http://en.wikipedia.org/wiki/Panopticon
.... just some notes on Doug's interview of Bill Ayers.
Wow. Great stuff. While I was listening, of course all I could hear were all the evolving issues between education and politics that super-saturated my brain forty years ago...
(among my part-time jobs to get through UCB, was working as an orderly at the student infirmary. It was a perfect model of the panopticon, and the congruency between a medical faculity and the unversity itself was stunning. Although I could see this intitatively and me and one of my political buddies laughed about it---curing our sink minds in revolt---we never formally explored that idea. It took reading Foucault many years later to remember all this...)
Those issues continued to evolve over the years, of course in my case, as intellectual themes rather than practice, since I never got to the working teacher level.
The point of these working ideas in progress were all directed at getting rid of the indoctrination, hierarchy, top down, of what Foucault called the panopticon. Ayers was just great on the theory of how to do that.
(What people who were not part of the era may not realize is that breaking down the established orders included academia itself, was an ever present constant of discussion. The unversity system is no friend to democracy...)
At some point in the late 70s I went on a math binge from problems in sculpture, and made one of the great discoveries of my own intellectual life, which is technically referred as linear algebra and group theory. Because I discovered these systems through the visual arts, I figured out that these contained the potential to massively overhaul mathematics education, and in a sense democratize mathematics. If that could be done in something of a systematic way, it would have a profound effect on developments in the sciences and open them up. This would in turn produce a far more literate and technically competent public, and in effect democratize the sciences and its allied technogies. These developments by turns democratizs the whole technological infrastructure of the society.
And the point? Wrestle the skills, knowledge, and productive capacity out from the grip of the capital technocratic elite, mass distribute them, and therefore start to dismantle the hiearchies of power that hold the political economy together.
Mathematics as a means to liberation? How is that supposed to work? (Now I am going to argue with myself). The first step is you have to re-conceive mathematics as a universe of symbolic forms. It is different than our more familiar universe which is almost entirely based on language. Mathematics does not derive directly from the world of language at all. It co-exists and co-evolves within the linquistic-cultural world. The key point of interface between language and mathematics is not rational thought. (Although you can develop an interface through symbolic logic.) The primary or foundational interface is the development of our spatio-temporal concepts which underly both the worlds of language and mathematics. Mathematics in this sense is a highly technical and systematic expression of spatio-temporal concepts, most of which reach far beyond even the plasticity of language to capture.
So, then by re-configuring mathematics education with this broad concept in mind, it seems possible to design curricula to open the mathematical world. In effect, this is a practical application of many of Levi-Strauss and Jean Piaget's ideas. That is the very bete noir of Foucault's generation, turns out to hold what I think is the key to the liberation Foucault sought. (Big philosophical irony here.)
Here's how that works. L-S and P were right about there being some form of structuralism to mathematical thinking. They just didn't quite get far enough into learning what those structural elements were. (I only barely perceive myself) The point is to use those structural elements to perform `open sesame' magic tricks with them. The trick is to teach and learn how to perform non-linquistic spatio-temporal thinking. It turns out with a careful analysis of higher level abstract algrebra, you can find all the tools you need.
Formally speaking, each of the algebras on sets, groups, and rings have classes of morphisms that take the algebraic representation to a corresponding space representation, often one that can be modeled in plain geometry. So then the symmetries and ordering properties of the algebra, can be seen, apprehended directly, and physically manipulated, played with. This is often dismissed as mere illustration. Not so. If done carefully, these 2-d and 3-d representations contain, in their spatial configurations, a true representation of the abstract forms themselves. There is a deep affinity between the concept of number and the concept of shape, and some elements of abstract algebra reveal some of those affinities to a truly remarkable depth.
The most well know examples are the rigid motions of a square on a plane, and a tetrahedron, in 3-space. In their algebraic representation, these motions correspond to various finite subgroups of the full finite symmetric group of permutations on an equivalent number of elements (letters, numbers, etc). The equivalence is obtained by one to one and onto map between the elements and the vertices and or midpoints (edges). The applications of these simple space models across mathematics and the sciences---and arts---is stunning. Learning these forms are why and how you get the open sesame effect. Here is a short, non-technical essay on why the study of groups is important:
http://www.math.uconn.edu/~kconrad/math216/whygroups.html
Now there is a whole history of this kind of new math teaching, but it has been miserably lacking in understanding the corresponding structures in mathematics, or teaching them. That's because ninety-percent of all mathematics instruction is done by teachers and educators who never got far enough in their own mathematics education to see these structures. And of course most also didn't take their anthro classes, their cognitive science, and any of the other background that would show them how to see and use mathematics in this way.
Again part of the reason for this lack is due to the fact that university mathematics doesn't teach these topics until upper division levels, and more often than not, saves the really good stuff for graduate school. In effect, the university curriculum cancels out almost all people headed into education. In turn anybody with a degree in math can get a much better paying job just about anywhere else than public education. So the whole system is self-reinforcing. The panopticon effect is quite real.
One of my oldest friends teaches eight grade algebra. I have tried for years to get him interested in higher algebra and adapting some of it to teaching at the elementary level. I've run into a brick wall! I tried to convince him for example to use Euclid's Algorithm to find the highest common factor. (For a hint at how far down the rabbit hole the Euclidean algorithm takes you, google it and/or Euclidean domains). But A. said, Forget it, Grimes.
What I discovered in these conversations is that he had his own battles to fight within the existing math curriculum and has used some higher algebra and number theory (via Pascal's triangle) to teach methods in factoring and the set-up method (binomial formula) that leads to introducing the quadratic formula. It turns out that while these ideas, presented as neat games, help the kids in their general understanding, these same games are seen as a trivial waste of time by his peers, especially the feeder high school teachers. What was at issue was a noticable number of his kids did not show proficiency levels in algrebraic manipulation and calculation required to quickly move into the high school curriculum. Of course A. blamed that problem on the elementary level. So it goes round and round.
Here is a very good article on Pascal's triangle:
http://mathdl.maa.org/mathDL/23/?pa=content&sa=viewDocument&nodeId=493&bodyId=685
(Side note. The connection to group theory is difficult to follow. To get from Pascal's triangle to group theory goes through number theory, congruences and modular arithemtic... Historically, that was the original path of discovery of groups via Galois. But a much more simple path came out of Kline's Erlangen program. Hence, that's the road I would advocate...)
For my buddy A., once No Child Left Behind hit the district, that was the end of any discussion on developing broad concepts like mathematical thinking. Fuck it, here's your fractions homework. The panopticon was re-modeled and made far more effective.
The right and especially the neocons have an absolute genius for creating these prisions of the mind. I don't know how they do it.
CG