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
Here’s a glimpse of the scenario as I understand it (which could be
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,
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:
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.
[mailto:firstname.lastname@example.org] 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
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.
Ed Wall wrote:
> 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