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

arithmeticof counting is not at all the same as thearithmeticof 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 offis 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.

You might decide on a different amount of wiggle room. That’s cool too. Deciding is personal. It’s an art.

Appendix:

If you end up wanting to pursue any of this with kids, here are a few things I found helpful:

http://www.democratandchronicle.com/story/watchdog/2014/08/20/rr-bridges-and-trucks/14338865/

http://safety.fhwa.dot.gov/geometric/pubs/mitigationstrategies/chapter3/3_verticalclearance.cfm

https://www.youtube.com/watch?v=o8Z1K37SHbE

http://www.truck-drivers-money-saving-tips.com/low-clearance.html

And here’s a “real-world” answer, courtesy of David Wees and Sky Wyatt:

UPDATE: September 2015. Somebody too tall drove under and broke the thing!

There’s even more to it.

1: “How long is the classroom?”, and we count the feet (unless we have an expensive tape measure).

2: There is a big difference between measuring quantities (weight, distance etc) and measuring movements ore displacements, which need signed numbers. Oooh, now we are going to have negative numbers, but not much if anything about WHY. We are not just sticking the negative numbers on the left of the numbers we have already, we are actually creating a whole new collection of numbers, the signed numbers, negative AND positive. This distinction is frequently overlooked, leading to student confusion (again!).

Thanks, Howard. There’s ALWAYS more.

Lovely post. Great brainstorm list!

I sometimes tell my classes this Nasrudin story when we’re talking about this kind of thing:

A guide was taking a party round the British Museum.

‘This sarcophagus is five thousand years old.’

A bearded figure with a turban stepped forward.

‘You are mistaken,’ said Nasrudin, ‘for it is five thousand and three years old.’

Everyone was impressed, and the guide was not pleased. They passed into another room.

‘This vase’, said the guide, ‘is two thousand five hundred years old.’

‘Two thousand five hundred and three,’ intoned Nasrudin.

‘Now look here,’ said the guide, ‘how can you date things so precisely? I don’t care if you do come from the East, people just don’t know things like that.’

‘Simple,’ said Nasrudin. ‘I was last here three years ago. That time you said the vase was two thousand five hundred years old.’

Retelling here: http://ninedonkeys.wordpress.com/2014/11/29/get-the-facts-straight-2/

I’ve just read, somewhere else, “Tyranny is the removal of nuance.” It’s why we need human teachers. There are always nuances, and it’s humanising, educative in the broadest sense, to spend time talking about them.

I love this story!!!

Wonderful post and I love the brain dump of questions that resulted from the 9’6″ sign. Can’t wait to hear/read more of these.

How many jobs have been lost because of that bridge? #estimation180 gag reel

Thanks, Andrew! I love the idea of an #estimation180 gag reel!

You know what a fan I am of Estimation 180! I have been thinking about the ideas in this post and the “reveal,” though. When you’re counting something discrete—Red Vines, almonds, etc.—no problem. We can know that answer. When the answer is a measurement, however, I wonder if you might want to expose the error or process a little more? I’m thinking about something like the bacon measurements. (Mmm, bacon.) When I look at a picture like this one, where you’re measuring the uneven edge of a piece of bacon, http://www.estimation180.com/uploads/1/3/8/8/13883834/183_answer_length.jpg, I think there’s an opportunity for kids to wrestle with the measurement more. Right now, they click on a button called “Answer,” and one answer is given in each measurement system, to the level of specificity you’ve decided is reasonable. There is one right answer: 9.75″. But when I look at the picture, I see a little piece of fat past 9.75″, and then an area that falls shy of 9.75″. So you must have decided 9.75″ was a reasonable measurement for that piece of bacon. And I think it is! I’m just wondering what message this presentation gives to kids? If a kid gave an answer of 9 7/8″, (closer to the maximum length rather than the average length), would that be wrong?

So one thought would be to give an interval as the “answer” instead of a single number. In this case, maybe 9 5/8″ – 9 7/8″? One thing that really came out of that article, to me, is how we need to legitimize the interval as an answer. When I think about regular life, I use intervals all the time. “I’ll be there in 15-20 minutes” is something I’d say. “I’ll be there in 17 minutes” isn’t.

Another idea would be to remove the type altogether, and just have the photo of the ruler and let kids figure out what it is? For teachers projecting on the whiteboard or smartboard, it would be hard for kids to see. But for teachers using ipads, that would be awesome. Maybe it could be an option? One button gives you the photo, and a second gives you the print? Just a thought.

Finally, I wonder about calling it something other than “Answer?” I worry that word, in some subtle way, turns this amazing thing you’ve created into a cooler version of the teacher’s guide with an answer key. What if it was labeled “Mr. Stadel’s measurement?” Or something else that describes what it is without making it seem absolutely right and error-proof?

I hope you understand that I’m sharing thoughts in the spirit of strengthening something I think is already brilliant! I love this resource so much, and I’m sure it gives students more opportunities to think and reason than most of what they encounter in math classes. I’m so grateful you have created and curated this great collection.

Tracy

Is the NCTM book you mentioned the yearbook edited by Harold Schoen?

A while ago, I had an idea for sharing (creating fair divisions) as another interesting extension within the topic of measuring/approximating/estimating/accuracy. Here is an outline I put together, but we haven’t yet done the exploration: http://3jlearneng.blogspot.com/2014/08/fair-sharing-exploration-post-warm-up.html

Joshua, Yes, Schoen edited the yearbook. Sorry it took me so long to get back to you!