I applaud James Cook for professing his opinion with some eloquence.
As I disagree with him on tone and some points, I (with respect for
his opinion) suggest the following:
Thinking is taught in the first, and every course.
Learning how to apply in some situations is taught in the
first (and every) course.
Learning how to make a proof and how to apply a result is
a skill/talent/goal that develops over time with practice;
both should be considered extensions of thinking.
Methods of analysis do not take a back seat to computations,
nor do computations take a back seat to analysis.
Real analysis is covered when an idea of how it is to be used is
Linear algebra is taught as needed: a course heavy on applications
with some theory (and where to find more theory and proofs when
needed) for those who need to use old tools and develop new tools
for application, and a course heavy on theory and light on applications
(and where to find more problems and possible room for application)
for those who need to advance the theory for the sake of both the
theory and the applications.
Generally the level of courses is better (not higher) because we
give people the information to decide what education will be best
for their intended growth and that of the society they live in.
Even if we don't know the proper direction for society, we should
make several directions available by the courses we teach and the
actions we do.
Main point: who needs a main point?
Rather than respond to the comments point by point, I
choose to explain the rationale and the nature of my
disagreement with the post of James Cook. Understand
first that I don't think James Cook is wrong. From his
perspective and experience, his answer may be as apt as any.
I see his answer as producing people who are
adept at checking and generating proofs, and I take his
"higher level" as meaning "more facility with handling
activities centered around understanding proofs".
Whether he intended this slant on it or not, in my view
this slant is wrong, and I attempt to correct it with
the list above.
Mathematics is not just proofs and understanding proofs.
Mathematics is about pattern, observing, creating,
convincing ourselves, convincing others, and building
something that is both beautiful and useful.
I want a student coming out of a mathematics course
to think "I could do that." That is not just doing
computations, or solving problems, or writing up
a proof of an assertion, or checking one's logic, or
trying to find a flaw in someone else's reasoning.
It is not a game of axioms and propositions.
Mathematics is an activity which both has the purity and
of intellectual play admired and pursued by the Greeks
(and others) of old, and has the power of being applied
by someone who would work in the physical world and
help society through such application. It may be
a pedagogical convenience to keep these two things
separate, but it is a mistake socially and perhaps
totally to do so.
I want a student coming out of a mathematics course to
think that they could imagine or create structures and
realize or apply them, check to see that they withstand
the brutal force of reason and reality, fix or modify
these ethereal visions and either apply them or give them
to someone else to use. Most importantly, I want them
to be able to tell other people about them in a way
So I prefer that education involve teaching the student
to think. Not "do I have to remember to write the
definition this way" or "is it pemdas or pedmas or
samdep" or "what chapters do I have to read the night
before the test". I want them to do mental simulations:
"I have to take a notion of continuity in one dimension
and generalize it to two dimensions. Can I think of
counterexamples I want to avoid so that I can frame
the definition to avoid them?" or "This notion of a
group acting on itself is a little too self-referential.
Can I use group of symmetries to simplify how to write
up this case-by-case combinatorial proof of this other
thing that interests me?" or "My uncle's going to ask
me to help tomorrow in the backyard: can I use any of
this to make things better/easier/cooler for us tomorrow?"
Even if the student will not be a theoretician, she or he
needs to explain and interact with others, and may need
a fragment of the theoreticians perspective. Similarly,
the student may not be an engineer, or customer support
worker, or salesman, or actuary, or one of a number of
other intended destinations, but they will need to know
that others have to apply math, and they would do well
to prepare explanations for those who aren't as adept
at theory. In short, all of the students need to be
teachers to some extent, for themselves as for others. I do not see this as a potential result from James Cook's list. I do see it in mine.
I will stop here.
End Edit: 2015.04.20
Gerhard "Not A Professional Mathematics Educator" Paseman, 2015.04.19