Critical Thinking in Math: A Focus on Mathematical Reasoning Competencies

Critical Thinking in Math

TC²’s approach to math embraces the idea that sustained quality mathematical thinking, or reasoning, is the key to the success of current and future generations of math students.

What Is Mathematical Reasoning?

A mathematical reasoning approach optimizes the learning opportunities for every student in the classroom. It empowers students with the capacity to independently detect the need for, and to use, a wide range of math reasoning abilities.

This approach ensures that students develop both

  1. a strong understanding of foundational math concepts and content, and
  2. the capacity to reason soundly about and with these concepts and content.
When we focus on reasoning competencies, students continue to learn important math concepts while also learning how to
  • deeply understand
  • appropriately act on, and
  • effectively communicate using those concepts.

What Are Mathematical Reasoning Competencies?

Eight key mathematical reasoning competencies underpin all math learning and are needed for student success in math. These are presented in A Math Pedagogy Designed to Empower Learners [PDF].

Competency Definition

Core competencies

Sound thinking or reasoning

Reasoning about the quality of one’s thinking

Reflective reasoning

Reasoning about the quality of one’s understanding, actions, and communication


Conceptual reasoning

Reasoning about what makes a concept what it is and how to recognize that concept

Detail-minded reasoning

Reasoning about which mathematical details to use and how to use them effectively


Connective reasoning

Reasoning about how to effectively connect ideas

Problem-managing reasoning

Reasoning about how to effectively identify and solve math problems (including how to effectively select, organize, and use mathematical strategies, tools, and resources)


Representational reasoning

Reasoning about how to effectively represent ideas

Language reasoning

Reasoning about how to effectively use the language of mathematics including its structure and symbols

"To provide each and every student equitable opportunities to improve their learning success in math, students need to learn how to reason soundly in a variety of ways through the application of critical thinking.

—Laura Gini Newman (2020)
A Math Pedagogy Designed to Empower Learners

To learn more about critical thinking in math with TC² check out:
Classroom Ready Materials
Professional Learning Resources
What Teachers Are Saying
Professional Learning

Classroom Ready Materials

Online Learning

Describing Trends in Data: Which data set should be considered linear in the trends it presents?
In this lesson, students learn how to use lines (curves) of best fit to help them effectively describe mathematical trends in data. Most suitable for grades 8–10.

Junior (7–9)

Grade 9 Student Lessons
These lessons were developed in partnership with the Matawa Education and Care Centre.

What is the best way to represent information to help you make financial decisions: a table or a graph? [PDF]
In this lesson, students compare different ways of organizing information to create a budget that will help them make the best financial decision.

Which best describes the trend in the data: a line of best fit or a curve of best fit? [PDF]
In this lesson, students consider different patterns in the data that describes the relationship between fish and seafood consumption and the year. They then make the most accurate prediction about fish and seafood consumption for the year 2030.

How well does an equation match a line of best fit, a table, and a description in words? [PDF]
In this lesson, students explain how well an equation given to them describes the line of best fit and the trend in the data. They then use the equation to make a prediction.

Elementary (K–6)

Coming Soon!

Secondary (10–12)

Coming Soon!

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Professional Learning Resources

Books, Articles, & Discussions

NEW! Assessing Mathematical Thinking: A Focus on Reasoning Competencies
This new title in our Quick Guides to Thinking Classrooms series presents a framework for effectively assessing and evaluating thinking in math. It shows how building math assessment practices on a foundation of essential mathematical reasoning competencies provides a clearly defined, manageable, and consistent way to focus assessments.

A Math Pedagogy Designed to Empower Learners [PDF]
Laura Gini-Newman outlines a new pedagogical approach to the teaching and learning of mathematics that is focused on building student capacity to reason mathematically through critical inquiry.

Critical Inquiry in Math Class
During TC²’s 25th Anniversary celebration, each month explored a different focus. April focussed on how we can bring critical thinking into math. This webpage introduces the focus and explores ways to enrich your classroom with critical inquiry in math.

Videos and Presentations

An Introduction to the Why, What, How, When, and Who of Assessing Mathematical Thinking
Listen to TC² math consultant Laura Gini-Newman as she explains why your assessments should focus on mathematical reasoning and offers a few tips on the why, what, how, when, and who of doing so.

OAME Talks
Listen to TC² math consultant Laura Gini-Newman as she shares her thoughts on Assessing Mathematical Thinking: Who, What, When, and How on the OAME Talks podcast (Season 5, Talk 41)

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What Teachers Are Saying

Critical Thinking in Math-Teachers

How Thinking Mathematically Changed My Teaching
Jocelynn Foxon talks about her experience completing the Math Lead Teacher Certification Program (MLTCP) offered by TC², and the work she did with teachers and students in supporting the implementation of this approach in the math classroom.

Helping My Students Take Ownership of Their Own Learning
Shamima Basrai talks about her experience embedding critical thinking in mathematics in her Grade 3/4 classroom.

The Perfect Fit for the Meandering but Wonderful Thinking Process of the Grade 4 Student
David Markus talks about his experience shifting to a critical thinking framework with his Grade 4 math students.

Generating Enthusiasm in the Math Classroom
Nina Perreault-Primeau talks about the transformation in her math classroom when she planned lessons focussing on student interest, creating authentic learning opportunities accompanied by sound critical inquiry questions.

But will it also work in math? [PDF]
Sarah Sommers describes what happened when she and her teaching partner took a critical inquiry approach with their Grade 5 students during a math patterning unit.

Basics vs Inquiry in Math? A critical inquiry approach can achieve both
Chris Achong talks about his experience with his Grade 9 math team implementing this approach to math learning—a comprehensive and balanced approach that improves the quality of every student’s capacity to think mathematically.

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Professional Learning

Math Lead Teacher Certification
This program supports teacher development of a rich understanding of the diverse role of critical thinking in math classrooms through 18 hours of personalized coaching and 10 hours of implementation support. Available as face-to-face and online sessions or a combination of both.

Book Professional Learning in Math
Learn how we can help you plan affordable professional learning in math facilitated by our experienced team. We consult and collaborate with you to develop custom, focused, and engaging face-to-face and online sessions to meet your math specific needs and grade level requirements.

Contact Victoria Campoli to learn more.