Mathematical discourse is essential for deep mathematical learning. We often spend a lot of time teaching our students how to talk to one another with sentence stems, practiced conversations, and teacher talk moves. However, before we can even teach students how to talk, we must make sure that our classroom is a place where students feel comfortable talking.
When students justify their thinking aloud or explain a strategy they have used, they are putting their ideas on display for their classmates. This can be intimidating for many of our students. Their ideas may be different from their classmates, they may have made a mistake, or they may have tried a unique strategy that no one has used before.
As teachers, we must set up a classroom environment where students feel safe enough and brave enough to share their ideas. They should be able to trust that their classmates and their teacher will treat their ideas with respect and curiosity.
There are a few things that can be done at the beginning of the school year and throughout the year to make a classroom a safe community where students feel comfortable sharing their thinking, having mathematical arguments, and questioning ideas.
1. Teach students to respect others' ideas, especially when they are different from their own. Respecting others' ideas is something that we encourage in our classrooms from day one and work hard to maintain throughout the year. We teach our students to treat everyone's ideas with respect, even if they may disagree. We reinforce the idea that respectful disagreements are not something negative, but are rather an opportunity to help us think deeper about a problem and learn more about the math. We use respectful sentence and question stems, such as "Can you explain why you ___________?" or "I respectfully disagree because ____________" to promote this discourse. We also teach respectful signals that show that they are thinking something different—for example, tapping a pointer finger on the forehead to indicate, "I'm thinking something different"—rather than a negative reply like, "You're wrong!"
Most important, our students know that we have multiple teachers in our classroom. Every student at some point becomes a teacher when they share their thinking or question their classmates. Students learn to ask questions when they disagree with something or when something doesn't make sense. They know that we (the teachers) are not the only ones that have valuable knowledge to share, and that we all learn from each other.
2. Celebrate mistakes and encourage risk taking. Many students come to us afraid of making mistakes or getting the answer wrong. We must help our students believe that their deepest learning can stem from their mistakes. We do this by celebrating "great mistakes" daily. We ask students who have made mistakes to share their thinking with the class and then celebrate them for doing so. We talk through the great thinking behind the mistakes and then thank the students for letting us learn from their mistakes.
When students are feeling unsure about a strategy or way of thinking, we encourage them to be brave mathematicians. If they feel comfortable making mistakes, then they become more confident in their ability to try new strategies because they know that they won't be looked at negatively for trying something new or difficult and struggling with it. We tell our students that if we've given them a math problem that is easy, then we haven't done our job as teachers! We apologize for that easy problem and promise our students that we will challenge them with something more difficult so that their brains can grow stronger.
3. Value multiple strategies and focus less on the answer. Mathematics is a subject that is creative and unique. Yet the way we traditionally teach math—through rote practice and memorization—keeps students from seeing the beauty of the subject. We teach math through problem solving, where students must find their own way to an answer using different strategies and learning to justify their reasoning. In our classroom communities, we focus more on how students got their answers and less on what the answer is. In one given lesson, we may have three or four different students share completely different ways of getting the same answer. We focus on helping students develop the skills they need to justify their answers and explain their thinking. We encourage students to compare strategies, make connections between representations, and celebrate the different ways our brains see mathematical problems.
4. Question everything. At the beginning of the year, we often see students erasing their papers the minute we ask them a question about their work. Teachers know that we must question our students' thinking to deepen their understanding; however, many teachers spend a lot of time questioning students who get the wrong answer and simply pass over or give a "Good job!" to the students who get the answer right. When we do this, we are training our students to believe that a question means they got the wrong answer.
In a discourse-rich community, students regularly justify their thinking and explain their ideas to others, regardless of whether they got the right answers. In our classrooms, we strive to question all thinking. When students get the right answer, we push them to "prove it." We want our students to be confident in their reasoning, and by pushing them with questions, they learn to justify their thinking and grow their confidence as mathematicians. When students become more used to our questions, they begin to answer them before they are asked.
Classrooms where students feel safe to make mistakes, share their thinking, and ask questions set the stage for productive mathematical discourse. Although it can be easy to want to dive into teaching students how to talk to one another, it is important to remember that there is work that must be done to build a trusting community before deep mathematical conversations can happen.
