[Xmca-l] Re: Davydov mathematics
Ed Wall
ewall@umich.edu
Mon Nov 3 13:16:29 PST 2014
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?
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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