Traditional mathematics classrooms were often quiet places. Teachers perceived their role as one of knowledge transmission. The belief was that a mathematics teacher was effective if students quietly listened to the teacher’s demonstration of rules and examples and then were successfully able to complete practice problems that were exactly like the demonstrated examples. The instructional emphasis was on students memorizing facts and developing proficiency with routine procedures.

##### Modern Goals for Learning Mathematics

Today, there are changes in how mathematics is used—for instance, technology can perform routine procedures and symbol manipulation—and the demands of the workplace put a premium on reasoning and problem solving. It’s important that students not only develop procedural fluency but also that procedural fluency is built on a foundation of conceptual understanding, so students can confidently draw on both their skills and understanding to solve problems they have never seen before. The instructional goal today is for students to be engaged in solving and discussing tasks that promote reasoning and problem solving.

##### Building Conceptual Understanding Through Math Talk

What does building procedural fluency built on a foundation of conceptual understanding look like? One feature of instruction that emphasizes conceptual understanding is purposeful discussion, or math talk, among classmates and the teacher about underlying mathematical concepts, their connection to skills and procedures, and problem-solving strategies. This type of mathematical discourse advances the learning of the entire class. Students who communicate with their classmates and the teacher about their thinking, solution pathways, and insights into how they solved a problem—or how to analyze another student’s solution—develop a deeper understanding of mathematics.

##### Math Talk Through Effective Questioning

Too often, the pattern of questioning in mathematics classrooms is that the teacher asks a question with a specific response already in mind, a student responds, and the teacher evaluates the student’s response in light of the teacher’s expected response. This approach to questioning gives students few opportunities to think, only engages a single student in the discussion, and provides teachers with very little insight into whether or not the student is developing an understanding of mathematics.

One way to generate more purposeful math talk is by asking questions that encourage all students in the classroom to become engaged in the discussion and to explain and reflect on their thinking. You can start by changing the typical pattern of questioning by asking more higher-level questions, including:

- “Why?”
- “Can you explain?”
- “How do you know?”
- “Do you agree? Why?”
- “Do you disagree? Why?”
- “Can you repeat what Student X said in your own words?”
- “Does anyone have something to add or a different interpretation?”

Asking these or similar questions of a second (or third or fourth) student after the first student responds—instead of evaluating the response yourself—engages more students in the mathematical discussion, focuses questioning on student thinking, presses students to communicate their thoughts more clearly, and communicates the expectation that all students will reflect on not only their own thinking but also on the thinking and reasoning of their peers. An additional benefit of this approach to questioning is that it communicates to students that mathematical truth resides in the validity and quality of the mathematical arguments and not within the teacher.

##### Talk Moves and Equity

It is not enough to simply ask more questions, even if you ask higher-level questions and change the pattern of your questioning. How we ask questions, whom we ask questions of, and how we respond to students all influence how students see themselves as learners of mathematics and as members of the community of learners. For example, it is important to reflect on the following:

- Are all student ideas heard, valued, and brought forth for further discussion?
- Who do you ask questions of?
- Whose thinking do you value?
- Do some students receive “easy” questions and others more demanding questions?

If only some students are consistently asked questions, certain students are always asked lower-level questions, or only some students have their thinking discussed, then through our questioning and math talk we are communicating to some students that we don’t believe they are capable learners and doers of mathematics—we are inadvertently damaging their mathematical identity.

As mathematics teachers, we have to appreciate that we teach more than concepts and skills. We also have a responsibility to support students in seeing themselves as capable of participating in and being doers of mathematics. When our talk moves in the classroom are intentional and purposeful, not only are we more likely to build student understanding of mathematics, but we can simultaneously help build strong student mathematical identities.

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