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*To*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Subject*: [Xmca-l] Re: units of mathematics education*From*: Ed Wall <ewall@umich.edu>*Date*: Sat, 1 Nov 2014 21:12:32 -0500*In-reply-to*: <CAGaCnpxO8HevPHmiudEsfmx+0G_48Bpc_-fJppLn4Lq4m6uLyw@mail.gmail.com>*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$@att.net> <C5E992C0-FEFD-4E41-8730-965752C16996@manchester.ac.uk> <54496BA0.6080104@mira.net> <803A57FD-ED25-490E-847E-71B4E2490A63@umich.edu> <5449BC4B.1020504@mira.net> <15A0050C-24E2-4D3D-A4BD-7C8FBE47F907@umich.edu> <5449F0B6.5040902@mira.net> <6073CDA7-B612-4CC2-AB79-312CE63F78BB@umich.edu> <1414280432557.55592@ucdenver.edu> <700B31E6-4D18-43A1-8357-47B8EAF5D08F@umich.edu> <1414337487568.10699@ucdenver.edu> <66AA7EEB-1F34-485D-9227-5F8EB31A56F8@umich.edu> <ced5ad95d0ae4d14ae2db1df8a8cd26e@SN2PR0601MB798.namprd06.prod.outlook.com> <4E081611-A3BD-46F3-AE79-BB41E08CCF87@umich.edu> <544DAD02.9070005@mira.net> <95546616-8723-4803-A0D9-72ECAF4F5143@umich.edu> <544DCA93.1050502@mira.net> <A20F993F-947D-4800-B5A2-896CAB22007A@umich.edu> <5452D88F.3090401@mira.net> <012a01cff51c$c6d1c780$54755680$@att.net> <CAGaCnpxO8HevPHmiudEsfmx+0G_48Bpc_-fJppLn4Lq4m6uLyw@mail.gmail.com>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>

Larry In the early grades most of the symbols are figural (1 especially although I've seen teachings treat the other recounting numbers in a figural fashion) as are things, for instance, like vertical addition. Perhaps you had something else in mind? Ed On Oct 31, 2014, at 11:55 AM, Larry Purss wrote: > Peg, Andy, Ed, and Martin > > This thread is exploring a theme at the edge of my ZPD but I do *sense* the > central key theme being pointed to. In this spirit I want to link up a few > ideas. > Peg, you wrote: > 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. > > Peg, this indicating *models* do NOT need to have a concrete world to *rise > to* > > I want to link this insight to Martin's notion of *modal simulation* as the > *act* of *seeing AS* > > NOT rising to the concrete and *modal simulations* [modes? models?] and > the notion of *figuring* as *thinking* through *images*. > THIS transformation from Davydov's actual concrete relations of [ = ] to > the level of *figural* simulations that ARE GENERAL and *transcend??* the > concrete actuality. > > Now I will move to *units of analysis* as Vygotsky wrote about this notion > in "Thought and Language". > > This TYPE of analysis [ of units] shifts the issue to a LEVEL OF GREATER > GENERALITY looking NOT at elements but at units. The unit of [verbal > thought] is the unit of analysis to study the interfunctional development > of the RELATIONS of thought and speech. > > The question I am asking is where the *relation* of the *figural* > [diagrammatic] *thinking* links with the unit of *verbal thought* and > moving to higher levels of > generality. The *figure* of the triangle or circle AS examples of THIS > figural level of thinking. > > My question now moves to Kant and his notion of *schemas* as transcendental > images and Peirce's engaging with THIS same level of schemas but NOT > seeing the *figural* as transcendent. > > I may be linking too many thoughts together [chaining] but I am circling > around this question of the relations of the > 1] actual concrete situational experiences > 2] verbal thought as a unit of analysis > 3] schemas as figural > > This movement of becoming more GENERAL and mathematical figuring as an > example of this continuing development of GENERALITY as showing how : > > "The mathematics operations + and - grow from these relations among > relations" > > [as an aside I am purposely using Umberto Eco's term *figural* to indicate > THIS form of *seeing as* [modal simulation??]. > I sense *figural* may be a term that is KEY to this thread on *seeing > relations among relations* > Figural as key to interfunctional movement towards greater generality > [beyond the sensual but not transcendental]. > Peirce returned to Kant's notion of *schemas* in his re-search for a > non-transcendental DIAGRAMATIC TYPE of knowing. [the triangle is an > example of Peirce's exploring this TYPE of logic]. Umberto Eco's notion of > the *figural* engages with exploring Peirce's relation to Kant's schemas. > > Vygotsky's unit of analysis [word meaning] also is moving in this > realm towards greater generality beyond sensual experiencing. THE FIGURAL > and its relation to word meaning seems central to this movement of > generality as modal simulation. > > I apologize if I am zigzagging as I try to connect the dots between the > concrete and the general > Larry > > > > > > > > On Fri, Oct 31, 2014 at 8:10 AM, Peg Griffin <Peg.Griffin@att.net> wrote: > >> 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/ >>>> >>>> >>> >> >>

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