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Re: [xmca] Numbers - Natural or Real?
Well, first of all, I wasn't responding to the Devlin article at all, Bill. I was responding to Andy's query about discourse and activity, Sfard and Davydov.
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 quantities, or OPERATORS. That, and the idea of part-whole relations, are the cornerstones of algebra, which Vygotsky considers the true goal of math education in elementary school (he's very ambitious, and in some ways his programme makes more sense in a Korean context than in the West).
We can easily see the difference when we consider how a given mathematical attitude considers the concept of INFINITY. If we consider infinity as a quantity, then there is no way to make sense of the statement that the infinity of whole numbers is larger than the infinity of even numbers (or the infinity of reals is larger than the infinity of naturals).
But if we consider infinity as an OPERATION, that is, the recursive operation of adding to the last in a series, we get something rather different, which can explain this (given enough imagination on the part of the child).
Bill, the reason I was pointing to PLAY as an important element in this was that I think the problem of teaching POTENTIAL as opposed to REAL is a recurrent problem in teaching. For example, when teachers want to teach "I can swim" in Korea they use a picture of a child swimming. But the result is that the children assume that "I can swim" essentially just means "swimming" or even "I am swimming" or "he is swimming".
Play allows the teacher to solve this problem. In play, a real move is understood by the child as an instantiation of the potential moves within the imaginary situation or within the abstract rules. So the child easily grasps that an actual move is simply an instance of a potential.
In the game I suggested, the main problem was showing the child how ANY number will do as a starting point, and how there is no ending point. Both of these are essential to teaching the point that each number is re-expressible in an infinite number of ways.
But if the teacher suggests a number, the children will often take this number as the only number that will do. The teacher solves this problem by setting up a game in which the children suggest a number, and see that it can be any number.
Similarly, if the teacher demonstrates a sequence (3/2, 6/4, 12/8 etc.) the children will see this as a complete or completable sequence. But if it is clear that the termination of the sequence is a LOSING move, then the idea of infinity as a process is within the child's grasp.
David Kellogg
Seoul National University of Education
--- On Fri, 7/1/11, Andy Blunden <ablunden@mira.net> wrote:
From: Andy Blunden <ablunden@mira.net>
Subject: Re: [xmca] Numbers - Natural or Real?
To: "Huw Lloyd" <huw.softdesigns@gmail.com>
Cc: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
Date: Friday, July 1, 2011, 5:54 PM
Well Huw, both the advocates of one or other method whose links I quoted believe there is a significant difference in choice of basic unit, and certainly from the point of view of the subject matter itself there is a difference: the concept of cardinal number is different from the concept of ordinal number, even though the non-mathematical adult probably never notices it. That distinction was one of the delightful insighsts I got from reading Anna's book. I knew the difference, but I just never reflected on it as something a child has to learn.
I guess (apart from my question about foundations which I am hoping the experience of a maths teacher will shed light on) I am still working through my lived experience of discovering that the 14 year old kids I was teaching in 1975 who could add, subtract, multiply and even divide, had no concept of numbers as representing quantity, and never knew which arithmetical procedure to use in which practical situation outside of a small range of repeatedly rehearsed scenarios. That is, after 8 years of British public education, they had still not made the leap talked about in the Devlin article, which forms the beginning in Davydov's approach.
Andy
Huw Lloyd wrote:
On 30 June 2011 09:52, Andy Blunden <ablunden@mira.net> wrote:
I don't really have an opinion on this matter, but I would be interested in listening in on those who may have an informed opinion.
I see two different approaches to the teaching of mathematics.
One takes the /Natural/ Numbers as the basic concept of the subject
The Other takes the /Rational/ Numbers as the basic concept of the
subject
The Davydov example, described in "Cultural-Historical Approaches to Designing for Development" describes quantity as the basic concept. Which seems sensible to me.
Both magnitude and multitude are variants of quantity -- a concept that entails a number along with a unit of measure.
One takes /counting/ as the basic Action
The Other takes /comparison/ of two lengths as the basic Action.
In both cases I would ascribe measuring as the "basic action", which includes the pattern matching found in counting apples.
Huw
--
*Andy Blunden*
Joint Editor MCA: http://www.informaworld.com/smpp/title~db=all~content=g932564744
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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