Kyle Webb drops a WCYDWT video on circle area and perimeter:
Academic Green Circumference and Area Problem from Kyle Webb on Vimeo.
First, let’s pay respect to how fast the video moves, how it sets a scene and establishes a problem in just 14 slides and 57 seconds. Webb knows his audience and its attention span. Also, none of this is stock photography. Every photo selected is of high bandwidth and relates directly to the problem. After 12 seconds, we have three different views of the lawn. After 15 seconds, a panoramic shot. I’ll begin my redesign 23 seconds in, when he mentions the lawn is 75 steps across.
This is really, really close to my textbook’s own installation of the problem. The text would ask a question like “how far is it around?” or something with a real-world spin like “how large would the ice rink be?” (standing in for “what is the area?”) and then it would explicitly define the only variable we need: 75 steps. My students would identify the formula and then solve.
This kind of instructional design puts students in a strong position to resolve problems the textbook draws from the real world but in no position to draw up those problems for themselves. This kind of instructional design also yields predictably lopsided conversation between a teacher and his students.
The fix is simple but difficult: be less helpful.
Let’s start here: is circle area just something math teachers talk about to amuse themselves or do other people care? If they care, why do they care? How do we convey that care to our students? Maybe someone needs to fertilize the lawn. Maybe someone wants to spray paint the dead lawn green in the winter. Without this component, the answer to the question “how far is it around?” is little more than mathematical trivia to many students.
So put them in a position to make a choice, a tough choice that’s true to the context of the problem, a choice that math will eventually simplify.
For instance: “how many bags of fertilizer should I buy to cover the entire lawn?”
Or, a little weirder: “how many cans of spray paint should I buy to cover the entire lawn?”
In both cases, we’re putting every student on, more or less, a level playing field. They are guessing at discrete numbers (ie. “fifty bags – no – sixty bags.”) and drawing on their intuition, which, from my experience, is a stronger base coat of for mathematical reasoning than the usual lacquer of calculations, figures, and formula.
This approach also forces students to reconcile the fact that the problem is impossible to solve as written. This is an essential moment. They need more information, but what? What defines a circle? Would it be easier to walk across the lawn’s diameter or around the lawn’s circumference? Which would be more accurate? Why is the radius difficult to measure? Did Kyle really walk through the center of the lawn or does he just think he did?
When you write “75 steps” on a photo, that conversation never happens.
My thanks to Kyle for jogging my thoughts here.
cornwalkerOctober 29, 2009 - 8:37 am -
Don’t forget that while the lawn is being fertilized you’re going to want to keep people off of it. You’ll need fencing, and some fencing stakes to support it.
Dan MeyerOctober 29, 2009 - 9:02 am -
Yeah, nice one there.
Kyle WebbOctober 29, 2009 - 9:18 am -
Thanks Dan for the great comments!! This was my first stab at something like this and I was hoping to get some constructive criticism like this. I did feel like it was a textbook-like problem put into a video but I wasn’t sure how to change it. I will definitely try my best to use your suggestion of being less helpful when I come up with my next video.
Once again, thanks for the constructive criticism!
Abby NoyceOctober 29, 2009 - 9:37 am -
How big a snowman can I make with the snow cleared off that pathway?
(Hmm, that’s a (long but) straightforward problem if you assume a snowman is made of three equal spheres, but trickier if the spheres are in the (more aesthetically pleasing) large-medium-small order. :) )
CalcDaveOctober 29, 2009 - 9:41 am -
The students will also have to think of “how long is a step?” in terms of standard units in order to figure out how many bags of fertilizer (or gallons of water for the ice rink) to get.
In calculus I use this as an example in optimization. The textbook would have things like, “A person walks 4ft/s on concrete and 3.7 ft/s on grass, what is the path a person should take to cross from one side to the other of this lawn fastest?” You can, in WCYDWT-spirit, take out the info at first and make the students ask for it. (College walking paths never seem to go straight or be very efficient. So, I would consider such problems on my walks from one class to another.)
Dan MeyerOctober 29, 2009 - 9:46 am -
@Kyle, Sure, no problem. These are fun thought exercises.
I’ll also point out that, with video, you’re basically obliged to layer on all the information at one point or another. You can’t send that video to a teacher in California and expect it to be useful if, at some point, you don’t mention “75 steps.”
This is why my preference, as that teacher in California, is to receive from you a zipped archive of all the great imagery, perhaps including my spray cans, my fertilizer bags, and cornwalker’s fence posts, and somewhere in the text of the e-mail mention, “oh by the way, whenever your students ask you for this detail: it’s 75 steps (or 55 meters, etc.) across.”
I’m trying to make a case for a certain separation of powers in curriculum design. I’m also trying to build an online forum to enable that separation, so teachers can share those resources easily. I hope you’ll let me sign you up for a beta account, Kyle, whenever we reach that point.
Kyle WebbOctober 29, 2009 - 10:14 am -
Just to make this clear; I should be sending a file full of images to be presented along with some information in text so students could have access to it if they decide they need it? That way, you as the teacher in California could present it how you’d like and allow the students to figure out a strategy to do so instead of me basically giving them the steps to solve it.
And yes, I would love to be a part of the forum with a beta account.
Dan MeyerOctober 29, 2009 - 10:32 am -
It’s like remixing music: it’s so much easier if you have access to the vocal track separate from each of the music tracks. In order to get a clean copy of the lawn, I had to go frame-by-frame until the name of the lawn had mostly disappeared.
I’m talking about making that process easier for teachers by being just a little less helpful.
Kyle WebbOctober 29, 2009 - 10:48 am -
Ok Dan, that makes perfect sense. When I make my next video I will try that approach and see how it turns out.
Joe MakoOctober 29, 2009 - 10:53 am -
If going off just the content in the video, at the 10 second mark, it shows a top-down drawing of the green, and it is not a circle, but an ellipse. The camera angles in the photographs give the illusion of a circle. (reminds me of when I thought the planets traveled in an ellipse around the sun from skewed renditions I saw)
Kyle WebbOctober 29, 2009 - 11:01 am -
You’re right Joe, it is an ellipse. That was intended to be something that the students may be able to pick up on and realize that there is a flaw with the question, and therefore their answers wouldn’t be exactly correct.
Jason DyerOctober 29, 2009 - 11:45 am -
I would want to make it really clear what a bag of fertilizer looks like. From your clipped photo, I imagine some students would be thinking small and some would be thinking large.
Dan MeyerOctober 29, 2009 - 11:52 am -
Sure. I included the “3 cubic feet” listing intentionally but, bytes being cheap and all, it’d be worthwhile to stand someone next to it for scale.
concretekaxOctober 29, 2009 - 5:12 pm -
Thanks for blogging on Kyle’s problem. I am the teacher that he created the video for. Comments that I would like to add is that this was for a 6th grade math class for whom area of a circle and concepts such as pi are fairly new concepts.
The students did recognize for themselves that “steps” was not an accurate measure and also as Joe Mako noted that it was not a true circle.
We had done a similar problem previously with figuring out how much wood chips needed for a landscape garden outside of my room. I posted about that here http://concretekax.blogspot.com/2009/10/wood-chips.html
For that problem the students were forced to come up with all information including researching the formulas on the internet, measuring the circle themselves, and performing the math. My impression was that since this was the first time the students were taught math in this way that many of them struggled. My guess is that with more exposure to this kind of teaching their problem-solving techniques will improve.
curmudgeonNovember 8, 2009 - 3:41 pm -
Something that I might do is to give them access to Google maps which has a scale in the lower right but limited resolution or Google Earth which has better resolution but has the ruler feature.
I’ll put some pictures on my blog to show the two options.