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These files are maintained strictly for historical reference, research, and recordkeeping purposes, and the content has not been altered or updated since archiving. Content on this site may not meet current digital accessibility standards. If you require an alternative, accessible format of any information found on this site, please contact Bruce Jones, bjones@ucsd.edu to request an accommodation.
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