If it were physically possible to fold a piece of paper in half 50 times (it's not), how thick would the resulting origami sculpture be? Quick! No fair calculating. What does your gut say?

If you have absolutely no idea, I'll tell you that a standard piece of printer paper, folded six times by high a school student with very little concern for symmetry or crease definition, has an average thickness somewhere between six and eight millimeters. How much will that increase over the next 44 folds? Any ideas?

Out of roughly 100 students, all guesses but one were under 10 feet. One girl said 100 feet, and she just about got laughed out of the classroom. The actual answer: a little over 100 billion meters. No, that's not a typo. In fact, if you could manage to fold the paper in half one more time, for 51 total, you'd have a stack that's taller than the distance from the Earth to the sun.

The math is pretty straightforward. Every time we fold our idealized sheet of paper, the thickness doubles. After 50 folds, that's 2^{50} times the original thickness, which, for the printer paper stashed in my room, is about 0.1 mm, or 0.0001 m. Multiply those two values together and be amazed.

I chose this exercise because it's a nice way to introduce recursive geometric sequences, and because it's a nice change of pace from the standard penny-doubling problem. But I chose it mostly because it's so incredibly counterintuitive. And *that*, I submit, makes a powerful argument for mathematics.

Every time we smack our students' intuition in the mouth, we implicitly answer the nagging question, *What is any of this good for?* Well for one thing, Mr. Thinks-the-answer-is-two-feet, It's good for critically examining our preconceptions, which are often comically off base. When we can spectacularly highlight the fallibility of the human gut, we create a tiny void that has to be filled with *something *useful. And the best candidate: mathematical inquiry. A surprising result forces students to shove a problem across the corpus callosum and give the analytical self an at-bat. Do that enough times in the classroom and we start to build mathematical habits of mind. Math starts to edge out intuition as the default setting.

So next time you're looking for an interesting problem to build a lesson around, give some extra weight to the ones with the most startling solutions. I conjecture that the dividends grow exponentially.

ChristopherTwo questions. (1) How impressive is the answer to

What is this good for?when the exemplar is predicting the result of a task that is literally impossible? (2) How do you balance the smashmouth trashtalk with a (presumably) equally strong impulse to invite students to mathematical inquiry, rather than turn them off by reminding them how terribly bad they are at it?Chris LustoYou're such a curmudgeon. To be clear, I would like to point out that I didn't actually, literally trash-talk anybody.

(1) First of all, I don't tell them it's impossible. It has to be phrased as a hypothetical because students are aware that you can't fold an 8.5 x 11 any more than six or seven times (which bears out with some in-class experimentation), but they presume that 50 is doable with a sheet as big as the room, perhaps, or certainly with one as big as the football field. (In fact, with a field-sized sheet, a forklift, and a steamroller, you can apparently get 11 folds.) I think that the answer to

What is this good for?is supremely impressive insofar as it's not at all obvious to most people justhowimpossible the exemplar is. In fact, it takes some mathematical reasoning about the questionjustto get to the conclusion that it's a ridiculous question to be reasoning about.(2) I don't think the point of any of this is to remind students how terrible they are at mathematical inquiry. In fact, the point is to remind them how terrible they are

withoutit. And just to make sure eventhatdoesn't bruise their egos, I basically blame it on biology. We had a nice conversation about how there are just some things that human beings are pretty awful at, as a general rule: generating and recognizing randomness, estimating probabilities, and reasoning about extremely large numbers. So I invite them to inquiry as a way of compensating for their evolutionary shortcomings, of transcending their (our) built-in limitations.Pingback: Fool me once, shame on… shame on you. Fool me… you can’t get fooled again. | Reflections in the Why

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