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[Xmca-l] Re: units of mathematics education



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
>
>
>