[BUG] Geometric vs parametric calculation

[Martin Jackson]

I think a big part of the issue here is the way textbooks are written 
and the way many students have trained themselves to use the 
textbooks.  (This observation is nothing new; many have recognized an 
issue here.) I know many students who, when left to themselves, are 
only willing to do problems by looking for the "best-fit" example. 
Most textbooks play into this by providing far too many examples. 
Good examples can play an essential role in a well-written text.  The 
style in which an example is written is also important.  A good 
author can take pains to show how the thinking (geometric and 
otherwise) evolves and where choices are made (pointing to other 
possibilities along the way).  This problem is further compounded by 
the existence of "student solution manuals" which provide even more 
examples.  These tend to be worse in that the writing is very terse 
so the solution appears as even more of a template.  As instructors, 
we should not pile on to this mess by providing any sort of solutions 
to exams and other assignments.  We should make sure that exams and 
other assignments are thoroughly evaluated in a way that encourages 
students to "use what they know."

When I got to vector calculus this semester, I tried to guide 
students away from the text we use here.  I supplied some handouts 
but I also assigned problems from the text.  This led to exactly the 
sort of thing Alex describes; a few very good students took up the 
geometric way of thinking while the rest looked for templates.  I did 
not do enough to wean those students away from the templates.  For 
the next time around, I plan to write my own problem sets so that I 
can tell my students to ignore the vector calculus chapter of the 
text entirely.  (I'm more content with the other parts of the text we 
currently use; it is the vector calculus material of this and most 
texts that I find harmful).

Martin



At 12:32 PM -0500 5/5/04, Smith, Alexander J. wrote:
>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".
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