For those of us, ahem, “of a certain age,” our memories of math class consist primarily of a teacher-led lesson, a group practice problem or two, and then individual practice in the form of similar problems either on a worksheet or in a textbook. Student participation came mostly in the form of the dreaded “Come to the board and show the class how you got the answer,” which really translated to “Come up and reproduce the steps the teacher just demonstrated by plugging in numbers from a new problem.”
Math education has come a long way since then. In the late 1980s and early 1990s, publications like NCTM’s Curriculum and Evaluation Standards for School Mathematics and Professional Standards for Teaching Mathematics began promoting the value of communication and discourse in mathematics education. In 2000, NCTM named communication as one of the five process standards in their Principles and Standards for School Mathematics. Ten years later, the Common Core Math Standards included “Construct viable arguments and critique the reasoning of others” as one of the eight Standards for Mathematical Practice. Currently every state includes some language, typically in their process or practice standards, about the importance of being able to communicate mathematically.
Why all the talk about communication in math class?
Research has found that having students actively engage in rich dialogue in response to open-ended questions increases learning. Studies on math specifically have found that students who “spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, [and] talked more using longer responses” showed higher achievement than those who did not. A study focused on students with lower performance on standardized math tests examined the impact of having students actively explain and discuss their mathematical ideas. The results showed “how students not considered to have extensive mathematics knowledge can forge new connections between mathematical ideas and representations, and extend their problem-solving strategies in ways that are directly related to their participation.”
Engaging in real mathematical discussion supports learning in multiple ways:
- Explaining their thinking helps students clarify, and sometimes revise, their thinking.
- Reflecting on discourse can highlight both what understanding students have and what misconceptions they hold.
- Comparing and contrasting multiple solution paths improves procedural skill and flexibility.
- Sharing ideas regularly improves confidence and help all students see themselves as mathematicians.
The challenge for teachers
Once the importance of mathematical discourse to learning is clear, the next question becomes how to foster meaningful discussion in math class.
The National Academies’ Adding It Up: Helping Children Learn Mathematics nicely describes the challenging role teachers have as facilitators of such classroom discussions. They state that teachers “need to decide who should get the floor in whole-group discussions and how turns should be allocated. Teachers have responsibility for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their colleagues. The point of classroom discourse is to develop students’ understanding of key ideas. But it also provides opportunities to emphasize and model mathematical reasoning and problem solving and to enhance students’ disposition toward mathematics. Therefore, discourse needs to be planned with these goals in mind, not merely as a ‘checking for understanding’ form of recitation.”
Enter the 5 Practices for Orchestrating Productive Mathematics Discussions
So how does a teacher start fostering mathematical discussions that are not merely recitations of how you solved the problem and also advance students’ thinking? Margaret Smith and Mary Kay Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions provides a framework to help teachers do just this. The book “identifies a set of instructional practices that will help teachers achieve high-demand learning objectives by using the work and thinking of students as the launching point for discussions in which important mathematical ideas are brought to the surface, contradictions are exposed, and understandings are developed or consolidated.”
Smith and Stein provide teachers with the following five practices to help them ensure classroom discourse deepens mathematical learning:
- Anticipating. When planning your lesson and the associated tasks, think about what approaches and responses you expect to see from students, what understanding each of these represents, and what questions you might ask students to further their thinking.
- Monitoring. During the lesson, actively observe student responses in real time and ask questions that clarify and extend student thinking.
- Selecting. As you monitor, you are also identifying which students or group of students you want to have present their thinking to the whole group.
- Sequencing. Once you have selected the students or groups you want to speak, determine the ideal order in which to have students present their responses. The order should be designed to help students make connections, expand their thinking, and maximize mathematical understanding.
- Connecting. As students discuss the approaches being presented, ask questions to help them both uncover the relationships between the various approaches and ideas discussed and make connections to “big ideas” in mathematics.
The goal of using these practices is to come into a lesson and discussion as prepared as possible to guide the conversation toward attainment of the lesson goals and greater conceptual understanding of math. This framework ensures students are not just conversing for the sake of conversing, but that their dialogue is being purposefully sequenced and facilitated to enrich learning of both the lesson goals and math in general.
