Jim Pardun sent me a video of a dog named Twinkie popping balloons in the pursuit of a world record. How you train a dog to do this, I don’t know. How there is a world record for this, I don’t know either.
What I know is that this video clearly illustrates the difference between math and modeling with math.
Our mathematical models, by contrast, arrive broken. “All models are wrong,” said George Box, “but some are useful.” And we see that in this video.
Twinkie pops 25 balloons in 5 seconds. How long will it take her to pop all 100 balloons? A purely mathematical answer is 20 seconds. That’s straightforward proportional reasoning.
But mathematical modeling is less than straightforward. It requires the re-interpretation of that answer through the world’s imperfections. The student who can quickly and confidently calculate 20 seconds may even be worse off here than the student who patiently thinks about how the supply of balloons is dwindling, adds time, and arrives at the actual answer of 37 seconds.
Feel free to show your classes that question video, discuss, and then show them the answer video. Or if your class has access to devices, you can assign this Desmos activity, where we’ll invite them to sketch what they think happens over time as well.
The difference between the students who answer “20 seconds” and “37 seconds” is the same difference between the students who draw Sketch 1 and Sketch 2.
You might think you know how your students will sort into those two groups, but I hope you’ll be surprised.
That difference is the patience that modeling with math requires.
BTW. I’m very interested in situations like these where the world subverts what seems like a straightforward application of a mathematical model.
One more example is the story of St. Matthew Island, which dumps the expectations of pure mathematics on its head at least twice.
Do you have any to trade?