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Re: [xmca] a minus times a plus

Mike, according to our wonderful correspondents, the number
line is the favourite "visual/practical image". I believe them.

People have rightly suggested the use of this to present
mutliplication tables in the form of a number line, which,
to start with, stop at 0. Then you ask: "what if we extend
this ...?" This looks great to me.

My only point is that the teacher would use this
practical/visual approach to get kids filling in the blanks
correctly ... *and then* abstract the general rule "minus
times minus = plus". You have to learn this as a rule at
some point. Question is when and on what basis is it to be
justified and understood?

There is no way you can introduce that rule prior to
practical/intuitive manipulations. If some appropriate
intuitive image is used to present the base regularity, the
kids will be blissfully unaware that stepping across the
boundary into a new domain of numbers, which you have
presciently placed as a continuation of the starting doamin
is an unwarranted, arbitrary and unjustifable logical leap.
And why not! But until kids have left behind them the
intuitive justifications, they have not learnt mathematics,
which is after all the object of the exercise, isn't it? Or
is it?

The great thing about Ed's first contribution, I think, is
that he showed how foundations-of-mathematics thinking can
mirror the operation used for teaching: i.e., extension of a
domain. And this is a foundation on which people can reflect
and learn about mathematics, as well as carry out the
required opeations when asked.

My only point is that kids need to first practice the
operations in the narrow and then the wider domain, and then
abstract the rule, and then much later learn the logical
meaning of domain-extension. At bottom, the answer to "why
is - x - = + ?" is "because we say so". It is no more
provable than "1 + 1 = 2" and intuitive demonstrations which
purport to prove it are actually fooling people. But maybe
it is necessary to fool people first and explain why later,
if they're interested.


To return to the example that we were discussing the other
day: e to the i pi = -1. This is justified in university
maths classes using the domain extension rationale. It is
difficult to see any other way of approaching it. And that
presupposes familiarity with 4 domains of numbers. But it
was fantastic at the time to already have the *Argand
diagram*. The Argand Diagram looks like Cartesian x-y
coordinates, and is understood as a representation of
"complex numbers" in which the domain of numbers has been
extended to include the square root of minus one, and
represented as a "2 dimensional number". With the Argand
Diagram a lot of the symmetries of this great equation
become visually very compelling.


Mike Cole wrote:

Andy-- Visual image PLEASE!!!
mike
On Mon, Apr 27, 2009 at 6:05 PM, Andy Blunden <ablunden@mira.net <mailto: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
        <mailto: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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-- ------------------------------------------------------------------------ 
    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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--
------------------------------------------------------------------------
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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