This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher. See my other posts and an introduction.
I wanted to teach data modeling. The students already knew the basic shapes of lines, hyperbolas, parabolas, cubic functions, square roots, etc, and how to graph them, but we hadn’t talked about what to do with real data.
The tempting approach is to give out 25 points and say “find an equation based on the functions you know that goes close to these points.” In fact, that’s what I did in my first year.
This is terrible.
In my second year, still burning from the bored stares of the first year, I convinced our facilities manager to climb something like nine meters up a terrifying ladder to hang a couple of ropes from the roof of our gym.
Later, I brought my class to the gym with bunch of weights between 2 and 10 pounds. I described a couple properties of pendulums, did the whole omg-its-going-to-smash-his-nose-oh-whew-he’s-fine trick, and finally defined “period,” complete with a tutorial of how to measure it (give all the kids stop watches, have everyone time it at once, and then give an official measurement so they can see if they were too slow or too fast or what).
I made sure NOT to say that the period of a simple pendulum is determined by its length alone, and that weights have no effect. They do not know at this point that period vs. length is a square-rooty function.
Then, I said, “please make your pendulum swing so that its period is within a tenth of a second of 4.5 seconds.”
This is great.
Here’s why: the students are conversationally fluent in this environment. They know that pendulums swing back and forth, they know how to tie knots, lift weights, push things, measure time and distance, and they consider themselves experts in this area. In fact, they are experts – they have been practicing physical interactions every day for the last 16 or 17 years! But they lack the technical precision to describe it well enough to predict specific measurements. When they realize they actually don’t know how to achieve a specific period, they will be intrigued. Some will think they do know how, but then be unable to do it, and they will be even more intrigued.
This is why you spend hours to create a real environment. They are good at reality, and feel comfortable in it. They have vocabulary to talk to each other about it. Compare this with discussing square root regression in class – they’ve learned about root functions and practiced for, what, 2 hours total (against 75,000+ hours of practice with physical motion and perception). They don’t feel comfortable talking about what they’ve learned yet, and they certainly don’t have the vocab to experiment with what they’ve learned by tweaking it in small ways. In the real world, they can easily say “let’s try pushing harder” or “add more weight!” and see results instantly.
Anyway, the students spend the next 40 minutes trying things, different weights, different initial velocities, different angles, and different pendulum lengths. Only the lengths make a difference. They naturally figure out which direction makes the period longer and which makes it shorter, and I jump on that opportunity to point out that they’re already fitting a pattern to what they’re seeing (aka translating reality into math).
They try a bunch of different lengths, but never get that close to 4.5 seconds.
“Why not?” I ask innocently. Some say they don’t know, others say they can’t reach high enough. I ask, “Well, how high will you have to go?,” again innocently enough. ”I could bring a ladder to the next class I suppose, but I don’t want to go to the trouble of bringing the wrong length. Those things are really awkward and weigh a ton.”
Guys, they are hooked right now. My question was so natural, so authentic that they want to find me an answer just as a matter of course. I obviously need that information to help them out, not just to be an obnoxiously unhelpful math teacher. As homework they look at their data, figure out how they have to organize it to get me an answer more precise than “it needs to be higher,” and bring that answer to me at the next class period. We go to the facilities office, get the appropriate ladder, and try out their answer, which turns out to be astonishingly close to 4.5 seconds (easily within our measuring precision). A cheer goes up!
Now, I’m happy, but not that happy, because every group used a line to model the data. It turns out that a line is a pretty good model for pendulums as long as you stay in a pretty small range. So I asked the kids, “how high would you have to be to get the period to be one second?” Their linear models told them something like 16 meters, and it was obvious (after converting to feet) that the ceiling was nowhere near 16 meters tall. We spent maybe 20 minutes at this problem, talking about domains of models, practicing using the models to predict various values, etc. A ton of normal curriculum at the board fell in to this discussion, and they got a lot of personal practice time (please always mix practice in to your lessons).
I suggest that we use a smaller pendulum to get more data. This required changing our models to be functions of the length of the rope instead of the height off the ground (if all of my kids graduated knowing only how to change the parameters of a function I would consider myself a success). I bring out a few pendulums of the 1-, 2-, and 3- meter variety (I brought some twine and a pair of scissors to class) and we get some more data points. CLEARLY not linear now. We get MORE practice fitting the same data with a different type of graph. Now we can compare models, learn about different kinds of errors, answer fun questions like “so how high do you think the gym ceiling is?” etc.
I hope I’m making the benefits of an immersive environment clear. When you are working with new mathematical concepts, students cannot play with them or experiment with them because they don’t understand them well enough. If you give them a framework with which they are comfortable, they can feel confident enough to experiment, think, discuss, and ultimately learn. Additionally, when students are asking questions of the environment instead of you, they are responsible for their own questions and data. You won’t accidentally give away the lesson by providing only and all of the pertinent data.
If you do this pendulum thing, I recommend you measure while you can reach the whole rope. I forgot until just after the facilities manager had climbed down from the ceiling, and I wasn’t about to ask him to go back up there. It needs to be a thin, light, sturdy rope that doesn’t stretch too much, and it can’t rub too much at the top when it swings. Please emphasize, organize, and enforce safety procedures. You can hurt someone with 15 pounds on an 8-meter pendulum (attached with knots by kids with Velcro shoes).