Struggle is not to be avoided. It is a natural part of the learning process and leads to increased understanding and long-term retention. Struggle occurs when students are asked to solve a problem for which there is no clear path to a solution. The goal of productive struggle is to provide students with opportunity to wrestle with mathematical situations, while avoiding feelings of frustration and overwhelm (Finkel, 2019).
The teacher’s role in productive struggle is to ensure that all students understand the task and feel that they can get started. Liljedahl’s (2021) work on Building Thinking Classrooms states, “if we are thinking, we will be engaged. And if we are engaged, we are thinking” (p.146). To create this optimal experience where struggle remains productive, teachers negotiate three key components within the learning environment.
1 | Communicate clear goals every step of the way
2 | Provide immediate feedback to the student
3 | Create balance between the doer's ability and the challenge of the task
-CBE, Supporting Productive Struggle in Mathematics
Ms. Nicole Byrne, grade 5 Learning Leader, led staff in professional learning on Wednesday morning looking at Peter Liljedahl work in Building Thinking Classrooms in Mathematics and the Global Math Project: Exploding Dots task.
At its core, Exploding Dots is a visual and interactive math game that focuses on place value and number systems (such as base-10, base-2, and beyond). It’s designed to help students understand how numbers work in a hands-on way, using a simple system of dots and "explosions" to represent numbers in different bases.
The task starts with a simple diagram consisting of a row of dots. These dots represent numbers in a particular base system. The goal is to shift, group, and explode these dots in a way that mimics how numbers are composed and decomposed in various base systems (like our familiar base-10 system, binary base-2, etc.).
Students begin by arranging dots in a row, and as dots accumulate in a specific place (like the ones place in base-10), they "explode" into a new place value (like moving from ones to tens in base-10). The "explosions" help students visualize how larger numbers form and how they break down into smaller units.
Through this activity, students see the connection between individual digits and their place value. For example, in base-10, 25 is made up of 2 tens and 5 ones, which the exploding dots visually demonstrate. As students explore different bases (like binary), they understand how different systems work for representing numbers. For instance, in binary, numbers are represented by 0s and 1s, and the dots explode when they accumulate in a "place" according to powers of 2.
Students can explore different base systems (e.g., base-10, base-2, base-5, etc.) and learn how mathematical operations (like addition and multiplication) work in these systems. Rather than just memorizing math rules or algorithms, students develop a solid understanding of how numbers behave. This foundational understanding is transferable to other areas of math, such as arithmetic, algebra, and number theory.
Sincerely,
Angela McPhee
Principal