Productive Mathematical Discussions  

April 17, 2014
I recently had the opportunity to attend the third annual Midwest Mathematics Meeting of the Minds (M4) Conference. This year’s conference was held in Omaha, and about 100 people involved with math education in Nebraska, Iowa, Kansas and Missouri were in attendance. One of the speakers was Margaret Smith, co-author of Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). During the presentation at the M4 Conference, Ms. Smith emphasized that, in order for mathematical discussions to be productive, the teacher first must have a clear mathematical goal in mind and select an appropriate task to be the basis of the discussion. Once these two foundational elements are in place, a teacher can then successfully utilize the Five Practices. These practices include Anticipating, Monitoring, Selecting, Sequencing, and Connecting.

When a teacher employs the practice of Anticipating, he or she considers what strategies students are likely to use to approach or solve a challenging mathematical task prior to asking the students to attempt the task. The teacher also Anticipates how to respond to the work that students are likely to produce, and which strategies are likely to be most useful in addressing the mathematics to be learned. In order to do this, the teacher will work the problem as many ways as possible and develop expectations of how students might mathematically interpret a problem. The teacher will also envision the array of strategies – both correct and incorrect – that students might use to tackle the problem and consider how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that you would like the students to learn. With these in mind, the teacher will write potential guiding questions that correlate with expected misconceptions (Smith & Stein, 2011).

The practice of Monitoring takes place while students are attempting the given mathematical task. During this time, the teacher will circulate the room as students work individually or in small groups, making notes about students’ mathematical thinking and solution strategies. The teacher might also ask questions to get students back on track as needed, help students clarify their thinking, ensure all students are engaged, and press students to consider all aspects of the task to which they need to attend (Smith & Stein, 2011).
The third and fourth practices, Selecting and Sequencing, are closely related and happen nearly simultaneously. The practice of Selecting is guided by the mathematical goal for the lesson and the teacher’s assessment of how each student’s solution contributes to that goal. The teacher will not take random volunteers. Instead, he or she will call on specific, pre-determined students/groups to present, let students know before the discussion that they will be presenting, or ask for volunteers but strategically call on those students whose ideas contribute to the goal. Sequencing is the practice during which the teacher makes purposeful choices about the order in which students’ work is shared so as to best highlight the key mathematical ideas. The teacher will not only consider which students can contribute to the discussion in a productive way, but also which students need to be heard from in order to ensure that all students have an opportunity to contribute on a regular basis (Smith & Stein, 2011).

Connecting, the fifth and final practice, is designed to help students see relationships between their solutions and other students’ solutions, as well as the key mathematical ideas. This is also when the teacher leads the students to make judgments about the consequences of different approaches (for example, accuracy and efficiency), attend to mathematical patterns, and develop key mathematical ideas as student presentations build on one another.

These five practices naturally tie-in with work already being done by math teachers in the ESU 10 area through the Adolescent Literacy Project. Mathematical discussions are an excellent way to foster students’ mathematical literacy, which directly supports both students’ mathematical knowledge and skills, as well as their overall literacy skills.

-by Emily Jameson, Professional Development Coordinator

Works Cited
Smith, M. S., & Stein, M. K. (2011). Five Practices for Orchestrating Productive Mathematical Discussions. Reston, VA: National Council of Teachers of Mathematics.