Finding the "Hidden Wows"

I was working with some middle school teachers-to-be who are noticing some of the aspects of algebra we tend to take for granted after years of practice and application, and they are noticing how difficult it can be to anticipate the strategies and struggles of students who are just learning algebra.
 
One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the Hidden Wows.

Suppose I am preparing a lesson for sixth graders to introduce problems of this form:

Solve: ax = b

 My planning might start out sounding a bit like this:

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.