Re: [xmca] a minus times a plus

[Mike Cole <lchcmike@gmail.com>]

Andy-- Visual image PLEASE!!!
mike

On Mon, Apr 27, 2009 at 6:05 PM, Andy Blunden <ablunden@mira.net> wrote:

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