Deepening Mathematical Learning with the Concept Circle Thinking Routine: Make.Math.Visible.

Concept Circles

The concept circle graphic organizer was introduced through Vacca and Vacca’s (2001)  work as a way to support student vocabulary development in language arts. A circle is divided into four sections with each section containing a word or phrase. Students are then invited to determine how the words are related to one another. Vacca and Vacca found that this learning process enabled students to deepen their understanding of vocabulary and enhance reading skills.

As I began to implement concept circles in my work for vocabulary development in mathematics, I realized that for mathematics instruction, I was not using this graphic organizer to its full potential. Yes, I did believe that it did deepened students’ understanding of mathematical language, but I was taken by its shape and wondered if I could modify its design and application for mathematics. I decided to  enlarge the concept circle and add a central circle in the middle (Marks Krpan, 2012).  Students could now use it to represent mathematical concepts and relationships using images, words and manipulatives. This was exciting! I found that by inviting students to engage in tasks where they placed manipulatives directly on the concept circles, they were able to develop key mathematical skills related to  discourse, visualizing, modelling, representing (and the list goes on).  The central circle provided additional opportunities for exploring mathematical relationships and patterns. You can download a copy of the concept circle  from my book Teaching Math with Meaning. Students do not have to use all of the sections, but having eight sections provides flexibility. I have found that concept circles are also helpful in supporting  Multilingual Learners.   

Concept Circle for Mathematical Learning

Concept Circle Thinking Routine (CCTR)

In order to support the implementation of concept circles, I have developed the Concept Circle Thinking Routine  to enhance student learning with concept circles tasks. The thinking routine includes four key stages: Planning the Thinking, Engaging the Thinking, Representing the Thinking and Debriefing the Thinking. Throughout the thinking routine, the teacher facilitates the thinking at each stage with insightful questions to assist students in reflecting on their learning and the representations they have created.

Concept Circle Thinking Routine

Among the many benefits for student learning, concept circles:

• Provide rich opportunities for students to visualize mathematical ideas and concepts

• Enable students to learn about the diversity of mathematical thinking and how one idea can be represented in many ways.

• Create meaningful assessment opportunities, offering  insight into student thinking and identifying possible misconceptions.

• Afford students contexts for meaningful mathematical discourse.

• Help students identify patterns and connections between mathematical representations.

There are many, many ways to use concept circles. More ideas can be found in my books Math Expressions: Developing Student Thinking and Problem Solving Through Communication (2012) and Teaching Math with Meaning: Cultivating Self-efficacy Through Learning Competencies (2018), and my new Concept Circle Thinking Routine Kit that will be released in 2026! This kit will include laminated, 3ft by 3ft double-sided concept circles and a detailed teacher’s guide. 

Examples of Completed Concept Circle Thinking Routine Tasks:

Below you will find completed concept circle thinking tasks from my own work and  from teachers who have shared their ideas online.

References

Marks Krpan, C. (2012). Math expressions; Developing thinking and problem solving through communication. Toronto, Pearson Education Canada.

Marks Krpan, C. (2018). Teaching math with meaning: Cultivating self-efficacy through learning competencies. Toronto, Pearson Education Canada

Marks Krpan, C. (2026) Concept Circle Thinking Routine Kit. Didax, Rowley, MA

Vacca, R.T., & Vacca, J. L. (2001).  Reading and Learning to Read. Boston, MA: Pearson.