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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Dale Cyphert <dale.cyphert@uni.edu>*Date*: Tue, 28 Apr 2009 08:00:11 -0500*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=LatL+oA+9ZSvEGl++NC9OPQqFyynyrkrD2W7Yhm4QOYaBOoDJQnCjqYo59H2btYEv n5PXGEYGj3tV22fosirPA==*In-reply-to*: <49F6DC22.4060002@mira.net>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <30364f990904271547o5b4df21eifca69bf8318483f2@mail.gmail.com> <2C46D7A7-AD94-441C-AABD-269045835E3D@umich.edu> <0193A85F-03A7-4811-B912-217722A5770B@uvic.ca> <C23F99F4-D4AC-463B-AF11-30585E74F7CF@me.com> <30364f990904271949r3cfef116m89d39ea419d112b2@mail.gmail.com> <p06240805c61c2a15a97a@[192.168.1.65]> <49F68AEE.40502@mira.net> <3B19033D3E2EC34C97DF364119A79A61C90CDD@EXVS1.its.uidaho.edu> <49F69438.4050900@mira.net> <18289BE6B2A646DFABDC1C342E7ED5A1@x2a4f0b4148df4> <49F6DC22.4060002@mira.net>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu*User-agent*: Thunderbird 2.0.0.21 (Windows/20090302)

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 withoutviolating someone's property rights?I am inclined to agree with von Neumann. In my limited experience ofmaths teaching I recall kids objecting to the rule change (and you arequite correct about this) by just having difficulty making sense of theextended meaning which underlies the rule change, of being led into whatthey thouht was forbidden territory. I don't know that "I'm changing therules here" would be my preferred way of explaining it. I think ourdiscussants are saying: "Hey, if you look at it this way ... it's notreally 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 tobegin with the negative... But for what it's worth.I just happened to have been looking for quite a while at kidslearning about negative numbers. My conclusion, in a nutshell: Thedifficulty is not with deriving the rule for minus-times-minus fromthose properties of numbers one wishes to preserve (while at the sametime giving up some others!) - this is easy! Rather, the difficulty iswith the fact that this is what one is doing when one wants to findthe rule and justify it.More specifically, for negatives to be accepted as numbers and be seenas objects in their own right some of the unspoken old rules of themathematical game need to change. From now on the process ofendorsement of mathematical statements (the process of labeling themas true) will be different from how it was so far. One will no longerappeal to any extra-discursive evidence and the only criterion for theendorsement will be consistency of a statement with the system offormerly endorsed math statements. Alas, nobody tells the kids as muchas that. Nobody is explaining that from now on, their mathematicaltalk will be incommensurable with their former mathematical talk.Well, try to explain such a thing to a kid! Or, for that matter, tothe teacher. Or even to a mathematician who is not particularlyphilosophically minded! In fact, 'explain" is not the word to be usedin this context. As von Neumann, a Hungarian-turned-Americanmathematician, once said, "One does not understand mathematics, one gets used to it" (of course, he did not meanthe whole of mathematics , but rather those special tacit turnaroundsthat 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 isto participate, participate, and participate again, and you will see thetransformed discourse gradually growing on you. This, of course, onlyif you really want this. And what if not? A good question. Is yourgranddaughter, Mike, motivated enough to persist? Is she confidentenough to be able to suspend disbelief while trying to overcomecircularities and looking for a reason? And if she is not, can anybody -you, for example - boost her confidence and motivation? I'm not surewhat would work, but I am pretty certain about what wouldn't. Theprospects of school examinations and of whatever people are going tomake of and with the grades are reliable confidence-, fun-, andmotivation-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 forLearning 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 weunderstand and convey concepts?I go back to Karpov & Gindis (2000) and the levels of problemsolving, an hierarchical arrangement that suggests to me that it isnot so much that we think differently but that perhaps we have cometo accept different levels of understanding... yet our level ofunderstanding could be developed:Symbolic or abstract Visual or visual-imagery Concrete or visual-motorKarpov, Y. & Gindis, B. (2000). Dynamic assessment of the level ofinternalization of elementary school children's problem-solvingactivity. In: C. Lidz & J. Elliott (Eds.), Dynamic assessment:Prevailing models and applications.(pp.133-154). Oxford, UK: ElsevierScience~em_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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**References**:**[xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] a minus times a plus***From:*Wolff-Michael Roth <mroth@uvic.ca>

**Re: [xmca] a minus times a plus***From:*Steve Gabosch <stevegabosch@me.com>

**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Jim Levin <jalevin@ucsd.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

**RE: [xmca] a minus times a plus***From:*"Duvall, Emily" <emily@uidaho.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

**Re: [xmca] a minus times a plus***From:*Anna Sfard <annasfar@math.msu.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

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