George Lucas Educational Foundation
Differentiated Instruction

A Simple Math Activity That Engages All Students

This engaging low floor, high ceiling activity boosts students’ confidence and helps them think creatively about mathematical relationships.

June 8, 2020
kazuma seki / iStock

As a rule, teachers seek economy in teaching. The more bang we can get for the buck when we’re with students, the better. Two years ago, I began looking for math activities that promote creative thinking, require persistence, necessitate the practice that many believe is required for learning, and could be given to the entire class. On top of that, I wanted to improve motivation and inspire a love of learning.

Impossible? As it turns out, it’s not as difficult as one might think.

I believe that all students can learn and, if not love, at least appreciate math—and researchers agree with me. Every day in my fifth- and sixth-grade combination class (in which more than a third of students have IEPs), we spend about 10–15 minutes focused on a math challenge. I’ve found that the key to deep, engaged learning is to provide a challenge everyone can participate in (meaning it has a low floor) and that has many different ways of being solved (meaning it has a high ceiling).

For instance, last week, the challenge was to use all the numbers in a given set (in this case, 1, 2, 3, and 4) to come up with the answer of 5. The students could use any operation or math symbol that they wanted.

Some of the results students came up with were: (3 + 4 – 2) ÷ 1; 12 – 4 – 3; and 23 – 4 + 1.

The first time someone came up with an out-of-the-box response (making 12 by combining 1 and 2, for example), there was an uproar of “That’s not fair! That’s cheating!” I had to remind them, “No rules, remember?” And they madly went back to their calculations.

Of course, while during this short time students work primarily at their desks, I make it a point to have students come up to the board to write down their solutions as soon as they discover them. This encourages discussion of how the solution works and reveals new possibilities to those who might have hit a wall. It becomes collaborative. Often, it allows me to reteach incorrect thinking about mathematical concepts or introduce new ones, making the learning both relevant and timely.

Let’s look at the ways in which this 15-minute activity can meet all the conditions I hoped for above, including requiring deep thinking and cognitive flexibility.

Creative Thinking and Cognitive Flexibility

Students know that there’s no set procedure, and they can’t just parrot something they’re supposed to have learned. As a result, they’re free to approach the problem in any way that catches their attention.

Those who may struggle with remembering procedural rules are able to use alternative methods to show what they know.

Thinking outside the box is encouraged. Students who come up with original ideas are celebrated, which opens opportunities for recognition for those who may not typically do so well with memorizing and repeating.

Practice and Perseverance

I find that when questions are posed that have only one answer, the class can get in the habit of allowing the fastest student to answer—without bothering to put in any effort. And why should they? Once the answer has been found, there’s no longer any need. With more than one way to solve the challenge, students know that they can still discover a solution, and I find that my students continue working longer than I even have scheduled time for.

Motivation

There are disparities between students in every classroom. Providing opportunities for creative thinking for students who don’t typically stand out can be a means for them to garner respect from their peers through open-ended challenges—which, in turn, increases their motivation to learn. These challenges, which empower students to access the tasks in ways that suit them, enable students to expand their own conceptions of what they’re capable of.

So, it seems deceptively simple—15 minutes a day of mathematical play. Give it a try for a week. Even in my virtual classroom, I have been able to engage students by posting a challenge as an assignment and asking them to respond in the comments section. I have gotten four to five responses in a row from students who couldn’t stop playing with the numbers, along with shouts of congratulations from their peers. By turning numbers from a “problem” into objects to play with, we can bring the joy and inquisitiveness into mathematics for all students.