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[Xmca-l] Re: units of mathematics education

Thanks for that, Peg. I have read Davodov's book on Generalisation http://www.marxists.org/archive/davydov/generalization/generalization.pdf but I have never known exactly how his idea were implemented beyond those germ cell operations comparing the size of objects. I lost interest in Davydov when I saw that (in my view) he failed to understand Vygotsky on scientific and everyday concepts.

Peg, recall that other interest of yours, in the Question Asking Reading: http://lchc.ucsd.edu/People/NEWTECHN.pdf where you suggest that reading is not just being able to decode a text, but in being able to "read the world" and consequently introduce the idea of a double mediation between the world and the child - by the text and by the adult, with all the mediations that go along with this. It seems to me that learning mathematics is like that as well. In the end, the child can manage the text without reference to the world, but if we are concerned with teaching and learning, we need a germ cell which can grow from its roots in the world.

Marx's unit of analysis/germ cell was genetic in that sense. He started with the exchange of commodities, and very quickly (still in chapter 1, volume 1) shows how money arises historically as a special type of commodity, and subsequently also how capital arises as a specific type of commodity relation. But the essential relation was the commodity relation.

Even though we do mathematics without natural objects, genetically, that is the germ cell. Each topic has to begin, surely, with a situation to be analysed until students have become at home in the world of mathematical texts in which the world is only implicit.

*Andy Blunden*

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-X= B+X



            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?):





            And they work out proudly that you keep the ? (don’t know)
            answer in the following situation




            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.


-----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 Blunden*

http://home.pacific.net.au/~andy/ <http://home.pacific.net.au/%7Eandy/>

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