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:

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:

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:


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.

Summary 

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.