Re: The state of education in the USA.

On Apr 21, 6:23�am, tgdenn...@xxxxxxxxxxxxx wrote:
<snip>
Thanks, this is a thorough, articulate, and objective response.

It is obvious that there is no software that will be able to respond
to a simple input in *all* cases and spit out *any and all* solutions
that we might desire. The question is, where are the useful limits
with respect to what most people encounter in their education or
profession?

And my fundamental argument is that, absent the algebraic skills, the
user will not know if the software is working, and, in extreme cases,
not understand the output.

This discussion always seems to start with the innumeracy of the
general population, (and perhaps detour through something like the
supposed shortage of civil engineers,) but end up with what
*mathematicians* do, and how they do it. Of course Tim, you, et al are
not solving problems that are completely folded into some software
package. If you were, you wouldn't think of yourselves as
mathematicians.

Which is why I and my colleagues are in demand with engineers and
other specialists to model, fix, and interpret the very results coming
from the software and experiments.

That's where the question of semantics or circularity
comes in. The article is an exploration of the capabilities of
(versions of) �Maple as applied to this physical system. Where do you
draw the line and say that "the software is being used to solve the
problem" --- how much human input or 'help' as the author says is
disqualifying? How have you defined "the problem" in the first place?
And to broaden the question, would you say that a civil engineer using
a program that designs trusses to meet input specifications �is 'doing
math'? What about someone using Maple to determine simple integrals?
What about looking up an integral in a table?

Nothing wrong with any of those things, provided one understands what
the results mean. Very few integrals in applications are exact. What
is more common is to exactly compute the integrals associated with an
asymptotic approximation of the 'real' integrand.

As for your main point, how much basic Algebra, Calculus and
Differential Equations did the author need in order to get Maple to
construct the results that you saw?

Your suggestion leads to the dreadful situation in Statistics, as used
in the Social Sciences and Education. Many studies in these
disciplines (I know from experience on graduate committees) come from
taking a survey of some sort, then applying every possible statistical
test (via software), and looking for any 'significant' result, while
ignoring the Bayesian concept that the tests which detect no
significance are themselves providing information. This stems in part
from the Statistics education given to those professionals, who are
not taught the underlying theory, since it requires too much hard
mathematics. They are, however, taught to use SPSS and other packages,
and can get results that are meaningful to them, regardless of
validity, which reinforces their belief that they don't need the
theoretical statisticians.

The point is, I *am* interested in the general population, and all the
people (like the potential civil engineers, or scientists,) who aren't
ever going to be working on the (currently) most intractable problems.
What would make them appreciative of and sympathetic to math and
science? What might encourage them to think that even they might be
able to find useful transformations from verbal or visual descriptions
to numerical, visual, and even symbolic forms? (My idea of "solve".)

This goes back to Euclid's comment to King Ptolemy. Who are you to
demand that the mathematics needed should be 'easy'? What *should*
make them appreciative, rather than hostile, is the simple fact that
the mathematics is both difficult and necessary. As my Chair stated,
he would be happier if people understood their limitations, and went
to an expert for computational help. But human nature pushes people
otherwise. In too many cases, the software gives the illusion that
they understand what is happening and drives people even further
towards the incorrect conclusion that higher mathematics is
unnecessary.

I would suggest that for most of what most people do, this can be best
accomplished if we share with them the productivity tools that are
becoming ubiquitous. �Yes, you guys can offer cases like the pendulum
where Maple et al may not (currently, since there's apparently no
formal proof of impossibility)

Actually, in some cases, there are such formal proofs.

meet *your* standard of utility.
But
obviously such tools are enormously useful *very much of the time*,
else there wouldn't such a variety out there that is pretty
expensive---hardly "the latest fad in math education"!

My problem with what is currently taught (K12 in particular) is that
it is neither fish nor fowl---it is some combination of mathematics
and applied mathematics and busy-work. �I suspect that a real aptitude
for *the subject of mathematics* �manifests itself quite early in the
majority of cases.

I fully agree, as I have said before. Such manifestation seems to
commonly appear in age 9-13. I am all in favour of extracting those
students (with their parents' consent), and working more intensely
with them. I also think that putting ever 'higher' standards (in
principle, but not application), is sheer torture for those students
with no ability. However, those students who do not learn should never
be put into a position (such as management) which requires
understanding of the models and output.

There's no reason not to have an elective sequence
that would serve those individuals, maybe online access, whatever. But
subjecting kids in general who grow up with computer games to peanut
problems is a lost cause, and counterproductive.. If you think about
it, they are already at least using truss-design software when they
play those games, just with different symbols.

The problem, as I have said before, is that students who do not gain
what they need at an early age generally seem not acquire those skills
later. The selection system described above, similar to the one that I
experienced, does indeed foster excellence, but is extremely cruel,
and does not give anyone a second chance.


This is a long post for me, and I hope it is sufficient for other
correspondents as a reply since �I've been a bit busy. I would like to
clarify the matter �of �"numerical solution" though since it seems to
me another example of the semantic point. Restating, the pendulum
problem which I proposed some years ago in a similar discussion to
support my position is as follows: Holding horizontal displacement of
the release point constant, increase �the length of the string from
its shortest point at the horizontal to achieve some arbitrary small
angle. At what length is the period a minimum?

If I now understand your question, the answer is the minimum is
probably when the length is the horizontal displacement.
After all, if the displacement is small *relative to the length*, we
have the asymptotic differential equation

y'' + (g/L) y = 0,

with solution y = A \sin[ \sqrt{g/L} (t-\phi) ]

and period 2 \pi \sqrt{L/g}.

Hence, for this simple model the smallest length which can be modelled
by this simplification will give the smallest period.


I would call plugging in numbers to the series approximation a
"numerical solution to the problem". �I understand that you might
disagree with that, but why exactly? �Are you *defining* solution as
finding an analytic expression for the minimum? Would an expression
tell the physics student more than a plot---which one would be more
'meaningful'?

Again, whence comes that approximation, and what is its domain of
validity? Without understanding those, one can generate all kinds of
seemingly nice plots and solutions that do not correspond to reality
at all. Exactly those questions led analysts like Abel, Riemann and
Lebesgue to correct some of the mistakes made by Newton et al.

-tg
<snip>

.

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