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Re: [xmca] a minus times a plus
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- Subject: Re: [xmca] a minus times a plus
- From: Andy Blunden <firstname.lastname@example.org>
- Date: Fri, 01 May 2009 13:20:04 +1000
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David, I gather from your careful explanation, that Mike
needs to build on whatever form of representation the child
has learnt so far. So if the kid has been introduced to
adding and subtracting and to negative numbers with a number
line, then subtracting negatives needs to rest on a number
line; but if they have become familiar with semantic
approaches to negative numbers, then multiplication has to
build on semantics. Is that right? Or shoud we follow FK's
advice and get the kids familiar with the multiplicity of
analogues of addition and substraction before moving on???
David H Kirshner wrote:
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.
From: firstname.lastname@example.org [mailto:email@example.com] On Behalf Of Andy Blunden
Sent: Thursday, April 30, 2009 8:31 PM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus
Here's a David Kellogg-style dialogue to illustrate but not
solve Mike's outstanding "why-problem":
Teacher: two minuses make a plus Peter.
Peter: Oh I see, so -3 times -4 is 12.
T: yes that's right.
P: Oh this is easy. -7 plus -3 is 10.
T: No, it's -10
P: But you said two minuses made a plus.
T: That's only for timesing, not adding.
P: I don't know if I can remember that. Why can't we have
the same rule for both?
T: Well, -7 plus -3 means you take away 7 and then you take
away 3 so altgother you take away 10.
P: I can see that.
T: and -3 times -4 means I take away the taking away of 3 4
P: What?! I can't do this. See you next week.
Mike Cole wrote:
Yes, right David. Very interesting.
I am left, however, without a practical procedure for help the teen who is
confusing addition/subtraction and multiplication (never mind division!).
The web has some nice number line demos that can really help with positive
and negative numbers along a single number line but the apps are
all addition/subtraction. Where is the app for multiplication??
On Wed, Apr 29, 2009 at 6:49 PM, David Kellogg <firstname.lastname@example.org>wrote:
In some languages, a double negative is an affirmative (e.g. the Chinese
hit song "Bushi Wo Bumingbai", which means "It's not that I don't
understand"). In other languages, a double negative is a negative (e.g.
French, which uses the "ne pas" construction and shows a fondness for
intensifying rather than negating double negatives in lots of other ways).
As the bastard tongue of bastards, English is somewhere in between. In my
examples, I deliberately cut out the following sequence:
a) It's worth nothing.
b) It's NOT worth nothing.
c) It ain't wort' nuttin'.
You can see that a) is a simple negative and b) is a CHINESE style double
negative, but c) is a FRENCH double negative.
Now, if we go any further (e.g. the kinds of triple and quadruple negatives
you get in something like "Nothin' ain't worth nothin' hon if it ain't
free") then we see that natural language (in numbers of negators over two
and even just with two negators) tends to use negation as an adverbial
intensifier and not really as a mathematical or logical operator.
Language is what it is because it does what it does. There is an expansion
of the Arab proverb which I well remember from my days on the street in
Algeria: "Me against my brother, me and my brother against my cousin, and
me, by brother and my cousin against you, you kafir (Kabyle, Jew, communist,
You can see that here the negation of the negation actually creates HIGHER
forms of solidarity rather than simply reversing the lower forms. You can
also see that none of them are particularly high. One can actually begin to
sympathize with Wolff-Michael's assertion, that Derek Melser claims not to
be able to see, to the effect that labor movements create solidarity by
fencing out rather than fencing in.
(I think what Wolff-Michael denies by this assertion is precisely that the
working class has historic tasks that are capable of uniting all the
oppressed and fencing out precisely those who might open the gates to the
oppressors. This is a fairly common form of denial, particularly among
academics, who are not always that careful about closing the political fence
gate after themselves.)
In order to get to the idea of negation as a reversible operator rather
than negation as an adverbial intensifier, we need a refined, more abstract,
more scientific model. This is why linguistic models really will muddle up
our mathematical understandings at some point, Mike, though I agree that
they are "bonnes a penser" at lower levels (and of course I am a hopeless
slave of language in the way I think about mathematics myself).
You know the hoary old linguist's joke about negation (and if you don't I
retell it mercilessly in my "Commentary" in the current MCA). A linguistics
professor explaining negation to a sleepy room of undergraduates: "A double
negation is a negation in French, but it's an affirmation in English. This
makes us rather doubtful of Chomsky's claim that language is based on
cognitive universals. However," he continued brightly, "there is no known
language in which a double affirmation is a negation!"
"Yeah," said someone in the back of the room. "Right."
Seoul National University of Education
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