Low Floor, High Ceiling

Gil Gallagher, Middle School Director
Middle School Director Gil Gallagher uncovers the benefits of the low floor, high ceiling concept of easy access points and expansive thinking in the classroom.
How do we hold high expectations for learning for all of our students while simultaneously accounting for the differences—sometimes vast!—that we know exist in their experiences, levels of skills, and neurological development? This question is central to our work as educators—and is a question that we pursue deeply at Field. Just as we pose challenging questions to our students, we also pose this question to ourselves. The answer informs practices that we can address in our classrooms. This inquiry is at the heart of differentiation, which I define as the practice of making systems function for individuals with varying skill sets. It is a question that I've been pursuing with great joy and sometimes frustration for the last 20 years.

I have enjoyed following Jo Boaler, a brilliant educator whose work is both deep and expansive. I could write a book about how her framework and methods have transformed my practice as a mathematics teacher and have deepened my thinking about teaching and learning in general. For now, I'm just going to focus on one concept that I have found particularly powerful in the classroom: the low floor, high ceiling task. For context, tic-tac-toe is a classic low-floor task. Chess has a very high ceiling and a high floor, too!

Here's an example of a task I used with students in the second week of school:

Leo, the rabbit, is climbing up a flight of 10 steps. Leo can only hop up one or two steps each time he hops. He never hops down, only up. How many different ways can Leo hop up the flight of 10 stairs? Provide evidence to justify your thinking.

The floor on this problem is low, meaning all of my students can access it and get started. They can imagine what a rabbit hopping up the stairs might look like, and they can see the pattern of hopping up 1 or 2 steps. They all have lots of familiarity with what a staircase looks like, and the number 10 is within reach for students with dyscalculia. In terms of providing evidence, students can all draw a workable picture of the staircase or write out a set of calculations to "show" how they approached the problem. Because I usually do this — and tasks like this — in pairs or threes, there is also the built-in support of their peers' prior knowledge and academic skills.

The ceiling on this problem is high, meaning that there are opportunities for all students to go deep into the mathematics behind this problem. I have seen students solve this problem by making the same drawing repeatedly and then looking for visual patterns in the pictures. Students run calculation after calculation looking for patterns in the arithmetic. I have seen students attempt to set up algorithms to anticipate ways that might or might not work. New questions arise when students work with their peers, such as, "wait, how does that work?"

While this isn't magic, what happens with students sometimes seems like magic. Students for whom mathematics has seemed like a secret society suddenly feel like members. Students who carry the idea that they can't do mathematics—an absurdity—see that they can and at a high level. Students who love mathematics get to experience and apply their love and skills to a task that meets them where they are and allows them to extend and connect.
 
Page Stites, the upper school director, and I talked about this idea the other day, and he mentioned a variation on this that I am now wondering about: low floor, high ceiling, wide walls. This extension reminds me of project-based learning: all students can enter where they are, get to as high a level as possible, and have space to explore and move around through the constellation of concepts and skills. Students literally and figuratively advance through this space and are grounded with room to extend upward and outward towards the ceiling and the walls. These are all within reach and just out of reach—a space between the familiar and the unknown. I believe this is where learning happens.
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