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

[John Denker <jsd@av8n.com>]

On 05/17/2012 01:00 PM, Robert Cohen wrote:
> You have to keep in mind that a lot of students are looking for ways to
> do physics without actually thinking about the physics.  Finding an
> equation that "works" is what they are looking for, in the same way that
> they are looking for a rule that works to solve the reciprocal of (1/x +
> 1/y) problem.

Quite so!

This is exactly what students have been trained to do.  It
would be silly to expect otherwise.

In particular, the mandatory standardized tests are what I 
call "game-show tests".  They are trivia contests.  Any 
question that cannot be answered in 45 seconds is not worth 
answering.

In college, the teacher can ask "Would you like to learn a huge
number of useless factoids, or a small number of useful principles?"
However, in high school, the only thing anybody wants to learn 
(or teach) are whichever factoids that can be used -- instantly 
and robotically -- to answer the standardized test questions.

>  In fact, I think every derivation one does in class is
> simply ignored by the students we are talking about (problems with
> proportional reasoning, scaling laws, et cetera).

That is a /fixable/ problem.  I see the main purpose of this long
and interesting thread is to figure out how to fix that problem.

Let me outline a possible way of getting started.  This is not
guaranteed to work with your students, but if not you may be
able to adapt it somehow....

First day:

"Let's start today by figuring out how far apart the molecules
are in a sample of ordinary air.  Probably nobody in the class 
knows that off the top of their head.  But we can FIGURE IT OUT.
So ... show of hands:  Who knows Avogadro's number?"  
  Call on somebody.  Write it on the board.  
"Show of hands:  Who knows how much volume is occupied by one 
mole of gas at STP?  This is something you all learned in 
high-school chemistry."  
  Call on somebody.  Write it on the board.  
"Show of hands:  What should we do next?" 
  (Crickets.)  
"How about we divide one thing by the other, to get the number 
of particles per liter."  
  Write it down.  
"Show of hands:  What should we do next?"
  (Crickets.)  
"Maybe we should have done the division the other way around, 
to get the volume per particle.
  Write it down.
"Show of hands:  What should we do next?"  
  (Crickets.)  
"This is telling us that on average each particle has a volume 
of yay much all to itself.  If we have a cube with this much 
volume, how long is one edge of the cube?  Maybe we should take 
the cube root."
  Write it down.
"Now, dear students, the point of this exercise was *not* to
teach you the distance between molecules.  I guarantee that
you will never see this question or this answer again in this
class.  The point is not to teach you physics factoids.  The
point is to teach you physics /style/.  New rule:  Whenever
you don't know something, FIGURE IT OUT."


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

Second day:

"OK everybody, please take out a piece of paper."
  (Wait)
"Ready?  Write down the mass of the proton, in SI units.
What is the mass of the proton, in kilograms.  If you
don't know, just write a question mark."
  (Wait)
"Anybody need more time?  If not, let's move on.  Turn
the paper over, face down.  On the back of the piece of
paper, write down Avogadro's number."
  (Wait)
"OK, time's up.  Let's move on.  I hope you remember the
value of Avogadro's number.  We just used it yesterday.
In fact, it's still sitting there on the chalkboard."

"So I have one more question for you:  How much does a
mole of protons weigh?"

"You've been had.  Given Avogadro's number, you should 
have been able to figure out the mass of the proton.
It's a one-line calculation.  Are you embarrassed?
You should be.  If it makes you feel any better, when
I was your age, I embarrassed myself by on this exact
same question.  I remember it vividly.  I promised
myself to make this mistake again ... not this mistake
or anything remotely similar."

"Note that this has got nothing to do with the mass
of the proton.  I guarantee you I do not remember
the mass of the proton.  I don't need to, because I
can FIGURE IT OUT whenever I want.  It is absolutely
not worthwhile for you to memorize the mass of the
proton.  You would just forget it by the end of June
anyway.  The thing that is important is for you to
get comfortable with the idea that you can FIGURE
IT OUT."

"You can throw away those papers.  I'm not going to
collect them.  I don't want to embarrass you any more
than necessary ... but I do want you to promise me,
and more importantly promise yourself, that from now
on, if you don't know something, you will FIGURE IT
OUT."

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

Third day:

Same gag, but the students work in groups.  They have 
to noodle out some problem that requires combining 
multiple disparate ideas.

Fourth day:

Same gag, smaller groups.

Later:

Same gag, more complex problem.  More steps required.

Yet later:

Figure it out, then figure it out again using a different,
independent method.  Use one as a check on the other.

====================
Bottom line:  

Thinking skills can be taught and can be learned.  It's
not easy, but it can be done, and it's well worthwhile.
  http://www.av8n.com/physics/thinking.htm