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RE: [xmca] a minus times a plus
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.
From: firstname.lastname@example.org [mailto:email@example.com] On Behalf Of Ng Foo Keong
Sent: Thursday, April 30, 2009 10:15 PM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus
what do you think of using coloured discs/chips (two colours) or coins?
unlike N or P which can be arranged only linearly, you can arrange
coloured chips in a straight line or in an array.
3 × (-2) means 3 times (3 rows of) two red chips.
[if you start off with pairs of red-blue pairs (equivalent to 0),
you can put blue on top of the red, but it doesn't matter]
-2 × 3 means twice take away 3 blue chips, or take away 2 rows of
3 blue chips, leaving you with 2 rows of 3 red chips.
by rotating the array of chips 90 degrees, you can see that that
is equivalent to 3 rows of 2 red chips.
2009/5/1 David H Kirshner <firstname.lastname@example.org>:
> Here's an instructional model that spans both additive and multiplicative cases.
> It builds on a typical additive representation for integers that works something like this.
> Let P be a marker for +1 and N a marker for -1. Then NP together are equivalent to 0. In this way, we can model any integer addition or subtraction. For instance
> -5 + 3 is modeled as NNNNN + PPP = NP NP NP NN = -2.
> (-6)-(-3) involves, simply, the removal of 3Ns from the following set: NNNNNN.
> For -5 - 3, we seem to have more of a problem, because we are give NNNNN and expected to take away some Ps. We do so by representing NNNNN as NNNNN NP NP NP. Now when we take away our 3 Ps, we're left with NNNNNNNN = -8.
> The extension to multiplication works by making everything an addition or subtraction from the starting point of 0. For instance
> 3 X (-2) simply means collecting together 3 groups of NN to end up with NN NN NN or -6.
> -4 X 2 means taking away 4 groups of 2. In order to do this, we have to have a whole bunch of Ps. So we start from 0 represented as (NP NP) (NP NP) (NP NP) (NP NP). When we remove our 4 groups of 2 we're left with NNNNNNNN or -8.
> -4 X (-2) is almost exactly the same, except we're going to remove the Ns and be left with PPPPPPPP or 8.
> I really like the fact that we can find a semantic representation that accommodates the full range of additive and multiplicative relations for integers. But I'm not prepared to go the next step and assert some special status for this representation as somehow central to the meaning of integer operations. From a conceptual perspective, this representation is as limited as all the rest. It captures some semantic aspects of the integers, but not all of them. For instance, 3 X (-2) from this perspective isn't commutative, -2 X 3 reflecting a completely distinct set of operations and relations. We can only make the case for the centrality of a representation if we can show how others can be built from it. In the meantime, we seem to be back at the initial impasse in which mastering the semantic realm of integer operations involves dealing with disconnected meanings spanning independently coherent representations.
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