I think that there are a couple of tensions underlying the aim of achieving
democratic approaches to learning and teaching mathematics. One comes from
Jay's note of the dual objects of learning math for understanding and learning
math for application. The other relates to Martin's note, which refers to
what math is considered to be "applied" is a function of the development of
society -- and Paul has mentioned this as well.
I don't think we can discuss these kinds of things well without some
ontogenetic background, that provides a sense of our personal trajectories.
It's ironic that as I am sitting here, I am wearing a t-shirt bought at MIT
that displays the four of Maxwell's equations in integral form. I would prefer
that they were in differential form, which is far more elegant and symbolically
parsimonious. And I know that the prior sentence might be a strange statement
for those who have not done that kind of mathematics. While I was never
considered talented in mathematics, I do love how, through my physics training,
I have learned to use mathematics in descriptive ways, and how that has enabled
me to design beautiful apparatus for physical experiment. My interest shows in
the high school integrated science/math/technology curricula I developed with
others at terc -- definitely my interest falls towards application. Yet what I
would think as applied in a broad sense, as in the application of fresnel's
equation to the creation of optical interference patterns, others would
definitely think is "way out there" in esoterica. A nod to Paul.
http://standards.nctm.org/document/chapter3/index.htm
I think on the one hand there is a tension between the activities of learning
and teaching for understanding and learning and teaching for application.
Drawing upon the nctm standards, seeing what is involved in teaching for
understanding, in seeing the deeper conceptual connections in the patterns of
mathematics, one must spend time on exploring these connections. This might
involve learning to reason through mathematical proof, and treating the issues
of mathematical representation. This type of activity is aimed at
understanding the "internals" of mathematics as a human heritage. In contrast,
learning and teaching for application may involve more of using mathematics to
be more informed in actions involved in engineering, shopping (consider buying
consumer electronics), or even physics. The two different objects of L&T for
understanding, or for application, each involve somewhat different actions, and
with the limited capacities of schooling, they compete for resources of space
and time as well as people and things. What is the proper balance between the
two? -- can I even ponder the answer well without the very students who are to
learn, who would in part constitute the activity system of the classroom, in
front of me?
On the other hand there is the tension in the reproductive functions of
schooling. What is applied now, as Paul and Martin indicate, may well not be
what will be applied in the future. Our increasingly complex and technological
society is morphing, and we grapple with the tensions between present and
future activity, not knowing what exactly the future holds. Looking backward,
at the time the Sherlock Holmes stories were written, few people new about
cryptography and few used it. Now however, with computer networks becoming so
prevalent, there are issues of information theft and vandalism, leading to such
pernicious crimes as identity fraud. Should we be teaching our children not
only base 2 mathematics, but also such things as mod 26 calculations (the basis
for lower case alphabetic (english) transposition cyphers) and mod 256
calculations (the basis for extended ascii set cyphers)? How about RSA
encryption (which my browser is using right now) and its basis in number
theory? Now there is a foray into esoterica!
It is just to say that I don't think there are simple solutions for democratic
mathematics, but that the whole conceptualization of mathematics for all is
fraught with problems, some of which, like the problem of trust that RSA
attempts to solve, are age-old cultural pathogens in new hosts.
Well, so I can't complain without at least offering some possibility -- and
perhaps what I am seeing is that more and more people are learning mathematics
throughout their lifespan. If we instead pose the problem of how do we prepare
people to be best prepared to continue to learn mathematics as they grow old,
then perhaps the problem is not one of pure vs. applied. Perhaps it is one of
Learning III? And what does that mean when it comes to math subject matter?
bb
=====
"One of life's quiet excitements is to stand somewhat apart from yourself and watch yourself softly become the author of something beautiful."
[Norman Maclean in "A river runs through it."]
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