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[Xmca-l] Re: Objectivity of mathematics

Huw & Anna,

I had forgotten, until I read the paper again, that Jenny and I based our analysis of the fractions class on three main figures: Rotman, Lachterman, and Sfard! All three see discourse at the heart of mathematics.


On Nov 7, 2014, at 5:57 AM, anna sfard <sfard@netvision.net.il> wrote:

> Hi Huw,
> Thanks for your thoughts. I agree with much of what you say. I would like to know more, though, about why you think that if you talked about problem solving in discursive terms, "you'd quickly end up with linguists reducing it to wording, and various kinds of "acquisitionists" thinking that this is where you're going." I do think about these processes in discursive terms and feel, on the contrary, that this is what guards me against objectification and acquisitionism. So why?
> And on this occasion, to the other debate, the one about "objective". If you assume the discursive stance, this word becomes an oxymoron.  Objective, as I understand it, means "mind independent", bound have a given form independently of one's tastes, values and judgments. But this adjective ("objective") refers to narratives, to what people say/think ("facts" are subcategory of narratives). So...
> anna
> -----Original Message-----
> From: xmca-l-bounces@mailman.ucsd.edu [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Huw Lloyd
> Sent: Friday, November 07, 2014 3:24 AM
> To: eXtended Mind, Culture, Activity
> Subject: [Xmca-l] Re: Objectivity of mathematics
> Hi Anna,
> Perhaps you could also assert that quantitative choices, predicated upon social commitments, offer a means to go beyond those tentative bonds formed in numerical rituals.
> Commitments, such as commitment to a task that makes it a problem, seem to be important.  Also, it seems to me that problem solving (mental searching etc) is something that should have a first class status in a theory about mathematics. The problem I'd have with referring to these processes as discourse is that I think you'd quickly end up with linguists reducing it to wording, and various kinds of "acquisitionists" thinking that this is where you're going.
> A second problem, for me, with fusing communication and cognition is the distinct role that communication has in mediating actions, rather than comprising the fabric of actions.  For me, the act of exercising that fabric, whether mentally or in relation to a present object, induces transformations.
> I don't think these issues conflict with your account, but perhaps there's quite a bit that is skimmed over (such as the bit about individualized discourse, perhaps).
> I enjoyed your paper.  :)
> Best,
> Huw
> On 6 November 2014 06:10, anna sfard <sfard@netvision.net.il> wrote:
>> Hi,
>> I have not been aware of this super-interesting (for me) thread, and 
>> now, when I eventually noticed  it, I cannot chime in properly. So I 
>> am doing this improperly, simply by attaching my own paper. Those who 
>> are interested enough to open the attachment will see the relevance of 
>> its theme to the present conversation. And although I mention Davydov 
>> only in an endnote, he is very much present. The theses I'm arguing 
>> for seem to substantiate his request for taking the quantitative 
>> discourse, rather than the numerical, as a point of departure for the 
>> process of  developing child's mathematical thinking (we cannot help 
>> it, but in our society, these two discourses appear in the child's 
>> life separately and more or less in parallel, with the quantitative 
>> discourse free from numbers and the numerical one innocent of any 
>> connection to quantities; at a certain point, these two discourses 
>> coalescence, thus giving rise to the incipient mathematical discourse; 
>> but at the pre-mathematical stage, quantitative discourse is 
>> meaningful to the child on its own, as it supports the activity of choosing, whereas numerical discourse is but a way to bond with grownups).
>> anna
>> -----Original Message-----
>> From: xmca-l-bounces@mailman.ucsd.edu [mailto:
>> xmca-l-bounces@mailman.ucsd.edu] On Behalf Of HENRY SHONERD
>> Sent: Thursday, November 06, 2014 3:11 AM
>> To: eXtended Mind, Culture, Activity
>> Subject: [Xmca-l] Re: Objectivity of mathematics
>> Ed and Andy,
>> Just a little while ago, while I was finishing the Moxhay paper, which 
>> seems to have produced an AHA! moment” regarding object-mediated 
>> action for Andy, I had my own AHA! moment, and it is this:
