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.

I agree completely. I tried to get across a similar sentiment in this post (http://samjshah.com/2009/12/10/phrases-actions-rituals/). The basic argument: do you always complete the square to find the zeros of a quadratic? Do you always rederive the quadratic formula? Or do you sing the little quadratic ditty you learned and apply it? I still say SOH CAH TOA and sing the quotient rule. And I think that’s A OK.

Sam

Comment by samjshah — May 30, 2010 @ 5:37 pm |

I do sing whenever I use the quotient rule!!! In fact, I made the claim this year to several students who I’d had for the previous three semesters that I knew exactly how much math they knew and I knew how to access it. When they didn’t believe me I started humming Old MacDonald and the three (almost instantly) said, “Quotient Rule.” Now, they believe me.

Thanks for linking to your post; I had forgotten all about the domain meter and I want to use that to teach precalculus this fall. Funnily enough, I’ve used a variation of that trick when teaching abstract algebra. I call it the Spectrum of Injectivity. On the right, I put Constant and on the left I put Injective. Taking a group (or ring, vector space, etc.) modulo some normal subgroup contained in the kernel of a homomorphism moves you from right to left. If you mod out the entire kernel, you end up at injective. I even use sound effects!

Here is a trick for multivariable calculus: curl(F) = grad x F, div(F) = grad . F

curl is the cross, div is the dot, hit ’em with grad and that’s what you got!

It’s alliteration, baby!

Comment by Adam Glesser — May 31, 2010 @ 4:49 am |

Although mnemonics work great for some people, they don’t work for all. I always have trouble remembering the mnemonics, and have to work them out from the things I do know. One of the reasons I loved math as a kid was that it was the only subject that didn’t require any memory work—a couple of little things that made sense and you could derive the rest as you needed it. I realize (now) that this is an unusual position, and most people don’t have as much trouble with memory work as I do.

I do now remember the quadratic formula, but as a kid it was easier for me to rederive it. Also that gave me something to do in the hour-long tests that covered only about 10 minutes worth of material.

Comment by gasstationwithoutpumps — June 11, 2010 @ 9:30 am |

Do I spot a banana slug in our midst?

You’re right about mnemonics not working for everyone (or least certain mnemonics not working for certain people), but math, as it is currently taught and practiced, does require some memory work. It is one thing to know how to derive the quadratic formula so as not to have to remember it, but the definition of is not something you can work out from first principles: it’s a definition. Another reason to consider a mnemonic for something that you know how to derive is that, sometimes, time is an issue. Of course, tests are an example, but I’m more interested in using math. Say someone is explaining to me a derivation. In the middle, they say, “…and, obviously, we can apply the quadratic formula to get the roots blah blah” and then they move on to the next step. If you can immediately picture the quadratic formula, plug in the values and get their roots, you simply move on to the next step in their argument. If, however, you have to derive the thing on the spot, you won’t. You’ll move on to the next step with them, but now with some uncertainty. Add up a few of those uncertainties and you end up lost. Alternatively, if someone asks you for a quick estimate and you have to derive a few formulas to do so, it won’t be a quick estimate.

Thanks for great comments and GO SLUGS!!!

Comment by Adam Glesser — June 11, 2010 @ 9:44 am |

I agree that a few definitions need to be memorized, but I never found mnemonics of much use for that. I realize that other people do, and I have deliberately tried to incorporate a few in my teaching, but I usually find the definitions easier to remember than the meaningless mnemonics that are supposed to help.

Yes, I’m a banana slug. Check out https://banana-slug.soe.ucsc.edu/

for a course that I’ve just finished teaching (and now have to grade).

Comment by gasstationwithoutpumps — June 11, 2010 @ 11:02 am |