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Topic: exponents

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Subject:   RE: exponents
Author: Susan
Date: Oct 18 2004
On Oct 18 2004, Alan Cooper wrote:
> On Oct 18 2004, Mathman wrote:
> I'm not trying to be argumentative
> or picky . . .
> . . .from observing empirically 10^5/10^2 =
> =10^(5-2) = 10^3
> and so on] it *follows* that 10^3/10^5 = 10^(-
> 2).  

I don't think so.
In fact, with the repeated
> multiplication definition, 10^(-2) does not even exist! It only
> makes sense if we *define* it somehow (and of course as 1/(10^2) is
> probably the only useful choice).

My concern is that, by saying
> "it *follows* that", we may cause some students to lose faith in
> their own (correct) understunding of logical implication.

I think
> we can agree that *I* am the one being "picky", and so I will try to
> avoid posting more on this, but will reply privately if you want to
> chat further.

I have to jump in here again.  Even after teaching the students why x^-1 is
1/x in the same fashion as Mathman does, two months later, even after completley
understanding the derivation, students will write 2^-1 = -2.  The logic
doesn't seem to click for all students, even when you combine it with scientific
notation.  And then we confuse them even more by telling them that f^-1(x) in
function notation is the inverse.  I think they see negative one as a negative
number and don't reinterpret is when it is used differently.  It's as if they
can't interpret READ according to how it is used:   "I READ the newspaper in the
morning" and "I READ the newspaper yesterday"...same word, but different
depending on the context.  I think we need to help students use the contextual
clues in mathematics too.  I am open to any and all ideas...bring them on!

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