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Re: [xmca] a minus times a plus

I was rather amazed at all the "logics" offered to explain multiplication with negative numbers. And quite happy to see Anna Sfard making it clear that sometimes "learning" these things is NOT a matter of understanding them -- which I think always means analogizing them to something else -- but of learning to use them on their "own" terms. Mathematics grew up together with physics, surveying, accountancy-of-old, etc., and so there are a lot of ways of "justifying" or making "plausible" how mathematics works in terms of other kinds of activities.

But there is another approach to mathematics: just postulate some rules and see what happens. You then pick the set of rules that helps you do what you want to do, and that choice may depend on how things work in some non-mathematical activity. Or it may depend on what leads to more "interesting mathematics" as defined in the culture of mathematicians (and so subject to historical change and cultural differences). In a way this is what lies behind some of the "explanations" offered here that argued that the negative-times- positive rule, as part of a larger rule set for working with negative numbers, led to desirable properties, whether correspondence to physical phenomena or more purely mathematical properties like closure, commutativity, etc.

I think it's important, however, to see, as Anna emphasizes, that there is a certain "arbitrariness" involved in this, or if you like it better: a freedom of choice. Yes, it's structure-and-agency all over again! Structure determines that some things fit into bigger pictures and some don't, but agency is always at work deciding which pictures, which kind of fit, which structures, etc. And behind that values, and culture, and how we feel about things.


PS. In my own thinking and teaching, I would start with justifying multiplying an original negative by some positive number of multiples as being just the usual increase of negative magnitude, or as repeated subtraction equivalent to one big subtraction. The un-intuitive bit comes when thinking of it the other way around: an original positive number "multiplied a negative number of times". That really makes no sense on its own, and you have to choose a criterion for setting the rule for how it works. The simplest criterion is that you get the same answer as if you did it the other way around (i.e. same as negative multiplied a positive number of times). That is, you go for extending the commutativity property (namely a*b always equals b*a) from positive numbers to all integers. Of course you don't HAVE to do that.

Jay Lemke
Educational Studies
University of Michigan
Ann Arbor, MI 48109

On Apr 28, 2009, at 12:12 PM, Anna Sfard wrote:

As a lurker, who never spoke on xmca before, i feel a bit awkward to begin with the negative... But for what it's worth.

I just happened to have been looking for quite a while at kids learning about negative numbers. My conclusion, in a nutshell: The difficulty is not with deriving the rule for minus-times-minus from those properties of numbers one wishes to preserve (while at the same time giving up some others!) - this is easy! Rather, the difficulty is with the fact that this is what one is doing when one wants to find the rule and justify it.

More specifically, for negatives to be accepted as numbers and be seen as objects in their own right some of the unspoken old rules of the mathematical game need to change. From now on the process of endorsement of mathematical statements (the process of labeling them as true) will be different from how it was so far. One will no longer appeal to any extra-discursive evidence and the only criterion for the endorsement will be consistency of a statement with the system of formerly endorsed math statements. Alas, nobody tells the kids as much as that. Nobody is explaining that from now on, their mathematical talk will be incommensurable with their former mathematical talk. Well, try to explain such a thing to a kid! Or, for that matter, to the teacher. Or even to a mathematician who is not particularly philosophically minded! In fact, 'explain" is not the word to be used in this context. As von Neumann, a Hungarian-turned-American mathematician, once said, "One does not understand mathematics, one gets used to it" (of course, he did not mean the whole of mathematics , but rather those special tacit turnarounds that happen in it every so often.) Or, translating this into vygotskian: when they change the rules and forget to tell you, all you have to do is to participate, participate, and participate again, and you will see the transformed discourse gradually growing on you. This, of course, only if you really want this. And what if not? A good question. Is your granddaughter, Mike, motivated enough to persist? Is she confident enough to be able to suspend disbelief while trying to overcome circularities and looking for a reason? And if she is not, can anybody - you, for example - boost her confidence and motivation? I'm not sure what would work, but I am pretty certain about what wouldn't. The prospects of school examinations and of whatever people are going to make of and with the grades are reliable confidence-, fun-, and motivation-suppressors.

Does it make any sense?
PS. if you want all this elaborated, you can look into my recent paper "When the rules change and nobody tells you" in the Journal for Learning Sciences

----- Original Message -----
From: Andy Blunden
Cc: eXtended Mind, Culture,Activity
Sent: Tuesday, April 28, 2009 8:29 AM
Subject: Re: [xmca] a minus times a plus

Emily, I quite candidly introduced my earlier message as
"the world's worst maths teacher". I developed this identity
partly by being given the task of teaching "New Maths" to
almost-innumerate kids in Brixton in the 1970s.

I was an Engineering PhD who could solve integral equations
but couldn't sing, and had no teacher training.

I was asked to teach for example, the algebra of
transformations of a figure in 3 dimensions (eg rotating by
90deg 4 times = null). This was not my choice. That was the
syllabus! But because of my own background, I couldn't
understand what they found so difficult. :)

Later I had a seminal chat with the English teacher who told
of how he only learnt to understand the workings of the
differential (those things on the back axle of motor cars
which allow the 2 wheels to go at different speeds), by
having someone tell him in words, and going over and over
those words. The diagrams meant nothing to him. My first
glimmer of thinking about thinking.

What sort thinking designed that maths syllabus?


Duvall, Emily wrote:
I think you bring up an important point, Andy. In what ways do we understand and convey concepts? I go back to Karpov & Gindis (2000) and the levels of problem solving, an hierarchical arrangement that suggests to me that it is not so much that we think differently but that perhaps we have come to accept different levels of understanding... yet our level of understanding could be developed:

Symbolic or abstract
Visual or visual-imagery
Concrete or visual-motor

Karpov, Y. & Gindis, B. (2000). Dynamic assessment of the level of internalization of elementary school children's problem-solving activity. In: C. Lidz & J. Elliott (Eds.), Dynamic assessment: Prevailing models and applications.(pp.133-154). Oxford, UK: Elsevier Science


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