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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] Where is thinking*From*: Jay Lemke <jaylemke@umich.edu>*Date*: Sat, 25 Apr 2009 15:54:56 +0200*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=jZQLvS+sIQN5OQFoHfktRfVYcDtS5VGUMeU6/unmakDyM/Ya8MdXVUF6ZPdk5YGft Yh6tKXYqJCEltMwOfR3JA==*In-reply-to*: <731CECC23FB8CA4E9127BD399744D1EC01847FE5@email001.lsu.edu>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <C6179756.1F6AE%packer@duq.edu><545862A6-9A98-43DC-9180-E09CE262AFA4@umich.edu> <49F26650.8010503@mira.net> <731CECC23FB8CA4E9127BD399744D1EC01847FE5@email001.lsu.edu>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

Simplicity, like beauty, is indeed in the eye of the beholder. JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On Apr 25, 2009, at 6:18 AM, David H Kirshner wrote:

How can you explain to someone the beauty of Goedel's proof of hisUndecidability Theorem? ... How do you communicate the beauty ofthese symbolic practices?I teach a course called The Culture of Mathematical Problem Solvingin which I seek to enable math teachers to connect with the cultureof their discipline. The mathematical dispositions I seek toinculcate include not only mathematical aesthetics, but also problemsolving heuristics, appreciation of logical nesting and paradox, andthe like. Dispositions, in my terms, are culturally specific formsof engagement (with problems and artifacts, with others, withoneself).In my own typology of pedagogical methods, I rely on two that arerelevant to inculcation of valued dispositions: "enculturationistpedagogy," and "acculturationist pedagogy." Enculturationistpedagogy involves the nurturing of the targeted dispositions withinthe classroom microculture. The teacher, using this method, needs toknow the target dispositions, as well as have a sense of theirdevelopmental precursors. The teacher supports the gradual evolutionof those dispositions by surreptitiously encouraging increasinglysophisticated forms of engagement. Students learn through theirenmeshment in the classroom microculture, rather than through anykind of explicit learning process. As more sophisticated forms ofengagement become normative in the classroom culture, the teachermoves on to the next level.Acculturationist pedagogy is appropriate for students who areidentified with the reference culture (in this case, mathematicalculture) and seek to acculturate themselves to it. The teacher, inthis case, needs to be--and to signify as!--a member of thereference culture. Instruction consists primarily of modeling thevalued forms of engagement for students who are eager to emulate theteacher's cultural practices.Combining these two instructional forms in my course, we spend agood part of the semester enmeshed in solving puzzle problems frommy personal collection. These non-technical problems embody specificdispositional characteristics, which we reflect upon collectivelyand individually. At the same time we are reading from thephilosophy and psychology of mathematics (e.g., Lakatos; Poincaré),interviewing mathematicians about their experiences, and the like.In this way, we have the opportunity to engage in mathematicalcultural practices even as we reflect on our own developing culturalorientations with respect to mathematics.David Kirshner -----Original Message-----From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Andy BlundenSent: Friday, April 24, 2009 8:25 PM To: eXtended Mind, Culture, Activity Subject: Re: [xmca] Where is thinking I've never heard this discussed before Ed. Very interesting. How can you explain to someone the beauty of Goedel's proof of his Undecidability Theorem? Like with Euler's equation and so on, it is one thing to give the "easy" explanation which appeals to intuition, but the mathematical formalism is something else again. How do you communicate the beauty of these symbolic practices, other than taking people through years of routinized practice exercises? Andy Ed Wall wrote:Martin and AndyThis is interesting (and an experience I had also withShrodinger'swave equation for a hydrogen atom although I was lured away by education) as I am trying with some of my students (who, so they believe, have neither strong interests or abilities in mathematics -elementary school teachers presently or on the way) to develop asenseof beauty within mathematics. Part of this is because theirstudents dohave sort of a sense and part of this is because I have beenwonderingif I can crank it up a notch (not all the way to tensor algebra -smile)and also, partially negate, their, these teachers, abominable mathematical experiences. I have begun to have a little success as without coaching (and an hour or so thinking, talking, and working) they seem to be able to distinguish on their own between two or so acceptable proofs - ones they, for the most part, understand and generate - as to the one thatsomehow is elegant (whatever that means although I happen to agreewiththeir choice). Assuming that taste is cultural (although there arewaysin which mathematics, one might say, isn't. I don't mean by thisPlatonic), I've been sort of bemused by the response. Anyway, itseemsthat Vygotsky would have been interested in 'intellectual' taste. Ed On Apr 24, 2009, at 5:20 PM, Martin Packer wrote:As an undergraduate I was in a class in which we solvedShrodinger's waveequation for a hydrogen atom (the simplest case, and I think theonlysalvable one) using tensor notation. I can confirm that it isbeautifulmathematics, and it almost prevented me from becoming apsychologist.Martin On 4/23/09 10:32 PM, "Andy Blunden" <ablunden@mira.net> wrote::) Yes, Ed, I found tensor calculus a genuine thing of beauty. After learning about e^ip=-1 a couple of years earlier only ijGlm=0 