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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Jay Lemke <jaylemke@umich.edu>*Date*: Tue, 28 Apr 2009 15:40:52 +0200*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=NST6r42FaCJ3NFgzDqBrtJApx4+sQ9Q+EiOVMLLAYFicdK1Qcm+8JJtEyd3MCXUXd 5WClRJQAjk+bFmKPf7zzg==*In-reply-to*: <18289BE6B2A646DFABDC1C342E7ED5A1@x2a4f0b4148df4>*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>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

JAY.

Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke 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 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 thesame time giving up some others!) - this is easy! Rather, thedifficulty is with the fact that this is what one is doing when onewants to find the rule and justify it.More specifically, for negatives to be accepted as numbers and beseen as objects in their own right some of the unspoken old rules ofthe mathematical 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 nolonger appeal to any extra-discursive evidence and the onlycriterion for the endorsement will be consistency of a statementwith the system of formerly endorsed math statements. Alas, nobodytells the kids as much as that. Nobody is explaining that from nowon, their mathematical talk will be incommensurable with theirformer mathematical talk. Well, try to explain such a thing to akid! Or, for that matter, to the teacher. Or even to a mathematicianwho is not particularly philosophically minded! In fact, 'explain"is not the word to be used in this context. As von Neumann, aHungarian-turned-American mathematician, once said, "One does notunderstand mathematics, one gets used to it" (of course, he did notmean the whole of mathematics , but rather those special tacitturnarounds that happen in it every so often.) Or, translating thisinto vygotskian: when they change the rules and forget to tell you,all you have to do is to participate, participate, and participateagain, and you will see the transformed discourse gradually growingon you. This, of course, only if you really want this. And what ifnot? A good question. Is your granddaughter, Mike, motivated enoughto persist? Is she confident enough to be able to suspend disbeliefwhile trying to overcome circularities and looking for a reason? Andif she is not, can anybody - you, for example - boost her confidenceand motivation? I'm not sure what would work, but I am prettycertain about what wouldn't. The prospects of school examinationsand of whatever people are going to make of and with the grades arereliable confidence-, fun-, and motivation-suppressors.Does it make any sense? annaPS. if you want all this elaborated, you can look into my recentpaper "When the rules change and nobody tells you" in the Journalfor 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? 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:Elsevier Science~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>

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