Great activity – thoughtfully sequenced. Thank you for sharing this. I think that the surprise you refer to is often the same reason that we appreciate great visual art. When you look at art, there are often layers that reveal themselves and surprise you. What appeared to be simple at first might actually have a depth and a novelty that we didn’t expect, and it’s this moment of joy at a violation of our expectation that sticks with us and makes us keep thinking about it. As a teacher, choreographing this moment, and observing this joyful lightbulb happen in students is one of the greatest gifts we can give to them. Mathematicians naturally understand this moment because we have had the chance to experience this joy. Most students’ opportunities for this experience are pretty limited by the idea that mathematics is a series of procedures that students need to be told how to do.

It’s interesting to me that you specify that the function is continuous for the first three, but not for the last function. The surprise is something of a magic trick here, a misdirection driven by an expectation you’ve created in the first three that the function will be continuous. I wonder whether we could think about two kinds of surprise: those arising from assumptions we’ve incorrectly conditioned into the universe of the problem, and those arising from genuinely counterintuitive bits of mathematics.

For example, Calkin and Wilf’s lovely paper on recounting the rationals (or even Cantor’s diagonal method) is, I think, the former. Students make assumptions about what it means to count infinitely many numbers, and those assumptions are wrong in a surprising way. Perhaps the definition of the zeroth power is the latter?

Certainly that sort of mathematical magic is powerful in the classroom – one of my favorites is to introduce logarithms by putting up a few values on the board and then make use of the arithmetic properties to compute more in my head. Not know how I’m doing that makes the students batty.

I had just come up with the observation that “most of the concepts of geometry are obvious to students (once they see them) and that most of the challenge comes from synthesizing multiple concepts within a problem. But now I have to seriously consider how I can surprise students in geometry. Thanks, Dan, you always throw a change up. I will be back with surprises.

(a) This is awesome. (b) I generated surprise and suspense around the IVT in a completely different way today, with a different goal. The context was a Real Analysis course so I wanted to motivate not the statement of the IVT but its proof. I had students work with a function defined only on the rationals, by f(x) = -1 when x^2 > 2 and f(x) = 1 when x^2 < 2. We applied the definition of continuity to this function and found that it was actually continuous at every point in its domain. We thus had a function with a "jump" that managed to be continuous anyway. How to account for this?