# [Xmca-l] Re: Davydov mathematics

```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 already
> >>>> 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'  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.
> >>>>>>>>
> >>>>>>>> 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 answer
> 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 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
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>> 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.
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>
> >>>>>>
> >>>>
> >>>>
> >>>>
> >>
> >>
> >>
>
>
>

```