Pocket strategies: Purposeful Incredulity

Think back to your school years. Think of that one teacher who managed to make every lesson fun and engaging, the one who had a canny way of making even the driest of topics seem worthwhile and interesting. 

How did they do it? 

Personality? Sure, probably. But I think this short post from Dan Meyer may have something to do with it, too. They made the content interesting, somehow. And it wasn't usually by embedding it into a game or puzzle or activity -- it was because they tapped into something genuine. They didn't just pose questions and show us how to find the answers, they made the questions seem worth investigating.

Dan's post calls it "developing the question."

http://blog.mrmeyer.com/2014/developing-the-question-why-students-dont-like-math/

A lesson I observed recently involved the theorem that "the sum of the vertex angles in any triangle is 180 degrees." The lesson involved an informal proof--an activity, wherein students created a triangle of arbitrary size and shape, cut it out, tore off the corners, and rearrange them to form...  a...

"What are we supposed to do with the pieces?" A student asked, mid-lesson. 

Just see what you can notice.

Eventually a few kids got the corners to form a straight angle. Word spread quickly, and before long everyone found they could do it. It's a nifty trick if you haven't see it:

Source: cutoutfoldup.com

But... now what? Because the question was never really developed, there wasn't much of a climax to build to, nor was there much relief when kids had figured out what to do.

But Dan's post got me thinking about that lesson again. How could we develop the question? One way I like to do it is by using what we might call 'purposeful incredulity'. I include it on my list recyclable "pocket strategies" for enriching, extending, or enhancing a traditional lesson.

Quadrilateral Hierarchies - Productive Struggle

An overarching goal for the semester is to help my pre-service teachers grow more comfortable with "productive struggle" and with persevering on challenging tasks. We worked at it for a long time earlier this semester on problems like the Chessboard Problem, the Cheesecake Task, and many others.

Last week, we moved into Geometry. After spending some time exploring the Van Hiele levels of geometric thought and the kinds of activities that help children progress to higher levels, it was time to put their own geometry knowledge to the test.

In this two-stage lesson, I first divided the class into five teams (of four) and assigned each team a shape class: rectangles, kites, rhombuses, parallelograms, trapezoids (inclusive definition). They were directed to produce a poster with a 2x2 grid with space for examples and properties of general and special members of their shape class.

Here's an example:

April Fool's Math: Pythagoras Who?

Some of my #ed331 teacher assistants have posted lessons they have taught -- to actual students! -- in which they supposedly prove the Pythagorean Theorem. You know:

For any right triangle, the sum of the squares on the legs is equal to the square on the hypotenuse. Sometimes folks just shorten it to a^2 + b^2 = c^2.

I know, right? Prove the Pythagorean Theorem!? No way.

Here's one example from @kayfayayyy's blog--she thinks she's going to be a math teacher one day--except here she is, showing her students a supposed 3-4-5 right triangle. Oh yes, very clever Miss Fayayyy (if that's even your real name).

You've learned your lessons well: just dangle some candies in front of your kids and they'll believe anything you say. You can read more about her lies at her blog if you like.

But what Miss Fayayyy doesn't know about candy is that it likes to play both sides. How about this 3-5-6 right triangle?

Go ahead, count the Skittles. 9 + 25 = 36? No way, man. There it is: a counter-example, in all its Wild Berry flavor glory. We must conclude that the Pythagorean Theorem is false.

Volume & Surface Area.. or, Wrapping Pancakes

We had pancakes tonight. And we had leftovers. So we decided to store them in the fridge to reheat later.

This is the way you are supposed to do it. 

But we just wrap them in foil and stuff them in the fridge. So, as usual, I pulled out a sheet of aluminum foil, eyeballing how much I would need and adding another inch or two for good measure.

Then I stacked the pancakes on top. It looked like this. (Ok so far...)

Two side-by-side stacks, and one extra on top. (That's odd...)

But then I folded up the foil, and that looked like this.