course is one of the places where these larger issues can effectively be addressed.

If you’ve read M&I, you know that it is filled with long chains of reasoning, much more so that other physics texts. One of the most beautiful is a 6 page explanation of how sparks work, and it’s a must read for physics teachers. I can’t tell you how amazed I was when I read this and realized that electrons don’t jump from one dome to the other with a VDG.

So I teach 9th graders, and they probably aren’t ready for the full-on sparks explanation just yet (though I did have 3 students work through this last year), so I’m trying with a slightly shorter, but still long chains of reasoning. Here’s an example.

We also tried something similar as kids were working on dropping coffee filters:

Me: So what do you think the drag force depends on?
S: the mass, heavier filters fall faster, so the drag force must be smaller.
Me: look as I hold this filter at rest in my hand—what is the drag force on the filter now?
S: Zero
Me: (moving the filter through the air) So what is the drag force now?
S: Not zero
Me: Did the mass of the filter change?
S: No, but that would mean the same mass would have two different drag forces.
Me: What does this tell you about whether mass affects the drag force?
S: I don’t think it does, maybe velocity does.

and so on, until we get to
Me: so how do you measure the drag force
S: Well, when it’s falling, the velocity is constant, so the net force must be zero
Me: so
S: There are only 2 forces, the drag force and the gravitational force, so these forces must be equal.
Me: Is this always true?
S: No, only when the velocity is constant.
Me: cool, so you just figured out how to measure the drag force.

This really is a core skill, and I’m trying to assess it more and more. For more on that, see my assessment.

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