could top it (excuse lack of sub- and superscripts and Greek letters). But it is not so much the mathematics that is at issue I think, when someone says "relativity is simple" but just how the mathematics is related to experience. Einstein himself wrote an introduction to the Special Theory which does the whole thing up to the variation of length with relative speed, without using mathematics. But tensors are a mathematics whose object is not physical relations, but differential equations. That's tricky! Any way, it's a long time ago for me too! Andy e^ip means the base of natural logatrithms raised to the power of the square root of minus one times the ratio of the diameter to the circumference of a circle, and it = -1 Beautiful. In ijGlm , G is a tensor of space-time, ij are subscripts and lm are superscripts. But I may have that wrong! Ed Wall wrote:Andy It has been quite awhile since I have taught a course in special/general relativity (about 20 years); however, the tensorcalculus is, I thought then, a nice way to go about it andbrings somethings to light that are important on the way to generalrelativity.Tensor algebra is actually somewhat straightforward by the way,butthatis a matter of opinion. However, all of this has now becomeperhaps abit off topic (smile) and you are correct that specialrelativity doesnot, at a certain level of understanding, require manipulation of tensors. Ed On Apr 22, 2009, at 10:40 PM, Andy Blunden wrote:Yes and No. I was using the word "metaphysics" in the wayPragmatistsuse it. Strictly speaking, of course, *all* thinking containsmetaphysical assumptions. So in that you and Kuhn are right andI waswrong. Perhaps I could stop using the word Metaphysics to mean thereification of thought forms into independently existingsubstances,and others stop using the word Ontology to refer to personalidentityformation? :)But I disagree with you about your Kantian conclusion that"science isa purely logical". It was this Kantian belief (along withEuclid) thatwas overthrown by Einstein. The Logical positivists were wrong ofcourse, because they interpreted the subject in Kantian terms,as anindividual person and their private psyche having direct accesstoeternal reason. Interstingly Einstein disagreed with Bridgman. Einstein said thatwithin the context of a consistent theory, not every entity inthetheory has to be subject to an operational definition. Einsteinright,Bridgman wrong. But I think Bridgman got the right ideanonetheless.Where Hegel and you are wrong, I believe, is the presumptionthat weare at the end of history (neither of you claim that of course,but itis a valid implication in both cases.) If the nature of timeand spacecan be deduced completely from a critique of the culturalpractices atany given time, e.g. in 1807 before the Michaelson-Morleyexperimentwas possible, then obviously the practices whose critique willallowthe Special Theory of Relativity to be deduced "by logic" i.e.,critique of practice, are impossible. If "science is a purelylogical"then that presumes that no further significant developments insocialpractices (such as the Michelson-Morlet experiment) can be made.BTW Ed, I think we have to treat the Special Theory and theGeneralTheory differently. There is absolutely nothing simple about the general theory and its tensor calculus! Andy Martin Packer wrote:Oh Andy, I'm going to have to disagree with you once again!At least, I'm going to disagree if by your statement here youmean tosay that Einstein was avoiding metaphysics. That was the interpretation thelogical positivists made, arguing that Einstein had exposedthe factNewtonian physics had hidden metaphysical assumptions, butthat, withhis operational definitions (Bridgman's term, but his ilustrations were from Einstein), Einstein had finally showed that science was a purely logical (orif you prefer practical) activity, free from metaphysics. Whata messthat has led us into! I'm on Kuhn's side on this issue: every scientific paradigm has metaphysicalassumptions embedded in its practices. So we don't havemetaphysicson the one hand and practice on the other. We have alternative kinds of scientific practice, each with their metaphysical assumptions. (The metaphysics ofEinsteinian physics include the assumption that space issomethingthat can be curved by a mass, for example.) The merits of each of the alternatives is what scientists spend their careers hotly debating. Even what *counts* as metaphysics is different from one paradigm to another. But that's probably what you meant! :) Martin On 4/22/09 8:17 PM, "Andy Blunden" <ablunden@mira.net> wrote:All Einstein did was, instead of regarding time and space as metaphysical entities existing independently of human practice, he closely examined the practice of measuring time and distance. That's all._______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca-- ------------------------------------------------------------------------ Andy Blunden http://home.mira.net/~andy/ Hegel's Logic with a Foreword by Andy Blunden: From Erythrós Press and Media <http://www.erythrospress.com/>. _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca-- ------------------------------------------------------------------------ Andy Blunden http://home.mira.net/~andy/ Hegel's Logic with a Foreword by Andy Blunden: From Erythrós Press and Media <http://www.erythrospress.com/>. _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**References**:**Re: [xmca] Where is thinking***From:*Martin Packer <packer@duq.edu>

**Re: [xmca] Where is thinking***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] Where is thinking***From:*Andy Blunden <ablunden@mira.net>

**RE: [xmca] Where is thinking***From:*"David H Kirshner" <dkirsh@lsu.edu>

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