Re: [xmca] a minus times a plus

[Ng Foo Keong <lefouque@gmail.com>]

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
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