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Subject:   RE: exponents
Author: Susan
Date: Oct 3 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.

I agree with Alan. The students have to learn WHY first and then practice!  When
they get stuck on a problem like (2x^2)^3 or (10x^3Y^5)/(5x^2y^7), we tell them
"When in doubt, spread it out!"  This helps them to at least have a back up
strategy if they can't remember the rules, or if they want to be sure it's
right.  At our school, we make exponents a "gateway skill".  We give short 10
question quizzes and require that they get 100% on one of them.  We have many
different versions of the quiz, and they just keep taking it and going over it
until they meet the requirement.  We find that this helps them focus on the
concept which looks easy, but isn't.  The quiz has close confusers on it, again
to help them focus on the process, so the problems look like this:  2^2 * 2^3,
2^x^2 * 2x^3,(2x^2)^3, (6x^9)(3x^3), (6x^9)/(3x^3).  We are trying to get them
to think about how coefficients operate versus how exponents operate, and always
reminding them " When in doubt, spread it out!"  I can't say that we have had
instant sucess with this strategy, but for now, it's working.  I think Gene's
idea of having them practice on a website that generates problems will be a good
resource for those students who need more practice, so we will add that to our

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