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RE: [xmca] Where is thinking

> How can you explain to someone the beauty of Goedel's proof of his Undecidability Theorem? ... How do you communicate the beauty of these symbolic practices?

I teach a course called The Culture of Mathematical Problem Solving in which I seek to enable math teachers to connect with the culture of their discipline. The mathematical dispositions I seek to inculcate include not only mathematical aesthetics, but also problem solving heuristics, appreciation of logical nesting and paradox, and the like. Dispositions, in my terms, are culturally specific forms of engagement (with problems and artifacts, with others, with oneself).

In my own typology of pedagogical methods, I rely on two that are relevant to inculcation of valued dispositions: "enculturationist pedagogy," and "acculturationist pedagogy." Enculturationist pedagogy involves the nurturing of the targeted dispositions within the classroom microculture. The teacher, using this method, needs to know the target dispositions, as well as have a sense of their developmental precursors. The teacher supports the gradual evolution of those dispositions by surreptitiously encouraging increasingly sophisticated forms of engagement. Students learn through their enmeshment in the classroom microculture, rather than through any kind of explicit learning process. As more sophisticated forms of engagement become normative in the classroom culture, the teacher moves on to the next level.

Acculturationist pedagogy is appropriate for students who are identified with the reference culture (in this case, mathematical culture) and seek to acculturate themselves to it. The teacher, in this case, needs to be--and to signify as!--a member of the reference culture. Instruction consists primarily of modeling the valued forms of engagement for students who are eager to emulate the teacher's cultural practices.

Combining these two instructional forms in my course, we spend a good part of the semester enmeshed in solving puzzle problems from my personal collection. These non-technical problems embody specific dispositional characteristics, which we reflect upon collectively and individually. At the same time we are reading from the philosophy 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 mathematical cultural practices even as we reflect on our own developing cultural orientations with respect to mathematics. 

David Kirshner

-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Andy Blunden
Sent: 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?


Ed Wall wrote:
> Martin and Andy
>      This is interesting (and an experience I had also with Shrodinger's 
> wave 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 a sense 
> of beauty within mathematics. Part of this is because their students do 
> have sort of a sense and part of this is because I have been wondering 
> if 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 that 
> somehow is elegant (whatever that means although I happen to agree with 
> their choice). Assuming that taste is cultural (although there are ways 
> in which mathematics, one might say, isn't. I don't mean by this 
> Platonic), I've been sort of bemused by the response. Anyway, it seems 
> that 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 solved Shrodinger's wave
>> equation for a hydrogen atom (the simplest case, and I think the only
>> salvable one) using tensor notation. I can confirm that it is beautiful
>> mathematics, and it almost prevented me from becoming a psychologist.
>> 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 tensor
>>>> calculus is, I thought then, a nice way to go about it and brings some
>>>> things to light that are important on the way to general relativity.
>>>> Tensor algebra is actually somewhat straightforward by the way, but 
>>>> that
>>>> is a matter of opinion. However, all of this has now become perhaps a
>>>> bit off topic (smile) and you are correct that special relativity does
>>>> not, 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 way Pragmatists
>>>>> use it. Strictly speaking, of course, *all* thinking contains
>>>>> metaphysical assumptions. So in that you and Kuhn are right and I was
>>>>> wrong.
>>>>> Perhaps I could stop using the word Metaphysics to mean the
>>>>> reification of thought forms into independently existing substances,
>>>>> and others stop using the word Ontology to refer to personal identity
>>>>> formation? :)
>>>>> But I disagree with you about your Kantian conclusion that "science is
>>>>> a purely logical". It was this Kantian belief (along with Euclid) that
>>>>> was overthrown by Einstein. The Logical positivists were wrong of
>>>>> course, because they interpreted the subject in Kantian terms, as an
>>>>> individual person and their private psyche having direct access to
>>>>> eternal reason.
>>>>> Interstingly Einstein disagreed with Bridgman. Einstein said that
>>>>> within the context of a consistent theory, not every entity in the
>>>>> theory has to be subject to an operational definition. Einstein right,
>>>>> Bridgman wrong. But I think Bridgman got the right idea nonetheless.
>>>>> Where Hegel and you are wrong, I believe, is the presumption that we
>>>>> are at the end of history (neither of you claim that of course, but it
>>>>> is a valid implication in both cases.) If the nature of time and space
>>>>> can be deduced completely from a critique of the cultural practices at
>>>>> any given time, e.g. in 1807 before the Michaelson-Morley experiment
>>>>> was possible, then obviously the practices whose critique will allow
>>>>> the Special Theory of Relativity to be deduced "by logic" i.e.,
>>>>> critique of practice, are impossible. If "science is a purely logical"
>>>>> then that presumes that no further significant developments in social
>>>>> practices (such as the Michelson-Morlet experiment) can be made.
>>>>> BTW Ed, I think we have to treat the Special Theory and the General
>>>>> Theory 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 you mean to
>>>>>> say
>>>>>> that Einstein was avoiding metaphysics. That was the 
>>>>>> interpretation the
>>>>>> logical positivists made, arguing that Einstein had exposed the fact
>>>>>> Newtonian physics had hidden metaphysical assumptions, but that, with
>>>>>> his
>>>>>> operational definitions (Bridgman's term, but his ilustrations 
>>>>>> were from
>>>>>> Einstein), Einstein had finally showed that science was a purely
>>>>>> logical (or
>>>>>> if you prefer practical) activity, free from metaphysics. What a mess
>>>>>> that
>>>>>> has led us into!
>>>>>> I'm on Kuhn's side on this issue: every scientific paradigm has
>>>>>> metaphysical
>>>>>> assumptions embedded in its practices. So we don't have metaphysics
>>>>>> on the
>>>>>> one hand and practice on the other. We have alternative kinds of
>>>>>> scientific
>>>>>> practice, each with their metaphysical assumptions. (The 
>>>>>> metaphysics of
>>>>>> Einsteinian physics include the assumption that space is something
>>>>>> that 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
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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/>.

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