Re: [xmca] a minus times a plus

[Mike Cole <lchcmike@gmail.com>]

I also find the "marching on the number line" idea helpful, Monica.
There ought to be several on-line ways to embody this way of thinking about
the problem, but I have not found them.
I personally, intuitively, find the  "a minus times a plus" easier to
understand than a  minus times a minus". For the latter,
something like the idea of the sign as an "operator" seems to be required.

But I'll bet your way of doing things could be a great outdoor game for
kids. How do you do multiplication?
mike

On Tue, Jul 21, 2009 at 1:23 AM, Monica Behrend <Monica.Behrend@unisa.edu.au
> wrote:

> 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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> >
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