I was kvetching on Twitter - I mean, consulting my PLN - about preservice teachers whose desire for fun, shiny lessons can distract them from finding lessons of substance. Nicora Placa, a math ed prof and teacher educator (with a fabulous blog), suggested Smith and Stein's framework for cognitive demand.
That's Margaret Smith, an all-star researcher at University of Pittsburgh, and Mary Kay Stein, who seems like her partner in crime at Pitt. Their paper(s) and book on the 5 Practices for Orchestrating Discussions has been very influential here at GVSU this fall. Several profs are using them to organize our first math for elementary teachers course. [Lengthy aside: book, NCTM's number one seller; MTMS article, very readable for preservice and inservice teachers; Christopher Danielson's 5 practices posts, start here; it synthesizes a lot of great research into worthwhile mathematical tasks and instructional practice that they have done with other partners.]
The cognitive demand article (full issue MTMS Feb 98; also available on JSTOR; Reflections on Practice: Selecting and Creating Mathematical Tasks: from Research to Practice, Smith and Stein, Mathematics Teaching in the Middle School, February 1998, v.3, n.5.) is great for a professional development discussion. They have several highlighted discussion questions that get to the heart of the matter. The article boils it down to four levels of tasks, with detailed breakout at each level.
(Click to see full size)
Luckily, there's a lot of overlap with out content. The fractions, not so much, but there's a lot of fraction language in the common core geometry and statistics, so it's been discussed.
Immediately, they went into test mode. Which is especially interesting as we don't take tests as such. (SBG, rather.) It took several urgings to get them to try the problems cooperatively, ask each other questions and share methods. I noted that they got very engaged - maybe I need more individual problem solving or even practice in class. (I'm reminded of our pre-assessment and how that's a tension for me.)
Afterwards, we talked about what did they notice about these problems. They focused on the subject of the content and the story problem aspect. Except they were impressed by the realism of these story problems. Many of them spent a significant amount of time on task C, and it was the most discussed at the tables.
Next I asked them to, in their groups, rank the problems on the amount of thinking required by students working these tasks.
- Task A was easy. The visual made it easy to extend the pattern. (Many skipped over the description.)
- The group that rated Task B the hardest had trouble making sense of the text.
- Task C had so much computation, calculators were required and the problem had extra unneeded numbers in it.
- Task E started to bring out some task analysis. It was easy with the picture, but the content was hard. People who had difficulty understanding the picture shared that, too.
- Task F was ranked easy because it was an easy pattern compared to the ones they had been doing in class.
- Task H was "just a rule"
It was astounding to me how ready they were for this discussion. They read for themselves, discussed in their groups, then rated the tasks by this framework within their groups. In particular I was pleased at how they discussed non-routine vs a difficult-to-execute algorithm in terms of difficulty, and the value they placed on connections and representation. They were able to make connections from their own experience to challenges students might face. For example, the symbolic rule for the train pattern (Task F) and going from the observation of 2 more each time to what that means in the rule. Unfortunately, some of them feel the solution to this will be in what to tell the students, but the majority saw the value in presenting cognitively demanding tasks.
The activity is obviously raw. I'd like to formulate some questions to support them finding some of these task characteristics themselves - but that's challenging without students to observe.
How might you teach a lesson on these ideas? Surely that's a teacher task with a high-level demand!