As a lurker, who never spoke on xmca before, i feel a bit awkward to
begin with the negative... But for what it's worth.
I just happened to have been looking for quite a while at kids
learning about negative numbers. My conclusion, in a nutshell: The
difficulty is not with deriving the rule for minus-times-minus from
those properties of numbers one wishes to preserve (while at the
same time giving up some others!) - this is easy! Rather, the
difficulty is with the fact that this is what one is doing when one
wants to find the rule and justify it.
More specifically, for negatives to be accepted as numbers and be
seen as objects in their own right some of the unspoken old rules of
the mathematical game need to change. From now on the process of
endorsement of mathematical statements (the process of labeling them
as true) will be different from how it was so far. One will no
longer appeal to any extra-discursive evidence and the only
criterion for the endorsement will be consistency of a statement
with the system of formerly endorsed math statements. Alas, nobody
tells the kids as much as that. Nobody is explaining that from now
on, their mathematical talk will be incommensurable with their
former mathematical talk. Well, try to explain such a thing to a
kid! Or, for that matter, to the teacher. Or even to a mathematician
who is not particularly philosophically minded! In fact, 'explain"
is not the word to be used in this context. As von Neumann, a
Hungarian-turned-American mathematician, once said, "One does not
understand mathematics, one gets used to it" (of course, he did not
mean the whole of mathematics , but rather those special tacit
turnarounds that happen in it every so often.) Or, translating this
into vygotskian: when they change the rules and forget to tell you,
all you have to do is to participate, participate, and participate
again, and you will see the transformed discourse gradually growing
on you. This, of course, only if you really want this. And what if
not? A good question. Is your granddaughter, Mike, motivated enough
to persist? Is she confident enough to be able to suspend disbelief
while trying to overcome circularities and looking for a reason? And
if she is not, can anybody - you, for example - boost her confidence
and motivation? I'm not sure what would work, but I am pretty
certain about what wouldn't. The prospects of school examinations
and of whatever people are going to make of and with the grades are
reliable confidence-, fun-, and motivation-suppressors.
Does it make any sense?
anna
PS. if you want all this elaborated, you can look into my recent
paper "When the rules change and nobody tells you" in the Journal
for Learning Sciences
----- Original Message -----
From: Andy Blunden
Cc: eXtended Mind, Culture,Activity
Sent: Tuesday, April 28, 2009 8:29 AM
Subject: Re: [xmca] a minus times a plus
Emily, I quite candidly introduced my earlier message as
"the world's worst maths teacher". I developed this identity
partly by being given the task of teaching "New Maths" to
almost-innumerate kids in Brixton in the 1970s.
I was an Engineering PhD who could solve integral equations
but couldn't sing, and had no teacher training.
I was asked to teach for example, the algebra of
transformations of a figure in 3 dimensions (eg rotating by
90deg 4 times = null). This was not my choice. That was the
syllabus! But because of my own background, I couldn't
understand what they found so difficult. :)
Later I had a seminal chat with the English teacher who told
of how he only learnt to understand the workings of the
differential (those things on the back axle of motor cars
which allow the 2 wheels to go at different speeds), by
having someone tell him in words, and going over and over
those words. The diagrams meant nothing to him. My first
glimmer of thinking about thinking.
What sort thinking designed that maths syllabus?
Andy
Duvall, Emily wrote:
I think you bring up an important point, Andy. In what ways do we
understand and convey concepts?
I go back to Karpov & Gindis (2000) and the levels of problem
solving, an hierarchical arrangement that suggests to me that it is
not so much that we think differently but that perhaps we have come
to accept different levels of understanding... yet our level of
understanding could be developed:
Symbolic or abstract
Visual or visual-imagery
Concrete or visual-motor
Karpov, Y. & Gindis, B. (2000). Dynamic assessment of the level of
internalization of elementary school children's problem-solving
activity. In: C. Lidz & J. Elliott (Eds.), Dynamic assessment:
Prevailing models and applications.(pp.133-154). Oxford, UK:
Elsevier Science
~em
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