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Re: [xmca] Numbers - Natural or Real?



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

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