How Thinking Mathematically Changed my Teaching

Published: Apr 14, 2020

This is the second 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 article, Jocelynn Foxon, numeracy coach with Pine Creek School Division (Manitoba), 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.

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

I am the Numeracy Coach for the division. I currently coach teachers in five K-8 schools, two high schools, and seven K-12 colony schools.

What made you interested in the Math Lead Teacher Certification Program (MLTCP) offered by TC²?

I was at a PD session with Garfield Gini-Newman where he was speaking about critical inquiry in the classroom. I asked him what this might look like in math and he put me in touch with Laura Gini-Newman and the MLTCP program.

How useful did you find the MLTCP as a professional learning opportunity?

Jocelynn Foxon

Extremely useful—I learned so much about how to not only guide students to think more critically, but also how to assess this thinking and provide rich engaging tasks in my math class. During this process I realized that I needed to lead teachers by example. Being an instructional coach gave me the opportunity to demonstrate and model how a critical thinking approach focused on the development of reasoning competencies can and does work in any classroom. Teachers were able to see the different style of teaching as well as see the effect it had on their students.

I also learned that if I were to work with a teacher again using this approach, I would like to work with them on developing their own lessons. One teacher had mentioned it was hard to know how to teach my unit because she was not a part of developing it. I thought that working with her in the future, she would be more involved in the process as we developed a unit on an outcome or strand. This program allowed me to not only learn how to become a better leader, but it also allowed me to test out an original unit with not one, but several classes. Doing this gave me the opportunity to collect immediate feedback from teachers and students, adjust where necessary, and then see if those adjustments made a difference. It was a great way to field-test the unit Laura and I developed in the MLTCP while also modelling the approach at the classroom level.

As you worked on completing the MLTCP, how did teachers respond? Why do you think they responded this way?

Teachers all had different responses; some were quite concerned about the ability of their students and if they could complete tasks that seemed to be “hard” while others were nervous about teaching in a different manner. I also had teachers who were concerned about having students struggle, but they learned quickly that, when students are given the opportunity and time to reason in math, this struggle would be productive. Then there were teachers that were also very eager to implement this approach in their classrooms and were excited to try the lessons right away—they saw an immediate positive impact on their students’ learning.

As you worked on completing the MLTCP, how did students respond? Why do you think they responded this way?

The response from students was amazing. At first, they were nervous about a new teacher and “new” math; however, it only took one lesson and students were enjoying it. Students who were generally quiet and didn’t participate much were giving responses and even helping other students. Students who were generally “good” at math and could quickly zip through an assignment found this more challenging and had to take time to stop and think as there wasn’t always immediately the “right” answer to find. I think that teaching in a way that allows students to explore within a sustained thinking structure gives all students the opportunity to learn high levels of math. And those students that initially struggle to understand procedurally or sequentially were able to find success using this approach of learning mathematics.

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

Enrich Your Thinking

In one lesson, students were asked to mathematically describe the changes that occurred between two geometric shapes both with the same area. Students had to use words and then try to come up with mathematical equations to describe these changes. At first, they were not told the area of the shapes, so they needed to think symbolically. During the “enrich your thinking” section of the lesson, students were offered a bank of equations with many options to consider using addition, subtraction, multiplication, and division. Some students even described one of the changes in While students were describing these changes and connected their word descriptions to the equations, they came to understand that cutting a shape in half was the same as dividing by two or that cutting a shape in half twice was quartering it or dividing it by four. However, if they cut a shape into unequal parts, they needed to subtract an unknown area x to create a new shape y. As a result of focusing student learning on how to reason soundly in mathematics, it was a common occurrence in our math lessons that students were able to achieve beyond grade level. These Grade 4 students were writing mathematical the shapes using the concept of fractions, even though they had not yet had a lesson on fractions. They described how, first, they cut the shape in half, then cut it diagonally but still in half. Next they re-attached the shape in a different way and finally rotated it to create the new shape. descriptions in algebraic form! In addition, they were thrilled and very impressed that they understood what that these algebraic representations meant; for example, that (shape 1 – x = y) rotate x and y (y’+ x’ = shape 2)!

What are some of the challenges you faced as you worked to complete the MLTCP and how did you overcome them?

Implementation time! We had originally planned that I would be working in classrooms for 4-6 weeks; however, in some classes, the implementation schedule was interrupted and in other situations, instructional time was shorter than we thought, so we needed to work around schedules to get the unit completed in the time we set aside. Unfortunately, this didn’t work in all classrooms and we were unable to fully implement the full unit in some. But this did not diminish the learning of the students and teachers.

Another challenge was that teachers did not initially fully understand the direction of the unit being implemented as they were not involved in planning it. They were not used to starting a unit with what the students would have to accomplish by the end of the unit (the assessment task). Teaching a unit in this way was new and they were unsure how to administer the activities so students would be successful. In response, I decided to model the unit in these classrooms. Ultimately, one teacher decided that it would be great to develop her own unit using this approach after observing the implementation so that she could teach it on her own; she saw the potential of the questioning methods and thinking strategies being modelled in her classroom.

