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[Xmca-l] Re: units of mathematics education


     Quite nice and almost perfect for what I had in mind as a teaching of mathematics (although I am inferring a bit as to the actions of the teacher).

      Number is actually in the works, the As and Bs are the usual precursors (counting numbers early on are just labels). Ordinality is being constructed (although that may not be a purpose of the teacher). The critical point is how to get to B=A and A=B. I one time asked some experienced teachers I was working with to prove that, in effect, if A<B then for any clay blob K, A+K < B+K. Since they had either forgotten or never knew how to get to B=A and A=B, they had real problems.
       You need this latter result, by the way, to, I think, justify the usual digit matching in the comparison of decimal numbers. This seems usually taken on faith. Kids sometimes trust teachers too much.


On Oct 27, 2014, at  12:15 PM, Peg Griffin wrote:

> Interesting about money not being in Marx’s unit of analysis, Andy.
> As I understand what educators following Davidov’s math curriculum were
> doing, numbers were not involved in the mathematics education germ cells.
> Here’s a glimpse of the scenario as I understand it (which could be wrong):
> 		Children start with strings or lumps of clay or what not
> (not easily countable).
> 		Alyosha’s object is greater than Borja’s.  Anyone can
> perceive the difference.  The mathematical recording of that is A>B.
> 		In the very same situation, one is less than the other,
> mathematically, B<A.
> 		It isn’t nice or fair that one child’s object should be
> greater than and another less than, mathematically recorded as A ≠B and as
> B≠A.  How to get to B=A and A=B?
> 		How to have a nice, fair situation?  The teacher and
> children work it about and discover the important operations that
> mathematics has for working on >, <,  ≠ and  = and the mathematically
> recordings with + and -.  The whole situation of transformations takes this
> nice set mathematically recorded as:
> 				A≠B
> 				B≠A
> 				A>B
> 				A-X= B+X
> 				A’=B’
> 				B’=A’
> 		Then of course there’s more fun when Katya’s in on it and
> transitivity pops in so that even without direct perceptual comparisons
> mathematics comes to the rescue so you can figure out stuff you wouldn’t
> know otherwise (do I smell motivation here?):
> 				A>B
> 				B>K
> 				A?K
> 				A>K
> 		And they work out proudly that you keep the ? (don’t know)
> answer in the following situation
> 				A>B
> 				A>K
> 				B?K
> 		It remains forever a ? for mathematics, maybe direct percept
> will help but current mathematics for the current situation takes a pass on
> it.  We might use mathematics to come up with some nice questions and
> suppositions and come to more or less likely answers but…
> 		And then you can get to precision with measurement tools
> that work for the kinds of objects and …
> It's apparent that mathematics can serve social justice sometimes.  As I
> understand it, the Davidov mathematics educators take it for granted that in
> “non-mathematics” everyday life learning, children learn counting
> (including the cardinality principle alluded to earlier in the discussion
> and others that I associate a lot with the work developed by Gelman,
> Gallistel, and her colleagues).
> And, as I understand it, the great day of the coming together of mathematics
> and counting doesn’t happen for the Davidov folks until later - maybe even
> fourth grade.  Mathematics of the type discussed above can start in the
> Davidov style Kindergartens.
> In the US where we start off with numbers right away, in fourth grade, there
> have been many children who are confident that 9>7 and 9-2=7 but can get
> nowhere with working out all those wonderful equivalences if there are no
> numbers - i.e., they count but don’t do mathematics.
> Of course, that’s what I understand but I could be wrong.
> Peg
> -----Original Message-----
> From: xmca-l-bounces@mailman.ucsd.edu
> [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Andy Blunden
> Sent: Monday, October 27, 2014 12:31 AM
> To: eXtended Mind, Culture, Activity
> Subject: [Xmca-l] Re: units of mathematics education
> Well, I think that if you make a decision that mathematics is *not*
> essentially a social convention, but something which is essentially grasping
> something objective, then that affects what you choose as your unit of
> analysis. Student-text-teacher is all about acquiring a social convention.
> Remember that when Marx chose an exchange of commodities as a unit of
> analysis of bourgeois society, he knew full-well that commodities are rarely
> exchanged - they are bought and sold. But Marx did not "include"
> money in the unit of analysis.
> Andy
> ------------------------------------------------------------------------
> *Andy Blunden*
> http://home.pacific.net.au/~andy/
> Ed Wall wrote:
>> Andy
>>     Asking that question was one of the dumber things I've done on
>> this list. Apologies to all
>>      Thanks for reminding me about pre-concepts. I've been thinking about
> something similar and wondering if this is part of what makes doing
> mathematics 'mathematical.' Historically, by the way, mathematics grew out
> of manipulating such material objects; however, there are indications that,
> at some point (and it may have happened more than once), there was sort of a
> leap.
>>      Mathematics is considered a science; for instance, of patterns or,
> as Hegel puts it, quantity. I agree for a mathematician symbols of various
> sorts are effectively 'things'.
>>       In the 80s some mathematicians (School Mathematics Study Group) in
> the US put together a formal curriculum - my aunt used it - which was a
> disaster (and a real pain for the kids involved). Indications are children
> learned little.
>>       So to add a little to a discussion that possibly has continued
>> far longer than it should. Mathematics may have a few characteristics
>> that may distinguish it from other disciplines such as
>>        1. A student has the ability, in principle, to be able to
> independently of teachers or peers verify a grade appropriate mathematics
> statement (not a definition although definitions admit, in a sense,  a sort
> of empirical verification).
>>        2. Solutions to problems are, in general, not subject to
>> social conventions (which probably is included in the above).
>> Amusingly, I believe in the US a state legislature once tried to set
>> the value of pi to 3.1417
>> However, I'm not sure how such would fit together into a useful unit of
> analysis.
>> Ed