To get an idea of what I mean, watch the video from 4:33 to 6:28 as Dan talks about the "structural layers" of a ski lift problem. As presented in the textbook, the problem lists out the specific steps that students need to take to solve the problem. What Dan does is take away the steps and the mathematical structure, leaving only a visual and a simple question: Which section is the steepest?
Stripping away the layers and leaving only two things -- the question and the scenario -- leaves ample room for discussion, debate, and ideas. (For my purposes here, question refers to the problem the student is asked to solve, in its simplest form. Scenario refers to the situation being modeled in the problem, in its simplest form.)
I admit, in my first three years of teaching, I've given my students so very many math problems that have self-contained instructions for how to solve them. It's not something that lends itself well to deeper learning, and that's something I'm trying to work on this upcoming school year.
I'm trying to look at math problems in this new light, which is to strip away the layers and leave just the question and the scenario. As Dan Meyer demonstrated in the ski slope problem, you certainly can create more room for mathematical discourse and problem-solving.
But... what if you go even further, and remove the question, leaving only the scenario?
Seeing Less: Baseball Diamond Racing Problem
Full disclosure: I am a lifelong Chicago Cubs fan. From time to time, I like to write math problems about baseball. Here's one such problem I wrote for a Pythagorean theorem unit a few years ago:
The problem itself is not really a bad problem; it does require students to recognize that the segments connecting 1st, 2nd, and 3rd base form the sides of a right triangle. (In fact, an isosceles right triangle.) Once this is realized, the student calculates the hypotenuse using the Pythagorean theorem (or by multiplying the leg length by the square root of 2), then finds the difference between the sum of the two legs and the hypotenuse.
What the problem doesn't do is leave terribly much room for discussion. The information needed to solve the problem is given. The exact path that Jeff and Peter each take during the race is described. The problem also takes great pains to mention that the baseball diamond is in the shape of a square and that Jeff makes a 90-degree turn at 2nd base, strongly hinting at the existence of a right triangle. Once students realize that there is a right triangle in the diagram (which is really the only major thing in the problem that there is to "realize"), the rest is calculation.
Now, let's strip away everything until we have only the question and the scenario:
While perhaps I may have taken away too much, there is plenty to talk about with this problem now. Students will have to decide what they need to find out in order to answer the problem, and ask questions accordingly. Of course, with how much information I took away, a few might be asking: "What does this diagram even mean?"
Which is a very good question.
Seeing Even Less: A Scenario Without a Question
Recall the question I posed earlier: What if you remove the question, leaving only the scenario? Let's do that now:
Now we just have a diagram of a baseball diamond, with emphasis on the distance from 1st to 3rd. We don't have a footrace anymore.
Ever since I wrote the original version of this problem, I couldn't help thinking that there had to be more math, deeper math involved with the baseball diamond scenario. There had to be more than just a Pythagorean theorem footrace problem with this. But, I couldn't see it.
I couldn't see it because my mind was stuck on the original problem and blocked my way to other possibilities.
Seeing More: The Third Baseman Problem
Yesterday, I was at Wrigley Field for the Cubs vs. Cardinals game. I was watching the players take batting practice before the game. In one moment, when I watched a ground ball dribble toward third base, watched the third baseman scoop it up and throw it to first, a question popped into my head:
"How hard does the third baseman have to throw the ball to get the runner out at first?"
That question fits perfectly with this scenario:
Now we have an entirely different problem, using exactly the same scenario as the racing problem, that involves a heck of a lot more math.
There is so much conversation that can go on here! What information do students need to solve the problem? What skills are required to find an answer? There are plenty of factors at play in this situation:
- We need to know how fast the batter is running to first.
- Consequently, we need to realize that the runner accelerates to his top running speed (thus putting quadratics into play).
- We need to know at what point in time the third baseman fields the ball; in other words, how far away from 1st base is the runner at that point?
- We need to know if any natural factors (such as wind) need to be accounted for in figuring out our answer.
- We need to know where the third baseman is when he fields the ball.
- We need to know when the third baseman throws the ball; he doesn't throw it at the same instant he fields it!
- We need a way to figure out how far away from 1st base the third baseman is when he throws the ball.
- We need to know how we're expressing "how hard" the third baseman throws the ball.
This problem is more mathematically rich and complex than the racing problem and would almost certainly result in different groups of students coming up with different yet justifiable responses. It's open-ended, rife with ambiguity; messy, just how real-world math tends to be.
I probably would not have come up with it had I not found myself in a moment where I was only observing the scenario -- a baseball diamond with an emphasis on the space between 1st and 3rd -- in the absence of a question.
The point is this: Math is freaking everywhere. If you're a math teacher, you know this all too well. The problem is that sometimes there can be so much math in a scenario, that we have a really hard time seeing it until we strip away everything except the scenario itself.
Dan Meyer's way of stripping away the layers of a problem until only a question and a scenario are left is a fantastic means of getting our students hooked into having patient, thoughtful conversations about math and problem-solving. I found his talk to be inspirational. Going further and removing the question, I think, can be a way to help math teachers look more deeply at a situation and uncover even more math that they weren't seeing before. The more math we can see in a scenario, the more complex the questions we can ask our students, and in turn the deeper their learning. But in order to do this, I think we sometimes have to make ourselves forget the question and just look at the scenario from a fresh perspective.
Is this true for every math problem? I doubt it. But seeing less really can be seeing more.