What were some of your key learnings from the MLTCP?

During this process, I learned so much about how to write unit and lesson plans that allow students to reason mathematically, while meeting the curriculum outcomes. Creating a math unit that promotes critical thinking requires purposeful planning.

  1. The first thing I noticed about writing a unit using this approach was that you needed to figure out exactly what question you want the students to answer at the end of the unit, as well as what task you want the students to be able to complete. These questions and tasks needed to invite students to reason mathematically and be provided to the students at the very start of the unit. Questions needed to be a decode, critique, or judge better or best. Tasks had students either performing to specifications, designing to specifications, or reworking a piece to better fit the requirements.

    Formulas for Perimeter and Area Once the overarching unit question and task are planned out, I was surprised to find that we could not just go ahead and plan the lessons as we normally would. I needed to stop and look at the reasoning competencies that were going to be involved in the unit. For example, I needed to look at the math processes the students would encounter and connect these to the mathematical reasoning competencies that would highlight the nature of the thinking students would be asked to do, as well as what the students needed to think about to better accomplish the tasks and answer the questions. Identifying the criteria for mathematical reasoning allowed me, as the teacher, to ensure that students were learning to reason in a quality way.

    After I had the unit questions and tasks planned out and I was ready to write the lessons, I needed to create a task and a question for each lesson (scope and sequence) and I needed look at background knowledge to ensure that I had a solid understanding of what information and content came before my lessons. This would allow me to identify any learning gaps that students may have, and also be able to write any needed review into my lesson to allow students to fill in those gaps while working through their own new learning.

    At first, I had quite a bit of trouble with this because I was not thinking about it critically myself. Once I started to really ask myself what I wanted the students to accomplish through the task, I could begin to develop the questions that would follow. When I began planning, I started with the concepts of perimeter and area, but as I dove into the scope and sequence I realized that what I really needed students to understand was the concept of space and how the mathematical operations could assist in mathematically describing the space of an object or shape. I also had to think about the natural sequence of thinking of the students. And I needed to guide them in a way that ensured they were still discovering this on their own. I wasn’t leading them to water, but rather providing paths that would get them there successfully on their own.

    Each lesson needed to be thought through using this critical thinking lens and each lesson needed to naturally follow from the one previously taught. Each lesson also needed to always iteratively return to the launch lesson task designed to support the students’ thinking about the unit overarching question and task.
  2. Interactive Euler Diagram Activity
  3. Another key learning was how to be a leader for my teachers. I realized that, just as our students have diverse needs and require flexibility, so do teachers. I learned how to work with them in their classrooms, offering support where needed, guidance when requested, and modelling when something was new or unclear. I learned that bringing a new model of teaching into a classroom can cause some stress and nervousness as teachers are unsure what is expected of them and doubt they can “teach it properly.” Understanding their position and modelling this critical thinking approach was a great way for them to see the value in this type of math pedagogy and to see results among their students. When a lesson would not go as smoothly as we would have liked it to, it was a great time for reflection and feedback on the lesson and why the activity worked or rather did not work the way we had hoped. By working alongside teachers, we were able to develop cohesive lessons that we could all be proud to teach our students.

  4. Finally, during this program, I was able to see first-hand the value of changing mathematical thinking in the classroom to a more structured critical thinking approach that not only invites students to reason mathematically but to learn how to do it well. Students were given the opportunity to see the value in learning a new math concept. By providing students with a task that may be challenging to complete and then providing enriching activities to scaffold their thinking, students are able to apply their new knowledge right away to a task they already knew they could not fully and immediately complete. It gave them purpose. They got to use criteria to come to their own conclusions and then apply their new learning and receive feedback immediately. Students who are given the opportunity to think about math critically see the value in what they were doing. I saw students who believed they were not good at math starting to enjoy learning, be engaged, and participate in high level mathematics with rich understanding. Learning how to reason soundly helped students learn to think mathematically in a variety of ways and apply it to real-life situations.

How have these learnings shaped your current teaching practices in mathematics?

Every day I make sure to implement critical thinking into my math lessons. I want students reasoning soundly, that is, thinking about their thinking, understanding language, and being able to reason effectively about why they did something, not just give me the answer they think I want or expect. Actually, unless they can provide evidence to support their thinking, I don’t believe they truly understand what it is they are learning. My goal for my students is no longer that they can complete a task, but rather that they can complete a task that requires them to reason mathematically and that allows them to do so in a way that makes the nature and quality of their thought process clear.

Have these learnings shaped the practices of others you worked with and if so, how?

Yes, I believe that working with the teachers through this process has changed the way they look at and teach math. One teacher in particular was quite procedural in her math class. This has worked for her for many years, her students know their math facts and are quite successful at math as a whole. However, while working through this unit, she saw the value in allowing students to reason about mathematics, dig deeper and thoughtfully apply the math in different and appropriate ways.