Re: [xmca] a minus times a plus

[Ng Foo Keong <lefouque@gmail.com>]

Teacher was using imprecise language (e.g. line 1), so
the concepts of × and + are conflated.  P (maybe even T) has
low knowledge base of the deep interconnections.  [see Ma Liping's
book "Knowing and Teaching Elementary Mathematics: Teachers'
Understanding of Fundamental Mathematics in China and the United
States" ]  Deceptively simple-looking, × has many meanings beyond
repeated addition (e.g. geometrical area, grouping by rows,
grouping by columns, counting in rectangular arrays, ... etc) that
are equivalent and interconnected.  P needs time to explore the rich
interconnections in the different shades of meaning for
'multiplication'.  Maybe T does not understand P's state of
understanding or "prior knowledge" enough.  and maybe T does not
know how to help P bridge from what P knows to what P is "supposed"
to be learning (Pedagogical Content Knowledge).

in practice, there is little curriculum classroom time available/spent
for deep meaning making.  worse still if there are classroom
management issues.  worse still if P comes from a family or
culture or country in which people expect to understand/get things
instantly (e.g. instant food, instant gratification).  if P was badly
taught by a previous teacher, there is a lot for P to unlearn.  also
P might be used to certain teaching styles which are different compared
to the current teacher.  P may have picked up certain epistemological
dispositions e.g. expecting T to hand out "understanding" on the
platter that he can simply "download".

in Singapore, the "solution" to this problem is to engage private
home tutors or send kids to extra classes in tuition schools.
the children "drill and practice" till they get good-enough grades,
but often not really understanding why maths clicks together.
and many of them still hate maths despite being good at it, or
because they are forced by their parents to work so damn hard for
a subject that "does not" allow them to express their natural
creativity.

F.K.


2009/5/1 Andy Blunden <ablunden@mira.net>:
> 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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