[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

[Xmca-l] Re: Davydov mathematics



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
> >> thread was about Davydov mathematics education.
> >>
> >> 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
> >>>>>> about
> >>>>>>>>>>> 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
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>> overshadow
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>>> other aspects of mathematics.  The example I gave earlier is
> >> about
> >>>>>> 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.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>
> >>>>
> >>>>
> >>
> >>
> >>
>
>
>