Category Archives: Desmos

Task Propensity & Rich Tasks

I wrote a few days ago about Desmos and task propensity. I’m interested in critically exploring the pedagogy and sequencing behind using Desmos activities effectively.  One challenge is the idea of task propensity: when presented with a conceptually-oriented task, teachers and students often focus too much on the thinking that will help them solve the specific task, rather than thinking that will help them connect what they are doing to new problems in the future.  If students are laser-focused on finding the solution to a problem, they are likely to lose the forest for the trees and miss opportunities to generalize their thinking.

Before I dive deeper into this idea, I want to explore the opposite perspective. Humans inevitably focus on the task at hand and push to the background thinking about how they will use it again in the future. This isn’t a problem unique to Desmos activities. And, if it’s in some ways inevitable, we might as well make the tasks that students are focusing on as rich, engaging, and meaningful as possible.

We talked a bit during the Desmos fellows weekend about ways that technology can help and hinder learning. There are lots of ways it can hinder learning — by isolating students, by presenting new distractions, etc. There are also lots of ways it can help learning — it can give immediate feedback, differentiate by providing multiple entry points and tiered challenges, and etc. But the heart of what I find most valuable in this technology is that I can give students richer tasks that create richer thinking with technology than without.

The phrase “rich task” is often thrown around but under-specified. I’m not sure I have a great definition, but I’d like to offer a few examples of what can make a Desmos activity a rich task.

Match My Parabola 

Here is a screen from the Match My Parabola activity:
Screenshot 2017-07-19 at 5.19.47 PM.png
I’m not sure I can even offer an alternative that is equivalent. Here, students can experiment and instantly see what graphs their equations produce. At the same time, once they figure out how to transform the quadratic function vertically or horizontally, they practice that transformation once more in a slightly different location. That sequencing — experimentation until students find success, and then immediate practice — is pretty hard to replicate elsewhere. Desmos does it smoothly and seamlessly.

Game, Set, Flat 

Here is a screen from the Game, Set, Flat activity:
Screenshot 2017-07-19 at 5.24.40 PM.png
Students have been introduced to the challenge: they need to be able to tell good tennis balls from bad tennis balls. They just saw a few examples of balls bouncing, with varying levels of bounciness, to help illustrate the difference. Before students work formally with equations, they have a chance to get a firmer grasp on the principles involved by choosing how high a ball bounces after each bounce. When they click the button, they see their model animated. It animates whether they create a close approximation of a bouncing ball or something silly and unreasonable. But this intermediate step helps students to visualize the problem and sets them up to make connections between equations and the objects they represent. Most importantly, this is something that is impossible to do without digital technology; this whole step is skipped in a pencil-and-paper lesson, and students miss out on the chance to do this thinking.

Burning Daylight 

Here are two consecutive screens in the activity Burning Daylight:
Screenshot 2017-07-19 at 5.32.26 PM
Screenshot 2017-07-19 at 5.32.56 PM
It’s easy to stay focused on the abstractions and equations when engaging with mathematical modeling. The first question would, in many paper-and-pencil lessons, be one that students rush past on the way to writing a quick equation and moving on. In this case, the media allows immediate feedback contextualizing the student’s thinking on the previous screen and reinforcing a connection between model and world that might otherwise be lost.


These are great tasks. And they’re great tasks because they have the potential to create rich thinking for students, in ways that aren’t possible in other formats.

One challenge that I’m interested in exploring is this idea of task propensity: to what extent do students, while working on these activities, focus on the tasks themselves without stepping back to consider how they can use what they’ve learned in new contexts in the future? That said, even if students have a hard time slowing down to make connections I would like them to make, the thinking that they need do to complete these tasks is richer, deeper, and more varied than the best replacements I could offer.

I want to offer this mostly as a check for myself. If I want to take a critical perspective on Desmos activities, I want to make sure I’m clear on exactly what they offer, what they are being compared to, and what my alternatives are.

Task Propensity

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.


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.
Screenshot 2017-07-15 at 7.02.24 AM.png
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.
Screenshot 2017-07-15 at 7.21.01 AM.png
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.