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*To*: <ablunden@mira.net>, "'eXtended Mind, Culture, Activity'" <xmca-l@mailman.ucsd.edu>*Subject*: [Xmca-l] Re: units of mathematics education*From*: Peg Griffin <Peg.Griffin@att.net>*Date*: Fri, 31 Oct 2014 11:10:14 -0400*In-reply-to*: <5452D88F.3090401@mira.net>*List-archive*: <https://mailman.ucsd.edu/mailman/private/xmca-l>*List-help*: <mailto:xmca-l-request@mailman.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca-l.mailman.ucsd.edu>*List-post*: <mailto:xmca-l@mailman.ucsd.edu>*List-subscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=subscribe>*List-unsubscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=unsubscribe>*References*: <1414042156116.36175@unm.edu> <54490772.2020805@mira.net> <49BC6E1396C8F94B92715533ED4D20DC1DAE749B@MBXP03.ds.man.ac.uk> <544911FA.3010808@mira.net>, <003a01cfeed5$9e928180$dbb78480$@att.net> <C5E992C0-FEFD-4E41-8730-965752C16996@manchester.ac.uk> <54496BA0.6080104@mira.net> <803A57FD-ED25-490E-847E-71B4E2490A63@umich.edu> <5449BC4B.1020504@mira.net> <15A0050C-24E2-4D3D-A4BD-7C8FBE47F907@umich.edu> <5449F0B6.5040902@mira.net>, <6073CDA7-B612-4CC2-AB79-312CE63F78BB@umich.edu> <1414280432557.55592@ucdenver.edu>, <700B31E6-4D18-43A1-8357-47B8EAF5D08F@umich.edu> <1414337487568.10699@ucdenver.edu> <66AA7EEB-1F34-485D-9227-5F8EB31A56F8@umich.edu> <ced5ad95d0ae4d14ae2db1df8a8cd26e@SN2PR0601MB798.namprd06.prod.outlook.com> <4E081611-A3BD-46F3-AE79-BB41E08CCF87@umich.edu> <544DAD02.9070005@mira.net> <95546616-8723-4803-A0D9-72ECAF4F5143@umich.edu> <544DCA93.1050502@mira.net> <A20F993F-947D-4800-B5A2-896CAB22007A@umich.edu> <5452D88F.3090401@mira.net>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>*Thread-index*: AQH+Xv9Oz+qV8FilwXZNhVM4bIuqNAFVKVrVAnmAF2kCqaEcYQI6O7pSAZkjTlACVEWyKAMwyA2WAhfMtPwCLK4aSAJq0L2sAeCPXCUCKasqNQJtMkNUAhuSfXIB+Wip9wIwBEENAbn9sFwCug8QwAGz5TXYAYFqV8EBOax82wIl17BkmntQcnA=

A small continuation that might help this along: In my understanding of the Davidov mathematics educators, it’s all about the objects (literal cloth strings or the clay etc.). They actually use Alyosha’s string and Borya’s string in their mathematical recording - they just use the letters when the strings get tattered or scarce or too troublesome or they are sick of being slowed down by drawing them so do what grown-ups do and give them names like A and B. And their direct perception of the cloth strings is crucial to using the initial symbols: = ≠ > < and the operation symbols + and -. Order doesn’ t matter for recording symmetric relations among strings (= and ≠ ). Put Alyosha’s string on top of Borya’s or Borya’s on top of Alyosha’s and perception remains the same; it is either = or ≠ no matter the ordering. But digging a little deeper into inequality gets to the non-symmetric relations recorded with the symbols > and <, perception supporting the demand that ordering matters for those symbols. The real cloth strings and the children’s perceptions make it that they CANNOT ever “see” or “feel” that “Alyosha’s string > Borya’s string = Alyosha’s string < Borya’s string.” That mathematical model (*A>B=A<B) DOES NOT have a concrete world to rise to! Instead, the children see/feel/perceive the strings and symbols having a relation among relations: A>B = B<A. Now it’s exciting. The mathematics operations + and - grow from these relations among relations and so on. So in my understanding, the answer of the Davidov mathematics educators to Ed’s question about “equal” would involve the following: The symmetry of equality is known (buttressed by direct percepts of objects in the world) only in the whole system with ≠ and the non-symmetrical relations > and < and the complex relations among their combinations. The cultural value of mathematics for me is not so much the specific answers folks can arrive at. I value two characteristics: On one end is the certainty of the “don’t know-no one can know” reached in some situations and the certainty of “NOT possible mathematical model” in some situations. At the other end is the persistence of mathematicians when they grasp these limits and gleefully set about re-phrasing, re-framing, what -if-ing, and re-presenting to push the edges of what could be known, what could be possible. -----Original Message----- From: xmca-l-bounces@mailman.ucsd.edu [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Andy Blunden Sent: Thursday, October 30, 2014 8:32 PM To: eXtended Mind, Culture, Activity Subject: [Xmca-l] Re: units of mathematics education Let's not let this thread drop, Ed. To my mind, understanding that mathematics is constrained by objective relations, and is not just a social convention, and therefore *reveals* objective relations, quite distinct from relations discoverable by "experimenting" in the world beyond the text, and opens the possibility for students to *explore and discover*. Such an experience has a very different content from that of acquiring a social convention. So I think it is important that the unit of analysis reflect this. Andy ------------------------------------------------------------------------ *Andy Blunden* http://home.pacific.net.au/~andy/ Ed Wall wrote: > Andy > > Nice and important points. Thanks! > > Ed > > > On Oct 26, 2014, at 11:31 PM, Andy Blunden wrote: > > >> Well, I think that if you make a decision that mathematics is *not* essentially a social convention, but something which is essentially grasping something objective, then that affects what you choose as your unit of analysis. Student-text-teacher is all about acquiring a social convention. >> >> Remember that when Marx chose an exchange of commodities as a unit of analysis of bourgeois society, he knew full-well that commodities are rarely exchanged - they are bought and sold. But Marx did not "include" money in the unit of analysis. >> >> Andy >> --------------------------------------------------------------------- >> --- >> *Andy Blunden* >> http://home.pacific.net.au/~andy/ >> >> >

**Follow-Ups**:**[Xmca-l] Re: units of mathematics education***From:*Larry Purss <lpscholar2@gmail.com>

**[Xmca-l] Re: units of mathematics education***From:*Andy Blunden <ablunden@mira.net>

**References**:**[Xmca-l] In defense of Vygotsky [[The fallacy of word-meaning]***From:*Annalisa Aguilar <annalisa@unm.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Julian Williams <julian.williams@manchester.ac.uk>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Peg Griffin <Peg.Griffin@att.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Julian Williams <julian.williams@manchester.ac.uk>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*"White, Phillip" <Phillip.White@ucdenver.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*"White, Phillip" <Phillip.White@ucdenver.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*David H Kirshner <dkirsh@lsu.edu>

**[Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] units of mathematics education***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: units of mathematics education***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: units of mathematics education***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: units of mathematics education***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: units of mathematics education***From:*Andy Blunden <ablunden@mira.net>

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