Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with Desmos tools. I first learned about task propensity through this paper, and you can read the rest of my series on the topic here.
One example where I see task propensity degrade learning in engaging, well-intentioned activities is Polygraph. For context, Polygraph is a Guess Who-like game, where one student chooses a rational function among 16, and their partner asks yes or no questions to figure out which function they chose.
This is aided by technology to make the setup and transitions effortless and let me watch through the Desmos teacher dashboard. For instance, check out these student responses in Polygraph: Rational Functions:
We pause the activity and talk about vocabulary, looking at questions that worked well and making explicit the language that students could use to talk more precisely about the graphs they see. The activity is well-designed to create an intellectual need for vocabulary, and it’s effective for many students. At the same time, I still see responses like this through the end of the activity:
There are some missed opportunities here, and these types of exchanges persist for some students every time I run Polygraph.
The Follow-Up Task
We finish with Polygraph, and the game and the computers go away. We maybe even wait a day for ideas to bounce around. Then I give students this:
I don’t think this is anything brilliant or revolutionary. But I do think that spending 15 minutes working on this task in class and completing it for homework, perhaps over a few days, is worth far more than extending Polygraph for those additional minutes. The practice is more focused and less haphazard. I can have students trade with each other and try to find counterexamples, and suddenly there is even more need for precision in language, and more opportunity for students to revise their language and feel a need to add to their vocabulary for talking about rational functions. There’s even a more rigorous logic in writing questions with a much more challenging goal, and some classes students ask questions of the form, “say yes if the graph has two vertical asymptotes, say no if the graph has one vertical asymptote, and say I don’t know if the graph has no vertical asymptotes”, which I think is awesome. This task doesn’t work without Polygraph, but I think Polygraph is incomplete without some type of follow-up task to consolidate and extend student thinking.
I love Marbleslides. Students love Marbleslides. They get to experiment with different functions, watch marbles roll around, and feel like they’re playing a game. Yet, despite the best intentions of Marbleslides’ design, students are guessing and checking far more than I would like. I love sequences of screens like these:
But too often students rush past these opportunities for reflection and consolidation of their learning to play with the marbles and the stars.
The Follow-Up Task
Enter the follow-up task. I have students do Marbleslides, and a day or two later I give them this:
I get a lot of bang for my buck with this task. Students need to articulate differences between sine and cosine functions, and then write a function when they can’t see a full period of what they’re trying to model. Finally, they have to do all of that thinking without being able to guess and check their way to an answer. While this type of task is possible using the Pause and Teacher Pacing tools in Desmos activities, separating this task from the activity lets me take my time formatively assessing students strategies, selecting different approaches, sequencing them for a discussion, and unpacking different choices students made in writing their functions. And all of this happens without feeling like I’m interrupting students from playing a fun game in math class.
In short, I’ve found that Marbleslides is insufficient. It needs some bridges to connect work on Desmos with the thinking I need students to be able to do without the support of technology. There’s no substitute for Marbleslides in the scope and sequence of my curriculum. There’s also no substitute for a carefully designed follow-up task to reinforce and consolidate the big ideas, to make sure students don’t lose the forest for the trees.
More broadly, I find Desmos activities enormously valuable, but I also find them incomplete. They are incredibly engaging opportunities to work at formal and informal levels, experiment with a low floor, challenge students by raising the ceiling, and make connections through rich multimedia experiences. However, Desmos activities are only one element of a coherent curriculum, and there need to be bridges between those two — bridges that explicitly return to the thinking students do during a Desmos activity, consolidate it, and build on it toward larger goals.