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Re: [xmca] Numbers - Natural or Real?
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- Subject: Re: [xmca] Numbers - Natural or Real?
- From: Tony Whitson <twhitson@UDel.Edu>
- Date: Sat, 2 Jul 2011 23:13:21 -0400 (EDT)
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As usual, when I haven't been following a thread, I don't want to butt in.
But ... the account of counting reminded me of discussion by Walkerdine
that might be of interest. It's on pp. 67-68 in Walkerdine, V. (1997).
Redefining the Subject in Situated Cognition Theory. In D. Kirshner & J.
A. Whitson (Eds.), _Situated cognition: social, semiotic, and
psychological perspectives_ (pp. 57-70). Mahwah, N.J.: L. Erlbaum.
(One reason I remember this passage is that the silhouette of 5 people
holding hands, which you can see at
only if you click on the image to get an enlarged view, is taken from an
illustration for my discussion of these Walkerdine examples, in my chapter
of the same book (at p. 111).) I hadn't made the connection before, but
the argument in my "Silhouette" paper in _Semiotica_ is actually germaine
to the argument that Walkerdine makes with her examples here.
Anyway, here is the Walkerdine:
 As Lave demonstrated, not only do calculations within shopping and
other practices exist in a different way than in school, but the
calculation is often not the purpose of the exercise. In my work examining
home and school practices involving calculation, the significant
transformation occurred to make the task mathematical only when
calculation became the target of the exercise. In home practices, this was
rarely the case.
Cooking practices at home had a cooking product as their aim.
When a calculation became the focus, certain semiotic shifts occurred. In
several examples of exchanges between mothers and daughters that I
analyzed, the mothers made certain discursive shifts when the focus of the
task moved from cooking to calculation as a product. These shifts
reoriented the focus of the task so that, as in school, the making of
cakes or some other practical task was no longer the product.
The product is the production of the counting string, an addition sum, or
whatever. To accomplish this shift, in the examples that I analyzed, the
mothers helped their daughters produce the move by the use of semiotic
chains in which new signs were constantly formed. For example, one mother
got her daughter to name people they were pouring drinks for and to work
out how many drinks by holding up one finger to correspond with each name.
This is the first relation of signifier to signified. In this case, we
might describe the names as signifiers that are attached to signifieds,
the people to which they refer. But here they have dropped to the level of
signifieds to be united with new signifiers, in this case iconic
signifiers, the fingers.
The next stage, which is not reached by this mother?daughter pair but is
by another, is for the fingers to drop to the level of signifieds to be
united with new signifiers, in this case spoken numerals. By this time,
any reference to people outside the counting string no longer exists
within the statement. In this case, the mother goes on to get the daughter
to unite fingers and numerals in small addition tasks of the form: "Five
and one more is . . ."
 I argued that discursive shifts such as this were central to the
accomplishment of the discursive transformation and to the repositioning
of the subject it entails. Statements of the form of "five and one more is
. . ." can refer to anything. All external reference and metaphoric
relations are excluded from the string. They create a discourse in which
there is no I, except an omniscient one (the God of mathematics) which can
describe the world as a book written in mathematics. I therefore argued
that discursive shifts, basic as they are in this case, produce the
possibility of huge shifts of subjectification and the production of the
man of reason, because for Lacan at least, semiotic chains are carried
along the metaphoric axis, and this no longer exists in school mathematics
In Lacan's analysis of unconscious processes, a signifier can drop to the
level of the signified and a new signifier/signified pair can be formed.
This is what Lacan meant by a semiotic chain. In examples such as these,
such chains form a bridge between one practice and another.
Walkerdine, V. (1997). Redefining the Subject in Situated Cognition
Theory. In D. Kirshner & J. A. Whitson (Eds.), _Situated cognition:
social, semiotic, and psychological perspectives_ (pp. 57-70). Mahwah,
N.J.: L. Erlbaum.
