One of the suggestions from Kratz's "How to teach math" (or rather, an
appendix) is that one should really think about the -goals- of a
course. Once you have clear, detailed goals, the course largely
follows from them -- and disagreements about teaching are often
largely about different goals -- hence why different sections of the
same course can be very different.

Note that if a department -cannot- come to common goals, there's a
pretty serious problem: in a very real sense, people will be teaching
-different courses-, not different sections.

Of course, once one has goals, different people can have different
opinions on how to achieve them and how to test/gauge them, etc.
but -separating- the questions of
- goals
- methods
is -deeply- important.
A -huge- problem that advocates of "reform" have is that they don't
separate their -goals- from their -methods-:
if they make it clear that they have the -same- goals as
traditionalists, but just quibble about -methods-, then they'll get a
fair hearing and much less resistance.
OTOH, if they have -different- goals, then they should make this clear
and discuss (and be prepared to defend) -why- they have these goals.
The nagging doubts that many people feel about advocates of reform is
that they're are -not- putting all their cards on the table:
differences of goals are disguised as differences of methods.
This feels slimey: it -can- be dishonest on the part of advocates of
reform, but it can also be unintentional, implicit.
Further, defenders of traditional methods would be well-served to
defend their -goals-, rather than just their methods:
methods are much more (potentially) fluid: whatever works
goals are presumably much more stable:
  other than reducing the value of manual calculations
  and proving new tools for experimentation and numerical analysis,
  technology has not changed the fundamental goals of math,
  which is -understanding- (of the math) and -rigorous thought-

Some of the explicit goals of -some- advocates of reform are the deep
cause of the rightful resistance to it, including:
- dumbing down
  that is, discarding content and lowering standards so everyone can
  pass

- lobotomizing
  
  I distinguish this from general "dumbing down", as it is different in type:
  rather than reducing what's required but keeping priorities the
  same, this removes key aspects.
  [It really is like the difference between the same person, only
   dumber, and a person who's been lobotomized but can still do fancy
   feats.]
  
  In particular, critical thinking is often axed.
  (sometimes on racist/sexist grounds, like "blacks don't think that way",
   which is bull, or "blacks shouldn't -have- to think that way",
   which keeps blacks down: math thinking is good thinking,
   and makes you rich, powerful, and fulfilled, both individually
   and as a group)

  EG, one can teach a dumbed down class that still requires thinking,
  albeit at a low level, and one can teach a -hard- class that's
  lobotomized, like requiring very complicated and worthless
  computations.
  
  That said, these are often combound.
  
- cost cutting
  People are expensive: the skills to be a good teacher, especially a
  math teacher (including mastery of math, mental organization, quick
  wits, good communication, empathy) are very saleable.
  Thus many advocated "reforms" are cost-cutting measures in disguise
  (submarines), trying to teach more students with fewer teachers.

  Efficiency is a good goal, but naive cost-cutting (cutting meat as
  well as fat) is (literally) suicidal.

  Key problem: costs are easy to measure with numbers, understanding
  much less so -- hence (stupid) people gravitate to numbers, while
  it can be quite difficult to defend something w/o numbers (however
  meaningless) to back you up.
  [To fight against this, have some ways to measure understanding,
   or at least get some numbers (however meaningless) out.
   EG, exit interviews, or maybe having client departments
   (physics, engineering -- meaning the professors) evaluate how
   well the math department is meeting desired goals.]

(note that large lectures (supported by TA sections) are probably the
-cheapest- way of teaching, short of not having teachers -- thus
suggested reforms can also be ways of -increasing- demand for
teachers!)

OTOH, some traditionalists have goals like:
- reinforce authority, esp. of elders/experts
  (learning as knowledge, passed from generation to generation)
- ego trip
  (I'm so much smarter than you)
- waste as little of my time as possible on teaching
...which are similarly vile,
and are often the -real- object of attack of advocates of reform:
the terms "sage on the stage" vs. "guide by the side" resonate b/c
arrogance is obnoxious and humility is nice.
That lecturing is -susceptible- to being an ego trip is a danger that
-should- be acknowledged, but which does not make it worthless.
(any more than knowing that driving is dangerous should be known and
borne in mind, but doesn't mean no-one should drive)

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Ultimately I should make this an outline (so easy to just look at
headings, etc.), with detailed elaboration, and (in very detailed
version) -motivation- for -why- certain things are goals.

