# [Xmca-l] Re: Objectivity of mathematics

```The way I like to think about it is how experienced problems structure
memory.  That is, how the awareness of the problem in getting the right
length strip can move to the beginning of subsequent attempts.  I think of
it as a process of exchange.  How memories are moved upstream through the
right kinds of action.

Those kind of effects are what I'd call "concrete generalisation", rather
than, say, "notational generalisation" which would be based upon patterns
perceived in the notation.  I'm guessing that its this latter sort of thing
that is often filling up that seemingly necessary gap.

Huw

On 6 November 2014 03:15, mike cole <mcole@ucsd.edu> wrote:

> Exactly, Ed.
>
> "Spontaneously" I was driving along and found myself of thinking about
> number lines and
> "a minus times a minus equals a plus. And for maybe the first time, sitting
> on a freeway in a car, I actually could arrive easily at the conclusion,
> "of course" without driving off the road or into another car. In the
> nonlinear way that meaning develops, I groked it and could have, at that
> moment, explained why on xmca.
>
> All very interesting. Makes one almost wish for traffic to sit in from time
> to time..... :-)
> mike
>
> On Wed, Nov 5, 2014 at 6:26 PM, Ed Wall <ewall@umich.edu> wrote:
>
> > Henry and Mike
> >
> >       That displacement you mention is part of why I've been doing the
> > wondering I mention below. I do think the teaching of mathematics should
> be
> > so that the gap, in part, is filled, as Mike put it, by imagination;
> > however, in many classrooms it is filled by anything but.
> >
> > Ed
> >
> > On Nov 5, 2014, at  7:21 PM, mike cole wrote:
> >
> > > Nice observation/connection Henry. I provokes the following thought.
> > >
> > > The result of a displacement, in the way I have been thinking about it,
> > is
> > > to create a gap in the connectivity/continuity of the experience, and
> > > filling that gap is a process of imagination, of seeing-as in a new
> way.
> > >
> > > mike
> > >
> > > On Wed, Nov 5, 2014 at 5:11 PM, HENRY SHONERD <hshonerd@gmail.com>
> > wrote:
> > >
> > >> 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
> > >>>>> :-)
> > >>>>>
> > >>>>>
> > >>>>
> > >>>
> > >>>
> > >>
> > >>
> > >>
> > >
> > >
> > > --
> > > It is the dilemma of psychology to deal with a natural science with an
> > > object that creates history. Ernst Boesch.
> >
> >
> >
>
>
> --
> It is the dilemma of psychology to deal with a natural science with an
> object that creates history. Ernst Boesch.
>

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