“What a teacher knows and what he or she does and believes have a major influence on how students learn. Most importantly, we know that these are dynamic behaviors and dispositions that (can) evolve over time.” -Adela Solís https://www.idra.org/resource-center/pedagogical-content-knowledge/
In our work, we spend a lot of time thinking about the kinds of learning experiences we want for teachers, and for the coaches, principals, and district leaders who support them. We come back again and again to a simple but powerful idea: we need to experience the kind of learning we hope to create for students. That’s why doing math together is often at the center of our work. It gives us a shared space to think, to wonder, and to reconnect with what it feels like to make sense of mathematics. It also helps us develop an important piece of knowledge, which is referred to as mathematics knowledge for teaching*. This knowledge is important because mathematics instruction requires teachers to possess deep and flexible understanding of the content they teach
The goal of our professional learning is to strengthen our ability to support students in making sense of mathematics. One of the most important commitments we make as teachers and teacher educators is that we do the math ourselves and engage in it alongside one another. This shared experience keeps our focus on the kind of thinking we want for students: reasoning, representing, conjecturing, and connecting. Even experienced teachers need time to sit with the mathematical ideas of a lesson.
When we do math together, we surface the many ways we each approach problem solving. Some of us may initially lean toward efficient procedures, while others may want to draw a picture to help them attend more closely to structure or relationships. By making these approaches visible and discussing them, we expand our collective understanding of the many different ways to make sense of mathematics. This can directly shape how we position students’ ideas in the classroom; not as right or wrong answers to evaluate, but as contributions for the classroom community to build on and connect.
Language becomes more precise as we engage in math as well. As we explain our reasoning to one another, we refine how we name strategies, describe relationships, and justify ideas. This shared attention to language strengthens our ability to support students in articulating and revising their own thinking.
Engaging in the math ourselves also helps us understand where the work invites deeper reasoning. We notice the points where a representation clarifies an idea, where a pattern begins to emerge, or where an assumption needs to be reconsidered. These are often the same moments our students encounter in their work. Experiencing them firsthand allows us to design learning opportunities that honor productive struggle and create space for students to develop understanding over time.
As we work through the math tasks, we naturally begin anticipating student thinking. We ask ourselves: What strategies might students use? What conceptions or partial conceptions might emerge? Where might students get stuck, and what might help them move forward? This anticipation is essential to our ability to support student sensemaking. It allows us to plan how we will elicit student ideas, interpret their thinking, and respond in ways that advance the learning of the group. We anticipate the questions we want to pose, the ideas we want to press on, and the moments where we might pause to consolidate understanding. Rather than reacting in the moment, we enter instruction with a clearer sense of purpose grounded in the mathematics itself.
Doing math together also sharpens our decisions about representations that are used by curriculum writers. We consider which model (number lines, diagrams, equations) best highlights the underlying mathematics. We experience how different representations reveal different aspects of a problem. This helps us be more intentional in the classroom, where these choices can either open up or constrain students’ understanding. Further, what isn’t apparent to us at first as we engage in the math task, isn’t often apparent to students either. The more we explore about the math and the tasks, the more we understand what students will need to grapple with.
Finally, engaging as learners helps us see the broader mathematical landscape. We identify the big ideas a task is designed to develop, consider the conjectures students might form, and anticipate connections to other concepts. This allows us to support students in building coherent understanding over time, rather than experiencing mathematics as a series of disconnected topics.
When we commit to doing mathematics together we align our professional learning with the learning we seek for students. Doing the math offers us insight, clarity, and a shared foundation for our work. It sparks curiosity about our students’ thinking. It keeps us grounded in what matters most: creating classrooms where students actively make sense of mathematics, drawing connections, testing ideas, and developing understanding that endures.
*Mathematical Knowledge for Teaching (MKT) is a construct developed by Deborah Ball and colleagues (Ball, Thames, & Phelps, 2008), building on Shulman’s (1986) notion of pedagogical content knowledge. It refers to the specialized knowledge that teachers need to effectively teach mathematics — knowledge that goes beyond simply knowing mathematics as a subject.
The ideas in this post are influenced by the following scholars: Deborah Ball, Linda Davenport, Jim Hiebert, Heather Hill, Geoffrey Phelps, Lee Shulman, and Mark Hoover Thames
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