Building A thinking Classroom - The First Curricular Thinking Task (Sort Of)

The past two days marked a big leap in my journey to build a "Thinking Classroom In Mathematics."  After three non-curricular thinking tasks (the first two of which I documented here and here), it was time to try our first curricular thinking task.

Non-curricular thinking tasks are tasks that involve numbers, logic, and complex thinking, but that don't necessarily address learning scheduled, grade-level content.  They're of great use when getting kids used to how to operate in thinking classrooms, and they're also fun to intersperse into the year just because the kids find them engaging and fun, and so they'll really do a lot of serious thinking about them.

Curricular thinking tasks, however, are designed to help students learn new, scheduled, grade-level content through thinking (as opposed to mimicking) 

Image taken from https://buildingthinkingclassrooms.com/14-practices/

To teach new content through curricular thinking tasks, Liljedahl has two main pieces of advice:

1. "Almost any curricular task can be turned from a mimicking task to a thinking task by asking a question that is a review of prior knowledge then asking a question that is an extension of that prior knowledge" (p. 28).
2. "It turns out that [most typical math problems] are excellent thinking questions - if they are asked before the students have been shown how to answer them" (p. 27)

He provides a brilliant example later in the book that really illustrates how this works.  Imagine for a moment that your Algebra 1 students had already learned how to multiply two binomials to create an expanded quadratic expression.  Liljedahl's example is

The next learning goal in the unit is likely to reverse that process - to understand how to factor the expanded expression back into (x+2)(x+3).  To turn this into a thinking task, Liljedahl simply suggests asking the students to figure out how to do that by leveraging their existing skill in this clever way (p.151):

And voila - students are thinking (which is "what you do when you don't know what to do" - p.19) rather than waiting to mimic.

From there, as students figure that out, they work on ever-so-slightly increasingly complex examples that challenge them to extend that thinking toward the most advanced version of that skill.  For the task above, Liljedahl presents this list of subsequent tasks:

If you're not a math person, the markings in red on these examples demonstrate where there is a new element of complexity.

This particular task, he explains, often results in students mastering an entire unit on factoring in an hour.  How so fast?  "The short answer is that, when students are not thinking, everything we teach them is difficult.  When students are thinking, however, ... a sequence such as this allows you to cover a huge amount of content in a single lesson" (p. 152).

So I tried it.

Sort of.

By virtue of the math standards changing in Georgia this year, my first chunk of content to teach this year - adding and subtracting fractions and mixed numbers - is largely something students learned last year.  Consequently, a lot of my students, I knew, wouldn't have to think to do it - they'd just have to remember.

I had two plans to address this.  

First, I showed an example using typical calculation to do this and asked who had seen this before.  Plenty of hands went up, so I asked that anyone who raised their hand to NOT use the typical method to do these.  Instead, I asked that they think of other ways they might approach this if simple calculation wasn't an option.  Most of them, as I hoped they would, gravitated to modeling visually, which a) I think shows genuine thinking and understanding, and b) I adore as a Common Core fanboy. 

The second adjustment I made was to have non-routine questions like this one from Open Middle ready to give to the students who blasted through the task.  This, I figured, would require thinking even if my curricular task sequence did not.

The process was clumsy, for sure.  I had to redirect a lot of students who ignored my directions and who were just plugging away with the routine algorithm.  I also had to work very hard to keep the groups working together, as many had a member who would try to split off and just do the task themselves, or who would only engage when it was his or her turn to hold the group's marker.

By and large, however, the students thought - at least at some point - and they stayed engaged.

The best moment came in consolidation time, when one of the meekest and quietest students volunteered to explain her group's thinking in front of the entire class:

Starting off with this largely-review topic, while not exactly on brand for a thinking classroom, did offer me a nice couple of days of testing out the curricular task process with fairly low stakes.  I was able to work out some kinks and to start to get a feel for how more genuine thinking tasks might go in the future.

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