RE: [xmca] a minus times a plus

["David H Kirshner" <dkirsh@lsu.edu>]

Andy,

I don't think we need to rely on the happenstance of "whatever form of representation the child has learnt so far." Our pedagogical responsibility is to get students comfortable with a variety of representations that span the semantic terrain of integer operations. We do need to be mindful of conceptual linkages and prerequisites as we develop a curriculum to do this. Unfortunately (as I suggested in my last post), there doesn't seem to be a master script for how to do this. We end up covering the semantic terrain with a patchwork of representations that overlap at some places and have disconnections at other places. Thus for students learning to work with and coordinate these representations is a matter of establishing correlational connections among representational forms and contexts of application. This linking together of semantics and syntax of mathematics is a messy business that lacks the coherence and elegance we like to associate with mathematical theory. 

Interestingly, this theme was addressed by Anna Sfard in conversation with Pat Thompson (a Piagetian constructivist scholar) at a conference I organized at LSU about 15 years ago. Needless to say she (and Pat) did a bang-up job of laying out these issues at a time when representational heterogeneity wasn't quite so well-recognized--thanks, Anna!

David


-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Andy Blunden
Sent: Thursday, April 30, 2009 10:20 PM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus

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???

Andy

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. 
> 
> David
> 
> 
> 
> 
> -----Original Message-----
> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] 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 
> times.
> P: What?! I can't do this. See you next week.
> 
> Andy
> 
> 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??
>> mike
>>
>> On Wed, Apr 29, 2009 at 6:49 PM, David Kellogg <vaughndogblack@yahoo.com>wrote:
>>
>>> Mike, Eugene:
>>>
>>> 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,
>>> Tunisian, etc.)!"
>>>
>>> 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."
>>>
>>> David Kellogg
>>> Seoul National University of Education
>>>
>>>
>>>
>>>
>>>
>>>
>>>
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-- 
------------------------------------------------------------------------
Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
 From Erythrós Press and Media <http://www.erythrospress.com/>.

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