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[Xmca-l] Re: units of mathematics education

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

-----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 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 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/