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RE: [xmca] a minus times a plus
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.
From: email@example.com [mailto:firstname.lastname@example.org] On Behalf Of Ng Foo Keong
Sent: 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
2009/5/1 David H Kirshner <email@example.com>:
> Foo Keong,
> Yes, you can increase the semantic span of this approach by changing the media, as you suggest. But the basic semantic limitation still applies. The negative sign in -2 × 3 is being interpreted as a subtraction [-2 × 3 = 0 - (2 x 3)]--very different from the negative sign in 3 × (-2). Thus the lack of a commutative interpretation of multiplication in this representation is not completely solved by arraying markers in a rectangular configuration.
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