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*To*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Subject*: [Xmca-l] Re: Davydov mathematics*From*: Huw Lloyd <huw.softdesigns@gmail.com>*Date*: Tue, 4 Nov 2014 01:05:36 +0000*In-reply-to*: <2471450A-10F3-4DF5-BFA4-331ED0BBFDE3@umich.edu>*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>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>

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:*Ed Wall <ewall@umich.edu>

**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>

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