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
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
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 different.
> --- On *Sat, 7/2/11, Andy Blunden /<firstname.lastname@example.org
> From: Andy Blunden <email@example.com
> Subject: Re: [xmca] Numbers - Natural or Real?
> To: "eXtended Mind, Culture, Activity" <firstname.lastname@example.org
> 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
> Speech", Chapter Six:
> > > "If we may say so, the assimilation of a foreign
> the level of the maternal language (rech) for the child as
> the assimilation of algebra raises to a higher level the
> 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
> raises it to a higher level of more generalized thinking, in
> same way the assimilation of a foreign language in completely
> diverse ways frees verbal thinking from the grip of concrete
> and concrete phenomena of language."
> > > David Kellogg
> > Seoul National University of Education
> > > --- On *Fri, 7/1/11, Andy Blunden /<email@example.com
> > From: Andy Blunden <firstname.lastname@example.org
> > Subject: Re: [xmca] Numbers - Natural or Real?
> > To: "Culture ActivityeXtended Mind" <email@example.com
> > Date: Friday, July 1, 2011, 10:53 PM
> > Can you give us your reference here David, in a
> > translation of Vygotsky?
> > andy
> > David Kellogg wrote:
> > > ... I don't think that quantity IS the basic
> > mathematics, though. Vygotsky is pretty clear about
this: just a
> > preschooler has to be able to abstract actual objects
> > groups in order to form the idea of abstract
> > schoolchild has to be able to abstract quantities
> > numbers in order to form the idea of RELATIONS between
> > or OPERATORS.
> > >
> > __________________________________________
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> *Andy Blunden*
> Joint Editor MCA:
> Home Page: http://home.mira.net/~andy/
> Book: http://www.brill.nl/default.aspx?partid=227&pid=34857
> MIA: http://www.marxists.org
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