Here's a few "Whys", that is, why we have specific goals.

- algebraic manipulations
Math is -key- to being able to do examples in other classes:
if you can do examples, then the math doesn't "get in the way", and
you can understand the concepts that they're trying to teach.
If you can't do examples/can't do the math, you'll end up wasting time
and energy struggling with it/explaining it.

- conceptual understanding (of derivatives/integrals, say)
Many concepts in other topics are really applied -math- concepts;
if you understand (and ideally -can recognize-; if not on your own,
then with some prodding) the math, the concept will be very easy.

EG: velocity in physics, marginal cost etc. in econ

This is a -serious- problem: in -intermediate- Harvard economics (a
class that -requires- calculus (IE, sophmores and above take it)!),
they spend some lectures explaining how marginal cost/utility just
mean "derivative".
-- and students appreciate this!

If the math department were doing its job, namely students understood
what derivatives were, this wouldn't be necessary, or rather one could
give a -much- abbreviated description, namely:
"Marginal cost is the derivative -- see how it's just the limit of the secants?"

[I don't know if they have this problem at UChicago.]

- meaning of functions
Anyone who does anything quantitative will be interpreting data
and formulae, and ideally understanding the -math- that's going on.

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math reasoning/expression:

math qua liberal art: good for the brain.
(logical thought, formal thought, geometric visualization,
 deductive proofs, abstracting -- these are all -deeply- valuable skills
 that are emphasized in math -- independentally of the actual content)

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Contextual learning

One trenchant criticism of service courses is that people learn better
-in context-. By this argument, physicists should learn math in
physics class, economists should learn math in econ class, etc.

There is -some- truth to this:
to some extent different areas need disjoint topics;
non-astronomers have little need for spherical trig,
while only mathematicians and physicists need $\SO(p,q)$.

For more specialized topics, and for brief topics that work fine in
isolation, math within a course is fine.

OTOH, the idea of disciplines is that many of these ideas (esp. in
linear algebra and calculus) are useful -generally-:
Calculus is just as true for chemists as for physicists or biologists
or economists.

Indeed, here "context" is useful to learn the idea -- even if you
don't care about the context!
As the example of particle (and projectile) motion in calc shows, a
-consistent- and -long-running- context works very well.
A context that's used just for one problem though is a waste:
(as in the rainforest EG that Krantz uses)
the setup costs of the context often outway the -math- understanding
gained (there's other problems too: often the math involved is stupid)

One -can- make the case for purely contextual learning; the main
criticisms are:
- it's antithetical to the university, to disciplines
  A key point of the university is that there are disciplines that are
  valid independent of context!
  To make all learning contextual is to destroy a foundation of the
  university.
  
- it's immature
  It's condescending (and indeed racist and classist, particularly
  towards blacks and poor folk): everyone can and should learn
  abstract subjects, and it's one of the most important things you can
  learn.
  Clearly it's not a great idea to be hardcore with 5-year olds, but
  18-year olds can be expected to deal with this, and indeed flourish:
  less prepared ones will just do so more slowly and need more
  guidance AT FIRST.
  
- the teachers are incompetent
  Ideally, math teachers know their math better than economists do.
  That is, economists know the (-current-!!) economic applications, but don't
  breath math in the depth mathematicians do.
  ...and this is -key- to moving fields forward: having people who
  really know their stuff.

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Note that the above are all -service- reasons (IE, non-math) for
people to do math stuff.
We mathematicians have our own internal reasons for people to learn
math well, which we often understand, at least intuitively.

Basic stuff like: "this is fundamental to differential geometry" or
whatever -- but it's good to clarify what is -actually- useful.

EG, Maxim playing with evolutes for his students is totally indulgent
and doesn't serve them well: it's a useless (but beautiful) topic that
has hardly any mathematical applications and far fewer outside it.
