Re: [Phys-l] vector inconsistencies

[John Denker <jsd@av8n.com>]

On 03/30/2007 05:59 PM, Bob Sciamanda wrote:

> I don't see your problem.

I'm not saying it's a big problem, but there is *some* sort of
problem.

Suppose I have two classes, the 10:00 class and the 2:00 class.
I show each of them a figure consisting of a bunch of vectors
radiating from a point:
  http://www.av8n.com/physics/img48/radiating.png
and tell them this is in the context of geography i.e. topography.

The question is, does this figure represent a peak or a pit?

It's tricky, because:

 -- The 10:00 class spent the last week studying electrostatics.
  They assume the vectors are force-field vectors, which point
  downhill, down the potential.  To the extent that the gravitational
  potential is analogous to the electrostatic potential, the figure
  "must" represent a peak, with high potential in the middle and lower
  potential elsewhere.

 -- In contrast, the 2:00 class spent the last week studying
  position vectors and displacement vectors.  Therefore they
  "know" that a positive vector goes /from/ a low-numbered place
  /to/ a high-numbered place.  Therefore the vectors "must" be
  gradient vectors, and hence the figure represents a pit.

NOTE:  There may be an established convention in topography, just
as there is an established convention (downhill) for force-field
vectors, and an established convention (uphill) for displacement
vectors.  The point remains that the students weren't born knowing
all the conventions in all the various fields, and if they have
to guess there is a very real risk of guessing wrong.

> You draw the vector to represent whatever it is that you want to express 

That's 100% true as far as it goes, but it's not the whole answer.
More generally, you draw the vectors however you want, but then you
must *explain* what you mean; the arrows are not self-explanatory.

I'm not saying such explanations are difficult, just that they are
necessary.  There is a two-step process:
  a) The first step is to recognize the ambiguity or potential
   ambiguity.
  b) Then it should be possible to express things in such a way
   as to resolve or avoid the ambiguity.

Step (a) is, in my experience, the hard part.

By way of analogy, ambiguous English sentences are a dime a dozen.
  http://www.ohiou.edu/~linguist/soemarmo/l270/Exercises/ambigs/ambigs.htm

I have written enough inadvertently-ambiguous sentences to know
how easily and naturally they arise.

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

To say the same thing yet again:  a gradient vector is a reasonable
thing, and a force-field vector is a reasonable thing, but they are
not the same thing.  That's the small problem.

The larger problem is that students by nature tend to overgeneralize.
Students (some of them, anyway) love rules.  If you don't give them
a rule, they will invent their own rule.  That's messy, because
every student must pass through a stage where they have learned
some examples but have not yet learned the counterexamples.  During
this stage they are in jeopardy of inventing overly-general rules.
The pseudo-rule that "we always draw vectors pointing uphill" is
just one example among many.

And that's what separates the expert from the student.  The expert
can see the counterexamples as well as the examples, and therefore
can formulate a more nuanced rule.

  FWIW I'm acutely sensitive to this right now, because yesterday
  I caught myself starting to make an inexpert generalization
  about uphill/downhill vectors.

The teacher's job is like the expert's job, only harder.  The teacher
needs to appreciate all the nuances, but for many reasons (including
pedagogical reasons as well as time constraints) it is not possible
to teach all the nuances at once.

Here's my strategy:
  A) Give the students some examples.
  B) Explain how the examples fit into some larger pattern or
   rule, even if the rule is imperfect.  An imperfect rule is
   better than no rule ... and the rule I come up with is
   likely to be better than rules the students come up with
   on their own.
  C) Give the students some appreciation of the /limitations/
   on the validity of the rule.  Doing this without getting
   bogged down in details is the hardest part.