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[Xmca-l] Re: units of mathematics education


      By ''formal arithmetic' I mean the usual US curriculum to which you refer to below; I wasn't talking about 'formal mathematics' when I mentioned Benezet. The point Devlin makes (and I'm not sure I entirely agree) is that the Davydov curriculum is about real number versus counting number. While Devlin and I both have problems with the usual US curriculum it is not entirely evident mathematically why one approach (counting number versus real number) is better than the other. 

       I am confused by the statement below concerning an example you gave 'earlier about US fourth graders.' The only example I remember was the one using the Davydov approach with participants Alyosha and Borja. 

       I would appreciate it if you would say a bit more about why "I don't know" is a 'mathematically' correct and 'impersonal'  answer in some 'little systems.' I would tend to think otherwise about "We can't know.' in some little (and some large) systems; however, I may misunderstand.

On Nov 2, 2014, at  9:42 AM, Peg Griffin wrote:

> Thanks for this and the Hawaii information, Ed.  I had looked into the
> Hawaii work before but I know nothing at all of Benezet, I'm afraid.
> I'm not sure what you (or Benezet) mean by "formal arithmetic," so I don't
> know what to make of the implication that the early Davidov mathematics
> educators were "something like" an approach that lacked it.  
> In my understanding, the Davidov mathematics is essentially all  about
> formal mathematics --symbols and systems of symbols are developed with the
> children for relations (=≠ ><) and operations (+ =).  Ignoring numbers
> until later allows teachers to avoid an epigenetic byway we often see in US
> elementary schools where counting relations among number symbols overshadow
> other aspects of mathematics.  The example I gave earlier is about the
> fourth graders in US schools who seem to understand > and < than relations
> in a little system of three mathematical statements but they do not
> understand that "don't know" is a mathematically correct answer in some of
> the little systems -- for them don't know is essentially a personal thing
> not a mathematics thing.
> PG 
> -----Original Message-----
> From: xmca-l-bounces@mailman.ucsd.edu
> [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Ed Wall
> Sent: Saturday, November 01, 2014 10:45 PM
> To: eXtended Mind, Culture, Activity
> Subject: [Xmca-l] Re: units of mathematics education
> Something like this - i.e. lack of formal arithmetic until 7th - (although
> the details are a little unclear) was done in the US in the 1920s by a Louis
> Benezet. My impression is that he was building on ideas of Dewey.
> Ed
> On Nov 1, 2014, at  8:48 PM, Peg Griffin wrote:
>> No move from numbers to x.  No numbers to begin with in mathematics 
>> education.  Kids count in everyday life but no numbers in the 
>> beginning mathematics classes.  It really is strings!  Not even rulers 
>> or tape measures of strings.
>> -----Original Message-----
>> From: xmca-l-bounces@mailman.ucsd.edu
>> [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Andy Blunden
>> Sent: Saturday, November 01, 2014 7:12 PM
>> To: 'eXtended Mind, Culture, Activity'
>> Subject: [Xmca-l] Re: units of mathematics education
>> Phew! So I was not the only one mystified by that expression. However, 
>> wouldn't the kids have been confused by it as well? Or would they 
>> react by
>> saying: "Hey, Teacher! That's stupid!"?
>> But certainly making the move to using letters only when the children 
>> are reaching out for some more convenient symbol seems the right way 
>> to go. I used to teach the first lesson in algebra by playing "Think 
>> of a number, double it,  ..., what's the number he first thought of?" 
>> with a classroom of kids and then introducing x for the number you 
>> first thought of. Vygotsky tells us to provide the symbol as a means of
> solving an existing problem.
>> How did Davydov make the move from numbers to x?
>> Andy
>> ----------------------------------------------------------------------
>> --
>> *Andy Blunden*
>> http://home.pacific.net.au/~andy/
>> Peg Griffin wrote:
>>> The * was an intrusion!  The expression is just paradoxical.  There 
>>> cannot be a concrete world such that "Alyosha's string is greater 
>>> than Boya's string equals Alyosha's string is less that Borya's string."
>>> (By the way, in case you want a smile on this November day,  my 
>>> favorite paradox is the pragmatic one: " Inform all the troops that 
>>> communication has broken down."  Can't remember who is the originator 
>>> of it, though!)
>>> -----Original Message-----
>>> From: xmca-l-bounces+peg.griffin=att.net@mailman.ucsd.edu
>>> [mailto:xmca-l-bounces+peg.griffin=att.net@mailman.ucsd.edu] On 
>>> Behalf Of Andy Blunden
>>> Sent: Friday, October 31, 2014 7:58 PM
>>> To: eXtended Mind, Culture, Activity
>>> Subject: [Xmca-l] Re: units of mathematics education
>>> Could you elaborate on what is meant by this passage, Peg? I am not 
>>> familiar with this use of * in mathematics, and I am not sure how the
>>>> and < relations are being evaluated here. Andy
>>> ---------------------------------------------------------------------
>>> -
>>> --
>>> *Andy Blunden*
>>> http://home.pacific.net.au/~andy/
>>> Peg Griffin wrote:
>>>> ...  That mathematical model (*A>B=A<B) DOES NOT have a concrete 
>>>> world to rise to! Instead, the children see/feel/perceive the 
>>>> strings and symbols having a relation among relations: A>B = B<A.