A nice and deep thought

*second in a series about how we avoid thinking in math class*

This September, I gave my 7^{th}-graders an elegant little problem about a 12-step staircase. You’re climbing from the bottom to the top, using combinations of single and double steps. The question is, how many ways can you do this?

I was stunned when some of my students offered answers almost immediately. “145!” one screamed, as if he had just gotten bingo. “Am I right?”

“Whoa, that was fast!” I said. “Why 145?”

“12 times 12, plus 1!” he announced. “Am I right?”

“But…” I hesitated. “But why 12 times 12? Why plus 1? Are we just doing random computations that sound like fun?”

He listened to my questioning with the same patience you’d give a friend’s mediocre guitar solo. Then he launched right back into his chorus: “So,” he said, “am I right?”

To him, at…

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