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Re: [xmca] a minus times a plus

Come home soon, Jerry!
At present we are mostly focused on subtraction as part of the summer camp
we are involved with at the center where we work,
so we are translating all sorts of math stuff into games. But algebra crept
into our discussion for the first time today because a number of the teens
who have gotten stimulus package money to work during the summer are
struggling way up there with algebra, so the topic will continue to

Wonder how we can make negative math fun. With subtraction we have a number
of successful activities going -- in the sense that the kids voluntarily
come and participate, but they are younger kids and when water baloons are
involved......   :-)).


On Mon, Jul 20, 2009 at 6:10 AM, Andy Blunden <ablunden@mira.net> wrote:

> 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 ... :)
> Andy
> Jerry Balzano wrote:
>> 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
>> -------
>> 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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> --
> ------------------------------------------------------------------------
> Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/
> Orders: http://www.erythrospress.com/store/main.html#books
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