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Dr. Banjamin Klaf points to a whiteboard while teaching a senior math class at Collingwood School
By Dr. Darien Allan and Dr. Benjamin Klaf (Bridge, Fall 2023 Edition)

Mathematics, in its essence, is the exploration and study of a set of beautiful, interconnected ideas – patterns. We live in a fabulously rich world with patterns of all kinds. As curious beings, we seek out, explore, discover, delight in and find comfort in an incredibly wide variety of patterns. Thinking mathematically about such patterns in many creative and useful ways helps connect us more deeply and authentically with the world around us. 

Over the last several years, our Senior School Mathematics Department has been working to incorporate various research-based practices in the teaching and learning of mathematics. Drawing heavily on the ideas in Dr. Peter Liljedahl's Building Thinking Classrooms, we are also integrating other findings from cognitive science research about how people learn. In our classrooms, students regularly engage in collaborative problem-solving where they suggest and test strategies, make conjectures and practice and refine their mathematical communication in a supportive and feedback-rich environment. Students are frequently asked to explain their choice of strategy and to justify their solutions. Time for reflection and practice is essential so that students consolidate their learning and test their understanding. 

Three students in navy blue sweaters practice a math problem on a whiteboard

We believe that students’ mathematical thinking and skills develop most fully when students engage in mathematical exploration and problem-solving which is authentic, developmentally appropriate and richly connected to the real world. An essential part of this skill development comes through teachers providing many opportunities and modes for our students to discuss, represent, understand, explore and answer mathematical questions. We find that students who consistently reflect on their learnings will begin to build their own connections and grow to become more confident and capable learners. 

For example, a Math 10 class was recently presented with a scenario involving an acrobat diver, a Ferris wheel, and a moving cart full of water. Students worked in groups to solve initial problems involving angular measurement, the horizontal and vertical position of the rider at a certain point in time and how long it takes to fall to the ground from a particular height. During group work, the teacher circulated to ensure everyone had the requisite understanding before proceeding. Next, students worked individually to determine when the acrobat should jump to land in the cart. Finally, students were asked to reflect on their solutions. Is it okay for the acrobat to jump a second earlier or later? What precision is needed to ensure the acrobat arrives safely? During the individual work, the teacher again circulated, asking and answering questions. We find this type of task allows students to practice the skills they are developing and encourages them to think critically about their problem-solving strategies and choices. 

Ultimately, we hope our guidance, support, feedback, encouragement and engaging coursework will help our students begin to find their own mathematical voice. And once found, we believe they will continue to use this mathematical voice to solve the big, important problems that help make the world a better place.

  • Mathematics