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[Xmca-l] Davydov mathematics

The article by Peter Moxhay is wonderful, Natalia! Thank you.
Despite my reservations (which would be relevant teaching and learning at a higher level), I am willing to pin Davydov's flag to my flagpole. It seems that the task of extending the idea set out so clearly here for arithmetic, to algebra, and beyond, is still a task to be solved, but I guess that any child who had acquired the concept of number by Davydov's method in primary school, is probably not going have trouble with algebra later on.

It would be an interesting exercise to render Davydov's method as a "unit of analysis", and that would perhaps indicate how the idea could be extended.

Also, to Haydi, it is worth noting that Davydov is an example of a CHAT theorist, i.e., someone who values and builds on both Vygotsky and Leontyev.
*Andy Blunden*

Natalia Gajdamaschko wrote:
Hi Dear All,
I am a lurker in this discussion thread on math education but find it very interesting! just to add to those two articles that Mike send of Jean Schmittau on Vygotsky/Davydov math curriculum, please, see attached another article Jean wrote with lots of good examples plus Peter's article. I use both of them in my class when it comes to discuss math curriculum done differently in my Vygotsky seminar. Cheers,

----- Original Message -----
From: "mike cole" <mcole@ucsd.edu>
To: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>
Sent: Sunday, November 2, 2014 1:45:28 PM
Subject: [Xmca-l] Re: units of mathematics education

As a small contribution to this interesting thread, two of Jean Schmittau's
writings. She has done a lot work with Davydov's ideas in math ed that may
give those following the discussion some useful info.

On Sun, Nov 2, 2014 at 12:03 PM, Ed Wall <ewall@umich.edu> wrote:


      By ''formal arithmetic' I mean the usual US curriculum to which you
refer to below; I wasn't talking about 'formal mathematics' when I
mentioned Benezet. The point Devlin makes (and I'm not sure I entirely
agree) is that the Davydov curriculum is about real number versus counting
number. While Devlin and I both have problems with the usual US curriculum
it is not entirely evident mathematically why one approach (counting number
versus real number) is better than the other.

       I am confused by the statement below concerning an example you gave
'earlier about US fourth graders.' The only example I remember was the one
using the Davydov approach with participants Alyosha and Borja.

       I would appreciate it if you would say a bit more about why "I
don't know" is a 'mathematically' correct and 'impersonal'  answer in some
'little systems.' I would tend to think otherwise about "We can't know.' in
some little (and some large) systems; however, I may misunderstand.


On Nov 2, 2014, at  9:42 AM, Peg Griffin wrote:

Thanks for this and the Hawaii information, Ed.  I had looked into the
Hawaii work before but I know nothing at all of Benezet, I'm afraid.

I'm not sure what you (or Benezet) mean by "formal arithmetic," so I
know what to make of the implication that the early Davidov mathematics
educators were "something like" an approach that lacked it.
In my understanding, the Davidov mathematics is essentially all  about
formal mathematics --symbols and systems of symbols are developed with
children for relations (=≠ ><) and operations (+ =).  Ignoring numbers
until later allows teachers to avoid an epigenetic byway we often see in
elementary schools where counting relations among number symbols
other aspects of mathematics.  The example I gave earlier is about the
fourth graders in US schools who seem to understand > and < than
in a little system of three mathematical statements but they do not
understand that "don't know" is a mathematically correct answer in some
the little systems -- for them don't know is essentially a personal thing
not a mathematics thing.

-----Original Message-----
From: xmca-l-bounces@mailman.ucsd.edu
[mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Ed Wall
Sent: Saturday, November 01, 2014 10:45 PM
To: eXtended Mind, Culture, Activity
Subject: [Xmca-l] Re: units of mathematics education

Something like this - i.e. lack of formal arithmetic until 7th -
the details are a little unclear) was done in the US in the 1920s by a
Benezet. My impression is that he was building on ideas of Dewey.


On Nov 1, 2014, at  8:48 PM, Peg Griffin wrote:

No move from numbers to x.  No numbers to begin with in mathematics
education.  Kids count in everyday life but no numbers in the
beginning mathematics classes.  It really is strings!  Not even rulers
or tape measures of strings.

-----Original Message-----
From: xmca-l-bounces@mailman.ucsd.edu
[mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Andy Blunden
Sent: Saturday, November 01, 2014 7:12 PM
To: 'eXtended Mind, Culture, Activity'
Subject: [Xmca-l] Re: units of mathematics education

Phew! So I was not the only one mystified by that expression. However,
wouldn't the kids have been confused by it as well? Or would they
react by
saying: "Hey, Teacher! That's stupid!"?
But certainly making the move to using letters only when the children
are reaching out for some more convenient symbol seems the right way
to go. I used to teach the first lesson in algebra by playing "Think
of a number, double it,  ..., what's the number he first thought of?"
with a classroom of kids and then introducing x for the number you
first thought of. Vygotsky tells us to provide the symbol as a means of
solving an existing problem.
How did Davydov make the move from numbers to x?

*Andy Blunden*

Peg Griffin wrote:
The * was an intrusion!  The expression is just paradoxical.  There
cannot be a concrete world such that "Alyosha's string is greater
than Boya's string equals Alyosha's string is less that Borya's
(By the way, in case you want a smile on this November day,  my
favorite paradox is the pragmatic one: " Inform all the troops that
communication has broken down."  Can't remember who is the originator
of it, though!)

-----Original Message-----
From: xmca-l-bounces+peg.griffin=att.net@mailman.ucsd.edu
[mailto:xmca-l-bounces+peg.griffin=att.net@mailman.ucsd.edu] On
Behalf Of Andy Blunden
Sent: Friday, October 31, 2014 7:58 PM
To: eXtended Mind, Culture, Activity
Subject: [Xmca-l] Re: units of mathematics education

Could you elaborate on what is meant by this passage, Peg? I am not
familiar with this use of * in mathematics, and I am not sure how the
and < relations are being evaluated here. Andy
*Andy Blunden*

Peg Griffin 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.