“What I’m going to ask you to do is watch a video. It’s a short video, made by me. You’re not looking for anything in particular.”
“Here’s the deal, I’d like you to watch it again, but this time, I’d like you to look for any quantities.” For every student in the room, it’s the first day of high school math class, and you’re reading a transcription of it. You can listen to it in full here (apologies for the low-quality).
“What’s a quantity?” a student asks. This is the first student question of the year.
“Great, what’s a quantity? I was hoping you would ask that. A quantity is anything you can attach a number to. I’ll give you two examples. One is the ‘number of trees in the image.’ That is a quantity. You can count it. The speed of the runner is another quantity. It’s tougher to count, but you can still put a number on it… Any quantity is OK. There are probably hundreds to choose from if not thousands. See if you can pick out one or two, or five or six. I’m going to play the video again, then I’d like you to share those quantities with the person next to you.”
10 students volunteer quantities they saw in the video to the whole class, one by one. As they speak, I type into a document projected at the front of the room, attaching their names to the quantities they say. Some kids don’t quite get what I’m going for when I say “quantity”:
Student: “There are 23 houses”
Me: “Can I say the quantity is ‘number of houses’? And I’ll put in parentheses that you counted 23”
Student: “Miles per hour that you hit the ball… er, speed”
Me: “Speed of… what? Speed of light? Speed of…”
Student: “Speed of the ball”
Kids start getting it. They shout out: “Strands of turf,” “Number of bases touched by the runner,” “Number of bases visible,” “Number of feathers on the Spartan helmet on the field.” Pretty soon it’s getting too easy and kids are jumping at the chance to say a quantity.
We do a second draft through our list of quantities. Without being too mean (it is the first day, after all), I’m pushing the students to attend to precision. Give me units. Be specific. Make sure the wording is right. “The bat bounced twice” becomes “Number of bat bounces.” “Speed of the ball” gets a unit (MPH, chosen over feet per second) and actually becomes “Speed of the runner, in MPH” because there is no ball, as a student points out. Our final list on the board looks like this, with students’ initials next to each:
1. number of bat bounces (JB)
2. distance run, yards (ZS)
3. steps taken by the runner (RE)
4. number of bases on the baseball diamond (OS)
4a. number of bases touched by the runner (OS)
5. number of houses in the frame (23) (SC)
6. strands of turf (LC)
7. number of bases shown (EE)
8. number of legs visible (CS)
9. speed of the bat, in miles per hour (HB)
10. number of feathers on the spartan helmet on the field (TD)
Now with our clean list of quantities, we separate them into quantities that change during the video, and those that stay the same. I also introduce one new quantity that is less visible: “time, in seconds.”
Now, all of these quantities are interesting to think about and explore. At this point, I had the class focus on just one: 4a, the number of bases touched by the runner.
I ask them for “the story of the number of bases touched, in relation to time.”
Before they actually dive into this, we have to attend to precision a bit more: Is home plate a base? What are we assuming about what happens when the runner goes off-camera?
Once we clarify that, I ask – numerous times – for them to tell the person seated next to them: “what’s the story?” I ask for “as much detail as possible.”
When we come back as a group, students start responding and though none of them were wrong really, I refused to be satisfied. “I want every detail possible. How many bases had I touched after 2 seconds? After 3 seconds? After 4 seconds? After 5 seconds? After 6 seconds? After 7 seconds? I want the full story. Leave no details out. What about 7 and a half seconds?” I’m sure this is frustrating. I basically asked them to do something that they didn’t have the tools to do. So…
“I’m going to give us a tool to tell this story in a little more detail.”
We are exactly 20 minutes into 9th grade math. Welcome.
This is the beginning of a unit called Graphing Stories, the brainchild of Dan Meyer. In the unit, students develop a conceptual understanding of functions by doing things like graphing changing quantities over time that they see in 15-second videos. I had taught Graphing Stories before, but without everything that you read in this post. It might seem odd to make an activity longer instead of cutting to the chase. Isn’t good teaching about getting more efficient? But Dan himself has advocated for additions like the ones I describe above, arguing that this pre-work is quite valuable. What did I buy with this 20 minutes? Students learned…
- what a quantity is.
- that quantities need units, which sometimes means we must choose between a few options.
- that being precise and clear helps people understand what you’re talking about.
- that people view the world differently, and your unique view of the world is valued.
- that many students will be contributing regularly to and driving the work we do in this class.
- that sometimes having a mathematical tool makes a task easier.
- that a graph is one way to tell a story.
As to the last point, I let Kurt Vonnegut convince them of that a few days later.