Nathan Garnett, via e-mail:
I showed students a few pictures of the Luxor Hotel in Las Vegas, and then I asked them how long they it would take to wash all of those windows. We did lots of math. I had a student clean a 2ft by 5ft section of the board (with windex sound effects) so we could get a cleaning time per 10 square feet. It was a blast.
Great work. We can do something similar with Pyramid of Pennies. Students are often curious how long it took to make the pyramid. So we give students enough pennies to make the top two layers. We time them. Then they use proportions to answer how long it would take them to make the entire pyramid.
And then we list all the reasons that answer is wrong.
It doesn’t account for bathroom breaks. For sleeping. For eating. For the fact that the top two layers are small and easy but the bottom ones require scaffolding and much, much more care.
Those exceptions aren’t reasons to not ask the question. Those exceptions make the question more messy, more meaningful, more like actual modeling, and less like textbook modeling where air resistance is neglected, the rates are constant, the men are strong, and the women lithesome.
We need more messy modeling tasks like Nathan Garnett’s.
hillbyMay 3, 2013 - 6:02 am -
No doubt we need more messy modeling. We also need that conversation about what can we assume or ignore in order to make a simple mathematical model. If we have that conversation in creating the model, then it’ll be easier to add in those corrections later.
CurmudgeonMay 3, 2013 - 2:40 pm -
SeanMay 4, 2013 - 2:38 pm -
“Those exceptions aren’t reasons to not ask the question. Those exceptions make the question more messy, more meaningful, more like actual modeling, and less like textbook modeling where air resistance is neglected, the rates are constant, the men are strong, and the women lithesome.”
One improvement to a lot of textbook modeling problems is to remove as much data as possible. But how much data removal is too much data removal? You’ve done such useful, provocative work exploring this, and I really liked Kate Nowak’s recent description of the Mathalicious process:
“…scaffold too little, and the lesson becomes too hard for too many teachers. Scaffold too much, and the lesson does too much of the cognitive work for the student, robbing them of the chance at the learning that comes from insight.”
Another improvement might be to invite criticism of the assumptions the authors make (e.g. “What are the risks of holding the rate constant? Why might they have chosen to neglect air resistance? What other variables might we add or discard?”)
Amy ZimmerMay 4, 2013 - 4:49 pm -
Great job Nathan! Love it!
Dan MeyerMay 5, 2013 - 10:52 am -
When we’re modeling in our own lives, we have to make decisions about what information is necessary and how we’ll get it. We start with no data at all, just a purpose that’ll eventually require data.
So on its own, if the question is “How much data removal is too much data removal in a modeling task?” my answer is “You can’t remove enough data.” (That’s only for modeling, though.)
Kate introduces another variable – teacher quality. Optimizing for both of those variables will lead us to different conclusions w/r/t scaffolding. I think we should consider the two questions separately, though.
One: what does mathematical modeling look like?
Two: how can we best enact mathematical modeling in the classroom with teachers of varying quality?
SeanMay 5, 2013 - 12:21 pm -
I liked this a lot and went googling around to see what other folks had to say. Found an old Jo Boaler paper with this gem: “School mathematics remains school mathematics for students when they are not encouraged to analyze mathematical situations and understand which aspects are central.”
Script one of your 3-Act pieces out word for word: every question you’d ask, every misconception you’d anticipate, etc.. That’d be top-tier PD for lots of us.