[Xmca-l] Re: Davydov mathematics
Ed Wall
ewall@umich.edu
Mon Nov 3 18:55:14 PST 2014
Huw
Thank you for your answer to my original question.
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
>>>> 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.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>
>>>>
>>>>
>>
>>
>>
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