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

[John Denker <jsd@av8n.com>]

On 05/16/2012 04:02 PM, Philip Keller wrote:
> I've never had a student gush over the algebra that leads to delta X
> = Vo t + 1/2 a t^2
>
> But I have seen many happy faces when we explain that the Vo t is a
> rectangle and the 1/2 a t^2 is a triangle, and the formula gives the
> area of the trapezoid on the V vs t graph.  In fact, before we cover
> that, I show them the geometric meaning of (x+y)^2 = x^2 +2xy + y^2.
> This is always met with happy astonishment and annoyance that they
> were not shown this picture in algebra class.

Yes yes yes yes.  This is IMHO a fine example of how to
answer the question that started this thread, i.e. how to
teach the students to think about things the right way.

> My point is that the algebra and the geometry dance together.  Gotta
> do both...

That's one way of saying it.

Even more generally, one could consider this an example of
a *connection* between one idea and another ... in this case,
a connection between algebra and geometry.  Continuing down
this road, one could remark that the ½ that shows up here is
the same ½ that shows up in
      E = ½ k x^2
and
      E = ½ C V^2
et cetera.  Now we have an N-way connection: several different
physical situations, all connected by similar logic.  Also the
logic is connected to the equation and to the geometry.

  The similarities here are deeply rooted in the physics and 
  math;  they are not merely metaphorical, not a Kiplingesque 
  just-so story.

On the other side of the same coin, we can also appreciate that 
in the context of compressing a gas, we dare not write
     E = ½ P V^2 
or
     E = ½ V P^2
because there is not a straight-line relationship between
pressure and volume.

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

Taking this up a level:  Remember the distinction between giving
them a fish and teaching them to fish.

It is 100% good and necessary to hand them the first fish, and
maybe even the first 100 fish ... but you know you have really 
accomplished something when they learn to catch their own fish.

Maybe 1% of them arrive knowing how to do this, and maybe 1% will
never figure it out on any reasonable timescale ... but the other
98% can be taught.  

At some point it is worth addressing the metacognition issue
directly.  Take a step back and say to the students:  See what
I did there?  I went looking for connections.  The connections
are there, whether I point them out or not ... and I will not
always point them out to you.  Every time you hear a new idea, 
from me or from anywhere else, you need to mull it over in your
mind, hunting for connections.

There is no distinction between memory and thought, because
 -- memorization is a thought process, and
 -- recall is a thought process.

Effective memorization depends on thinking about the connections.
This takes time and effort.  Recall depends upon using the 
connections to dredge up the ideas that you need to solve the 
problem at hand.

So, class, let's practice right now:  We have already seen
  ½ k x^2
  ½ C V^2
  ½ a t^2
and the corresponding straight-line diagonal picture.  Let's
see if we can brainstorm up some other examples in the same
family.  Any suggestions?

On the other hand, not every square law is the area of a
triangle.  Can you think of some counterexamples?

This is how the game is played.  Learning how to do this --
how to hunt for connections and disconnections -- is the
most important thing you could possibly learn, far more
important than monkey-shooting or the phases of the moon
or the mass of the proton or any other domain-specific 
factoids.