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Re: Non-Euclidean Arithmetic
Posted:
Sep 9, 2012 1:25 AM
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On Sat, Sep 8, 2012 at 11:34 PM, Joe Niederberger
<niederberger@comcast.net> wrote:
>>The lack of fluent proportional reasoning in a large percentage of students all the way up through adulthood is one of big problems that must be addressed,
>
> SO you are going to address it by assuming kids just learning to multiply two integers are ready to be told "you just scale one number by the other"?
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> Let me know how it works out.
>
> Joe N
When would you teach this model? Never?
Regardless:
Kids are smarter than you think.
In my post
http://mathforum.org/kb/message.jspa?messageID=7886496
I did not say what you said I say. I said that teaching the model of
repeated addition is fine, but that scaling also should be taught, but
I did not say that we should say "you just scale one number by the
other" when teaching it.
Kids are smart enough to see that if we start at point 1 and make that
length 3 times longer we end up at point 3. Nothing wrong with using
repeated addition here to count up to the target point, and I think
Devlin is fine with that. They are smart enough to then see that if we
start at point 5 and make that length of 3 times longer then we end up
at point 15. Again, nothing wrong with using repeated addition here to
count up to the target point, and I think Devlin is fine with
that.count up to the target point. Finally kids are smart enough to
see the same pattern being done: "Stretching a length of 5 out to make
it 3 times longer is the same pattern as "stretching" a length of 1
out to make it 3 times longer. 15 is to 5 as 3 is to 1.
If you think 7 year old kids are too stupid for this, then OK, go
ahead and believe it. I don't believe it.
Side note: This teaching of patterns, or mathematics as patterns, is
in line with the theme of Devlin's book "Mathematics: The Science of
Patterns: The Search for Order in Life, Mind and the Universe".
(Scientific American Library)
I reiterate that Devlin did not actually say don't teach or use
repeated addition. His position is only that we should teach that
although these two operations of multiplication and repeated addition
get the same results, that does not mean therefore that they are the
same operation.
His main beef I think is teaching that multiplication *is* repeated
addition is the same as teaching that they are not different
operations but are the same operation simply because they get the same
results. A great example to illustrate this idea that different
operations can get the same result is 2 plus 2 equals 2 times 2.
His other beef I think is that repeated addition even as a model does
not generalize to the later real numbers well - and not at all when we
get to the point of multiplying two irrationals. And so we need a
another model to add to the repeated addition model, where this other
model holds up without having to change at all through the multiplying
of two irrationals.
I reiterate what I said in my post above, especially this:
"With repeated addition as a model we have to keep changing the
definition of repeated addition as we go to more and more general
number systems and to the breaking point when we hit multiplication of
two irrationals. But with scaling we don't have to change the
definition of scaling at all. Even for multiplying two irrationals e
and pi, where 1, e, pi, and e(pi) are viewed as distances from 0, we
use the same exact model of scaling we use for multiplying two
naturals: We have these ratios as a model: pi is to 1 as e(pi) is to
e, and via commutativity we have e is to 1 as e(pi) is to pi. Or if we
wish to use equality on fractions, pi/1 = e(pi)/e, and via
commutativity we have e/1 = e(pi)/pi. And we can model this by
construction on a graph: Connect 1 on the x-axis to pi on the y-axis,
and, parallel to that drawn line segment, connect e on the x-axis to
the y-axis, and this point on the y-axis is e(pi). And via
commutativity we have the other way: Connect 1 on the x-axis to e on
the y-axis, and, parallel to that drawn line segment, connect pi on
the x-axis to the y-axis, and this point on the y-axis is e(pi).
No one is saying don't use repeated addition *as a model* in the
naturals, and no is saying don't use the repeated idea later with the
redefinitions. But just don't exclude the one and only model that
needs no redefinition all the way up through multiplying two
irrationals, especially the one and only model that has the added and
probably vastly more important benefit of teaching proportional
reasoning from day one via reasoning on equal or equivalent ratios or
fractions (which is a great foundation for working with percentages -
and I mean using the equivalent ratios or fractions x/100 = y/z.)
The lack of fluent proportional reasoning in a large percentage of
students all the way up through adulthood is one of big problems that
must be addressed, and a least including this scaling way of modeling
multiplication using equivalent or equal ratios or fractions from day
one is a ready-made way of attacking this problem."
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