Productive Struggle and Deep Learning in Math

Some examples of low-floor/high-ceiling tasks that support this kind of identity-building include asking students how many rectangles they can find on a grid, exploring how many ways they can represent a fraction visually or numerically, using visual patterns to predict what comes next and justify the rule, determining how many different shapes can be made from a set number of tiles, or examining a “Which One Doesn’t Belong?” image where every choice can be defended with reasoning. Each of these tasks is simple enough for anyone to enter, yet each invites deep thinking and diverse strategies from students of all strengths.

In our math classes, our opening activities are followed by tasks that encourage students to employ the very skills they’ve warmed up through the low floor/high ceiling activity. The central task or tasks in a lesson are consistently accessible for nearly every student, and are designed to include an element of mystery, but they are usually more focused on developing a specific skill or concept. These activities give students an opportunity to engage in productive struggle as they build their conceptual understanding. It’s strikingly different from learning a procedure from a teacher or classmate and employing it. Our classes have a different structure that has multiple advantages, but we do still teach procedures and a favorite procedure to learn in fifth grade is long division.

Prior to the IM curriculum, fifth graders often told me that they couldn’t divide multi-digit numbers because they hadn’t learned long division. But after a year of the IM curriculum I prompted the fifth graders I taught to “use what they knew about division” to solve a multi-digit division problem and they were able to use subtraction, estimation and a bit of multiplication to solve a problem that had previously seemed impossible. Of course, they still wanted to learn long division and needed it to solve problems more efficiently, but it was exciting to see students develop an understanding of the nature of long division in real time. Prior to having the experience of puzzling through a multi-digit division problem, long division seemed to be a bit of magic. After solving a multi-digit division problem using logic and their math skills, students could see long division as what it is, an efficient algorithm.