deciding – Becoming the Math Teacher You Wish You'd Had

Maya (7) asked me about this sign in the coffee shop drive-thru the other day:

And so, we had a lovely conversation about clearance. Once she understood the notation, she asked, “Does that mean if your truck is 9’5″, you could drive underneath?”

That’s just gold.

I said, “Well, it depends how confident you are in the two measurements. Are you sure the clearance is exactly 9’6″? Are you sure your truck is exactly 9’5″?”

She thought about it a while, and decided that was too close for comfort. We brainstormed a nice list of sources of error while we waited in line:

What if the tires weren’t as inflated when you measured, and then you filled them up with air?
What if the person measuring made a mistake?
What if the tape measure was a little bent?
What if they didn’t get to the very top of the truck, like an antenna or exhaust pipe sticks out or something?
What if there’s ice on the road, so your truck sits a few inches higher? (There was ice, in fact. Happy Thanksgiving.)
What if they’ve paved the parking lot since they made the sign, so now the road is higher?
What if whatever is overhead is curved, and the 9’6″ is the highest part, even though it should be the lowest?
What if it was really almost 9’6″, so they just went to the next closest number to make it even, but it’s really a little under. And what if the truck is really a little over 9’5″, but they went to the next closest number to make it even too? It could be a lot less than an inch!

That last one was Maya’s, and it’s my favorite, probably because I’ve been spending time with the (unfortunately out of print) 1986 NCTM handbook Estimation and Mental Computation lately. It’s a worthwhile read about an important and neglected topic, and I hope an update is in the works this (ahem) century. Peter Hilton and Jean Pederson caught my eye with this quote in their article, “Approximation as an Arithmetic Process”:

It is an extraordinary triumph of the human intellect that the same number system may be used for both counting and measurement. Nevertheless, we argue that the arithmetic of counting is not at all the same as the arithmetic of measurement; and that the failure to distinguish between these two arithmetics (which one might call discrete and continuous arithmetic) is responsible for much of our students’ confusion and misunderstanding with respect to decimal arithmetic and the arithmetic of our rational number system (57-58).

That’s a doozy, and it’s been rattling around in my head for a while. The authors point out that, in counting, 3.00 = 3, but in measurement, 3.00m ≠ 3m. Do we make a big enough deal about this distinction in class? I’m thinking no, not by a long shot.

Hilton and Pederson make the case that all measurements are approximations, and we should teach them that way. This is a really interesting idea. What would happen if, from the outset, we were honest about the inherent uncertainty and decisions made in every measurement? What if we stopped referring to the “actual” measurement, and instead talked about the closest approximations we could get, or a range of reasonable measurements?

Thus there are situations in which we round off in view of the accuracy we NEED and situations in which we round off in view of the accuracy we can genuinely GET—and, of course, situations in which both kinds of consideration apply. The technique of rounding off is a familiar one, but we emphasize that the technique, easy in itself, is based on the nontrivial, nonalgorithmic art of deciding the appropriate level of accuracy (58).

Teaching the “art of deciding the appropriate level of accuracy” is a far cry from a worksheet with the directions, “round off to the nearest hundredths place.” The art of deciding involves the idea of authorship, which runs headlong into the myth of math as objective truth. The art of deciding means there is always an interval of reasonable answers to a measurement problem, not just one “right” answer printed in the back of the book. The art of deciding is empowering; the opposite of regurgitation.

As a student, I loathed the dreary lessons on significant figures, uncertainty, and measurement error. They were full of hierarchical rules and tedium and a complete absence of intuition and sense-making. Booorrriiiing. As an adult, I’m baffled that we somehow make boring this element of mathematics that involves a little bit of mystery, the explicit acknowledgment that sometimes we can’t know no matter how hard we try, and an honest admission that we make decisions when doing math. How do we kill it? We mask the interesting parts because we, the teachers, are the ones practicing the art of deciding. We are the ones who choose whether to measure to the nearest 1/2 inch or millimeter or gram. We leave the trivial implementation of our decisions to our students, but we keep the decisions for ourselves. Not good.

So, I left the deciding to Maya. Would you drive your truck underneath if it was 9’5″? She said no. I asked, what if your truck was 9’3″? She said no.

“I don’t think I’d be comfortable going under unless my truck was under 9′. Then I’d be confident I wouldn’t hit.”

She’d decided that 6″ is a reasonable margin of error, so she wouldn’t end up like these guys.