I want to mention a project I carried out a few years
ago. I wanted to teach the standard differential
geometry course as a course on the history of geometry,
motivated by the classical problems, ordered by the history
as the appropriate material at the appropriate time.
I found that the course worked, the students were able
to do deeper things by the end than in previous courses and we
all felt that we could understand a few ideas in a
grand tradition, motivated, developed, and we could even
say what it meant to call differential geometry Geometry.
I found the contrast of this experience with standard
treatments of the subject a different sort of lesson in
the history of mathematics, that is, how do we teach in
the twentieth century? (See H. Pulte's edition of Jacobi's
lectures on analytic mechanics for Jacobi teaching the history
of the subject as his opening lecture). A misuse of history
or simply a lack of it in our courses and textbooks is more
about the twentieth century than it is about instructors
and authors.
This forum at the turn of the century is about how this
is changing and it doesn't happen immediately.
John