I first heard the phrase “student discourse” as a central tenet of student learning in mathematics from Deborah Ball in 1991. At the time, she was the Chair of a NCTM Task Force I participated in on *Professional Standards for Teaching Mathematics*. I remember thinking, “Why not just use the word ‘communication’ instead?” But I learned soon enough that student discourse is so much more than just the nature of student communication during the lesson.

As my teaching of mathematics matured throughout the 1990s, I developed clarity around the use of structures designed to support two primary types of student discourse that support learning for all students:

**Whole-group discourse:**The teacher leads the discussions and models how to do a problem. He or she calls on students one at a time to respond to questions as the mathematical task solution unfolds and deflects student responses back to other members of the class. During this type of discourse, the teacher is doing most of the speaking, feedback to students occurs one student at a time, and the significant conversations taking place in class are of the teacher-to-student variety, as other students sit and listen (or not). Although this type of discourse is necessary at times during the lesson, it rarely can achieve the student active engagement expectations for learning mathematics.

**Small-group discourse:**The teacher facilitates student discussions as they work together, usually in teams of three to four, with a time limit and specific directions or prompts provided for discussions and sharing. Students collaborate with other peers and process together through mathematical strategies for an assigned mathematics task or discussion prompt. During this type of discourse, the teacher leaves the front of the room and observes students*doing*work*together*, provides meaningful feedback on what he or she hears and sees from the student team discourse, and supports the use of meaningful conversations between students as they act on the mathematical task.

Fast-forward two decades later, and the *National Board for Professional Teaching Standards* highlights this importance of peer-to-peer student discourse in the third edition of its Mathematics Standards:

Accomplished teachers deliberately structure opportunities for students to use and develop appropriate mathematical discourse as they reason and solve problems. These teachers give students opportunities to talk with one another, work together in solving problems and use both written and oral discourse to describe and discuss their mathematical thinking and understanding. As students talk and write about mathematics—as they explain their thinking—they deepen their mathematical understanding in powerful ways.

Meaning, teachers should facilitate student discourse to create and support a classroom-learning environment that values reasoning and sense-making from the *student’s point of view.*

##### Formative Student Learning

How you ask questions and facilitate discourse during your lesson has important implications for whether or not your mathematics instruction promotes a *formative* and differentiated learning culture built to develop student understanding.

For a mathematics lesson to be effective, the lesson *forms* the students’ thinking through a reflective process, with designed student *action*. You can do great checks for understanding from the front of the room, but the learning of a mathematics task also needs to be part of a more formative process of learning for every student. Otherwise, the moment a student gets stuck, he or she will shut down and stop listening during the lesson.

In 2009 and 2012 published works, John Hattie describes formative learning as part of a process of feedback to students with subsequent action on that feedback. The teacher and/or student peers provide the feedback during the working of a mathematical task. Accomplished teachers of mathematics understand it is best for students to experience the learning of mathematics through an environment where students are inspired to *embrace their mistakes* through the use of student-engaged explorations and discussions with peers.

As students make errors during the task, they receive feedback on those errors and take immediate action to correct their mistakes in reasoning. Students begin to view reflection and refinement of their in-class work as something they* do *in order to focus their energy and effort for future learning. A great place for this type of student reflection occurs during small-group discourse, as your students work together on various problems or tasks you provide, and you walk around the room evaluating student understanding based on what you see and hear. You provide meaningful feedback to students and student teams—and expect them to act on that feedback as you leave the team.

You and your students share the responsibility for successful implementation of in-class formative assessment practices. When your students can demonstrate understanding through reasoning mathematically *with one another*, they connect to the essential learning standard for the unit and can reflect on their individual progress toward the learning target of that day’s lesson. You support students’ lesson progress by using the immediate feedback during the daily classroom conversations and then expecting your students to act on the feedback provided.

##### Differentiated Student Learning

What is a differentiated response to learning during a mathematics lesson? It is a differentiated response for each mathematical task you present to the students (and not a different lesson for every child). The entry and exit points of each mathematical task you present to help students learn the mathematics standard for the lesson can vary based on the needs of each student team.

The use of small group student discourse feeds into differentiated instruction during the mathematics lesson. As you tour the classroom, see and hear student discussions, mistakes, and possible solution pathways, you do much more than listen. You use prepared prompts that can scaffold down an entry point into the mathematical task presented for the student teams that are “stuck” so to speak. These “un-stucking” or assessing prompts help students to persevere and enter into the mathematical task with their peers.

You utilize these assessing and advancing prompts that support student team access or entry points to the task (thus unsticking them), as well as prompts to help student teams completing the task to go a bit deeper. Differentiation in the mathematics lesson is differentiation on the entry points into the task for support or the exit point to advance student thinking.

Promoting productive small-group student discourse will allow your students to find their voice and actively engage. Students can self-assess how they are working together as you provide feedback to student teams on their ability to work together. During a lesson, students should reflect on their depth of *conceptual understanding. *As you plan your lesson design, it might help to ask the following questions with your teaching colleagues:

- How did my students demonstrate their understanding of the most essential concepts of the lesson through the mathematical tasks presented?
- How did my students reflect upon and engage with information, arrive at conclusions, and evaluate knowledge claims?
- What kinds of questions and conjectures were proposed (by my students) during the lesson?
- How did my students express their ideas, questions, insights, and difficulties?

Promoting and using small-group discourse throughout your lesson will foster student perseverance while maintaining rigor. It will authentically promote a community of student learners who see each other as valuable resources as they communicate about their ideas and it provides multiple opportunities for students to reason and make sense of the mathematics. Plus, it allows you, as the teacher, to monitor students’ initial responses to the mathematical tasks presented as part of the lesson.

As your mathematics lesson design is entering into the next decade, a primary purpose will be to actively engage all students in demonstrating evidence of meeting the essential learning target for that day. To honor this purpose, carefully consider the nature of the student discourse you will expect during the mathematics lesson.

As you facilitate small-group student conversations and support student listening, engagement, and learning when using a mathematical task, you will need to establish clear norms for how students are to behave and engage with each other in the task. I’ll cover those norms in an upcoming blog post.

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*Dr. Timothy D. Kanold **is an author of *Into Math*. Learn more about the HMH solution for K-8 students *here*.*