# [Xmca-l] Re: Davydov mathematics

```You're welcome, Ed.

On 4 November 2014 02:55, Ed Wall <ewall@umich.edu> wrote:

> Huw
>
>
> Ed
>
> On Nov 3, 2014, at  8:02 PM, Huw Lloyd wrote:
>
> > On 4 November 2014 01:19, Ed Wall <ewall@umich.edu> wrote:
> >
> >> Huw
> >>
> >>      I referring to, one might say, a mathematical derivation. You might
> >> say our conversation is, unfortunately, incommensurable.
> >>
> >>
> > Hi Ed,
> >
> > That's perfectly consistent if man is abstracted from mathematics.
> > "Nothing human is alien to me".  I expect you'll need to find a source
> for
> > number somewhere, however.
> >
> > Best,
> > Huw
> >
> >
> >
> >> Ed
> >>
> >> On Nov 3, 2014, at  7:05 PM, Huw Lloyd wrote:
> >>
> >>> Ed,
> >>>
> >>> I'm referring to a psychological derivation.  The image as manifest in
> >> the
> >>> act of measuring.  I suspect your 1x1 square is similar, but I'm happy
> >> for
> >>> you to disagree.
> >>>
> >>> Huw
> >>>
> >>> On 4 November 2014 00:17, Ed Wall <ewall@umich.edu> wrote:
> >>>
> >>>> Huw
> >>>>
> >>>>    You have a very different understanding about the nature of number
> >>>> than I. In a sense, as soon as I draw the diagonal of a 1 by 1 square,
> >> that
> >>>> number (to the dismay of the Greeks) is no longer derived from
> >> measuring.
> >>>> Perhaps you think I'm talking about some sort of 'Davydov
> mathematics.'
> >> The
> >>>>
> >>>> Ed
> >>>>
> >>>> On Nov 3, 2014, at  4:53 PM, Huw Lloyd wrote:
> >>>>
> >>>>> On 3 November 2014 21:16, Ed Wall <ewall@umich.edu> wrote:
> >>>>>
> >>>>>> Huw
> >>>>>>
> >>>>>>    I am interested infer instance, thinking about the difference
> >>>>>> between mathematics and physics. 'Meaningful quantification' or
> >>>> 'meaningful
> >>>>>> activity' seems to be too large a label to detect differences. That
> >> is,
> >>>> one
> >>>>>> aspect of a 'unit of analysis', as I have gleaned from the
> >> conversation
> >>>> on
> >>>>>> the list, its minimality. Hmm. perhaps I need to ask what do you
> mean
> >> as
> >>>>>> regards 'quantification' re the mathematical?
> >>>>>>
> >>>>>>
> >>>>> I mean that an understanding of number is concomitant with competence
> >> in
> >>>>> the application of units of measure.  That number is derived from
> >>>>> measuring.  But not just any old measuring, measuring that solves a
> >>>>> meaningful problem.
> >>>>>
> >>>>> The Moxhay paper that Natalia sent covers some of this.
> >>>>>
> >>>>> I don't think a label is used to detect any differences at all, which
> >> is
> >>>>> why I called it a label.  Your unit of analysis will depend upon what
> >>>>> processes you're studying.  If you want to study how students
> construe
> >> a
> >>>>> situation in order to undertake a task, then it makes sense to study
> >>>> their
> >>>>> competence at that task over time via, for example, an analysis of
> how
> >>>> they
> >>>>> construe and structure that task.
> >>>>>
> >>>>> Best,
> >>>>> Huw
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>> Ed
> >>>>>>
> >>>>>> On Nov 3, 2014, at  2:38 PM, Huw Lloyd wrote:
> >>>>>>
> >>>>>>> Hi Ed,
> >>>>>>>
> >>>>>>> One can characterise physics by its interest in physical processes.
> >>>>>>> Physics employs quantification as a means to study these processes.
> >>>>>>>
> >>>>>>> I merely offer "meaningful quantification" as a label.  That is,
> >>>> engaging
> >>>>>>> with the meanings redolent in problems resolved through
> >> quantifying.  I
> >>>>>> am
> >>>>>>> also paraphrasing Gal'perin's "meaningful activity".
> >>>>>>>
> >>>>>>> Best,
> >>>>>>> Huw
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>> On 3 November 2014 19:54, Ed Wall <ewall@umich.edu> wrote:
> >>>>>>>
> >>>>>>>> Huw
> >>>>>>>>
> >>>>>>>>    How does 'meaningful quantification' distinguish between
> >>>>>>>> mathematics and, for instance, physics?
