# [Xmca-l] Re: units of mathematics education

```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.
>
> 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
>> 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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