Mathematical discourse has been articulated as one of the Common Core Mathematical Practices: construct viable arguments and critique the reasoning of others. Sounds stuffy and maybe even intimidating, right?
My teaching is heavily influenced by John Seely Brown and Daniel Pink, who encourage teachers to incorporate more creativity and "playful thinking" into the classroom. I want students to feel liberated to try on ideas, think with erasers, and generally play with ideas until they "get it." This kind of atmosphere helps students to be comfortable with mathematical discourse. It builds their confidence and spirit (so that their default is to think, "This is hard, but I am smart, and I'm going to keep at it until I learn it").
Break it down.
I don't think you can just jump in and take off, asking students right away to know how to test expressions for being "always, sometimes, or never (ASN) true", for example. They wouldn't know how to form an opinion (in math we call it a conjecture) because they don’t know how to test their assumptions or conclusions yet (gather evidence to support their conjecture).
Instead, tear apart the techniques of discourse into small pieces that students can learn. Build those simple skills into more complex techniques. By being deliberate, you can eventually develop a classroom of students who can think and debate effectively about mathematics.
Seven months into the school year, my students have used the ASN strategy multiple times. They no longer struggle over building their conjecture or evidence. Instead, they can focus on the math being analyzed—and they actually seem to enjoy the lessons.
Here a student ponders over where to categorize an expression. She must place it in one of the three categories—and write a defense of why it belongs there.
Incorporate playfulness and highlight insightful moments.
When students debate and play with math ideas, they create and identify a foundation of understanding. Listen for moments when students are connecting the new to the old. You'll hear things like, "Oh this is like when we learned _________, right?" That kind of conversation helps everyone. Celebrate those statements, and consider capturing them on chart paper displayed in your classroom.
Here students experiment with paper models of a more complex Algebra/Geometry problem: finding points on a circle and midpoints of line. I help them connect to prior learning so the abstract and symbolic problems become more familiar.
Display the problem… without the question.
An easy way to start off is to display the problem without the actual question. Sounds simple, doesn't it? But for a short period of time, it forces students to do two things: find all the relevant information that the problem offers and to generate possibilities for what the question might be.
I'll warn you that students are typically pretty frustrated with this approach…. they just want to know what you’re going to ask of them so they can get to it! But forcing students to spend time harvesting information pays huge dividends later on.
Slowing down is key. This part is more discussion than discourse, but it builds a common understanding among classmates. It's also a strategy that levels the playing field for students who need more think time or scaffolding of the open-ended problem. By the time you reveal the problem, most students thoroughly know the "given" information and the context—which are the keys to making "hard" problems easy!
Pair talking strategies and listening strategies.
Activities that build mathematical discourse give voice to students—rather than a teacher talking them through a challenge, they work it out themselves.
But talking must be partnered with listening.
As teachers, we must make sure that all students have a chance to think for themselves, not allowing less capable students to continually rely on someone else. (This is why giving students private time to think about some problems—to solve or struggle with them on their own—is very important.)
Incorporating listening skills can be especially difficult at first. Here are the tried-and-true techniques I require students to use:
- Restate what the person before you has said.
- Explain how your idea connects with something that has already been said or learned.
- Summarize the discussion so far, and then add your ideas.
While the tasks are difficult for my students, they later articulate the value of practicing these skills when we have a class "debrief." They discuss what they think they have learned from listening, what they did well, and what they didn't like. Each class builds a set of strategies for overcoming the things that are hard, and they celebrate their successes.
Taking time to break apart mathematical discourse (see an example in this Tch video) into more accessible pieces is essential to an effective learning process. As we begin to facilitate this kind of learning, teachers must master new techniques for helping students achieve and gain academic confidence. It's a growing and learning journey for all of us!