>>     Some years ago, after teaching Intro to Linguistics many times, I 
>> decided that the most important property of human language that 
>> clearly sets it apart from what we know about other species’ ability 
>> to communicate is what is called DISPLACEMENT: the ability to use 
>> language to refer to things removed from the here and now, including 
>> imaginary happenings or things. The Davydov tasks in the Moxhay 
>> article give children the same problem of displacement by requiring 
>> that they figure a way to compare two objects removed from one another 
>> in space, and, effectively, in time. And I am wondering if this 
>> touches on the other threads I have been following: L2 and the 
>> Blommmaert/Silverstein. Does the need for standardization in 
>> measurement of the objects in the world today find its way into L2 
>> teaching and language policy? The blending of qualitative and 
>> quantitative research methods come to mind, to my mind at least. 
>> Moxhay’s article ended with a comparison of  Classroom A and B that  
>> certainly was a blend of the two methods, though the ways in which the 
>> dialog broke down in Classroom B (a qualitative issue, I would think) 
>> was only hinted at. That would have required a narrative. So, the interplay of narrative and dialog, objects mentioned by David K. I know I have bitten off more than I can chew.
>> Henry
>>> On Nov 3, 2014, at 10:51 PM, Ed Wall <ewall@umich.edu> wrote:
>>> Andy
>>>     What you say here fits somewhat with some of the thinking I've 
>>> been
>> doing, but, in part, it is at the point of symbol manipulation that 
>> things seem get complicated for me. Also, I find myself wondering 
>> whether teaching mathematics, in effect, as mathematics or even 
>> Davydov-style is just the things you list. There seems to be more that 
>> is needed (and I could be wrong about this) and I have yet to factor 
>> in something  like those pre-concepts you mentioned earlier. So I need 
>> to do a little reading/rereading on the symbolic question, think a bit 
>> more about the space the teacher opens up for studying mathematics, 
>> and factor in those 'pre-concepts' before I can reply reasonably to what you are saying here.
>>>    I admit that I tend to complicate things too much (smile), but 
>>> that
>> may come from thinking about them too much.
>>> Thanks
>>> Ed
>>> On Nov 3, 2014, at  10:45 PM, Andy Blunden wrote:
>>>> Particularly after reading Peter Moxhays' paper, it is clear to me 
>>>> that
>> teaching mathematics, Davydov-style, is orchestrating 
>> concept-formation in a particular domain of activity, and that what 
>> the children are doing in forming a concept is a system of artefact-mediated actions: "For Davydov,"
>> he says, "a theoretical concept is itself a /general method of acting/ 
>> - a method for solving an entire class of problems - and is related to 
>> a whole system of object-oriented actions." Pure Vygotsky, and also 
>> equally pure Activity Theory except that here the object becomes a 
>> "theoretical concept," which is characteristically Vygotsky, the point 
>> of difference between ANL and LSV! Just as in all those dual 
>> stimulation experiments of Vygotsky, the teacher introduces a symbol 
>> which the student can use to solve the task they are working on.
>>>> So the unit of learning mathematics is *an artefact-mediated action*.
>> The artefact is introduced by the teacher who also sets up the task. 
>> At first the symbols is a means of solving the material task, but 
>> later, the symbol is manipulated for its own sake, and the material 
>> task remains in the background. This is what is special about 
>> mathematics I think, that the symbolic operation begins as means and becomes the object. C.f. Capital:
>> the unit is initially C-C' becomes C-M-C' and then from this arises M-C-M'
>> - the unit of capital.
>>>> Andy
>>>> -------------------------------------------------------------------
>>>> --
>>>> ---
>>>> *Andy Blunden*
>>>> http://home.pacific.net.au/~andy/
>>>> mike cole wrote:
>>>>> That is really a great addition to Andy's example, Ed. Being a 
>>>>> total
>> duffer here i am assuming that the invert v is a sign for "power of" ?
>>>>> You, collectively, are making thinking about "simple" mathematical
>> questions unusually interesting.
>>>>> The word problem problem is really interesting too.
>>>>> mike
>>>>> PS - I assume that when you type:  There is, one might say, a
>> necessity within the integers is that 5 x -1 = -5.   you mean a SUCH not is?
>>>>> mike**2
>>>>> :-)