Two key perspectives in math exposition are:
looking forward/looking back

More precisely:
- the perspective of someone just learning the material,
- the perspective of someone who knows the material and much more besides.

worm's eye VS bird's eye (god's eye? ;-)
student VS teacher

One should be aware of both, but not pander to one at the exclusion of
the other.
A totally "student-oriented" exposition is easy to follow like a mine
shaft is easy, but leaves you lost at the end: where are we? where
have we gone? what's the point?!?
A totally "teacher-oriented" point is fun for the teacher, but leaves
the students lost and disinterested.

The goal should be to bring students to the teacher's level, and
beyond! (or at least show them -how- and -where- to go beyond)

student-view:
- no perspective
  For a student, often what they are learning -now- is the -most-
  important thing in the world. Not a bad thing (you should care about
  what you're learning!), but you can lose perspective and spend too
  much time on unimportant things, to the exclusion of valuable
  things.
  (e.g., have you ever cared if a space is lindeloff?
   semi-locally simply-connected? (yes, this latter is useful in
   understanding $\pi_1$, but not elsewhere)
   Cauchy mean-value theorem (outside of intro calc)?)
- often close to "historical"
  ...that is, questions you ask when you "follow your nose" are
  generally the questions people -did- ask back then -- and the
  questions asked back then often -feel- natural to students
  ...but beware of limitations of "historical" approach: it was the
  great mathematicians of the day asking these questions, so
  historical questions are often rather
  difficult/intricate/technical/sophisticated
- "intuitive"
  proceeds -forward- from what they know; don't have

teacher-view:
- perspective
  The teacher knows (or should know!) what material being covered is:
  - important for the future/applications
  - an interesting result on its own
  - scafolding:
    useful for understanding the theory, but not useful on its own
  ...also, which avenues lead to interesting math, and which are more
  sterile.
  (e.g., Abel summation leads (eventually) to Tauberian theorems, and
  a nice proof/understanding of the prime number theorem...but this
  takes a while)
  (e.g., asking "on what sets can a function fail to be
  differentiable" leads to set theory, which is interesting, but the
  actual answer is not too interesting -- but answering it leads to
  interesting stuff!)
  likewise, the complex numbers are much more important than the
  quaternions: 19th century American mathematicians especially
  over-valued the quaternions -- for now, the quaternions are perhaps
  best seen as an important example, on the edge of usefulness:
  still useful (e.g., $\Sp(2n)$, spin, etc.), but not as
  key/mainstream as $\bC$
  (the octonions seem to be best seen as "sporadic", in the sense of
  exceptional Lie groups etc., to which they are realated)
- unmotivated
  ...often a teacher knows that a result is important for later study,
  but it feels unmotivated to the student, as it's not something they
  care about (yet!).
  This should be addressed from both sides:
  - the student should trust the teacher, and study the material
  - the teacher should make the material compelling, rather than
    presenting it as "learn this or else"
    ...but this doesn't mean that it needs to be "relevant"
    (e.g., the relevance of group theory to a calc student is unclear,
     but you can give heuristics as to why it should help answer
     questions about polynomials, and how it's intrinsically
     interesting (in the case of platonic solids, say...))
- organized
  teachers should give -some- outline of a theory, some coherent
  picture (this is the biggest failure of calculus instruction: it
  presents a bag of techniques, as in a "Solving ODEs" course),
  which is something students have trouble doing on their own.
