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RE: [xmca] a minus times a plus

Hi, having been an XCMA lurker and a post maths teacher and maths educator, I have followed this discussion with interest - and certainly concur with the conceptual issues around this! 

Two analogies that have somehow promoted Eureka moments for my students and student teachers have been:

1. Marching forwards (+ add) and backwards (- subtract) along number lines. We reckoned that we defined the +4 as walking four steps in the direction of the larger numbers, and -4 as 'turn around' and walk four steps! We had great fun and I still remember a trainee secondary teacher from a Pacific Island country doing that with much aplomb! That can be extended to the multiplication. Somehow though then we wanted to not step but jump in lots of 4! So we turned into frogs!  

2. Money ... so -4 X 4 = four dollar debt by 4 ... means you owe 16 dollars i.e. -16. So -4 x -4 means four dollar debt and four times repaid ... +16. This one is not so neat, but can make the point too.

Cheers

Monica Behrend
Research Education Adviser (International)
The University of South Australia
Adelaide, SA Australia




-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Ng Foo Keong
Sent: Tuesday, 21 July 2009 11:38 AM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus

from my teaching experience, the main confusion is between (in our conventional
notation)  -4 x -4 = 16  and   -4  - 4 = -8.

learners must know the difference in meanings between the two.

in mathematics, the definitions of operations and the use of symbols
are actually
arbitrary, and it is possible to explore alternative systems of
algebra (like what
professional mathematicians / algebraists do) as long as they make sense and are
consistent.

i'd say, just like non-Euclidean geometries, alternative algebras would be
interesting topic for exploration away from the boredom of "normal math",
but students still need to be able to discern different meanings and use
the appropriate different symbols.  the systems that students propose/explore
should make sense, be well-defined and be consistent (non-contradictory) -- what
the community of mathematicians would require.  explorations give
students a sense
of agency and creativity and builds their identity as active contributors of
mathematical knowledge (instead of being passive followers of algorithms).

F.K.




2009/7/20 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
>
> -------
>
>
> 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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