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We used the number line. A negative number just means going backward on the line, and multiplication was (still) just adding a bunch of times, so multiple jumps backward on the number line, and a negative result is pretty obvious. I'm mystified, actually, why someone here called this a "rule change"...the beauty of it seems to be that there is no change at all...very logical. Multiplying a negative number by a negative number took a tiny bit of faith as we had to think about multiplying as something that could go in either direction, but -8 x -8 was just "going left" on the line eight times BACKWARD (or to the right), so we wound up at 64. Technically, this doesn't really work by counting out the number line, but that Mr. Walsh was just too dreamy for me to care about little details like that.
A similar problem came up for me when I was trying to explain debits and credits to a group of retail clerks. They needed to be able to issue customer satisfaction credits to customers, crediting the cash drawer with a charge to an expense account. The negative numbers gave them fits (and my cash drawers never balanced) until I drew them a little map to show how the "invisible money" was moving from place to place.
dale Dale Cyphert, PhD Associate Professor and Interim Head Department of Management University of Northern Iowa 1227 W. 27th Street Cedar Falls, IA 50614-1025 319-273-6150 dale.cyphert@uni.edu Andy Blunden wrote:
Anna, is there any way you can share your paper on the list without violating someone's property rights?I am inclined to agree with von Neumann. In my limited experience of maths teaching I recall kids objecting to the rule change (and you are quite correct about this) by just having difficulty making sense of the extended meaning which underlies the rule change, of being led into what they thouht was forbidden territory. I don't know that "I'm changing the rules here" would be my preferred way of explaining it. I think our discussants are saying: "Hey, if you look at it this way ... it's not really a rule change, it's just going a bit further."So I am inclined to think of a cycle of participate - accept the change - participate again - then reflect on the change.Hope to see you here again, Anna! Andy 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? annaPS. 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 AMSubject: 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? Andy 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-motorKarpov, 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~em_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca
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