Getting the conversation rolling
There are two critical steps to take before implementing the five practices: defining your learning and selecting the appropriate task to meet the goal. Smith and Stein devote an entire chapter to these two steps, highlighting that, “Teachers often think of lessons in relation to the activities that they will ask students to do rather than what students will come to know and understand about mathematics as a result of having engaged in the lesson.”
Given the importance of these two steps, let’s dig into them, starting with defining your learning goal. The authors define the learning goal as one that, “clearly identifies what students are to know and understand about mathematics as a result of their engagement in the tasks that will be used in the lesson.” The learning goal should be a statement of mathematical understanding and is different from a performance goal, which tends to be more procedural. Here is an example of the difference between a learning goal and a performance goal for a lesson on understanding and modeling fractions:
- Learning goal. Students understand and can explain that a fraction represents partitioning a number line or a shape into equal parts and copying or iterating a number of those parts. So, 2/3 represents partitioning a number line or shape into three equal parts and taking two copies of 1/3.
- Performance goal. Students can name fractions represented by a given model and create a model to represent a fraction.
You can see from this example how much richer the learning goal is.
The next step is to identify a task or set of tasks that will support the learning goal. A well-defined learning goal makes it easier to determine the appropriate task.
The book includes a task analysis guide designed to help you determine the level of cognitive demand a task requires. It also provides different versions of the same task to illustrate what contributes to the complexity of a task and how teachers might modify a task to make it richer and more worthwhile. This could involve asking students to go beyond just calculating the answer to finding and generalizing patterns. Smith and Stein refer to this as “algebrafying,” borrowing a term from researchers Maria Blanton and James Kaput. Algebrafying involves “transforming problems with a single numerical answer to opportunities for pattern building, conjecturing, generalizing, and justifying mathematical facts and relationships.” The authors describe additional ways to modify problems, including asking students to represent and explain procedural tasks in multiple ways or giving them a procedural task and asking them to think of a real-world situation that it could describe.
Putting the five practices in action
The rest of the book delves into the five practices, and it is designed to maximize reader engagement.
Each chapter includes rich sample tasks, implementation narratives, transcripts of classroom discussions, and samples of student work. This is the book’s greatest strength. Through examples and narrative, the authors show how the practices play out in a classroom, rather than just telling you what to do. The chapters that detail the practices themselves include practical suggestions, called “Try this,” for applying the content from the chapter within your classroom. Sprinkled throughout each chapter are short active engagement prompts designed to encourage teachers to think about the material relative to their own classroom practice. These prompts may ask readers to try problems on their own or relate the content to a lesson or a classroom experience.
The appendices contain a variety of resources including:
- A list of web-based sources for tasks and lesson plans
- A completed sample lesson plan that incorporates the five practices
- A completed sample monitoring chart for one of the tasks given in the book
- A guide to help use the book as part of professional development or a professional learning community with discussion questions for each chapter
Planning for the big picture
For me, the power of the five-practice approach lies in the pre-thinking and planning it encourages teachers to engage in—and the fact that it pushes you to think beyond a narrow lesson goal toward a broader learning goal. When I taught, I often took the time to consider what misconceptions or common errors my students might produce. However, I did not think about follow-up questions I might ask based on these misconceptions or errors. I also did not think about which students should present, and in which order, to build from a certain type of thinking to one that led all students toward a deeper understanding of the learning goal and its connection to bigger ideas in mathematics.
The five practices provide a practical structure for making those instructional moves more intentional. Rather than leaving rich mathematical discussion to chance, they help teachers plan for conversations that uncover student thinking and make connections across strategies, representations, and important mathematical ideas. Frameworks like the one provided in 5 Practices for Orchestrating Productive Mathematics Discussions can help you transform discourse from a classroom activity into a powerful vehicle for learning math. Ultimately, the goal is not simply to get students talking more, but to engage them in the kinds of mathematical conversations that deepen understanding, build confidence, and help them all see themselves as capable mathematicians.
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