Saturday, May 22, 2010

Kidding on the square

Lessons unlearned

My teaching career began with sporadic part-time assignments in the seventies and finally became my full-time profession in the eighties. Of course, with decades of teaching experience under my belt, nothing can surprise me anymore.

Yeah, right.

During a prealgebra final exam, a student came up to me with a question. My students know that they can ask me questions at any time, even during tests. It's possible, of course, that I might respond by saying, “I can't tell you that,” but it doesn't hurt to ask. Sometimes all a student needs is the tiniest nudge, after which the light comes on in their eyes and they demonstrate their knowledge instead of their confusion.

That's the theory, anyway.

In this instance, my prealgebra student comes up to me with a question about finding the area of a rectangle. She points to a figure on her exam, and says, “Dr. Z, is this a square?”

I'm sure you know what happened next. I calmly prompted her to give me the definition of a square, after which she quickly perceived her mistake and returned to her desk prepared to solve the problem.

Uh, no.

The first thing I really said, as best as I can reconstruct it, was “Gak!” After recovering my equilibrium and taking a deep, calming breath, I added, “Why do you think it's a square?”

She wasn't sure. She just wanted to know which area formula to use. Which was better, A = LW or A = s2?

Maybe it was my fault. I had focused on the notion of length times width for the area of rectangles, letting the area of a square fall out as a special case of no special interest. Besides, we were already used to the word “square” to indicate the product of a number with itself. Did we have to belabor it?

Evidently we did.

One of my students has made it to adulthood without ever learning the definition of a square. I remain capable of surprise.

I was pleased when—without undue prompting on my part—my student decided that she wanted to use A = LW to find the area of the rectangle. As I discovered later while grading the exam, she had proceeded to add L and W to get her answer. She gave me half the perimeter instead of the area.

Oy. I am a really bad teacher.


Anonymous said...

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for some interesting ideas on how to teach math better ;)

kai said...

I never cease to be amazed at how basic stuff you are forced to teach your students—can they even read for the most part?

Zeno said...

Sometimes so am I, Kai. However, my school is an open-admission community college, so any adult with any level of education can sign up. To address that population, we offer anything from arithmetic (yes, basic math with positive integers and fractions) to differential equations and symbolic logic (transfer curriculum for students going on to four-year schools). It makes for quite a varied teaching experience and I am one of the faculty members who teaches the entire gamut. (A few of us huddle toward one end of the spectrum or the other, but no one is a specialist at my level -- although I prefer calculus to anything else.)

As it turns out, reading skills are a big problem for many students. That's why we have introductory reading classes and English as a Second Language. One message we always try to convey to our students is to read the damned instructions. Lots of them just plunge in without reading the problem. This can be quite tragic when the instructions say, "Set up but do not try to evaluate a definite integral for the arc length of the curve."

Sili said...

Gak, indeed.

And I to think I' interviewing for that kinda job.

Zeno said...

No kidding, Sili. It's a great job. But it does have its dumbfounding moments.

Good luck with your interviews!

Karen said...

The mindset that you solve a problem by plugging numbers into a particular formula, without having the formula make sense to you, has always baffled me. It seems to turn math into a game of match-up, where you try to match a formula in column A to a clue in column B. It's the equivalent of trying to match actors to the titles they've played in, or politicians to their indiscretions, or some such. There's very little math involved!

João Paulo said...

Dear Zeno,
I received an e-mail today from a friend that works in a courtroom in Lisbon concerning the use of simple math by a judge. I immediately thought of sending it to you but I don't see an e-mail address anywhere. Can you tell me where to send it? Or, if you prefer, send me a message to

Interrobang said...

Karen -- Please, will you teach me the heuristic that allows you to match a formula to a problem? I don't care if the formula makes sense or not; I mean, do they ever? I'd be able to get a lot of practical use out of such a scheme. I'm never going to be able to understand it, so being told "you need to understand it in order to use it" is really frustrating, especially since you don't, really. Propositional calculus essentially works by a similar heuristic (if it looks like this, you proof it in thus-and-so way) and it's brilliant. Algebra? Incomprehensible.