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*To*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Subject*: [Xmca-l] Re: Davydov mathematics*From*: Ed Wall <ewall@umich.edu>*Date*: Mon, 3 Nov 2014 19:19:09 -0600*In-reply-to*: <CAG1MBOEGDx5E8w7gR5huNcu=cN94PVaa20Y0OygOjVzY1iyNuQ@mail.gmail.com>*List-archive*: <https://mailman.ucsd.edu/mailman/private/xmca-l>*List-help*: <mailto:xmca-l-request@mailman.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca-l.mailman.ucsd.edu>*List-post*: <mailto:xmca-l@mailman.ucsd.edu>*List-subscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=subscribe>*List-unsubscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=unsubscribe>*References*: <1414042156116.36175@unm.edu> <012a01cff51c$c6d1c780$54755680$@att.net> <008801cff5f4$8d61b590$a82520b0$@att.net> <003301cff63f$27973ff0$76c5bfd0$@att.net> <86D8D1D6-E3A2-4FA2-9F6D-0A55359E31D8@umich.edu> <002801cff6b3$8e5dbb00$ab193100$@att.net> <4FD6099D-A5CB-4A9B-911D-D4B2E192E724@umich.edu> <CAHCnM0Dsnfu94PT=HUxiyKXqT-LsVXTjviYmz+E-mw0Waj83EQ@mail.gmail.com> <1216709733.58605697.1414966336813.JavaMail.zimbra@sfu.ca> <54571DEA.2080802@mira.net> <CAG1MBOFuLRvK9qP3ubJm2_fnEyMdF3B=8odqkYA-BXhPm_v2dg@mail.gmail.com> <4B9E54C2-F101-459A-9AAB-EF957C736B96@umich.edu> <CAG1MBOGbZjOXOFm+dT526ZHhyrKNjZqb_SevsWS=db1bN5G5hw@mail.gmail.com> <4CD1C6B6-D222-4040-96EB-BDD244B125D2@umich.edu> <CAG1MBOHG0+QUN3w1jdhfEucTMakB0P-ixk1wwcrUmqMKsvAm5Q@mail.gmail.com> <2471450A-10F3-4DF5-BFA4-331ED0BBFDE3@umich.edu> <CAG1MBOEGDx5E8w7gR5huNcu=cN94PVaa20Y0OygOjVzY1iyNuQ@mail.gmail.com>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>

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. >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>> >>>>>> >>>>>> >>>>>> >>>> >>>> >>>> >> >> >>

**Follow-Ups**:**[Xmca-l] Re: Davydov mathematics***From:*Huw Lloyd <huw.softdesigns@gmail.com>

**References**:**[Xmca-l] Re: units of mathematics education***From:*Peg Griffin <Peg.Griffin@att.net>

**[Xmca-l] Re: units of mathematics education***From:*Peg Griffin <Peg.Griffin@att.net>

**[Xmca-l] Re: units of mathematics education***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: units of mathematics education***From:*Peg Griffin <Peg.Griffin@att.net>

**[Xmca-l] Re: units of mathematics education***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: units of mathematics education***From:*mike cole <mcole@ucsd.edu>

**[Xmca-l] Re: units of mathematics education***From:*Natalia Gajdamaschko <nataliag@sfu.ca>

**[Xmca-l] Davydov mathematics***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: Davydov mathematics***From:*Huw Lloyd <huw.softdesigns@gmail.com>

**[Xmca-l] Re: Davydov mathematics***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: Davydov mathematics***From:*Huw Lloyd <huw.softdesigns@gmail.com>

**[Xmca-l] Re: Davydov mathematics***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: Davydov mathematics***From:*Huw Lloyd <huw.softdesigns@gmail.com>

**[Xmca-l] Re: Davydov mathematics***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: Davydov mathematics***From:*Huw Lloyd <huw.softdesigns@gmail.com>

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