Clearly, the student was delighted that his calculations matched what happened in the video. I said to him, “E., so you learned a little something about volume today, eh?” His response, without hesitation: “Always go with the fat one.”

Sam Shahshares his struggle to make related rates interesting and finds some solutions:

Then I asked our esteemed volunteer to use one breath to blow up the first balloon. Taped it up. Again, for two breaths. Taped. Et cetera until we got a total of six balloons taped. Then I asked what things are measurable in the balloons. Bam. List.

Bruce Ferrington asks, "How Tall Is Our Class?" a question which is surprisingly involved:

So, how were we going to measure the height of a class? Well, the kids came up with three suggestions.

I was walking with my wife along the River Corrib in Galway last weekend when we got into an argument that lasted the rest of the walk. I'll present our two arguments and some illustrative video. Then I'd like you or your students to help sort us out.

Argument A: It would be much harder to swim to the other side of the river in the fast-moving water as in still water.

Argument B: It would be just as easy to swim to the other side of the river in the fast-moving water as in still water.

I hope this gets as out of hand for you and your students as it did for me and my wife.

This excellent question exhibits a quality that is not found often in math curricula: it has the "specificity sweet spot": it is specific enough for a student to answer, but non-specific enough for every kid to agree on the answer. Students making different assumptions will have different responses, thus creating a real mathematical argument.

The whole reason I created this 3-act task was to have students realize that not everything is linear. The students watched the first act and I encouraged them to make guesses, just a number from their gut. Many students used linear reasoning for their guess. I heard this over and over again today: "Since he's a little lower than half the height of the pole, he must have a little less than half of the points."

The model wasn't what they expected. Oswald locates a similar flagpole model on a fan wiki. It wouldn't be the worst idea at all to have students graph that relationship (step function!) and compare it to the incorrect linear model they anticipated.

We watch the third act (the answer). Once we are getting close to the third jump, the students are hooked, their eyes are glued to the screen, and one students rubs his palms together and says, "Here we go!". When they see that the answer is 400, one student stands up, throws his pencil down, and complains that he was so close.

Love this also:

One of my students (who hardly says boo in class) threw his pencil down when he saw the third act. If that isn't passion, I don't know what is. They were in to it, they wanted to know why, they asked where those numbers came from, they made guesses, they tried to figure out why, they took pictures of the board before leaving class. One student even said that he was going to post about this on some gaming forum.

Me: Now … the strange thing about this elephant parade is that each elephant is a bit smaller than the one in front of it, and each one has an additional leg.

Griffin: How many legs does the last one have?

Let’s pause for a moment. This was his question, not mine. Real world be damned, this is a habit of mind thing.

Jimmy Pai, on students who dislike applied math problems:

Have I been focusing too much on applying mathematics and expanding the concept of "relevance?" Have I been expending too much energy on looking for relevance when I should play off of the interesting and awesome world that is mathematics? I had one student who became more disengaged throughout the year as everyone else was loving the relevance and exploring their own questions.

These are important posts. They remind me that it's a flawed theory of student engagement that leads you to draw two circles, one labeled "real" and the other "fake," and put material into either one with any kind of objective confidence.

Seven-legged elephants are real to Griffin. Numbers are real to Jimmy Pai's student. Star Wars isn't real, but for millions of viewers, it feels real. The techniques of storytelling can make the unreal seem real, but let's all agree to bear in mind that what's real to one student isn't necessarily real to another.

My favorite quote [from this article] is “the quality of a task need not be judged by its relation to real life but in relation to how it engages students’”

It's fascinating introspection on what counts as scholarship, how status is awarded, and how we define academic relevance in the 21^{st} century. It's particularly interesting given that he's a tenured professor at an elite university. He's throwing stones from inside the glass house, basically. On a personal level, having felt forced to maintain something of a firewall between my Internet advocacy and my work at Stanford, it was cathartic to hear that one of my professors isn't just aware of Twitter but understands that it complicates and enriches his professional existence.

There are plenty of interesting moments throughout the 20-minute talk. Here are a few I wanted to transcribe and collect:

On publishing:

After I received tenure … I had an opportunity to reflect upon the way that my own values had become changed by the culture of the university in which I was rewarded for publishing in the most prestigious journals, whether or not those journals had any effect on anyone else except for the small number of people reading those journals.

A foundation supervisor asked him about his theory of change. His response:

Our theory of change is that we will produce materials that are better than those that are commercially bought and the commercial companies will start to see that people are downloading our materials for free and they will start to copy what we are doing and imitate what we are doing.

On the waning relevance of universities:

Our worry is that, as we continue to produce only our refereed journal articles, we are not understanding the profound changes in how information is disseminated in modern society. The university, particularly professional schools that are supposed to be producing knowledge for practitioners, are being left behind. We are becoming less and less relevant to the people who most need our knowledge — teachers and students and principals and decisionmakers in the field.