Critical Thinking in Math: A focus on mathematical reasoning competencies
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.
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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
- a strong understanding of foundational math concepts and content, and
- 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.
| 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 |
|
Understanding |
|
|
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 |
|
Acting |
|
|
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) |
|
Communicating |
|
|
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
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Classroom ready materials
Online Learning
Describing Trends in Data: Which data set should be considered linear in the trends it presents? [ZIP]
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.
Grade 9 Student Lessons
These lessons are part of a larger collection of lessons created for 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.
Professional learning resources
Assessing Mathematical Thinking: A Focus on Reasoning Competencies
This 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.
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.
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.