>I agree, teaching math in context using applications is motivating and
>wonderful. However, is the only mathematics worth knowing and teaching
>applied mathematics? Isn't mathematics as a human accomplishment in =
itself
>worth studying? Certainly many other disciplines that we make part of =
the
>core curriculum include topics that are not necessarily "real life" and
>"useful to solve problems".
True enough. However it's possible to have it both ways: =20
have generic, pure math considerations inform your treatment=20
of the applications -- just make sure the applications=20
aren't bogus. Better to stick with "pure theory" than=20
invent pseudo-applications.
=46or my money, a book like Dr. Jay Kappraff's "Connections:=20
The Geometric Bridge Between Art and Science" (McGraw-Hill,=20
1991) is indicative of how we might enrich the curriculum,=20
especially for those not on a precalc/calc trajectory (the=20
non-precalc track can be especially mediocre I think, as=20
it's too often built around the model of a student who=20
couldn't hack calculus -- as if students might not simply
_elect_ to study other topics, even if perfectly able to
do calculus if they wanted to).
Based on sections in this book, I've recently been chatting=20
up continued fractions, recursion, Fibonacci numbers, Pell's=20
series, Roman and Greek architecture, the golden mean,=20
Platonic polyhedra. Someone else brought up a way to do=20
continued fractions using 2x2 matrices, even made a link=20
to eigenvectors. But applications and demonstrations are=20
never far behind.
I think there's a big difference between pseudo-applications=20
(phoney word problems) and getting clear information about=20
how some of the math you're learning actually does get used=20
-- even if you don't go there in practice. Making the=20
connections, showing the relevance, may ultimately be more=20
important than developing a proficiency that has only=20
1 chance in a 100 of ever proving useful to someone's=20
career.
=46or example, I think it's useful to study the history of=20
math, and realize how finding a generic solution to a 3rd=20
or 4th degree polynomial (after finding the quadratic=20
equation for a 2nd degree equation) was a real focus for
a lot of thinkers.
What mathematicians have deemed "prize worthy" is a good=20
hook, helps give students a sense of the timeline. Writing=20
of these matters, Carl Boyer remarks "The solution of the=20
cubic and quartic equations was perhaps the greatest=20
contribution to algebra since the Babylonians... The=20
solutions of the cubic and quartic were in no sense the=20
result of practical considerations, nor were they of=20
any value to engineers or mathematical practioners=20
('A History of Mathematics' pg. 287, John Wiley and
Sons, revised 2nd edition, 1991). =20
What these discoveries did is help further invent our=20
system of algebra, and led to new categories of symbol,=20
for negatives, for irrationals, for imaginaries, with
surrouding formalizations. Having a sense of math as
evolving, ever changing, is part of what adds to its=20
appeal. It's not "fixed in stone" and handed down from
one generation to the next as some unchangeable code.
Maybe some students don't really understand this?
But after giving a talk about such a topic, I wouldn't=20
turn around and drill students in actually chugging=20
through the algorithm for solving a 3rd degree polynomial=20
over and over. To know that something exists, was at=20
one time highly sought and highly prized (is prized=20
to this day), is not the same thing as needing to=20
develop the a new proficiency yourself. Just know=20
what's out there, and appreciate how hard won so much
of it has been.
Along these lines, I think more time on philosophy=20
of math, with some focus on various notations, exercises,=20
can open doors, shed light. A book like 'Philosophy of=20
Mathematics: an introduction to the world of proofs=20
and pictures' by James Robert Brown (London: Routledge,=20
1999) could be used in a sort of overview math course. =20
Here you get actual examples of mathematics (e.g. Chapter 3)=20
mixed with language _about_ mathematics. I think one=20
ingredient missing from math appreciation is precisely=20
this: we don't give students enough exposure to meta-
mathematics, language that uses math as its subject=20
matter (like music appreciation language is about music).
That's a real shame. Even in the early grades, we could=20
be doing so much more in this regard.
>Besides, we all know that the mathematics used in applications is often
>developed using pure mathematics that isn't "useful". One of the =
reasons
>that we have so many contrived word problems is that textbook writers =
are
>trying to meet the call for problems with context in situations where =
there
>are few if any applications.
I'd prefer they just present straight exercises in these=20
cases, without the pretense. Students understand if you=20
need certain interim results before you're ready to tackle
an application. You need div and curl before you tackle
Maxwell's Equations. Fine. But it's OK to start with the
road map, and a simple explanation of the ideas involved,
at the outset.
Like, I have no problem saying to a 9th grader: here's a=20
ballpark idea of what the calculus is like... and then I=20
launch into 20 minutes talking ratios, changes, snap-shots,
summation, limits. You give a feel for the territory, with=20
no pressure to demonstrate immediate competence. Just take=20
the mystery out of it. Make it at least discussable.
I think too much of the curriculum takes a "just trust=20
us" approach, phasing in concepts over several years that=20
might not finally "click into place" until much later.
I think a lot of students are children of students who
don't accept "just trust us" as legitimate. They look=20
back and see a lot of stuff they could have learned but=20
didn't, because no one really laid out a road map and=20
said, this path will take you in this direction, this=20
path in that direction.
>I'm still wondering about intermediate algebra. What skills/techniques
>should we eliminate from this curriculum under the situation in which we
>must work -- our students must succeed in the next math course or in the
>other courses for which this is a prerequisite. Lets get away from the
>theoretical for a bit and talk reality. Next semester, what should I do
>differently?
>
>--Laura
Maybe the realistic answer is "there's nothing much you=20
can do, acting alone". You're trapped in a system as=20
much as the students are.
I'm just wondering to what extent our sense of students
slacking off traces to a greater sense of skepticism about=20
the whole design of this "obstacle course". If we really=20
respected their time and intelligence, would we mock them=20
for not meeting our standards, or would we revisit the
standards ("to change" does not always mean "to lower").
When a student asks "What's this good for?" or "when am=20
I going to need this?" that's often a sincere plea for a
demonstration of relevance. I don't think it's any answer
at all to say "You need this because the next course up
the line requires this as a prerequisite" or "you're
going to need this on the test next week."
Kirby
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