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

[John Denker <jsd@av8n.com>]

On 05/15/2012 10:40 AM, Bill Nettles wrote:
> I have found very few students who can do proportional reasoning when
> they arrive, and they are very resistive to using it to solve
> problems.  For example studying  Kepler's Laws, having them work
> examples using ratio & proportion in class, I ask them on a test to
> find the period of an asteroid orbiting the Sun at a distance of 3.5
> AU.  They ask me what the mass of the Sun is. I tell them they don't
> need it. They stare at me like a calf looking at a new gate

I would love to see more discussion of this point.

There is a problem here that we should be able to solve.  I
don't pretend to fully understand the problem, let alone know
the solution(s), but I know this is important.

Oddly enough, this topic is not covered in the typical physics
text ... even though it is more important than 99% of the stuff
that is covered.

Some miscellaneous thoughts:

Remark:  It is interesting -- and encouraging -- that even though
they ask for the mass of the sun, they don't ask for the mass of
the asteroid.  That suggests that they have internalized the
principle of equivalence, or something like that.  Not bad!

Question and/or possibly-constructive suggestion:  Students
probably don't consider cosmology important to their daily
lives.  So perhaps when broaching the subject of proportional
reasoning and scaling laws, would they do better if we start 
with something closer to home?  For example:  Consider a 
skateboarder starting from rest at the top of a half-pipe.
Neglecting friction, calculate the gee-force (in gees) he will 
experience at the bottom.  It turns out you do not need to know 
the radius of the pipe.
  Note that the skateboard problem has direct application to
  cars, and has life-and-death application to airplanes.

Another remark:  Perhaps I'm missing something here, but I don't
think proportional reasoning is a substitute for algebra, or in 
any way easier than algebra.  It is *part* of math, and not
even the easiest part.  For example, I don't remember the exact 
form of the Kepler 1-2-3 law.  I don't even try to remember it, 
because I can rederive it when needed.  For me, the effort of 
rederiving it is negligible compared to the effort that would
be required to memorize it verbatim.  However, rederiving it
requires algebra.
  I decided long ago that if I'm going to go to the effort of
  learning something, I will learn things that have the most
  power and generality.  If I know algebra, dimensional analysis,
  the law of universal gravitation, and a couple other similarly
  broad ideas, I get the details of the 1-2-3 law for free.

I am quite aware that it is possible to perform proportional
reasoning without algebra.  Euclid did it and Galileo did it, 
to great effect.  Similarly, I am quite aware that it is
possible start a fire by rubbing two sticks together ... but
I would rather not do so.  There exist more modern techniques,
which are considerably more convenient.

I reckon trying to get by without algebra is a fool's errand.  
High-school algebra is a prerequisite for high-school physics, 
not to mention college physics, and it would be madness to
pretend otherwise.  I am not interested in hearing pseudo-
Piagetian claims that students "can't" do algebra.  That's a
lame excuse.  There's a big difference between "never bothered 
to learn it" and "can't possibly ever learn it".

============

Maybe I'm off base here, but my first instinct would be to 
/not/ require students to remember the exact form of Kepler's 
1-2-3 law.  That seems only fair, given that I don't remember 
it myself.  Similarly, I would not expect them to memorize
the loop-de-loop scaling laws.  Instead, I would want them
to learn that:
  a) There are lots of scaling laws.  They are very
   convenient and very powerful.  They have have been central
   to physics for nigh on 400 years, and will remain so.
  b) If you have never seen the scaling law that you need,
   or if you have seen it but forgotten it, you can figure
   it out on the spot.  In simple cases you can use dimensional
   analysis.  In more challenging situations, you might need
   non-dimensional scaling.
       http://www.av8n.com/physics/dimensional-analysis.htm
       http://www.av8n.com/physics/scaling.htm

=============

Again I recommend picking some random adult non-scientists
and asking them what they remember from physics class.  Most 
likely they will say "Oh, let me think ... I remember there 
was something about monkey-shooting, but I don't remember 
the details."

We can do better than this.  We /need/ to do better than
this.  It is an enormous waste of time and money to teach 
stuff that students will not remember and/or not find useful.