# Re: [xmca] a minus times a plus

Nice one Jerry. There was actually some talk about this early on in the discussion, not around Martínez, but what anyone who studied mathematics, as opposed to elementary arithmetic, understands. I think the idea that an algebra can be freely invented (eg by changing the sign multiplication rule) and a meaning for it, if any, found later, is far out of reach for someone who has not yet grasped the reason that -x-=+ is a useful rule at all. I think the learner has to build up a familiarity with a reasonable base of possible algebras and their usefulness, before being introduced to an overview of them. Learning to walk before you run is an old fashioned idea, I know ... :)
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Andy

Jerry Balzano wrote:
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Hi Mike, from across UCSD campus ... actually from across the country since I'm currently in NY ...
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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)
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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:
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A student in class asks the math teacher: "Shouldn't minus times minus make minus?"
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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."
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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?
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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.
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Jerry B

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On Jul 16, 2009, at 9:18 AM, Esther Goody wrote:

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```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,
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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
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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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Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/
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