Those who know me well know that I like to be provocative and that thinking
about what I write, rather than just agreeing or disagreeing with it, is
usually the response I hope for.
There is a lot to think about if the question is --
How abstract and formal a view of the nature of mathematics should we
present to some/all students at different stages of their educations?
My idea of democratic education is (1) we don't scare students, esp. those
who lack the intellectual self-confidence encouraged in children of
upper-middle class families, off from mathematics or science or other
subjects at an early stage by presenting them in their most abstract or
formal guises; and (2) we provide pathways of access to these ways of
thinking, in this case, about mathematics, for all those who are inclined
to pursue them, without expecting that everyone will do so.
I don't regard the formalist approach to mathematics as intellectually
superior. It is a specialized approach, and a difficult one, alien to many
people's ways of thinking. I do not believe that it is more powerful, or
that mastering it is more empowering, if the goal is doing things that make
any practical difference in the world. It is perhaps empowering as a route
to an extremely specialized occupation with long-term value to society.
There is a mystique about "pure mathematics" (as in the film Good Will
Hunting recently) from which it derives its status. Along with that
mystique goes a lot of mystification and romanticizing. It's a technical
skill that requires an unusual form of intuition to do well, and to
initiates what is done has a special kind of beauty somewhat akin to that
of great music. I would treat its place in the curriculum in much the same
way: some practical fundamentals of music theory for all, some music
appreciation for all, the option to pursue the matter further for all who
are interested, and almost certainly advanced training for very, very few.
The difference in the case of mathematics is that there is a great deal of
"applied mathematics" that can be learned, understood quite thoroughly in
its own terms, and used, probably in a larger number of occupations than
(though perhaps not as well paid as?) "popular music" (if that is the right
analogy). Access to these skills is potentially very practically empowering
for much larger numbers of people, much more widely distributed by social
class and gender, than is now the case. This makes for a reasonable
educational goal.
Kay O'Halloran did a wonderful dissertation and has since published several
articles on, among other things, differences in how math is taught to boys
vs. girls and to upper-middle class vs. working class students. The math
that was taught was in both cases of the "basics" and "applied" variety,
but the more privileged groups were expected to understand what they were
doing, while the others were just expected to do it. This points the way to
what is needed to even out the elitism in much math teaching.
Getting very specific on this, when I say I don't favor an emphasis on math
as a formal system (beyond the basics, such as knowing what a formal system
is and playing with a few such systems in simple terms; or knowing what
"proof" is and a few of the issues around decidability and how you can know
whether you've proven something or not), I am not saying that I don't agree
with the finding of a lot of research along the lines of O'Halloran's work
that what all students do need is to see mathematical objects as abstracted
from, and so applicable to a wide range of, specific applications, and not
to be limited to only seeing them as concrete and instance-bound.
Mathematics does manage to be the most "decontextualized" of our semiotic
inventions, and students should all understand the basic trick by which
this is achieved. But that does not mean studying all mathematics in terms
of formal systems, or studying mathematics and meta-mathematics as a pure
mathematician would.
I am not sure how many people even on this list have much of a sense of
what pure mathematics really looks and feels like. I'd like everyone to at
least have a taste, as everyone should try their ear on a Bach fugue, or
try to write at least one non-trivial computer program in a higher-level
language. It's "educational".
But my basic question was about what sorts of difference to the teaching of
mathematics particular views of the ontology of mathematical "objects"
make? And my provocation was that I don't think they ought to make much
difference at all, because they only really matter at the level of thinking
deeply about formal mathematical systems, and I don't think that ought to
be part of general education -- not because it should be reserved for an
elite, but because it is a matter for those who have decided they want to
devote themselves to a particular specialization.
A final note on the larger issue of intellectual elitism. Subjects like
advanced mathematics and theoretical physics draw their elitist reputations
from the specific cultural theory that to do them well requires greater
intellectual "ability", that not everyone CAN do them, and those who can
are somehow better. In other cultures, the theory is that it is a matter of
work, effort, discipline, and interest/desire, and that if you have the
perseverance and the self-discipline you will master them. All "masters"
are equally to be admired, whatever arts they have mastered. We, on the
other hand, are taught that "dull" kids become auto mechanics because they
can't do anything "harder", and "bright" ones become scientists and
mathematicians because they are just born "smarter". Both the hierarchy of
skills, and the folk theory (legitimated by what is to me a very
unscientific psychology) of "degrees of intelligence", seem to me to be
very historically specific ideological supports of a social system that
recruits the skills it needs, in the numbers it needs, and with the
differential pay-scale required to maximize the profit of exploiting the
labor of all of us while keeping the overall social system relatively
quiescent.
JAY.
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JAY L. LEMKE
PROFESSOR OF EDUCATION
CITY UNIVERSITY OF NEW YORK
JLLBC@CUNYVM.CUNY.EDU
<http://academic.brooklyn.cuny.edu/education/jlemke/index.htm>
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