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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Mike Cole <lchcmike@gmail.com>*Date*: Sat, 2 May 2009 15:11:44 -0700*Delivered-to*: xmca@weber.ucsd.edu*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:reply-to:in-reply-to :references:date:message-id:subject:from:to:content-type; bh=oSzq2UHc4kqWLe35hCvQGwAr4qsPfbPlEqmG/zrJtFs=; b=IPMAju1rCg7MVJaXjJte0NZyp/4JoVNDWM25N/32NxRY9lkGMC0EqJRuAXZmqM8RLT c8xo45XnH2kc0E2xU3s/Tx22XWCsUBt/C0wWQ3UYapq7cpZAvqaK7uqt6fQsm7rQTEUs NQcPon3gjqPyzJNNLPG7SXhEZa6K+u9m8S0YM=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:reply-to:in-reply-to:references:date:message-id :subject:from:to:content-type; b=rpDkt7qWrXdTFKYuMHE6vF1w8zCyyAsoeSQlYJmJeF6boOMp3bEkIePz+L9HYWAybe +Fz9hdiw+gKidfuhzqvbyHfA5QIDAkEWHfUXKpvaiMiJW4TKP3im5BUEkRKOznqwxPx9 P30wstETnY7B3Oo+C0lfTxzZMJe9XO7ZV072E=*In-reply-to*: <E0584F84-9159-4E06-9B53-C6130F59CD5E@umich.edu>*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*: <384808.62393.qm@web110310.mail.gq1.yahoo.com> <30364f990904300839g29d5f7dbh3c016e17ed25bfa0@mail.gmail.com> <49FA50CE.1030102@mira.net> <731CECC23FB8CA4E9127BD399744D1EC018B6AA3@email001.lsu.edu> <14a6419f0904302015x4418174cm7f8a835049d83d9a@mail.gmail.com> <731CECC23FB8CA4E9127BD399744D1EC018B6AB2@email001.lsu.edu> <14a6419f0905011246l70e7a63ck66134f5fe0011f87@mail.gmail.com> <731CECC23FB8CA4E9127BD399744D1EC018B6BD5@email001.lsu.edu> <14a6419f0905021213ncb88568vd6fc71abf1614c12@mail.gmail.com> <E0584F84-9159-4E06-9B53-C6130F59CD5E@umich.edu>*Reply-to*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

Build it Jay, and we will keep coming. mike On Sat, May 2, 2009 at 2:24 PM, Jay Lemke <jaylemke@umich.edu> wrote: > Forgive me, but I find myself nearing the limits of language as a medium > here. I need some pictures, if not an animation! > > What ever happened to the plans for a next-generation xmca where we could > have more than text?? > > JAY. > > Jay Lemke > Professor > Educational Studies > University of Michigan > Ann Arbor, MI 48109 > www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke> > > > > > On May 2, 2009, at 9:13 PM, Ng Foo Keong wrote: > > to David Kirschner:- >> >> How about my 4 quadrants and smiley faces array example? >> >> 2 × 3: first put the 2 rows × 3 column rectangular array of smileys on >> the first quadrant (x positive and y positive) put a right-pointing arrow >> along the lower edge (representing +3) and an up-pointing arrow along the >> left edge (representing +2). the smileys have an smirk on the left >> corner of the lips, so that it looks like an arrowhead indicating a >> clockwise turn -- there are 6 clockwise (+) smileys. >> >> when you do mirror reflection of the above in y-axis, you get 2 × -3 >> 2 rep. by up-pointing arrow along an edge >> -3 rep. by left-pointing arrow along an edge >> and the result is 6 anticlockwise (-) smileys i.e. -6 >> >> when you do mirror reflection of the above in x-axis, you get -2 × -3 >> -2 rep. by down-pointing arrow along an edge >> -3 rep. by left-pointing arrow along an edge >> and the result is 6 clockwise (+) smileys i.e. +6 (upside down doesn't >> matter) >> >> when you do mirror reflection of the above in y-axis, you get -2 × 3 >> -2 rep. by down-pointing arrow along an edge >> 3 rep. by right-pointing arrow along an edge >> and the result is 6 anticlockwise (-) smileys i.e. -6 >> >> when you do mirror reflection of the above in x-axis, you get 2 × 3 >> the original array. >> >> '-' is reified to mean 'opposite', the result of a mirroring process:- >> left, as opposite of right (+) >> down, as opposite of up (+) >> anti-clockwise, as opposite of clockwise (+) >> >> F.K. >> >> >> 2009/5/2 David H Kirshner <dkirsh@lsu.edu>: >> >>> Foo Keong, >>> >>> I am fully supportive of efforts to create a master narrative that >>> organizes all of the semantic representations of integers into a coherent >>> scheme. This can be the basis for a curriculum that conveys something of the >>> systematicity and intellectual rigor of mathematics. Unfortunately, I've not >>> yet seen that done for integer operations. Of course, process/object >>> reification--a la Sfard, Schwartz, Tall, Harel, etc.--is a wonderful >>> resource toward that effort. But the argument needs to be framed in the >>> particular, not the general. Although processes cohere into objects which >>> later participate in higher level processes, these reifications follow >>> specific trajectories. A classic example is the reification of an expression >>> as a sequence of instructions (e.g., 3x + 2 meaning take a number, multiply >>> it by 3, and add 2) into an expression as the result obtained through that >>> process. As Sfard noted, only in mathematics does the recipe become the >>> cake. So for the case at hand, it would be necessary to argue that the >>> process of negation as take-away compresses into the object of negative as >>> location (for example on a number line). Even then, the scope of the >>> negation remains non-symmetric. The negative in 3 × -2 applies to the "2". >>> The negative in -2 × 3 applies to the 2 x 3 (in the representation I >>> introduced earlier that we're now discussing). >>> >>> But I think we're on the same team. >>> >>> David >>> >> _______________________________________________ >> xmca mailing list >> xmca@weber.ucsd.edu >> http://dss.ucsd.edu/mailman/listinfo/xmca >> >> >> > _______________________________________________ > xmca mailing list > xmca@weber.ucsd.edu > http://dss.ucsd.edu/mailman/listinfo/xmca > _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**References**:**RE: [xmca] a minus times a plus***From:*"David H Kirshner" <dkirsh@lsu.edu>

**Re: [xmca] a minus times a plus***From:*Ng Foo Keong <lefouque@gmail.com>

**RE: [xmca] a minus times a plus***From:*"David H Kirshner" <dkirsh@lsu.edu>

**Re: [xmca] a minus times a plus***From:*Ng Foo Keong <lefouque@gmail.com>

**Re: [xmca] a minus times a plus***From:*Jay Lemke <jaylemke@umich.edu>

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