Re: [Phys-L] widget rate puzzle ... reasoning, scaling, et cetera

[John Denker <jsd@av8n.com>]

On 01/02/2015 02:21 PM, Robert Cohen wrote:

> An ohmic resistor of resistance 10 ohms allows 1 A of current to flow
> when a voltage of 10 V is applied across it. What is the resistance
> of the SAME resistor when  a voltage of 100 V is applied across it?

    Aside: As a practical matter, I'd be a careful about
    applying 100 V to any of the 10 Ω resistors I have 
    lying around the lab.  Suggestion: The question would
    work just as well using mA and kΩ instead of A and Ω.

It's a fine question.  I has some characteristics of what
I call "St. Ives" questions.  The defining property is 
that a great deal of the information given in the 
statement of the problem is irrelevant.
  http://en.wikipedia.org/wiki/As_I_was_going_to_St_Ives

Another question of the same ilk is:
  "How much dirt is there in a hole 2 ft wide by 3 ft
   long by 4 ft deep?"

   Answer:  https://answers.yahoo.com/question/index?qid=20080312154842AAC72LA

You can create more questions of this ilk by starting
with a typical end-of-chapter exercise and adding
extraneous information.

=====

By way of contrast, note that the original widget rate
puzzle was *not* of this ilk.  The only way you can get
the "conventional" answer is to assume that all the given
information is relevant, and all other information is
irrelevant.

I call this type of question "a spherical cow in the ivory
tower".  Any attempt to apply common sense or critical
reasoning to this type of problem will just get you into
trouble.

Different games are played by different rules, and I'm OK
with that, within reason ... but it is unfair to students
to change the rules in the middle of the game without
warning.  It is unfair to give them some questions where
they must assume all the given information is relevant,
then suddenly switch to questions where they must not.

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

On the other hand, the resistor question is analogous to
the widget question in the following way:  Both of them
invite the student to apply the wrong scaling law.  Both
of them look like we might be able to apply the raisin-bread 
scaling law  -- just increase everything in proportion --
but we can't.

> Students frequently have trouble with it.

Sure.  This is practically the definition of a "sophomoric"
mistake:  Overconfidence resulting from inexperience.  This 
includes assuming that a rule that applies in one case 
"should" apply in all cases.

> How would you help students who have difficulty with this question?

I'd tell them it is an easy mistake to make, especially
if you are not experienced ... or not careful.

The only cure is to get some perspective on the situation.
There are lots of scaling laws in this world.  Not all of
them follow the trivial raisin-bread pattern.  That is, if
you increase one variable by a factor of x, it does *not*
mean that all the other variables go up by the same factor.

It is fairly easy to construct additional examples that
illustrate this point.  Here's one I cooked up just now:
  There are two samples of helium/neon mixture:  One is
  at STP.  The other has twice as much helium, twice as
  much neon, twice as much volume, twice as much energy,
  and twice as much entropy;  what is its pressure?

(The pattern here is that all extensive variables scale
like x to the first power;  all intensive variables scale
like x to the zeroth power.)

After they've seen 100 examples of nontrivial scaling laws
they "might" stop assuming that every scaling law is trivial.

I would also tell them that if they see a spherical cow
in an ivory tower, they should recognize it as such and
*label* it as such.  That is, if you have to make assumptions
to solve the problem, document the assumptions.

In summary:
-> Not everything scales like raisin bread ingredients.
-> Not every simple question has a simple answer, so
 be careful.  Don't be overconfident.  Check the work.
-> Document the assumptions.