July 6, 2012

In Defense of Shrtcts

Filed under: Tricks of the Trade — Adam Glesser @ 10:41 pm
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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?


May 30, 2010

Shortcuts, Mnemonics and Why They Ain’t So Bad

Filed under: Classroom Management — Adam Glesser @ 4:20 pm
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In my previous post, a commenter bemoaned the use of shortcuts in adding fractions. This isn’t the first time I’ve read a person criticize the use of a shortcut technique or mnemonic. Very often, their point is that if a student learns only the shortcut and they don’t understand why it works or what they’re doing, what is the point? This seems reasonable to me. However, there is a difference between knowing how something works and knowing how best to do it. For instance, I spent a great deal of time studying Sudoku solving techniques (and we’re not talking about the kind of puzzle you find in a Will Shortz book; we’re talking about techniques for puzzles you would never solve with pattern-based techniques unless you were reading this). My wife, meanwhile, would work the easy and intermediate level puzzles and occasionally something harder. Put us in a competition: she would blow my socks off because while I’d be trying hard to find these nice patterns, she would be plugging in a number and working it until she solved the puzzle or found an error. The latter technique is what Sudoku enthusiasts call T&E (trial & error) and is frowned upon, except in competition where it is the method of choice for the tough puzzles because it is ridiculously fast. Dan Meyer made a similar observation in the context of solving systems of linear equations (or, more precisely, in not solving them when guess-and-check is easier).

So, a big difference between my examples and the usual shortcuts is that we aren’t usually telling our students to avoid the standard algorithm by guessing; instead, we are suggesting a different algorithm. Nonetheless, the point is the same. Why have a student use a slow method when a faster one is available? Is it just so that we can lie say to ourselves that they really understand what is going on? Anyway, my feeling is that the existence of these shortcut methods is not as important as looking for them. Consider Kate Nowak’s recent post about a student suggesting a shortcut method. Even though the method was dead wrong, the subsequent discussion is extremely valuable.

By the way, if you’re into shortcut mental arithmetic and beyond (can you tell that I am?), there is a fantastic book called Dead Reckoning that goes (far) beyond the methods of every other book on mental math I’ve ever read. We’re not just talking about multiplication and division tricks, but also ways of estimating roots, logarithms and trig functions.

What about mnemonics? These don’t give you a faster way to compute things, simply a way to remember them. Shouldn’t we get our students to remember things because they make sense that way instead of using some bizarre story, silly rhyme or reference to a non-existent indian chief? Ideally, yes. In practice, mathematics isn’t designed to facilitate this process. Our notations and naming conventions aren’t perfect and even when they’re pretty good, it doesn’t always help the student (does the word scalene really tell you anything if you don’t have a classical education that includes Greek?). No, we often need these mnemonics until the meaning becomes internalized. For students who don’t ever need to use the term again (and, thus, won’t have the required repetitions to internalize), the mnemonic may be the only thing they remember. I can’t remember much French from high school, but I do remember the verbs that are conjugated with être in the passé composé. Why? Because I learned a song to remember them.

Look, I’m not trying to say we should avoid teaching meaning or attempting to get those messages across. However, I think we are going about it the wrong way when we fail to account for the way our students learn and remember. This next week I’ll go deeper into this topic with a look at WCYDWT and storytelling.

P.S. Thanks to my loyal subscribers for making my first week of posting a success. I appreciate all of the comments and helpful pieces of advice.

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