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[Xmca-l] Re: Davydov mathematics

I am teaching a class in Immersion this semester, and every week we do
a lesson from the Korean curriculum in English. Last week we had to do
a mathematics lesson, and almost everyone chose fractions.

I usually start off the homework with an example (sometimes not a very
good example, though!) and my lesson was essentially Davydov method:
the kids had a big tank of banana milk and they had to measure it with
impossibly small cups, but I supplied a bowl that held exactly six
cups. From this we derived the idea of the bowl as a fraction of the
tank and the cup as a fraction of the whole (and, as I understand it,
it is this idea of part-whole relations that Davydov considers the
entry point into algebra, because  instead of starting off with a
"known" like zero or one as you do when you count or when you do an
arithmetical equation like 6 x 17 = ? you can start off with an

Nobody likes to copy my lessons, but nobody likes to stray too far
from them either. The favorite method was to substitute something for
the banana milk, and the favorite subsitute was pizza. But this led to
unhealthy lessons such as, you and your friend want to divide a pizza,
so you cut it in two and each of you takes half. This led to an
obvious absurdity when one grad decided to substitute an entire
watermelon (and, absurdity on absurdity, used a picture that showed a
watermelon in cross section instead of a whole). But what was really
unhealthy about the lesson was the regression in mathematics to fit
the low level of English.

In class I pointed out that the way people actually do eat pizza is a
better fit for the lesson. You cut your pizza into eighths or twelfths
or sixteenths, depending on the size of the pizza. Everybody takes one
and eats it more or less simultaneously, and then everybody takes one
when they are hungry and eats it, and then you try to figure out who
is entitled to or wants the last piece, on the basis how many have
been eaten.

As you can see the difference between the two lessons really is
twofold: in the case of starting with the whole pizza and dividing it
by the number of eaters, you start with a known and proceed to the
unknown, while in the case of the more naturalistic pizza-consumption
mode, you start with an unknown and proceed to the known.

But more importantly, because my grads are all TESOL teachers, they
have a natural tendency to reduce everything to naming: when we have
named 1/2 by its proper English name, we have the whole key to
fractions. Unfortunately, with everyday words, that is exactly what we
do not have--you are much better off working with sixteenths, because
that forces the children to work with a system, to think of two
numbers and an operation instead of just a lexicalized expression like
"half" or "quarter".

This week we are doing science lessons, and I can already see that the
science lessons come in two definite types: 'What" questions ("What is
solar altitude?") and "Why" questions ("Why does it it get cooler
instead of hotter when the sun comes down?"). Sure enough, the grads
who used half pizzas and quarter pizzas and assumed that once you have
learned these opaque and lexicalized everyday words you have mastered
the system of concepts behind them all fall into the first category.

Of course, the term Vygotsky uses is really something like "academic
concept" rather than "scientific concept". This is sufficient ot
explain to me, or at least to the linguist in me, why they come in
hierarchies, why they are definite and exhaustive, why they are
morphologically complex, and why they are born in laboratories but
like to hide in classrooms....

David Kellogg
Hankuk University of Foreign Studies

On 3 November 2014 12:47, Andy Blunden <ablunden@mira.net> wrote:
> Mike, thank you for the two Schmittau articles on Davydov maths teaching.
> The first was very brief, but useful.
> The second was very helpful, in that it did put a lot of meat on the bones
> of my sketchy understanding of what VVD's maths program meant, but I have
> some problems with it, which people on the list could probably help me with.
> (1) The abstract claims that the order of learning (first arithmetic and
> then algebra) traditionally used is reversed. I found this an astounding
> idea. But when I read, this seems not to be what actually happens. The
> children are doing complex arithemetical task like dividing by 3-digit
> numbers, and still haven't actually got to algebra. Though
> (2) there is talk of a schematic kids are offered to use to structure
> problems. It would help to know what this schematic is.
> (3) There is a fault in the PDF, causing some mathematical symbols to not
> show, which (I think) is making some of the examples incomprehensible,
> (4) though I find telling a kid who says 14-4-4=14 is making an error a bit
> rich as it seems to me an equally valid answer to an ill-posed problem.
> (5) Finally, Schmittau assumes that by "pre-concept" Vygotsky meant
> "complex". I thought this at first, until a few years ago David Ke kindly
> corrected me, and indeed this is not the case. Although merely a question of
> terminology, a rather crucial one, as it is pre-concepts which are the basis
> for learning mathematics and "complexes" lead to set theory only. I note
> that Paula Towsey in her work, also distinguishes "preconcepts" as a
> particular formation, not simply a name for the whole bunch of concepts
> arising prior to the formation of theoretical concepts. (Though I did
> appreciate Schmittau's rare distinction implied in the use of the term
> "theoretical concept" instead of the more usual "scientific concept.")
> Can anyone help?
> Andy
> ------------------------------------------------------------------------
> *Andy Blunden*
> http://home.pacific.net.au/~andy/
> mike cole wrote:
>> As a small contribution to this interesting thread, two of Jean
>> Schmittau's
>> writings. She has done a lot work with Davydov's ideas in math ed that may
>> give those following the discussion some useful info.
>> mike