This task propensity entices teachers and textbook authors to capitalize on procedures that can quickly generate correct answers, instead of investing in the underlying mathematics while accepting that fluency may come later.

(source)

The article linked above is a thought-provoking perspective on why some conceptually-focused math reforms have been unsuccessful. The authors explore the idea of task propensity, or the tendency of teachers and curriculum writers to focus on features of specific tasks rather than  the underlying mathematics that may be used in new tasks in the future. Teachers may have great, conceptually oriented tasks that can elicit mathematical thinking, yet if they only focus on teaching students how to solve those specific tasks that thinking is unlikely to transfer to new problems down the road.

I’m hanging out with some great folks at the Desmos fellows weekend, and I’d like to share two contrasting cases:

Case 1
We spent some time yesterday mingling and doing math together. I spent much if working on this problem from Play With Your Math with a great group of teachers.

I won’t spoil it; this is absolutely worth exploring, and after what was probably an hour of work I have plenty more to learn. The most important feature of my learning was that, in a relatively short period of time, the group I was working with established the answer to the question as it was posed. We then went further, and explored different conjectures and directions to extend the problem. The vast majority of our learning came after we had solved the problem, and depended on our interest in creating new problems to further our thinking. In other words, we avoided the temptation of task propensity to fixate on the problem at the expense of additional learning.

Case 2:
I have really enjoyed both playing and watching students play Marbleslides lessons like this one. Students have to transform various functions in order for the marbles to get every star when they are launched.

This is one of my students’ favorite things to do in class, and is far more engaging for them than any other lesson I have on rational functions. At the same time, I find that students often learn less than I would like from the activity. They spend most of their time focused on the task at hand — getting all of the stars — and less on what I want them to learn — general rules for transforming rational functions. This is not to say that no learning happens, just that students can fall victim to task propensity and lose the forest for the trees.

I am looking forward to my Desmos fellowship and what I will learn from a great group of teachers and the stellar folks at Desmos. One of the important questions I have is around when Desmos is the appropriate tool to use, and when other tools will work just as well or better. One challenge I have with many activities is task propensity; that, while Desmos is a powerful tool for generalizing thinking, that generalization does not happen if students are too focused on the specific features of a task to make connections to broader mathematical ideas. I hope to do some writing over the next few months to explore this idea and try to better understand when Desmos is the right tool, and how to use it effectively.