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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?