It seems to me that differential geometry is a topic which is especially
well adaptable to teaching from a pedagogical-historical point of view,
while, say, abstract group theory is not. I have invented the term
pedagogical-historical to describe the kind of history one might use in
teaching a course which is basically a mathematics course, and not a
specialized course in the history of mathematics.
For example, in teaching the theorema egregium of Gauss, one could talk
about Gauss's motivation for developing a theory of surfaces, say something
about the mathematical environment in which he was operating, something
about Gauss as a person, etc. This would not involve the student or
teacher -- or teacher! -- in reading the original work of Gauss in any
language, the original or a translation. Just something from a standard
work on history of mathematics would be sufficient. The main thing would
be to deliver the historical material with some reverence and enthusiasm.
Come to think of it, maybe that's the way the "purely" mathematical part
should be delivered, too?
But something like abstract group theory, or even linear algebra, seems to
me to have such diffuse and diffused historical roots that it would be
harder and too much off-course to try to use an historical-pedagogical
approach. (I've decided that my term pedagogical-historical commutes.)
Gordon Fisher gfisher@shentel.net