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Re: [xmca] a minus times a plus

From the worst ex-maths teacher in the world ...

Certainly I think Ed's explanation of "why" minus numbers behave the way they do when included in operations that make intuitive explanation impossible is right. I.e., you ask that regularities that applied in the domain so far ought to be retained when the domain is extended by adding a new group of numbers. There is no meaning for "multiplying by a negative number" that can be reliably deduced from intuitive definitions of "multiply" and "negative". So the rule is that you can grasp the idea of "multiply" intuitively through the idea of repeated addition, just as you grasped the idea of addition by repeated counting. And you can grasp the idea of negative numbers in some equally intuitive way (there are several options), but not a way which can be fitted into the idea of "repeated addition".

So you take Ed's advice and rely on some general rule or visual image that worked before and require that it still work for negative numbers. In that way you move out of the bounds of intuition into mathematical thinking, guided no longer by plausible intuition, but by a mathematical rule.

That still leaves open the question as to whether you can teach general rules and mathematical reasoning to someone who has had no practice in applying the rules whose jutification you claim to achieve by this "rule extension" rationale that Ed exlained.

I was of a generation that learnt my times-tables by rote and had my first lesson on real mathematics in my last years as an undergrad. 15 years later, and then 6 years later was asked to teach "modern mathematics" to 13 year old kids who couldn't count and had no idea of what "1/2" meant except a 1 a stroke and a 2. I was not a happy chappy at the time. I blame Piaget and his "Genetic Epistemology" and a whole lot of absurdity that went down in the early 1970s.

I say: learn to ride your bike, and then learn dynamics to make sense of it afterwards.


Mike Cole wrote:
Great!! Thanks Ed and Eric and please, anyone else with other ways of
explaining the underlying concepts.
Now, we appear to have x and y coordinates here. If I am using a number line
that ranges along both x and y axes from (say) -10 to +10 its pretty
easy of visualize the relations involved. And there are games that kids can
play that provide them with a lot of practice in getting a strong sense
of how positive and negative positions along these lines work.

What might there be of a similar nature that would help kids and old college
professors understand why -8*8=64 while -8*-8=64?

Might the problem of my grand daughter, doing geometry, saying, "Well, duh,
grandpa, its just a fact!) arise from the fact (is it a fact?) that
they learn multiplication "facts" before they learn about algebra  and
grokable explanations that involve even simple equations such as
y+a=0 are unintelligible have become so fossilized that the required
reorganization of understanding is blocked?


On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu> wrote:


    It is simply (of course, it isn't simple by the way) because, the
negative integers (and, if you wish, zero) were added to the natural numbers
in a way that preserves (in a sense) their (the natural numbers) usual
arithmetical regularities. It would be unfortunate if something that was
true in the natural numbers was no longer true in the integers, which is a
extension that includes them.  Perhaps the easiest way to the negative x
positive business is as follows (and, of course, this can be made opaquely
precise - smile):

3 x 1 = 3
2 x 1 = 2
1 x 1 = 1
0 x 1 = 0

so what, given regularity in the naturals + zero) do you think happens
next? This thinking works for, of course, for negative times negative. The
opaque proof is more or less as follows.

Negative numbers are solutions to natural number equations of the form (I'm
simplifying all this a little)

                     x + a = 0    ('a' a natural number)

and likewise positive numbers  are solutions to natural number equations of
the form

                    y = b          ('b' a natural number)

Multiplying these two equations in the usual fashion within the natural
numbers gives

            xy + ay = 0

or substituting for y

      xy + ab = 0

so, by definition, xy is a negative number.

Notice how all this hinges on the structure of the natural numbers (which
I've somewhat assumed in all this).


On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:

 Since we have some mathematically literate folks on xmca, could someone
please post an explanation of why

multiplying a negative number by a positive numbers yields a negative
number? What I would really love is an explanation
that is representable in a manner understandable to old college professors
and young high school students alike.

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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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