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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*: Sat, 2 May 2009 22:45:17 +0200*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=FfTHe8FkPJXhhCCFDx8W0+ZEgdKIsXnHP0ZBxv6SgkJEpCBBvfyFXzRU7ktl7CPmN yaGk/khp3/W6pNP2dGsRw==*In-reply-to*: <731CECC23FB8CA4E9127BD399744D1EC018B6BD5@email001.lsu.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>*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 May 2, 2009, at 12:10 AM, David H Kirshner wrote:

Foo Keong,I am fully supportive of efforts to create a master narrative thatorganizes all of the semantic representations of integers into acoherent scheme. This can be the basis for a curriculum that conveyssomething of the systematicity and intellectual rigor ofmathematics. Unfortunately, I've not yet seen that done for integeroperations. Of course, process/object reification--a la Sfard,Schwartz, Tall, Harel, etc.--is a wonderful resource toward thateffort. But the argument needs to be framed in the particular, notthe general. Although processes cohere into objects which laterparticipate in higher level processes, these reifications followspecific trajectories. A classic example is the reification of anexpression as a sequence of instructions (e.g., 3x + 2 meaning takea number, multiply it by 3, and add 2) into an expression as theresult obtained through that process. As Sfard noted, only inmathematics does the recipe become the cake. So for the case athand, it would be necessary to argue that the process of negation astake-away compresses into the object of negative as location (forexample on a number line). Even then, the scope of the negationremains 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 Iintroduced earlier that we're now discussing).But I think we're on the same team. David -----Original Message-----From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Ng Foo KeongSent: Friday, May 01, 2009 2:46 PM To: eXtended Mind, Culture, Activity Subject: Re: [xmca] a minus times a plus So the negative sign in -2 × 3 is being interpreted as a /process/ whereas the negative sign in 3 × -2 is being interpreted as an /end-product/ (i.e. after taking away the blue chips from the zero-pairs, you get 2 red chips; -2 = 0 - 2). as an advanced learner i don't feel that these are different, because (using Anna Sfard's theory) i have /reified/ the process, compressed it as it were until i can treat it like an object without any problems. for a beginner, there is still a very wide gulf between the process and the end-product. is there another way out? is it the representation that is the problem, or should educators put more focus on the learner's learning experiences? F.K. 2009/5/1 David H Kirshner <dkirsh@lsu.edu>:Foo Keong,Yes, you can increase the semantic span of this approach bychanging the media, as you suggest. But the basic semanticlimitation still applies. The negative sign in -2 × 3 is beinginterpreted as a subtraction [-2 × 3 = 0 - (2 x 3)]--very differentfrom the negative sign in 3 × (-2). Thus the lack of a commutativeinterpretation of multiplication in this representation is notcompletely solved by arraying markers in a rectangular configuration.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

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**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>

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