I've been reading a lot of math blogs lately and one thing I have come across quite a bit is that people are opposed to math tricks. Maybe I am misinterpreting the expression math tricks, but I always thought they were good. They were little ways of making a complicated example easier.
My AP calculus students are bright kids but have never been taught some of the little tricks to make their lives easier. For example, we were writing the equation of a tangent line last week. The slopes all came out to be fractions. The kids did not know an easy way to change the format and avoid working with fractions. They never knew that tanx/sinx can be rewritten (sin x/cos x) (1/sin x) instead of working with the complex fractions. They don't know the special triangles and insist on getting sin 45 from their calculator instead of figuring it out.. It never occurred to them to use the graphs of the trig functions to get the values of quadrantal angles instead of relying on a calculator.
Tricks don't replace knowledge. Tricks are part of knowledge and they make expanding knowledge easier.