Re: [xmca] Numbers - Natural or Real?

[Bill Kerr <billkerr@gmail.com>]

My reading of the Devln article is that:

There is an important difference b/w starting from counting (leads to focus
on natural numbers) and starting from measurement (leads to focus on real
numbers). The former is represented by western curricula (eg. USA), the
latter by Soviet curricula.

Devlin suspects that starting from measurement may be superior ("...if the
Davydov approach is in some sense inherently superior - and when taken as a
whole I think it may well be") but he qualifies this heavily by:
(a) more research needs to be done
(b) good teachers of either approach will do better than poor teachers of
either approach - the main issue is the  need to improve teacher training

Devline says it is hard to argue that one approach is better grounded in our
natural and built environment than another:

"Thus, the Davydov curriculum is grounded in the real world, but the
starting point is the continuous world of measurement rather than the
discrete world of counting. I don't know about you, but measurement and
counting both seem to me to offer pretty concrete starting points for the
mathematical journey. Humans are born with a capacity to make judgments and
to reason about length, area, volume, etc. as well as a capacity to compare
sizes of collections. Each capacity leads directly to a number concept, but
to different ones: real numbers and counting numbers, respectively"


There are other aspects of maths education in schools that would have to be
taken into account before a fair comparison could be made. eg. some students
need lots of repetition. It appears that the Davydov approach as implemented
does not include a lot of that. That doesn't disqualify the approach as a
different approach, more repetition could be built in.

David Kellog said that play is important:

"I guess what I would say about this is that it is not really about
discourse or activity, but rather about play. Of course, both discourse and
activity CAN be seen as play and indeed ARE seen as play by children. But
the key component is not so much mediation as contradiction"


I would agree that relevant play and contradiction is important in all
teaching but how is that an argument against Devlin's assertion that there
are two different approaches to beginning maths teaching?

Andy Blunden said that teaching in Britain using textbook recipes does not
work very well. This is true but in itself does not demonstrate that a
beginning approach based on counting could not work well if done by a
teacher with a profound understanding of maths fundamentals who incorporates
relevant play and does not rely entirely on the textbook.

I have read the Devlin article but haven't yet read Anna Sfard's work or the
Jean Schmittau article (thanks Larry for making that available). That puts
me behind with regard to some of the theory discussed here. I wouldn't
comment on other peoples work without reading them first. In this regard I
have the impression that there has been some comment here about Devlin's
article without reading it first.


On Sat, Jul 2, 2011 at 10:24 AM, Andy Blunden <ablunden@mira.net> wrote:

> **
> 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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