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 those tools. I first learned about task propensity through this paper, and you can read the rest of my thinking on the topic here.
I think the best example of task propensity is Marbleslides.
Students solve challenges like the one above by rewriting the function so that when the balls drop, they capture all of the stars. I love Marbleslides and I use variations on it often. At the same time, I find that a subset of students — usually the students who are already struggling — learn less than I would hope through these tasks. They are likely to solve Marbleslides challenges through trial and error without paying attention to the structure of the mathematical objects they’re working with, or they get frustrated and use functions outside of the family I want them to learn about.
Marbleslides offers one paradigm for what Desmos activities can look like. These activities are incredibly engaging — students love them, and are often asking for more. They let students see math as a dynamic process, learning about objects that make sense and follow certain rules — and learning those rules is what learning math is all about. They are valuable activities and I’m glad I am able to use them.
But in this post I want to offer an alternate perspective that tries to avoid the challenge of task propensity. I spent a bunch of time this summer thinking about polynomials. My polynomial units often feel flat and uninspired and I wanted to add a wider variety of activities. I’ve previously used this Desmos activity.
Students solve challenges where they need to build functions that meet certain conditions. It can be great for certain features of polynomials, but can also suffer from some elements of task propensity. Students just end up fiddling with different functions until they find something that works, and in the process they may or may not learn what I want them to learn about polynomials.
I wanted to design a new activity, one that falls after Polygraph at the start of the unit but before students start doing more formal algebraic work. My goal was to bridge some of the gaps between using vocabulary to describe polynomial functions and writing polynomial functions that meet certain conditions. I also wanted to write something humble. There’s nothing very flashy about this activity, no high-engagement tasks that students will want to keep coming back to. I want this activity to provoke useful thinking, and to do so using tools like sketching and interactive Desmos graphs that are impossible with a pencil and paper or whiteboard and marker. And I want it to stay laser focused on a few key ideas that I want to get across. I’m having trouble clearly articulating what I like about this activity that I don’t find in some others, but I’ll try to lay out what I was going for below. The activity is linked here if you’d like to play along.
Below are the first two screens. They are meant to give me a rough idea of how my students conceptualize polynomials.
I would use teacher pacing here, so that students can only work on these two screens and can’t go further ahead. Then, I would pause and project a few examples anonymously to discuss why they are or are not polynomials, and show students a few different ideas of what the function could look like. Nothing crazy, just trying to see where student thinking is and help them do some informal work sketching and seeing sketches of polynomials.
The next four screens are also formative, and are more focused on multiplicity, where I find many students get tripped up when working with polynomials. I would use teacher pacing on these screens as well so that students can’t go ahead.
The goal here is to explore how even and odd multiplicity influence a function, and see how well students can sketch a function that has specific characteristics while connecting multiplicity to other properties of that function. Nothing too crazy, but also something that I can only let students experiment with informally through a Desmos activity. They’re also meant to be really carefully focused on an informal understanding of multiplicity, making sure students are doing the right thinking on these screens. There’s a great opportunity to share different students’ graphs with the whole group and discuss both the specific properties and how they come together to create the larger function.
Next they would likely finish the activity at their own pace. I don’t want work through the potential management challenges of continuing teacher pacing. I’m watching the dashboard and looking for two things: interesting disagreements or misconceptions to surface and discuss at the end of the activity, and where student thinking is more broadly as I figure out what to do after this activity.
I think this is far from perfect and some of its rough edges could be smoothed out. But this lesson sticks with some values I want to try to use more often with Desmos activities. It isn’t trying to tour through an entire concept in 45 minutes. It isn’t supposed to be my most engaging lesson. Instead, the goal is to be laser focused on an important development in student thinking — reasoning flexibly about polynomial graphs and the vocabulary we use to describe them, without getting into algebraic notation. By staying really focused and living in that specific place, I’m trying to avoid some of the challenges of task propensity. There are no fancy challenges that students have to work through. The focus is on sketching and explaining their thinking. There is less emphasis on guess-and-check than many other Desmos activities. And I built this activity thinking specifically about how I want to use teacher pacing and other Desmos conversation tools in order to create useful moments of formative assessment and class discussion.
I don’t mean this to be a criticism of other Desmos activities, just a change in emphasis for me on what is missing in some of my pedagogy. It’s also meant to be something that complements what I’m already doing, rather than replacing other activities. There’s a time for engagement and excitement, and there’s a time for humble activities that zoom in on specific goals and focus on getting all students to meet those goals.
I would love feedback. Is this a distinction worth making? Is this activity really just a mess? Where else might this type of thinking be useful? Where could I go further?