Thinking about Instructional Routines in Mathematics Teaching and Learning

  • December 2018

    Routines are an essential part of mathematics classrooms because they give structure to time and interactions, letting students know what to expect in terms of participation, supporting classroom management and organization, and promoting productive classroom relationships for teaching and learning. Additionally, curriculum design and teacher planning are strongly influenced by considering which instructional routines will best support specific learning goals. Having clearly defined routines for interactions and discourse supports students’ engagement in mathematical practices and their learning of mathematics content.

    There are instructional routines known to support the development of mathematical proficiency that include conceptual understanding, strategic competence, adaptive reasoning, productive disposition, and procedural fluency. If the goal in mathematics teaching and learning is to support student success with mathematical proficiency, then we must be explicit about using instructional routines that focus on student engagement in activities that support reasoning and sensemaking, communication with and about mathematical ideas, making meaningful connections, building procedural fluency from conceptual understanding, and productive struggle.

    Observational studies dating from the 1950s through the early 2000s have documented that many mathematics classrooms employ a familiar instructional routine in which students are expected to mimic procedures demonstrated by the teacher. In this routine, students often take notes on the demonstrated procedure, and are then expected to apply by rote what was shown to them on a set of similar problems. Curriculum materials built around this routine structure questions to become progressively more difficult, but often the materials do not expect or encourage students to draw on their funds of knowledge gained outside the mathematics classroom. This instructional routine, common to many mathematical classrooms, is described by some as “initiation–response–evaluation (IRE).” It is a teacher-centered instructional routine with teacher-initiated explanations and questions, student responses to the teacher, and teacher evaluation of correctness. Little emphasis is placed on students explaining their thinking, working through mathematical ideas publicly, making conjectures, or coming to consensus about mathematical ideas as a community of mathematical thinkers.

    Related to the IRE approach is the “I do—we do—you do” instructional routine that is used across several content areas. “I do—we do—you do” is sometimes described as a gradual release of responsibility (GRR) model. Typically, the GRR model has three phases: “I do”—where the teacher demonstrates procedures before students attempt to solve problems on their own; “We do”—students are guided by the teacher to model the procedures demonstrated; and “You do”—where students practice the procedures demonstrated.

    Many administrators require that teachers use “I do—we do—you do” as an instructional routine in all subject areas. This requirement for mathematics teaching is challenging because it is not clearly understood how this instructional routine supports students’ development of the strands of mathematical proficiency and student engagement in the standards for mathematical practice. While “I do—we do—you do” might be effective for supporting proficiencies and practices in other content areas, I argue that “I do—we do—you do,” as practiced in many mathematics classrooms, focuses on doing processes and procedures with little understanding of how and why they work or the appropriate use of different processes and procedures and how they can be applied in varied mathematical situations. The focus is on mimicry and memorization rather than deep mathematical thinking and understanding, flexible use of mathematical concepts, communication of mathematical arguments and justifications, and developing a positive disposition that values connections between mathematics and students’ identities beyond the classroom. I think it is important that mathematics teachers use instructional routines that not only build procedural fluency through conceptual understanding but also support strategic competence, adaptive reasoning, and productive dispositions.

    An adaptation to the “I do—we do—you do” instructional routine that attempts to address this concern is “You do—we do—I do.” McCaffrey’s (2016) blog post Rethinking the Gradual Release of Responsibility Model explains “You do—we do—I do.” “You do”—the teacher gives students a task to see what students know and understand. The task should have multiple entry points, have varied solution paths, and focus on mathematical processes. The teacher monitors the classroom for strategies and asks probing questions. “We do”—after working on the task independently, students collaborate with peers in pairs or small groups. On the basis of the monitoring of the classroom while students worked independently, the teacher is purposeful in putting students in pairs or small groups. Additionally, this might be an opportunity to orchestrate a productive mathematical discussion. “I do”—the teacher engages in instruction, pulling together the mathematical ideas that arose during “you do” and “we do.” Also, the teacher’s instruction connects and deepens the mathematical understanding of students. “You do—we do—I do” provides opportunities for students to engage in the mathematical practices that deepen their understanding of mathematical content and the practices.

    Below are some NCTM resources focused on instructional routines that support the effective mathematics teaching practices as well as mathematical proficiency and the standards for mathematical practice.

    • Clark, Jancey. Transforming the Culture of Math: Routines for Making Thinking Visible. Online Journal of Teaching Children Mathematics: Oct. 23, 2017.
    • Marin, Katherine Ariemma. "Routinizing Mathematics Vocabulary: The Vocab Grid." Mathematics Teaching in the Middle School 23, no. 7 (2018): 395–398.
    • McCoy, Ann C., Joann Barnett, and Emily Combs. High-Yield Routines for Grades K–8. Reston, Va.: National Council of Teachers of Mathematics, 2013.
    • Newell, Christine, and Chase Orton. "Classroom Routines: An Invitation for Discourse." Teaching Children Mathematics 25, no. 2 (2018): 94–102.
    • Sliman, Emily. "Visible Thinking in High School Mathematics." The Mathematics Teacher 106, no. 7 (2013): 502–07. doi:10.5951/mathteacher.106.7.0502.
    • Sword, Sarah, Ryota Matsuura, Al Cuoco, Jane Kang, and Miriam Gates. "Leaning on Mathematical Habits of Mind." Mathematics Teacher 111, no. 4 (2018): 256–263.
    • Yeh, Cathery, Mark W. Ellis, and Carolee Hurtado. Reimagining the Mathematics Classroom: Creating and Sustaining Productive Learning Environments. Reston, Va.: National Council of Teachers of Mathematics, 2017.

    I encourage you to share instructional routines and resources that are supportive of effective mathematics teaching and learning. Then please share your reflections on

    Robert Q. Berry III
    NCTM President