# Re: [xmca] a minus times a plus

```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:-
>>
>>
>> 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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>>
>>
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```