Have any of you seen the video/film MAA did of Polya teaching a group of students at Stanford in the ’60s – “Let Us Teach Guessing”? He gives them a very nice problem about 5 planes subdividing 3-space into regions and the way things progress during the lesson is masterful. His accent is sometimes hard to parse, but it’s very much worth the effort.

Someone recently put it up, but I’m waiting for permission to share the link. I hope everyone interested in the issues being pursued here gets a chance to see it.

That said, I have a specific question for Dan: you say

“You look at the modeling standard and look at what it says. (This terrifies me by the way.) “Identifying variables. Formulating a model. Analyzing and performing operations. Interpreting the results. Validating the conclusions.” That’s your ladder of abstraction there. [I won’t exactly sign on off that now, FWIW, but let’s see where he’s going with this. -dm] And what do we have students doing? Just that middle rung.”

Could you elaborate on what terrifies you exactly in what you’re citing and why? Not that I don’t think there are things to be terrified by, but I’m not clear as to what you’re pointing there and don’t want to jump to conclusions about your reasons, either. Thanks.

I had my first day with students today and we used your Gatorade problem with our Algebra I classes for the second year in a row. I was surprised when more than half the class thought the 6-pack was going to be the better deal, but it was that much more powerful when they proved themselves wrong with the math!

The new technology that was added from last year’s Gatorade problem to this year’s was the introduction of the document camera. It is amazing how such a simple tool can change the conversation about math in your classroom when 23 ninth graders are all looking at and analyzing their classmates’ work.

Debbie:

Hi Dan, this post makes me wonder a couple of things. Firstly, are any publishers actually listening to you? Is there a project under way to produce a textbook along these lines?

I’ve been advising Pearson on their new CCSS textbook but I’m afraid, as a sub-contractor of a sub-contractor, I’m pretty easy to ignore.

Secondly, we keep seeing the same examples — the water butt, the basketball shot et cetera. Great questions which I feel exemplify your thoughts so clearly, but do you find that you are spending less time on this element of classroom teaching?

There’s never enough time for everything, unfortunately. The more I give talks about curriculum design, the less time I have for designing curriculum. I release a couple of new tasks every month at threeacts.mrmeyer.com, though.

Secondly, we keep seeing the same examples – the water butt, the basketball shot et cetera. Great questions which I feel exemplify your thoughts so clearly, but do you find that you are spending less time on this element of classroom teaching?

So instead of guessing about the basketball once, you have many many opportunities to guess (different shots). Not just multiple guesses as more information is added, but multiple problems with slight variation (not sure if you already intended that).

I’m also encouraging more tools so that you can literally get the right answer through bulk force. So for the basketball problem maybe there’s a way to create an additional arc through a simulation. You can see if that arc matches the current trajectory and goes in. As you progress through more problems the tools get more complex, so maybe you have to type in the calculation required to provide the estimated arc and see if it matches up.

Does that make sense?

Imagine if we never exposed our students literature, to Shakespeare or to Milton or to Vergil or to Homer. They’re part of the reason that you learn the formalism and the operations of the languages that you’re learning.

What you’re suggesting, I think, is that way we write curriculum is equivalent to the pablum that ends up as examples in English (or foreign language!) textbooks. It’s stripped of the immediacy, tension, drama, and truth and resonance of good literature. We can do better.

I stayed up till 2:00 in the morning to watch Curiosity land because it was the answer to a heck of a math problem. Even though I didn’t do the work to solve it, the drama was killing me. I *had* to know the answer.

And I think it’s also worth noticing that you just told the people who put their hands up early, “oh, that’s the rock hitting the side of the well.” Sometimes it’s okay to just tell people stuff about the context.

It does make perfect sense to ask them to guess first. It’s even a pretty decent hook with minimal work and could be a smooth transition into the lesson.

I’m excited to get started. What great timing to post this simple idea at the beginning of the school year.

Thanks