I’ve written often in the past about low-floor high-ceiling tasks. This week I’m thinking about what exactly I mean when I use those words. I started thinking because of this problem from Play With Your Math:
I’ve had so much fun playing with it. No spoilers here. But there’s something about the problem that invites exploration and curiosity.
Second, this recent tweet thread from David Wees:
The words “low” and “high” have the connotation of ability. One might imagine an elevator. A low-floor high-ceiling task is then one where the elevator can go “lower” to pick up certain low students, and also go “higher” to accommodate certain high students. This metaphor seems likely to reinforce fixed ideas about ability. I also worry about any task where different students work with and learn different math. The more time students spend doing different math, the harder it is to bring the class together for a productive discussion.
A second metaphor might be a spider web. Rather than presenting a task as a linear sequence of strategies, a task might have several different entry points and exit points. Here, one strand is not “higher” or “lower” than another. Instead, they might illuminate different representations of a problem or connect a problem with different mathematical ideas. There is still the challenge of bringing together different perspectives, but if those perspectives are all connected to the same central ideas they support different student approaches, rather than positioning some as better than others.
In David’s thread, he argues that teachers should emphasize student knowledge, rather than student ability. Ability is often seen as fixed, while knowledge can change. On the Play With Your Math blog, Joey Kelly describes three principles they use to design problems:
- To make success attainable
- To make space for curiosity
- To shelter from inaccessible questions
I think these principles are particularly useful through the lens of knowledge. If I give students a task because I like the problem and think it’s interesting, that task probably isn’t going to go very well. If, instead, I focus on finding problems where success is attainable, I frame my planning through the lens of what my students know and can do. When students feel successful in class, they’re more likely to take risks, share ideas, and enjoy math. Discussions of low-floor high-ceiling tasks often focus more on the task than the students. Which is inevitable, because it’s much easier to talk about tasks than students, but also probably unproductive.
I have a third metaphor that might be more useful. I first saw it in a talk by Andy Gael.
This is a curb cut. Curb cuts were originally designed for wheelchair users, but they benefit everyone. People pushing strollers, transporting large objects, walking with other mobility issues, or even walking home intoxicated have an easier time getting around with curb cuts. And curb cuts aren’t seen as some niche accommodation for people with wheelchairs; they are ubiquitous, and they’re just seen as normal.
Returning to math problems, curb cuts provide an on-ramp to a problem for all students. The Play With Your Math problem above has a lot to do with prime factorization. But rather than asking a question about prime factorization, it provides an on-ramp by giving students a chance to play and experiment first. We can formalize their understanding of prime factorization later, but that won’t be an obstacle to entering the problem. The problem avoids making assumptions about what students already know. Know nothing about prime factorization? No problem. At the same time, the opportunity to explore and extend the problem offers space for curiosity. I can provide access without reducing the interest of students who already know some things about prime factorization.
I like the metaphor of a curb cut because it focuses on getting someone from where they are to where they are going. Curb cuts work for everyone, and they provide access rather than transforming the destination. They also don’t make assumptions or separate people based on their mobility. Most people don’t even notice them. Similarly, an effective task doesn’t need to be some crazy complex production that offers different options for different students. Instead, tasks often need simple changes that provide access to more students. I’d love to spend more effort finding curb cuts, and using this metaphor to guide how I design tasks that engage all students.