Mnemonics and Acronyms are BAD (Best to Avoid Discussing)

Please Excuse My Dear Aunt Sally, SOCATOA, All Students Take Calculus, FOIL, ...

It seems math education is riddled with acronyms and mnemonics.  Students love them because they afford an easy way to remember what otherwise might be a challengingly complex concept.  What is a little more surprising is that many teachers are also very found of these tricks.  Last semester I had a student who seemed to have a mnemonic for everything, and she claimed that her high school math teacher showed her dozens of them.  What a shame, I thought.

Now I understand that students will not be able to advance very far in mathematics without being able to remember the order of operations, but there must be a better way.   Consider the "All Students Take Calculus" example.  If you are not familiar with it, this is the trick to remember which trigonometric function is positive in which quadrant.  Starting in quadrant I, all three trig functions (sine, cosine, tangent) are positive.  In the second quadrant, sine is the only positive one (Sine = S = Students), in the third tangent is the positive one, in the fourth cosine is.  Okay, so this definitely works, and I'm sure there are students who know that sine and tangent are negative in quadrant IV but cosine is positive because of this, if for no other reason.  But there's the problem: if for no other reason!

Look, let's be honest.  Students do not need to know which quadrant tangent is positive in.  It is unlikely they will every need that fact on the job, nor be asked about it in a job interview.  There are no exclusive math parties where that particular piece of knowledge is required to gain entry.  This of course does not mean we shouldn't teach the topic.  Determining the sign of a trig function based on the quadrant of its angle is a perfect exercise in understanding the meaning of the trig functions.  Students should already know which of x and y are positive in a given quadrant.  Students should already know the definition of the trig functions in terms of x and y.  Putting these pieces together is exactly why we teach trigonometry at all.  Its to develop that kind of thinking.

On the other hand, there are some formulas or concepts which simply need to be memorized.  For example, I would not expect my students to be able to derive the quadratic formula each time they need it.  If they can think up a little jingle to help them remember which constant belongs where, more power to them.  Additionally, I must admit I was surprised to hear that some schools are avoiding mentioning FOIL.  I would probably not teach students that method of multiplying two binomials at first, but if they already know the trick, it seems a waste not to use that simply label to remind them what's going on.

So how should we approach mnemonics such as these?  Depending on the particular example, my answer might change, but in general, I avoid introducing these tricks completely.  I might ask my class how they plan on remembering a particular formula or concept.  If a student suggests a mnemonic, I act surprised, as if I have never heard that one before.  I then go through it and "check" that it works.  This reinforces the original, conceptual basis for the fact, as well as challenges the students to think creatively about the subject.  Once the class knows a mnemonic, I tend not to be a hard-ass about it.  I would never respond to a student suggesting we FOIL an expression with, "what's FOIL?  do you mean doubly distribute?"

Mathematics as weightlifting

Every math teacher has heard it: "Why do I need to learn this."  Like nails on a chalkboard!  I find the question especially difficult to answer as a pure mathematician.  I enjoy mathematics particularly because it is not something that can be used -- the abstractness is exciting.  But students don't want to hear this.  Increasingly, students go to college for the purpose of getting a higher paying job, and have little patience for anything they will not need to know in the workforce.

To be clear, there are two very different questions of this type.  One is an honest question: "What can this mathematics be used for?"  This question deserves an honest and careful answer.  The other: "Why do I need to know this if I won't ever use it in my particular job?"  It is this latter question I want to answer here.  Of course, I realize that often students pose the question as rhetorical, as an exhibition of their frustration.  In that case, the following can be used to motivate them to get back to work.

A few years back I was watching a baseball game, and the commentators got to talking about a particular player and how dedicated he was.  They said that he spent even more time lifting weights than his teammates did.  The whole idea of baseball players weightlifting was perplexing to me: why not spend that time at batting practice, or running sprints?  Baseballs are not that heavy, and it's not like once reaching second base the runner must bench-press his bodyweight.  But of course upon further reflection, it became clear that while the players do not need to use the particular weightlifting skills they work so many hours on, doing so makes them stronger over all.  It makes them better athletes.  Similarly, doing mathematics makes (future) scientists better thinkers.

Baseball players are not the only people who lift weights.  Most everyone can benefit from weightlifting as it is a great way to stay fit and healthy.  Similarly, mathematics is a great way to stay mentally fit and healthy.  Like weightlifting, it can be difficult when you first start out, especially if you are not doing it correctly.  But once you get the hang of it, not only can you lift more and more weight, but it will become enjoyable.  In fact, there are people who enjoy weightlifting so much, they do it professionally. 

Probability is hard

There is a nice article on the ScienceNews.org site about the Tuesday birthday problem:

I have two children, one of whom is a boy born on a Tuesday.  What is the probability that I have two sons?

