[BUG] Geometric vs parametric calculation

[Smith, Alexander J.]

I mentioned this last semester...and I think it really cuts to the need
for the Bridge Project. 

1. Our textbook (Thomas) first does the surface area differential with
the 1/cos(gamma) formalism (project the surface into a coordinate plane,
etc)

2. Next the textbook does the surface area differential with the
parametric formalism (area of an infinitesimal rectangle calculated from
the cross product of two infinitesimal tangent vectors).

3. The instructor points out that the surface area differential for a
sphere or cylinder or the "cap" of cylinder is easy to write down using
the most basic geometrical reasoning.  This observation is backed up by
textbook examples. 


4. Students get homework problems (verify the divergence theorem for a
region enclosed by a cylinder, etc)...and most are just adamant about
using the 1/cos(gamma) game.  "B" students will go for the parametric
approach, and only a few of the "A" students will use the geometrical
reasoning approach.



This is of course one of the problems the Bridge project tries to
address, as I understand it. In spite of the Group Activities, "use what
you know" and stuff with calculating without parameterizations, etc.
students gravitate to the oddest thing.."close your eyes and use an
equation".