On Sun, 3 Jul 2011, Andy Blunden wrote:
David, you cast doubt on the ancient idea that mathematics is the science of
quantity and said that Vygotsky was clear on this. If Vygotsky is so clear,
then you wouldn't need to go to an English translation of an Italian
translation to find Vygotsky refuting the idea that mathematics is the
science of quantity. But your re-translation doesn't say this anyway. The
colon was a typo.
But let's take up the interesting point you raise anyway, even though it does
not say what you claimed it said, it is nonetheless interesting and
Am I right here? A child learns to survey the perceptual field and point to
things one after another reciting "one," "two,"three," ... and then remember
the number they say as they complete the practice. This is called "counting."
And I think it is a way children learn to abstract the units from a
collection in their perceptual field - pointing to each ion turn and saying
the next number. So I think they don't first abstract the actual objects and
then abstract number from this. Learning the practice of counting is how they
learn to abstract units from a whole.
Now, and this is the wonderful thing I learnt from Anna. Just because the
last number I said on completing counting wa "Five!" does not mean that I
know that there are 5 things. In fact, "Five" is a property of my counting
action; but I have to be taught to see "5" as a *property of the collection
of actual things*. AND then I have to learn that "5" is a *quantity* (a
cardinal as well as the last ordinal).
So there are two big conceptual leaps involved *after *I learn to abstract
things *by counting* them, before I get to the concept of quantity ... and
the beginnings of a type of mathematics (since other types of mathematics
will grow from other types of quantity).
So Bill, I think the position may be this (and please, I am way out of my
comfort zone here, but the July 4 holiday will be over soon and maybe the
cavalry will come to our rescue.) Your kids can't see any 2s in the 5 of 54,
because they see the 5 as an ordinal. They can see 2 2s in 4, because they
have been told so countless times, But they haven't been able to generalise
that knowledge because 5 does not "contain" 4, it is just the number "after"
4. OK? What do you think? Does that make sense?
David Kellogg wrote:
I don't understand this, Andy. The short answer is "Sure".
What is YOUR short answer supposed to mean? In particular, what does the
colon mean? I'm afraid the emoticons that we use in Korea are a little
--- On *Sat, 7/2/11, Andy Blunden /<email@example.com>/* wrote:
From: Andy Blunden <firstname.lastname@example.org>
Subject: Re: [xmca] Numbers - Natural or Real?
To: "eXtended Mind, Culture, Activity" <email@example.com>
Date: Saturday, July 2, 2011, 5:33 AM
So the short answer is ":no."
David Kellogg wrote:
> Sure, Andy!
> This is from Luciano Meccaci's translation of "Thinking and
Speech", Chapter Six:
> > "If we may say so, the assimilation of a foreign language
the level of the maternal language (rech) for the child as much as
the assimilation of algebra raises to a higher level the childs
arithmetic thinking, because it permits the child to understand
any arithmetical operation as a particular case of algebraic
operations, furnishing the child a freer, more abstract, more
generalized and at the same time more profound and rich view of
operations on concrete quantitites. Just as algebra frees the
thinking of the child from its dependence on concrete numbers and
raises it to a higher level of more generalized thinking, in the
same way the assimilation of a foreign language in completely
diverse ways frees verbal thinking from the grip of concrete forms
and concrete phenomena of language."
> > David Kellogg
> Seoul National University of Education
> > --- On *Fri, 7/1/11, Andy Blunden /<firstname.lastname@example.org
> From: Andy Blunden <email@example.com
> Subject: Re: [xmca] Numbers - Natural or Real?
> To: "Culture ActivityeXtended Mind" <firstname.lastname@example.org
> Date: Friday, July 1, 2011, 10:53 PM
> Can you give us your reference here David, in a pubished
> translation of Vygotsky?
> David Kellogg wrote:
> > ... I don't think that quantity IS the basic concept in
> mathematics, though. Vygotsky is pretty clear about this: just a
> preschooler has to be able to abstract actual objects away from
> groups in order to form the idea of abstract quantity, the
> schoolchild has to be able to abstract quantities away from
> numbers in order to form the idea of RELATIONS between
> or OPERATORS.
> xmca mailing list
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