Several times, I have written about handy shortcuts that bypass some of the tedium of calculation.The frequency with which readers derided these tricks or mnemonics surprised me. For instance, my post on a shortcut for logarithmic differentiation ended up being the topic of a Facebook debate on sound pedagogy. One of the participants claimed that he would never teach the SPEC rule since he would rather the students know how to use logarithmic differentiation. It seemed to me that this is equivalent to not teaching the power rule because you would rather the students know how to evaluate limits and to utilize the binomial theorem.
Let’s face it, anyone who disagrees with using shortcuts or mnemonics probably should add, “at least in addition to the shortcuts and mnemonics that are already in common use.” Our entire system of notation is designed (sometimes poorly) to make the meaning easier to remember and computations easier to perform, e.g., decimal notation versus Roman numerals. There is nothing wrong with observing that a long computation reveals a pattern or an invariant that allows for a more direct route to the answer; this is a process embraced by mathematicians (I’d love to know what percentage of math papers simply improve on the proof of a known result).
Am I wrong or is there a misconception that teaching a shortcut implies not teaching the reason behind the shortcut? When I was in ninth grade, I did a project on mental arithmetic. The teacher gave back my draft with a comment asking for justifications of the tricks I was using. I learned so much about algebra trying to complete that assignment, perhaps more than I would learn in an entire high school algebra course. Make learning the inner-workings a priority, and the shortcuts arise naturally.
The June 2012 issue of the Notices of the AMS contains a provocative article by Frank Quinn. Amongst other things, he stresses that work on an abstract and symbolic level is important. Of course, there are lots of ways of incorporating abstraction into a class. Wouldn’t it be doubly beneficial if the result of that work was a faster way to perform a calculation?