[Xmca-l] Re: Davydov mathematics

Mon Nov 3 12:38:09 PST 2014

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.
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>
> >>>>
> >>>>
> >>>
> >>>
> >>>
> >>>
> >>
> >>
>
>
>