Historicism

3 topics are taught to kids:
- old
- basics (of a more complex theory)
- hot (often technical details of a hot theory)

What -should- be taught, of course, are:
- core areas
  (fundamental to the field)
- useful (in outside applications)
- future directions
  (to the extent known)

The problem with traditional teaching is:
- old
Old topics are often taught -historically-.
This is particularly a problem in calculus:
people don't use modern understanding or even interpretation:
they simply use -what was known at the time-,
and indeed the perspectives -of the time-,
ignoring centuries of refinement.

- basics
This is a problem in point-set topology:
the basics of topology are easy enough that they can be taught to kids
in a 1-semester course, but you don't get any appreciation for where they come from.

A related issue is the "tools" class, which often teaches a necessary tool -before- it's been motivated elsewhere (or -after- it would've been useful).

[NB: Tom Braden taught simplicial approximation,
 and I did a DSP on de Rham cohomology
 ...these are both EGs of technical tools,
 exactly the -wrong- thing to teach at these levels.
 Instead we should've explored hyperbolic geometry, say.]


Galois theory is an example of a topic taught right, generally:
- it's a fundamental and old question
  (?? you can't solve the quintic ??)
  ...and it motivates some easy field theory and group theory
W/o Galois theory as a guiding light, intro algebra would be like point-set topology (and indeed Math 250 had much that feel; some refinement of linear algebra was the only redemption/organization in 250a, and the hope of algebraic geometry (someday) motivated 250b, but neither had much coherence).
