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[Xmca-l] Re: units of mathematics education
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
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
Peg, this indicating *models* do NOT need to have a concrete world to *rise
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
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
[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
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
> 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
> 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
> -----Original Message-----
> From: email@example.com
> [mailto:firstname.lastname@example.org] 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 Blunden*
> 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
> 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
> 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/