> >>>>>>>>
> >>>>>>>> Ed
> >>>>>>>>
> >>>>>>>> On Nov 3, 2014, at  11:57 AM, Huw Lloyd wrote:
> >>>>>>>>
> >>>>>>>>> Andy,
> >>>>>>>>>
> >>>>>>>>> I haven't been following the recent threads, so this may have
> >>>>>>>> been
> >>>>>>>>> covered.
> >>>>>>>>>
> >>>>>>>>> 1) Algebra in the sense of variables, is introduced by labelling
> >>>>>>>> concretely
> >>>>>>>>> given particular lengths.  E.g length A is larger that length B,
> >>>> using
> >>>>>>>> the
> >>>>>>>>> familiar notation A > B etc.
> >>>>>>>>>
> >>>>>>>>> 2) For an elaboration of mediating schemas, see the works of
> >>>> Gal'perin.
> >>>>>>>>>
> >>>>>>>>> 3) For units, I think this is going to depend on the creative
> >> extent
> >>>>>>>>> applied to the notion of concept.  One could say that any
> >> conceptual
> >>>>>>>>> knowledge was incomplete if the subject was not able to derive
> the
> >>>>>> means
> >>>>>>>> to
> >>>>>>>>> transform situations (to have some notion of a concept of
> concepts)
> >>>>>> which
> >>>>>>>>> would be required to construe new situations in terms of the
> >> concept.
> >>>>>> I
> >>>>>>>>> think the origins of that go back to the social understanding
> (not
> >>>> mere
> >>>>>>>>> understanding).  For mathematics, one could label that
> "meaningful
> >>>>>>>>> quantification".
> >>>>>>>>>
> >>>>>>>>> Best,
> >>>>>>>>> Huw
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>> On 3 November 2014 06:17, Andy Blunden <ablunden@mira.net>
> wrote:
> >>>>>>>>>
> >>>>>>>>>> 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
> >>>>>>>>>>
> >>>>>>
> >> ------------------------------------------------------------------------
> >>>>>>>>>> *Andy Blunden*
> >>>>>>>>>> http://home.pacific.net.au/~andy/
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> 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,
> >>>>>>>>>>> Natalia.
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>> ----- 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.
> >>>>>>>>>>> mike
> >>>>>>>>>>>
> >>>>>>>>>>> On Sun, Nov 2, 2014 at 12:03 PM, Ed Wall <ewall@umich.edu>
> >> wrote:
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>> Peg
> >>>>>>>>>>>>
> >>>>>>>>>>>>  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'
> >>>>>> 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.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Ed
> >>>>>>>>>>>>
> >>>>>>>>>>>> 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
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> don't
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> 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
> >>>>>>>>>>>>> formal mathematics --symbols and systems of symbols are
> >> developed
> >>>>>>>> with
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> the
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> children for relations (=≠ ><) and operations (+ =).
> Ignoring
> >>>>>>>> numbers
> >>>>>>>>>>>>> until later allows teachers to avoid an epigenetic byway we
> >> often
> >>>>>>>> see in
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> US
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> elementary schools where counting relations among number
> >> symbols
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> other aspects of mathematics.  The example I gave earlier is
> >>>>>>>> the
> >>>>>>>>>>>>> fourth graders in US schools who seem to understand > and <
> >> than
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> relations
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> in a little system of three mathematical statements but they
> do
> >>>> not
> >>>>>>>>>>>>> understand that "don't know" is a mathematically correct
> >>>> in
> >>>>>>>> some
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> of
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> the little systems -- for them don't know is essentially a
> >>>> personal
> >>>>>>>>>>>>> thing
> >>>>>>>>>>>>> not a mathematics thing.
> >>>>>>>>>>>>> PG
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> -----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
> >> -
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> (although
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> the details are a little unclear) was done in the US in the
> >> 1920s
> >>>>>> by
> >>>>>>>> a
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>> Louis
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> Benezet. My impression is that he was building on ideas of
> >> Dewey.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Ed
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> 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
> >>>>>>>>>>>>>>
> >>>>>>>>
> >> ----------------------------------------------------------------------
> >>>>>>>>>>>>>> --
> >>>>>>>>>>>>>> *Andy Blunden*
> >>>>>>>>>>>>>> http://home.pacific.net.au/~andy/
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Peg Griffin wrote:
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> The * was an intrusion!  The expression is just
> >>>>>> 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
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>> string."
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>> (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*
> >>>>>>>>>>>>>>> http://home.pacific.net.au/~andy/
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> 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.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>
> >>>>
> >>>>
> >>
> >>
> >>
>
>
>

```