[Xmca-l] Re: Davydov mathematics
Ed Wall
ewall@umich.edu
Mon Nov 3 11:54:52 PST 2014
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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