Thursday, July 1, 2010

Curriculum Makeover and Problem 11

It's summer and I now have time to sit and think about things. This time it's math curriculum makeover. We start with Dan Meyer's Rethink of a textbook problem. (seen at right).
"A water tank is in the form of a regular octagonal prism. The base octagon has side length 11.9 cm.  Then lateral edge is 36 cm.
a) What is the surface area of the base? 
b) What is the volume of the water tank?
c) If you pour water into the tank at a rate of 1.8 oz/sec, how long will it take you to fill the tank?"
Dan's take: "claims real-world but its only clip-art or a line drawing. Specifies the exact method of solution, and only gives useful information." So he filmed himself with a garden hose filling the tank with a timer available. Show the film and let the students ask "How long will it take to fill?" Here's a textbook editor's discussion of this problem.

WCYDWT: Water Tank from Dan Meyer on Vimeo.

There's a couple of things about Dan's treatment of the question that bugged me but I hadn't really thought about it till now.

First, of course, is that subtly, he changed the question to be ONLY about part c. There was no calculation of area or volume, just an approximate measure of time. If I am doing proportions, then it would be okay but, to me, the real point was ignored. Let me be clear on this: that video is cool. It worked for my students in the same way in worked for Dan's. It's a good way to start a section on proportions or rates.

BUT. I find the immediate solution method his students (and mine!) choose as being too simplistic and hands-on. Elementary and Middle school students need tangible, hands-on material and filling a tank with a hose is a fine way to bring them up to higher levels of thinking but it's not the point of algebra; high school students should be beyond that kind of simplicity.

Algebra is more abstract, more at an arm's length from the immediate numbers of a problem. The idea of algebra (to me) is to model something without having to count squares, count seconds slowly or tick off every single mark. I don't like it when the only solution that students come up with is "guess-and-check" or "counting squares" if there's a better analytical solution.

The video only works because the tank in the problem is small but even then the video was too long. (I'm not sitting through 8.5 minutes watching a tank fill up.) What if this were a town water system or a swimming pool? What if this "real world problem" is too expensive for guess and check, which is what a real world problem SHOULD be? (If it weren't too expensive or time consuming, you WOULD guess and check or count squares.)

I agree with Dan's comments elsewhere when he says that problems should be solved in an appropriate fashion and that often textbooks will invent something just for the rather lame use of the topic du jour. I get that argument. I don't think problem #11 is such a problem.

It could have been stated with more pizzaz and could have been given more "connection" for the students, but AZZA math teacher, I should provide that myself. "I need to know the volume in liters because I have to add salts to my salt-water tank in exactly the right proportions -- the fish are delicate and EXPENSIVE. I can't guess and check with $100 fish."

Second, the givens should be real-world, too, and I think Problem #11 does that. Physically, you can measure the outside edge, the height. Had the problem given the apothem or the radius of the circle enclosing the base - how would anyone measure that accurately? Diagonal through the center - yes! Now you're thinking.

Showing all those measurements would be a good thing but ultimately you have to decide where this problem is in the scaffolding and plan accordingly. A few extra numbers go a long way in allowing the students to explore without having the tank in front of them but they also confuse the hell out of those who are just getting the idea.

Bottom line: Keep the video but don't use it for just a volume calculation. Use it for proportions or rates.


  1. I totally agree, and all that, but looking at the problem again, the thing the thing I'd really like to do is to make the same sort of film, but with a conical tank, instead or a prism-tank. Rather different thinking and calculating required there...I wonder where I could find a conical tank...

  2. Turn your classroom dunce-cap upside down?