The article does a very good job of discussing the solution, as well as why it is difficult (and why "Tuesday" has anything to do with the solution).  Commenters have pointed out that there is an additional problem: what the meaning of "one of whom" is.  It could mean "at least one of whom," which is the strictly correct mathematical interpretation, or "exactly on of whom," which is what most people would read if they are not being careful. 

And we wonder why students hate word problems!  Problems like this are often offered as examples of how counter-intuitive probability (especially conditional probability) can be.  And probability can be counter-intuitive -- just see the Monty Hall Problem.  However, problems like these two children birthday paradoxes are often confusing mostly because of the wording.  We should be very careful to remove all ambiguity from the question and let the interesting mathematics stand on its own.

Random examples

Some examples are better than others - most of the time.  For example, when first teaching the product rule, it is not a good idea to use $xe^x$: since the derivative of $e^x$ is $e^x$, students don't see the form of the product rule explicitly.  That said, there are times when the technique being taught are so general, that they would work equally well with any example.  In cases like these, I like to construct a random example, with the class's help.

Suppose I wanted to show my students that the Taylor series for any polynomial is simply the polynomial back again.  If students have not thought about this yet, the result can be rather surprising (after all, the Taylor series for the other standard examples look nothing like the starting function).  What I don't want to do is look down in my notes, carefully copy down a polynomial and start from there.  While this would definitely be an example worth sharing, I fear that students would not be impressed.  Of course I, the powerful math professor, could come up with an example of a function which is identical to it's Taylor series.  Just another example, they would think.

Instead, I write on the board "Find the Taylor series for $f(x) =$" and then dramatically point at a student and demand, "What's your favorite number!?"  After regaining his or her composure, the student will say, perhaps, 7.  I write $7 x^3 +$ and then repeat with another student.  Continuing in this fashion, the class and I together come up with a random polynomial.  And wouldn't it be amazing if this random polynomial happened be it's own Taylor polynomial?  Surely that cannot be a coincidence.

Random examples like these can be used all over the place, although it is important not to use them as a substitute for a well written lecture.  As with any interaction with individual students in class, this technique will keep students alert.  Most find it humorous (especially when a student can't remember their favorite number).  And most importantly, when used correctly, the random example can drive home the fact that the mathematical technique can be used in any case, not just the special cases the professor has prepared.

Latex on Blogger

Apparently, I have just enabled latex on Blogger.  I did so following the instructions found here.  If this is working, then $e^x$ will appear instead of $!$e^x$!$.

Mathematics through puzzles

I love mathematical puzzles.  I still remember the first one I ever heard.  It was the nine weights puzzle, where you have to find the heavy weight by using a balance scale only two times.  I was in forth grade.  I remember thinking how clever it was; how simple; how elegant.  I can't be sure, but I suspect that puzzle got me on my way to being a mathematician.

Students like these puzzles too.  I usually try to give a few of them as extra credit over the course of a semester.  What I need though, are some really good ones.  I need puzzles that are not only clever, but also remind students about the mathematics we study in class.  To keep track of such puzzles, I've started the Math Puzzle Wiki.  I would love to find more puzzles, so if anyone has some good leads, please send them along, or add them to the wiki.  And of course, feel free to use any of the puzzles you find there. 

Grading is great

This semester I happened to teach three section of Trigonometry, and had an undergraduate grader.  He was supposed to grade 15 hours a week (5 hours per class) which meant he could easily grade both homework and quizzes for me.  This left me with only exams to grade.  Sounds great right?

Turns out, not so much.  The grader has done a fine job (although having to get through so many papers meant the students didn't get feedback very quickly).  The problem is that I didn't get a chance to grade their work on a regular basis.  This in turn held me back from teaching as effectively as I could have.

Apparently I have taken for granted the importance of grading students' work - not because students need to be assessed or get feedback - but because regularly grading allows me to monitor students' progress.  Yes, some students ask questions in class, but sadly most do not.  Many students will pretend to understand a concept as to not appear ignorant.  This semester in particular I have been finding it very difficult to know when I have covered a topic enough so that the majority of my students understand it.  I think the reason is, for the first time, I have not been grading the weekly quizzes.

Today I did so for the first time this semester (my grader had a busy week, so I had him just do the homework).  It only took me about an hour all together, and going in to tomorrow's lecture, I know that I need to review the polar form of complex numbers, while I should probably not spend much more time discussing the different ways to write vectors. 

The point is this: grading, while often a tedious chore, is a great way to ensure that students are getting the most out of course.  It is a simple and effective way to take the mathematical pulse of the class.  Plus, students appreciate when you get quizzes and exams back quickly, which is easier to do if the grading is done by the professor.  From now on, I will grade the quizzes.  And I will like it!