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[Phys-L] bathwater +- baby (invert and multiply)



On 05/17/2012 01:00 PM, Robert Cohen wrote:
I've gone to professional development sessions for K-8 math teachers and
it is easy to see that they are also looking for "tricks". The clearest
example of this is with dividing fractions. They teach that one should
"invert and multiply". This allows students to get the right answer
without understanding what they are doing (i.e., without understanding
proportions and ratios) -- as long as they can remember the "invert and
multiply" trick. Some teachers are surprised to learn that there are
"other" tricks for dividing fractions, like getting a common denominator
and then cancelling the denominators. However, they see no use for
those "other" tricks -- why memorize more than one trick when one trick
alone will work?

I emphasize that I agree with the main point of that argument,
namely that dirty tricks are bad.

However, I feel obliged to object to the example. The idea
of "invert and multiply" is not a dirty trick. If you want to
call it a trick, that's OK, but it is about the most un-dirty
trick I can imagine. It is only half a step removed from the
axiomatic definitions of "division" and "inverse".

This does not change the main point of the argument, insofar
as the teachers are learning the trick in a dirty way, without
seeing its connection to the axioms of arithmetic ... but we
need to see that as a problem with how the trick is taught
and learned, not as a problem with the trick itself.

Please let's rescue this baby from the bathwater.