[Phys-L] Re: student difficulties with velocity as a vector?

[jsd at AV8N.COM (John Denker)]

Robert Cohen wrote:
...
> Further thoughts?

1) I'm sure there are more than one "right answer".  Different students
  are having different difficulties.

2) In this thread there have been many suggestions.  These should be
  considered only hypotheses (even though some of them have been stated
  overly strongly, as if they were "facts").

3) The accumulated set of hypotheses is almost certainly incomplete.

4) Even so, we have not nearly enough data to discriminate among the
  available hypotheses.  I suspect collecting more data would be a very
  fruitful direction to proceed.
  1) Can they handle vectors in general, e.g. position vectors as opposed
   to velocity vectors?
  2) Do they know the terminology, e.g. "horizontal velocity" means
   "horizontal component of the velocity vector"?
  3) Are they paying attention at all, e.g. can they answer even the
   most trivial of questions about what is being discussed?
  4) Do they understand about separation of variables, i.e. that the
   vertical acceleration and velocity are independent of the horizontal
   acceleration and velocity?  This is is not trivial or obvious to
   some students.  It is a reasonably direct consequence of the linearity
   of the Euclidean vector-space axioms and the laws of motion ... but
   intro students typically have no clue about "linearity" or "Euclidean
   vector-space axioms".  If they understand vectors at all they probably
   think in terms of direction & magnitude which is a grossly *nonlinear*
   way of describing vectors.  Just because the people on this list think
   of "vector space" and "linear space" as practically synonymous doesn't
   mean the kids do.
  *) etc. etc. etc.

I suggest multiple rounds of testing.  The second round may suggest what
questions need to be asked on the third round.

Returning to item 4:  Personally I'm torn ... I don't know "the" best way
to introduce vectors.  There is a consensus among mathematicians that the
"linear Euclidean space" axioms are "the" way to go, and there is merit in
that approach.  Meanwhile, there is also merit in the "direction & magnitude"
approach.  Each approach is optimal for capturing the physics in some
situations but not others.

One thing's for sure (as "sure" as anything can be in this business) is
that until the kids can see vectors *both* ways, passing back and forth
between the linear idea and the direction & magnitude idea, they don't
really "understand" vectors, and you're going to get weird inconsistent
results when trying to apply vector ideas.