Low-Floor/High-Ceiling Tasks & Other Takeaways
Our curriculum specialists recently attended a workshop with Jo Boaler, Cathy Williams, and their youcubed team. We left energized by three key messages and affirmed by recognizing how The Math Learning Center addresses them.
Math is visual
The Math Learning Center grew out of a project funded by the National Science Foundation to improve the teaching of mathematics. Our founders developed a philosophy that emphasizes building a deeper conceptual understanding through the use of visual models.
When we launch a task during Number Corner or unit work, we use manipulatives and a variety of visual models to invite students to make sense of the mathematics through materials, drawings, and mental images. In a Bridges classroom, you’ll see students gesturing with their hands or using models to explain their ideas.
The learning environment is filled with wonder, curiosity & risk taking
Bridges teachers often share stories of their students’ sense of wonderment and inquisitiveness. They nurture a community of learners by establishing rules for discourse. In a Bridges classroom, students ask questions, share ideas, and edit their thinking in an environment that values mistakes as a sure sign of learning. Direct instruction is replaced by posing a visual and asking, “What do you notice?” Or by posing a problem-solving situation and trusting that students already have entry points to begin their representation and productive struggle.
Problems are not always solved in one session, and that’s okay! The teacher and students continue to puzzle because we are never really “done” with math understanding. Even weeks later, students often bring up a connection to a topic. We know that when we step aside and trust they can, our students will amaze us.
Low-floor/high-ceiling tasks work for everyone
By definition, a LFHC task is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Tasks present possibilities for the participants to do much more challenging mathematics. At the workshop, curriculum team members made connections between LFHC design and our Work Places. When teachers use the Assessment & Differentiation chart for extensions and support, they encourage multiple entry points, solution methods, and representations.
In Grade 2, Unit 3, the Work Place Base Ten Triple Spin is a low-floor/high-ceiling activity. Students begin by determining whether they are playing for the greater or lesser number. Players then spin for a digit (2–7) and decide whether the number will represent ones, tens, or hundreds place. Once a 3-digit number is generated, they determine who has the greater or lesser number.
Some students may choose to build or sketch the place value model as they create the 3-digit numbers first and then spin to establish whether winning requires the greater or the lesser number. Other students can extend the game by building two 3-digit numbers, finding the sum of the two numbers, and then figuring out who has the greater or lesser sum. Another variation might include finding the difference between the two 3-digit numbers players created before they spin to determine whether the winner is the player with the greater or lesser difference. Base Ten Triple Spin poses endless possibilities for students to create even more variations. Students have multiple access points to work at their instructional level, sharing ideas and representations with peers.
Target One Thousand in Grade 3, Unit 4 is another great example of a low-floor/high-ceiling task. Students choose eight cards and arrange six of them to create two 3-digit numbers which are then added together to get as close to 1000 as possible, either over or under. A teacher can easily have students choose 6 cards and create two 2-digit numbers that get as close to 100 as possible, or have them choose 10 cards to create two 4-digit numbers that get as close to 10,000 as possible.
Bridges provides about three dozen Work Places at each grade level that are suitable for everyone, allowing the high flyers to explore and challenge themselves while less confident students stay with the original directions and develop their understanding. Dice, cards and spinners often level the playing field. The workshop reminded us yet again that when all students have access to meaningful work, in a classroom that embraces a growth mindset, everyone raises their game.
Pia Hansen is the Director of Professional Development for The Math Learning Center