[Xmca-l] Re: Davydov mathematics
Ed Wall
ewall@umich.edu
Mon Nov 3 17:19:09 PST 2014
Huw
I referring to, one might say, a mathematical derivation. You might say our conversation is, unfortunately, incommensurable.
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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