# [Xmca-l] Re: Davydov mathematics

```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
>>>>>>>>> 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
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>> 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.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>
>>>>
>>>>
>>
>>
>>

```