Basics vs Inquiry in Math? A critical inquiry approach can achieve both

Published: Jan 10, 2019

Introduction by Laura Gini-Newman, TC² Math Consultant

There is currently a raging debate in math educational communities that suggests there are only two options for how students can learn math—rote memorization or discovery. The fortunate reality is that there is a third that combines the benefits of both these options and empowers students with the most essential tool for success in math—the capacity to reason effectively. This is the first Ideas and issues article in a series where math teachers and leaders share their experiences and learning using a critical inquiry approach in their classrooms. In this first article, Chris Achong, math department head at Markville SS (York Region District School Board) 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.

What role do you play in helping to implement effective teaching and learning practices in mathematics?

Chris Achong

I am the Head of the Mathematics Department at Markville Secondary School. In this role, my responsibilities include the facilitation of curriculum implementation, mentoring teachers, as well as the support and implementation of school improvement plans. I am currently working with TC² to develop and implement a critical inquiry approach to math learning aligned to the Ontario Grade 9 math curriculum. I have been working with a team of three teachers teaching five sections of the Grade 9 academic course.


Given all the math initiatives and resources available to support student learning in math, what made you interested in a critical inquiry approach?

At the end of the 2016-2017 school year, our math department reflected and agreed that we wanted our students to improve their capacity to think critically while nurturing their growth mindsets. We needed a pedagogy to support these two goals. TC²s critical inquiry approach aligned well with our board and school improvement plans focused on Modern Learning, deep inquiry-based co-learning with students using authentic tasks and critical thinking. It also aligned with our use of Non-Permanent Vertical Surfaces to support student collaborative problem-solving. TC²’s critical inquiry approach gave us a method to frame problems that clearly invited students to engage in quality thinking about the curriculum while also providing us with clear success criteria for monitoring and assessing thinking. Critical inquiry complemented our use of Vertical Surfaces to ensure our students engaged in quality mathematical thinking.

On a personal level, critical inquiry is a topic I had been interested in since teacher’s college when I was exposed to the TC² approach. I appreciated the fact that critical thinking was clearly defined and could be implemented through specific types of tasks. At Markville SS, we embarked on implementing critical thinking as a school back in 2011. At the time there were limited examples on how math was supported. More recently, however, I have attended TC² professional development opportunities that have been focused on math including sessions sponsored by the Ontario Teachers’ Federation, online webinars and a summer institute all delivered by Laura Gini-Newman on Sustaining Inquiry. These have helped me delve deeper into how to implement a critical inquiry approach to learning in math in a sustained way that nurtured a growth mindset in students.

As you worked on implementing a critical inquiry approach in math classrooms, how did students respond? Why do you think they responded this way?

Student work

Students responded favorably to critical inquiry questions and tasks. Critical inquiry invites students to use their math knowledge and understanding to solve problems that focus on process rather than the final answer. It also invites students to collaborate with each other, to share ideas, and make mathematical decisions. This focus on mathematical decision making naturally requires students to consider many different options—ways of doing math, using tools, and solution methods—allowing students at varying levels of understanding an opportunity to immediately participate and feel safe and confident in sharing their ideas. In many cases students who are typically hesitant to contribute to class discussions are now able to have a voice amongst their peers as they work through thinking tasks. This includes students with weaker background knowledge as well as students who are proficient but lack the confidence to participate in regular open class discussions.

Can you share an interesting example of how a critical inquiry approach to learning mathematics helps students build math foundations as well as thinking abilities?

At the start of our Linear Relations unit, we invited students to connect various representations of linear relations to help them make connections among the different forms presented in words, tables, graphs, and equations. Typically, each of these forms are taught individually and then pulled together in a final lesson, but in a critical inquiry approach, students were presented with all of the different forms at the beginning to determine the properties of a linear relation. Instead of feeding information to students about linear relations and then asking them to apply it, they were asked to make connections right from the start and throughout the unit, continuously applying a fundamental mathematical idea as they practiced solving rich, contextualized problems. To be mathematically successful in a critical inquiry classroom, students are required to draw on a full range of background knowledge to detect patterns, analyse and synthesize them, and apply that knowledge to problem solve, thereby co-creating and deepening learning every step of the way.

