Hi Mike, from across UCSD campus ... actually from across the country
since I'm currently in NY ...
by my count, this topic has accumulated 147 emails since your original
April 27 posting (this one would be #148) ... quite a fecund topic, and
not bad on the longevity meter either! (nearly 3 months)
I just this morning ran across another remarkable connection to the
topic that I had to tell you and everyone else about as I was in google
bookland, pursuing cross refs to -- of all things -- WIttgenstein's
Lectures on the Foundations of Mathematics. It's a rather fascinating
book called Negative Math, by Alberto A. Martínez, and the online "book
overview" starts off, believe it or not, just like this:
A student in class asks the math teacher: "Shouldn't minus times minus
make minus?"
There's a chapter in the book with the seemingly heretical title, "Math
is Rather Flexible", and as if to demonstrate this via a kind of tour de
force with an exceptional resonance for our discussion, Martinez asks
"can we construct a system in which, say, -4 x -4 = -16? Actually, yes
we can."
This raises the question: Is such a book good for students or bad for
students? It seems terribly subversive, doesn't it? I can imagine more
than one math teacher applauding it "in principle" but panning it in
practice for fear that it "might confuse" a student who was "having
enough trouble learning the (correct) rules". But (on the other hand)
perhaps if we had a more playful, less rigid attitude about "the rules",
we would engender a less fearful attitude in students about them.
Perhaps books in the spirit of Martinez' Negative Math would be a proper
antidote to such (apparently!) unproductive approaches to thinking about
teaching and learning mathematics?
The book is in our library at UCSD, and I'd be more than happy to "play
with it" with you when I return (beginning of August), if you like. In
the meantime, the Google Books link is here.
Jerry B
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On Jul 16, 2009, at 9:18 AM, Esther Goody wrote:
Dear Mike,
Hope you have caught up with sleep since Alaska!
Until now I have not looked at the "a minus times a plus" topic in XMCA,
supposing it would be about word games or something. Now I see it is
about
'How and What to teach in school maths'? This is something I stumbled
into
in my northern Ghana classrooms. The first Spencer Foundation grant was
about differences in learning literacy skills in L1 and L2 in high and
low
authority classrooms. However a large section of the middle year
report was
about reasons why kids were not learning school maths in the upper
primary
grades in village schools.
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