Longtime NW tutor Sara shares an “a-ha!” teaching/learning moment she experienced with her son. It’s so wonderful when we learn as much as our own children and students!
Meters and Tracks
My 7-year-old son and I watched my older child compete in a track meet. It was a chilly day, and we’d been standing outside, huddled under our coats, for about an hour. As we stood at the fence waiting for the next event to begin, my son sighed, looked up at me and said, “Mom, I’m bored.”
The teacher in me always looks for moments like these to squeeze in some mental activity, so we started talking about the track.
“Do you know how many meters long the track is?” I asked. He shook his head. “It’s 400 meters,” I said. “So when a runner does one lap around the track, he runs 400 meters.”
That seemed to make sense to him, so I figured I’d start off with an easy problem. “All right,” I said, “so if a runner does two laps around the track, how many meters is that?”
I’ll never forget the look of surprise on his expressive face. Shaking his head, his brow knit with genuine confusion, he responded, “What? Mom, I don’t know. We’ve studied meters before in school, but we’ve never studied tracks!”
This time, it was my turn to be surprised. My son was in second grade, so I knew he had the arithmetic skills necessary to compute 400 + 400. But for some reason, he didn’t recognize that this was just an addition problem. I smiled and crouched down next to him so that we could talk through how to get the answer. In the back of my mind, I was at work on a problem of my own: why was he so baffled by what I thought would be a simple problem?
The answer, I found, lies in the concept of the structures of problems, explained by Daniel Willingham in his book Why Don’t Students Like School?.
Word problems, such as the one I gave to my son, all have two elements to them: their surface structure and their deep structure. We adults recognize immediately that the ‘laps around the track’ problem is, at its core, an addition problem; that is, its deep structure is an addition problem. The fact that we’re talking about meters on a track does not change what we do to solve the problem. Actually, we could quite easily think of dozens of ways to “disguise” problems with the same deep structure:
- If there are 2 gallons in 1 jug, how many gallons are in 2 jugs?
- You need 3 eggs to bake 1 cake. How many eggs do you need for 2 cakes?
- There are 5,280 feet in 1 mile. How many feet are in 2 miles?
- If there are 20 ziggiblops in 1 splutzik, then how many ziggiblops are in 2 splutziks?
- There are 12 # in 1 ^, so how many # are in 2 ^?
The “disguise” is the surface structure of the problem. And, as you probably noticed in the last two examples, the surface structure need not even be familiar (or pronounceable) for you to recognize the deep structure and solve the problem.
The disconnect that arose for my son is that while he had the skills and knowledge necessary to solve the problem, he got hung up on the surface structure of the problem – laps around a track, which he had never formally studied – and failed to recognize the deep structure – addition, something he’s been practicing for months on end.
Being able to apply knowledge to new and unfamiliar problems is called transfer, and it’s something that we teachers and parents try to encourage in our kids. But, as you may have noticed with your own kids, knowledge transfer isn’t easy. There are reasons why it’s tough, and there are things we can do to promote knowledge transfer – but we’ll get into those another time. For now, see what you can do identify the surface and deep structures in problems you encounter – and have patience with your kids (or yourself!) during the countless repetitions of problems that will be necessary for successful transfer of knowledge.