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Topic: exponents

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Subject:   RE: exponents
Author: lanius
Date: Oct 4 2004
On Oct  2 2004, Alan Cooper wrote:
> On Oct  1 2004, PWR wrote:
> . . .To me, the rules are both logical
> and easy,
> but my students usually seem to find them too
> numerous and confusing. . .

Perhaps you find them easy *because*
> you find them logical.
I believe that if we insist on students
> understanding and explaining "rules" rather than memorizing and
> reproducing them, then they, like us, may see that there is really
> only *one* rule of exponents - namely that a whole number power
> corresponds to repeated multiplication - and that the rest all
> follow quite easily from this.

Once whole number exponents have
> been fully understood, then the definitions of negative and
> fractional powers can be seen as natural and convenient choices of
> notation (which they are) rather than facts of nature (which they
> are not).

The failure to clearly identify such distinctions is
> actually one of the most confusing things about our subject, and
> perhaps explains the sense of frustration and anger with which many
> people leave it.

So Alan, this is very intriguing. Would you please explain it further. What do
you mean by not a "fact of nature".

I know that at one point I lumped rules, conventions, and notations all into one
basket. You couldn't have 4/6, for example (as if that weren't a perfectly good
number). How do we help students differentiate the mathematically true, what I
think you are calling the facts of nature, from conventions or symbolic
expedients, however convenient they might be. BTW, does anyone know the history
of exponential notation?


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