There’s a water tower right next to the building I teach in. Naturally, I had my trig students figure out how tall it is.

They were lying on the ground and measuring angles.

They were borrowing tools from other teachers (the science teacher has something she uses to see how high her 7th graders’ rockets go; the PE coach has a long tape measure for when he makes lines on the field).

They were mad that I wouldn’t let them climb the fence so they could get to the base of the water tower. (Dude, that’s not our property!)

They were making estimates before they took measurements.

They were recognizing when an answer they came up with wasn’t reasonable, working to figure out what went wrong, then trying again to correct it.

They were enjoying the nice weather.

They were noting that the ground isn’t completely level and trying to compensate for that in their measurements.

They were drawing pictures to represent their work.

After they’d gathered measurements and performed their calculations, I let a student call the city to find out the actual height of the water tower. Most of the students were within 8 feet of the right answer (which was 216 ft). One group was way off, but they realized that they hadn’t done a good job of determining the angles of elevation, so we got to see how much accuracy matters.

I love doing things like this. It seems to me like the students really feel like they own the mathematics when they tackle a problem like this and reach a solution.

However, I’m thinking about Dan Meyer’s recent post on pseudocontext. If we can just call the city to find out how tall the water tower is, what’s the point?

Well, it’s fun. It’s a chance to go outside. I think it’s significantly more engaging than the example in the textbook where you have to figure out how long the rope is that’s holding the tent up (the example is labeled as “Real World Application: Entertainment” – *really?* entertainment, because it’s a tent? yeesh).

But is the water tower activity flawed because there was an easier way to get the answer? My gut tells me no, but I’m still working on why.

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Not pseudocontext at all. It’s a perfectly natural way of figuring something out. The issue isn’t really, is there an easier way? But – would you ever actually solve it in this way? You’re also not forcing artificial constraints onto it. A water tower is a perfect example because they can’t just go and measure it with a ruler. They have to take various factors into a account. They need to identify all sorts of errors that might be affecting their results.

A rope/tent? You would NEVER use trig, you’d just go over and pull out a tape measure. That’s pseudocontext. They’re forcing a method into a situation that would never require that method.

So…don’t worry. This is a great example.

Thanks, Jason! That’s a good way of thinking about it. You guys (and the conversation over on Dan’s post) are really helping me understand the idea of pseudocontext much more clearly.

I used to do something similar with a domed theatre near my school- I agree that it sparks curiosity in a meaningful way in that it’s much more fun to figure it out than it is to call city hall. And what if you someday want to measure or at least estimate the height of something to which city hall doesn’t have the answer, like a tree in your backyard that may or may not fall on your house in the next big storm?

Good point – thanks! And my students are still talking about the exploration, several days later, and that curiosity and interest are key, I think.

The benefit of the phone call is being able to check the work in the practice situation, figure out how to adjust, and then go measure something new. (That no one knows how far/tall/wide it is.)

Oooh, I love that idea! I hadn’t even thought of extending it like that. Thank you!

Just to follow up – I had them figure out how far away from the water tower they actually were when they took their angle measurements. One kid, after finding two distances that were about 45 feet apart, wrote on her paper, “Hmm…so I guess our angles were REALLY off, because we were supposed to be 30 feet apart for the two measures.” Thank you so much for this idea!

I’ve been struggling about just what Dan means when he says pseudocontext. I thought I knew, but I am realizing that I am thinking about it differently–which is fine, especially since it’s making me think really hard about this.

Anyways, I wanted to quote you some Jo Boaler to show you just how awesome this problem is:

“A realistic use of context is one where students are given real situations [check] that need mathematical analysis [check], for which they do need to consider (rather than ignore) the variables [uber-check].”

Thanks! I am loving the conversation that’s going on about pseudocontext (including your recent post); it is really helping me develop my understanding. Is the quote you posted here from What’s Math Got to Do with It? It’s been a while since I’ve read any Boaler, but I think I need to. :)

I was in Sweden this summer and my cousin showed me how to do this. You hold a stick in your hand that is the same length as your arm. You then walk back until the top of the stick lines up with the top of the water tower or whatever you are measuring. At that point the height of the object in question is equal to your distance from it. Is that what the kids finally figured out? See my blog for cool Book Summaries. DrDougGreen.Com.

Best,

Douglas W. Green, EdD

That’s not how they did it (there’s a fence around the water tower, so they couldn’t get to its base), but I will have to share that method with them!

What they did was to measure the angle of elevation to the top of the water tower, move away from it a known distance, and measure the new angle of elevation. From that, they could use tangent ratios to determine (1) how far they were from the base of the water tower to begin with and (2) the water tower’s height.

All life is a problem searching for a solution. Presenting problems that actually exist in the world give students actual real world problem solving experience. While asking if there is “an easier way to get the answer?” is a good thing, it is far better to develop in students the ability and skill to be “patient problem solvers” as Meyer likes to say.

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[…] all. But contexts, used appropriately, can be really awesome. See Amanda’s super-cool lesson on finding the height of the water tower next to her school. Personally, I think Amanda’s lesson satisfies Jo Boaler’s criteria for good use of context, […]