Re: [Phys-L] proportional reasoning, scaling laws, et cetera

[Bill Nettles <bnettles@uu.edu>]

> -----Original Message-----
> From: phys-l-bounces@mail.phys-l.org [mailto:phys-l-bounces@mail.phys-
> l.org] On Behalf Of Brian McInnes
[snip]
> However the way these real world things are treated take the problems way,
> way from reality.  The momentum of the bowling ball doubles (how?); the
> kinetic energy of the ball doubles (how?); the radius of the track doubles
> (how?); the rain immediately halves the
> coefficient of friction (how?);  the mass of the car doubles (how?).
> All these "hows" are non-trivial.  Certainly they all  convey to students the
> message that the physics they are being taught and being examined about
> has little (nothing?) to do with the real world.

I agree that we could word the situations more carefully.  Instead of "the mass of the car doubles" we could say 2 cars, one with twice the mass of the other, etc.  Overall, the student who is distracted by the "how" questions when we speak of changes in the setup hasn't learned to discern what's important and what is not in solving a problem.  I was discussing with a colleague the silliness that is seen on final exams, and he was lamenting his students trying to find a "use" for every number given in a problem.

Regarding my Kepler's 3rd Law example--that was mentioned strictly as one example of the resistance.  When trying to introduce the concept of proportional reasoning I usually start with eggs and money: If 2 eggs cost a quarter, how much do 16 eggs cost?  Most students are shocked that I would ask that question, because it seems easy, but then they take at least 30 seconds to get a right answer (not all, but most). 

For the Kepler problem, T^2 = ka^3 where k depends on the orbital center, I show them the derivation for a uniform circular motion orbit, then explain that later some of them will do the elliptical problem, but that the result is the same.  I explain that there are two or more ways to do the problem, but if you can do it with proportional reasoning, you have a powerful tool that makes the job easier.  I WANT them to use proportional reasoning, and the Kepler problem is fresh on their minds. I TELL them multiple times they will use PR every day even if they never do another derivative or integral. (Even if they don't use proportional reasoning for this propblem, they still don't need the mass of the <name your orbital center> because eventually that would disappear in the algebraic manipulations; PR just makes it disappear faster.)  But most students are number-crunchers (especially engineering students) and they simply have a hard time avoiding plugging in numbers...I'm trying to break them of that, too, by NOT supplying that mass. 

Short summary: The Kepler's Law problem was a fresh example of the resistance to using proportional reasoning. They were required to know the proportion law, just like they are supposed to know that acceleration is prop to net force.  Many students are going to forget that, too, but I surely don't want students to forget it for a physics test.

I agree that PR and algebra go hand-in-hand.  PR is just a special case of algebra where the "y-intercept" is zero, and even it's not zero, PR is still useful and powerful.
Ultimately, one tool of a functional and educated person is the ability to do proportional reasoning. And regarding those who refuse to learn it, Mama said if you can't say something nice ...