Using a critical inquiry approach also challenges students, in a supportive environment, to pay very close attention to details. In our Solving Equations unit, students were asked to evaluate the usefulness of a variety of equations to solve a Pythagorean Theorem based problem. In order to provide a consistent and full analysis of the equations, students had to discuss all aspects of each one. The task invited students to ask each other questions, to clarify their thinking, and to conjecture as to the importance of each of the equations provided in relation to the problem. This led to more thorough analyses requiring students to think about what they already knew and understood to be mathematically true (accurate math knowledge) about each equation, and all the mathematical details (evidence) given in each equation—two basic considerations that underpin all sound mathematical reasoning.

As you worked on implementing a critical inquiry approach in math classrooms, how did teachers respond? Why do you think they responded this way?

Grade 9 Math Teachers

Implementing critical inquiry in the classroom initially generated a sense of discomfort for teachers in my department, myself included. While some questions, problems, or tasks were easily tweaked to explicitly invite mathematical thinking, other tasks were more challenging to generate. One of the reasons for this discomfort was the use of unfamiliar vocabulary required to make mathematical thinking clear for students. For example, qualifiers (or key words used to determine criteria) such as most important, most similar, fully consistent, or fully simplified are often used to explicitly invite students to make mathematical judgments or decisions. Qualifiers encourage students to think about the quality of their responses and to justify them, not to think only about whether they have arrived at the correct response. Teachers and students experienced some difficulty unpacking these terms as they are not typically used in math classrooms.

What are some of other challenges you faced as you worked to implement a critical Inquiry approach to math learning?

Making clear to students what the process of critical thinking involves is difficult if it is not clearly understood. There is a learning curve that teachers have to experience as they work to become proficient in creating critical thinking tasks, nurturing a critically thoughtful learning environment, and developing assignments or assessments that allow students to demonstrate the quality of their thinking. This ultimately requires us to mark assessments both for procedure or correctness and for the quality of thinking presented by the students. Students can’t reason effectively in math without a strong and deep knowledge and understanding. They need content and thinking to be successful.

Implementing a critical inquiry approach also involves a thoughtful sequencing of lessons and lesson topics. This allows students to experience thinking opportunities that are logically connected and build to the successful completion of a provocative, authentic, and complex task or problem. This does not always align with individual teachers’ ideas of how units traditionally unfold—units often based on how math textbooks have been laid out. As a result, teachers may initially experience a lack of confidence when implementing a critical inquiry approach as this creates uncertainty in how lessons should unfold.

Were you able to overcome those challenges? If so, how?

To overcome these challenges, collaboration among teachers was vital as well as support from the school administration. This created opportunities to share ideas when creating or tweaking questions or tasks, verify with each other that the questions or tasks genuinely invite critical thinking, and give confirmation that our questions or tasks met this goal. Collaborating with TC² facilitators such as Laura Gini-Newman provided essential insights to the work and much needed guidance and clarification in the learning process. Having PD sessions and regular informal meetings on site allowed for shoulder to shoulder support so that teachers’ concerns and questions could be addressed.

Finally, working closely with Laura to deepen my own understanding has helped provide me the leadership capacity to support implementation. Recently, my Grade 9 math team, Laura, and I decided that implementation would be easier if Laura and I collaboratively developed a critical inquiry unit that our team would implement and continuously assess for both impact on student learning and ease of implementation. We all agreed that this collaborative joint venture would result in a unit design that every teacher on our team will deeply understand, is confident implementing, and will meet the needs of all of our learners.

Has there been value in working to implement a critical inquiry approach to math learning?

There has been great value in using a critical inquiry approach. Students are engaged to think and assess the quality of their own thinking, make decisions, and understand the importance of collaboration. In addition, critical inquiry provides time for students to practice important math content. For example, from the beginning and throughout the Linear Relations unit, students connect all the ways to represent this type of relationship so that they deeply understand the concept of a linear relation in any form. Most importantly, critical inquiry is based on making sound mathematical judgments. Although students are invited to consider a range of possibilities, these must be mathematically sound and ultimately students are encouraged to select the most sound option given the nature and context of a problem. Critical inquiry moves beyond correctness to sound reasoning.

This approach has encouraged the teachers in my department to reflect on their own pedagogy and deepened their understanding of how to effectively teach and assess for mathematical thinking in our classrooms. But there is still much work to do to gain further proficiency in this framework. Although there have been learning bumps along the way, these have served to strengthen the learning of both of our teachers and students while nurturing a growth mindset in all of us.