[BUG] a different perspective

[Tevian Dray]

The following comments are based on a recent "debriefing" of an
instructor and TA, after completion of a course using Bridge Project
materials.  I've included some comments of my own; further discussion
is welcome.

*Parameterization*

In retrospect, the instructor would have chosen to emphasize
parameterization more than was actually done.  This instructor has
no problem viewing parameterization as something you only use if
you have to, but feels that sometimes you have to.

The specific example which triggered this concern was an exam problem on
finding the area of an ellipse.  The problem was worded so as to expect
students to use Green's Theorem, and most students had little difficulty
writing down an appropriate form of the Area Corollary.  But very few
students were then able to parameterize the ellipse, and could therefore
not complete the computation.

(It is an interesting question whether "use what you know" helps here,
since the trig substitution needed to do the final integral is
equivalent to the parameterization.  But it still might be worth asking
whether students are more comfortable with trig substitutions than with
parameterization.)

*Pace*

The instructor also noted that the second half of the course moved more
briskly than the first half, and suggested moving more quickly through
the early material to leave time later for a more complete discussion of
the theorems.

(I pointed out that a traditional course doesn't always have time to do
more than state the theorems, so that I regard the very fact that this
instructor thought there could have been time to do more to be a
feature, not a bug.)

Yes, this early "review" sets the tone for the course (more geometric,
fewer templates, use of curvilinear coords).  But as the TA pointed out,
students need to review this material anyway.  Increasingly, at least at
this institution, students are not taking their math classes "in order",
resulting in gaps between when they take multivariable calculus and when
they take vector calculus, causing many students in our vector calculus
course to be "rusty".  (This effect would not of course be present at
schools which teach this material in a single course.)

It is perhaps also worth noting that the brisk pace at the end may only
be possible /because/ of the extra time spent on the early "review".

*Order*

Several suggestions were made about possible changes to the order in
which the material is covered.  One was to consider talking about the
"ant" derivatives (Lesson 3 of the Study Guide) /before/ discussing
differentials and the gradient.  (My own sense is that this might be
more appropriate in a Calculus III class covering both partial
derivatives and vector calculus.)

Another suggestion was to consider talking about area elements
immediately after discussing line elements, and in particular before
discussing conservative vector fields.  The TA noted that learning about
area elements really helped the students to understand dr-vector, so
that the earlier this could be covered, the better.

On the other hand, this order might affect the central role currently
played by the master formula (relating df and grad f), which is used
both to "review" gradient and differentials (and hence should be
introduced early) but also to motivate the fundamental theorem for
gradients (and hence should now be covered later).

*Curvilinear coordinates*

Finally, the question was raised as to whether the use of curvilinear
coordinates (and vector fields...) is essential to our approach.  I
would say yes -- but perhaps not for the obvious reasons.

I do /not/ feel that one of our goals is to get students to understand
curvilinear coordinates.  Rather, we want students to understand what
vector calculus means, which I like to describe as the geometry of
vector calculus.  I believe it is much easier to develop geometric
intuition by taking advantage of high symmetry where possible.  The use
of curvilinear coordinates is an excellent way to do that, but should
not be viewed as an end in itself.

Tevian