Re: [xmca] a minus times a plus

[Jerry Balzano <gjbalzano@ucsd.edu>]

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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