Re: [Phys-l] [math-learn] Re: Free Versus Pedantic Thinking

["Ralph A. Raimi" <rarm@math.rochester.edu>]

On Wed, 27 Feb 2008, Richard Hake wrote:

> ABSTRACT: GS Chandy implied, in his Math-Teach post, that Alexander
> Calandra's 1968 version of the "The Old Barometer Story" carried a
> good moral for math teachers: recognize sound and creative "outside
> the box" thinking rather than only pedantic thinking....

	To me the moral of the barometer story has always been obvious, 
and quite different from what is implied above, and it surely had this 
meaning for us in 1943, when I first heard it while taking an elementary 
physics course at the University of Michigan.  It was not that the list of 
alternate replies to the barometer question represented "outside the box" 
thinking that should be emulated by us in our progress towards the 
practice of science; it was that the exam question was a fatuous attempt 
to make physics "meaningful" by concocting a phone "real-life" application 
of the knowledge that atmospheric pressure diminishes with altitude above 
sea-level.

	The student who gave the "wrong" answer was calling attention to 
this feature of the question by taking it literally:  "How can the height 
of a building be determined by the use of a barometer" (This might not be 
the exact wording, but is one possibility.)  How can it *be* determined? 
Well, if you put it *that* way, I can think of a dozen answers.  But if 
you only want to test me on whether I understand that atomspheric pressure 
declines with altitude, why not *say* so -- instead of larding it with 
"practical value"?

	Another way the question might have been posed by the fatuous 
professor is "How would you measure the height of a building ...?"

	How would *I* do it?  Let me count the ways...

	Or perhaps he would ask, "How would I (your professor, a real-life 
building measurer, one supposes), measure the height ...?"  Aha, a 
mind-reading problem.  Who can be sure what is in the professor's mind if 
he doesn't tell us?  And so we concoct a ludicrous answer, which is all he 
deserves.

	In math this happens all the time.  They ask kids, "Find the fifth 
term of a sequence that begins 2, 5, 8, 11, ..  ."  Well, the answer is 
"2", is it not?  "Two?  no credit!"

	But the sequence is 2,5,8,11,2,5,8,11, ..., where the given four 
entries are repeated endlessly in the given order.  Is this silly?  Maybe 
it is an unusual continuation, but at least it is well-defined, and 
finding the 53rd term is possible given the data, something that cannot be 
done given only the date of the original statement.

	The battle against fatuous examinations is an important one.  The 
lesson of the barometer story, and of the sequence story (which I admit 
doesn't suggest equally imaginative "answers" to my limited mind), is that 
in science and mathematics it should be mandatory for teachers to speak 
precisely, to avoid both ambiguity and what they imagine to be 
candy-coating for the kiddies.  Especially on examinations.  Asking 
the student to read the mind of the problem setter is out of bounds, and 
asking him to concur in the educationistic foolishness of "real-life" 
trappings which are no such thing is infuriating.  Does anyone believe for 
one minute that someone has ever measured the height of a building by 
means of a barometer?

	In the schools sometimes they fancy they are teaching children to 
"behave like a scientist", when they spend hours having them set up a 
silly "experiment" the import of which they can understand in two minutes 
if it is explained to them.  They will learn more by learing to read about 
such things in a book.

	*Children don't read enough.*

	They should know that "real scientists", when faced with a 
problem, do not immediately construct a Parafissotron and take 
measurements.  The first thing real scientists do is find out if that 
problem has been solved by someone else, and how.

 Ralph A. Raimi        Tel. 585 275 4429 or (home) 585 244 9368
 Dept. of Mathematics, Univ.of Rochester, Rochester, NY 14627
      <http://www.math.rochester.edu/people/faculty/rarm/>

     "Algebra is conducive to symbolic reasoning." (PSSM, p.345)