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specific goals for calc class:
(this is a bit of a laundry list, as befits an outline;
 I've tried to highlight key points with ***)

+ algebraic skills
  - be able to apply derivative rules:
    + linearity
    + Leibnitz
    + chain rule
  - be able to take derivatives of
    (these are all EZ)
    + elementary functions
      mostly meaning rational functions, plus radicals,
      in x and exp/log/trig
      [NB: few people bother taking derivative of complicated
       algebraic functions, like ``a y s.t. p(y)=0, where
       p(y) is a high degree polynomial'' -- but this is
       damn easy by implicit differentiation.
       OTOH, it's hard to have a good feel for such functions,
       at least at first.]
    + functions defined by integrals
    + formal power series
  - application: min/max
    + and also constrained min/max,
      EG, Lagrange multipliers
      (but not linear programming/quadratic programming)
  - higher dimensions
    + take derivatives of R -> R^n
      (as paths/velocity)
    + take derivatives of R^m -> R
      (as partial derivatives)
    [Not here, but in a several variables class:
     + Jacobian (should do here, but need linear algebra
       for this to make any sense)
     + applying chain rule in several dimensions
       (now requires matrix mult!)
     + what else?]
  - some integration:
    + parts
    + u-sub
    + partial fractions
    [but -not-: trig subs, reduction formulae -- except
     maybe to see that we can reduce sin/cos to rational functions
     via tan]
  - convergence of series
    + convergence tests
    + computing radius of convergence
  - some ODEs
    + separable
    + homogeneous linear?
    + solution as power series?
+ understanding of functions ***
  - specific functions
    + polynomials, rationals
    + elementary transcendental
      - trig **
      - exp/log **
      (- hyperbolic trig)
  - recognizing concepts as EGs of functions
    + sequences as functions
    + characteristic functions of subsets
  - behavior of functions from a graph **
    + max/min
    + dynamics?
  - -meaning- of functions ***
    as I say, "make the equations sing"
    (tricky to state what this means)
  - be able to translate between: ***
    + formula
    + graph
    + table of values
  - functions as first order objects [this is -hard-] ***
    + equality of functions
      (-esp-: identically zero vs ``having a zero'')
    + sequences of functions
    + function spaces
    + maps of function spaces/operators on functions
      (especially linear ones)
      - evaluation at point
      - multiplication by fixed function
      - composition by fixed function (not linear!)/precomposition (linear!)
      - diff/int as operators
    + functions as -solutions- to equations (esp. diffy q, but
      also $f^2 = x$)
+ understanding of limits
  - of functions at infinity and at finite points
  - of sequences
  - rigorous epsilon definition
  - meaning of continuity
  - dangers of interchanging limits
+ understanding of derivatives/integrals ***
  - specifically
    + of various functions (EG, power rule, sin'=cos, exp'=exp, log'=1/x)
    + of rules (product rule, chain rule)
    [+ by parts/u-subs essentially formal tricks,
      but have good interpretations: u-sub is change domain,
      by parts...urk...]
  - generally
    + physics (velocity/accel/distance)
    + integral as -signed- area
    + fund. thm (as obvious, and how it relates to formal defn of continuity)
    + as local approximation/Taylor series
      (and how continuous/differentiable/2-times diff
       = approximable by const, linear, quad)
+ understanding of numbers *
  - rationals as fractions
    (and why equiv to repeating decimal; really a pre-calc topic)
  - real numbers as equiv classes of cauchy sequences
    (and why decimals are a special case of this)
  - complex numbers as easy algebra
+ formal reasoning **
  IE, being able to work with abstract concepts, given only axioms,
  esp. for:
  - factorial/binomials
  - convergent series
  - continuous functions
  - continuity as a formal property
  - apply derivative/integral rules -formally-
+ logic ***
  (more basic than formal reasoning in particular complicated cases)
  - implication (if...then)
  - equivalence (if and only if)
  - quantifiers (for all, exists)
  - basic model theory (implication = subsets, etc.)
  - contraposition
  - contradiction
    ...which are -not- the same thing!
    the difference is kinda an aesthetic issue,
    but it's a meaningful difference
  - induction
+ etc.: many more
