Re: [Phys-L] effective teaching of scaling/ratio use

[John Denker <jsd@av8n.com>]

On 12/18/2014 12:38 PM, Bill Nettles wrote:

> My quandry is how to get students to actually use it.  I demonstrate
> it in class, I give exercises and show solutions, but when it comes
> time for a test or quiz, about half my students act like they have
> never seen the technique, even when I tell them "use ratio and
> proportion like you did on the homework and classwork."

Question:  Are they paying attention to everything else, just
not paying attention to scaling?  That would be rather odd.
There's got to be more to the story;  we just need to figure
out what it is.  See (*) below.

> What methods have any of you found which are effective in having 
> students grasp the concept of scaling and apply it to a "new" 
> relationship they haven't been drilled on before?

In addition to the various other answers that have been
given, I am reminded of the proverb:  Utility is the best
mnemonic.

I would hope that the eleventeenth time that scaling makes
the problem very much easier it would begin to sink in.

I was recently reminded of the classic "ball and ring" demo.
It is nominally about thermal expansion, but the thing that
makes it interesting is the scaling argument:  The inner
diameter gets /bigger/ along with all the other linear
dimensions when the ring expands.  If you think about the
expansion the wrong way it is seriously counterintuitive, 
but if you think in terms of the scaling the right answer 
pops out immediately.  Just now I added this to my list 
of examples:
  https://www.av8n.com/physics/scaling.htm#main-linear

A similar example concerns thermal expansion of a railroad 
track:
  https://www.av8n.com/physics/scaling.htm#main-earth-belt

There's nothing special about these examples.  There's never
going to be a silver bullet.  Instead there's millions and 
millions of little examples.

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

(*) In most ways, scaling is easier and more directly useful
than a lot of the other stuff that goes on in physics class.

There is however one way in which it is slightly tricky, or 
at least not what people are expecting:  Some students have 
an overly narrow idea of what "algebra" means.  They think 
that algebra is 100% dedicated to "solving  for X".  In
contrast, I prefer to think of algebra as a /language/.

In particular, you could say that "addition is commutative"
but it's hard to explain in words precisely what that means.
OTOH you can explain it very precisely by an algebraic rule 
that says:
	x + y = y + x        for all x and y     [1]

My point is that for some people, the train of thought goes
off the rails when they see a rule like that.  They ask
"what is x"?  They observe, correctly, that you cannot 
"solve for x" in equation [1].  They are slow to realize
that solving for x isn't the point.  The numerical value of 
x isn't the objective.  The *rule itself* is the objective.
The part about "for all x and y" is a subtle but super-
important part of rule [1].

It may be that heretofore the students have mostly been 
given equations and told to solve for x.  Now when you 
say you want them to create their own equation, and then 
not solve for x, that is at least two jumps removed from 
what they are used to.

Among other things, there is an element of originality
and creativity involved in cobbling up a new rule.
Everybody says they want independence and liberty and
freedom and creativity, but for a lot of people, when
they actually get some of that they freeze like the
proverbial deer in the headlights.

I'm not saying this is what the problem is, but it is
a hypothesis you can check easily enough.  See if they
can handle rules like equation [1] that don't involve
scaling.  See if they can effectively use algebra as a
/language/.
  *) Explain "commutative" as an algebraic rule.
  *) Ditto for "associative".
  *) Ditto for "distributive".
    -- does multiplication by scalars distribute over vector addition?
    -- does the vector dot product distribute over vector addition?

This hypothesis fits some of the given facts:  Demonstrating
the process in class doesn't help, if they are still
expecting somebody else to come up with all the rules.

This is part and parcel of my hypothesis that if they are
having trouble with scaling to any nontrivial degree, it's 
almost never the root cause;  it's virtually always a mere
symptom of some deeper problem.

Possibly constructive idea:  If you think they are suffering
from the deer-in-the-headlights problem, then rather than asking
them to cobble up a rule ab_initio, give them a list of rules
to choose from.  I despise multiple-choice tests, and you 
want to move away from this as quickly as possible, but as a
temporary crutch it might help them get started.  By way of
analogy, when you teach somebody to write computer programs,
you don't ask them to start with a blank sheet of paper; you
give them a small working program and ask them to do a remix.

Along similar lines:  You could give them a partially-worked-
out scaling law and ask them to finish it.  For example, the
vertical stiffness of a solid wood beam is proportional to the
width, proportional to the /cube/ of the height, and proportional
to the (x) power of the length.  Find x.  This addresses the
idea of scaling but keeps the students in the familiar "solve
for x" groove.  Again you want to get out of this groove as
quickly as possible, but as a diagnostic and/or as a temporary
crutch this trick may have some value.

Another idea:  Sometimes /graphical techniques/ help.  If you're
looking for a power law, plot the data on log/log paper.  The
power law will jump out at you.  Some students are happy with
the equation per se, and some others can look at the equation 
and visualize what it means ... but the other 98% need help
visualizing things.