Thinking about Instructional Routines in Mathematics Teaching and Learning
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.
I encourage you to share instructional routines and resources that are supportive of effective mathematics teaching and learning. Then please share your reflections on MyNCTM.org.
Robert Q. Berry III
For example, students who encounter complex and challenging situations using rules of thumb (i.e., using linear, circular, quadratic and logarithmic equations with nonlinearities) are likely to exhibit lower levels of engagement than those students who encounter complex or demanding situations using rules of thumb (i.e., using logarithmic, square, or exponential equations in logarithmic and/or linear form). Additionally, rules of thumb can be difficult to learn once students are accustomed to them, making them less attractive as learning aids, according to another study (Stroessner and Fenton, 2014: 31). Students also have difficulty with complex mathematics problems once they have established how they wish to solve them (https://oddstwister.com).A key factor for good curriculum design and schoolwide teaching is encouraging students to feel good about participating in and making decisions related to their learning
Amy Lucenta and I have developed a set of instructional routines specifically designed to develop the mathematical rpactices in all students. These "Routines for Reasoning" have been intentionally designed to ensure ALL students have access to the mathematical thinking, reasoning, and communicating. Essential stratigies that ensure access for English learners and students with learning disabilities include ask-yourself questions, sentence frames and starters, annotation, and the Four 4's. For more on these routines check out www.fosteringmathpractices.com
Grace, Thank you for sharing. I have used your work with my students.
I would like to leave this for discussion. Learned in Student Teaching and sub experience.
1. Warmup- with a partner if necessary.
2. Students or teacher answer warmup and homework questions.
3. Guided Practice using presentation software and socratic approach, inviting participation. No Talking, I was tolerant of real whispering if they were on-task.
4. Homework Time. Both of my Master Teachers preferred to leave ample time in class for students to make significant progress or complete the homework in class. One master teacher even had a "Music Monitor" playing rock music on a desktop computer. Partner work was allowed in both environments.
David, Thank you for sharing your instructional routine. As move forward professionally, I would invite you to incorporate more routines into your teaching practice.
From your mouth to everyone's ears.
My only quibble is that I'm not so sure the issue is any different in other content areas. Children in literacy would also benefit from being able to solve problems independently in their own reading and writing rather than mimicking the teacher.
The sad, little identical paragraphs (topic sentence, detail, detail, detail, conclusion) that come home with my elementary school daughter every week just make me sad.
Amy, I appreciate your feedback. I am not knowledgable about GRR in other content areas. While I have some expertise in mathematics education, I did not want to give the impression that I have a broad knowledge base in other content areas. When my children were younger, I shared similar thoughts as you regarding writing.
I have worked as an instructional coach with both administrators and math teachers to secure an understanding of instructional routines which support a problem-based classroom. As you have alluded to, suggesting that an "I do-we do- you do" approach removes problem-solving from a math class must be accompanied by alternate routines. I have used Van de Walle's Launch-Explore-Summary model where the summary is generated from student work as described in the five practices model from Smith, Hughes, Engle, and Stein.
However, knowing that an "I do-we-do-you-do" instructional model is very effective in a literacy class, I have been supporting my math teachers on how to use this when teaching students how to write strong justifying statements associated with their mathematics. Knowing when to use a math-based routine and when to use a literacy-based routine in a math class is a complex endeavor which requires a great deal of coaching support for most teachers.
Ken, I am a big fan of Van de Walle and Smith et al's work. I am interested in situations in which you use "I do-we do-you do" in mathematics.
I love that you are bringing up the topic of instructional routines! I agree that the I-R-E lesson strategy that many people use is insufficient to meet the students' need to learn mathematical processes (and it often fails at its intended goal of students learning simple mathematical processes as well!).
It is helpful for people, who may be unfamiliar to alternative instructional routines, to have examples of instructional routines to draw on. Almost all of us were taught using I-R-E so it is hard for us to imagine a different mathematics classroom. This is why the non-profit organization I currently consult for created these videos of instructional routines in action: https://curriculum.newvisions.org/math/course/getting-started/instructional-routines/
In the workshops we run we start with teachers playing the role of students in the routine so that they can get some experience and empathy for what the routines feel like from the student perspective before unpacking the routines themselves as teachers. So far most teachers who experience the routines, unpack them, and rehearse them with other teachers (in our workshops) have at least tried the routines in their own classrooms.
It turns out that just knowing the steps to the routines themselves is not enough, there are all sorts of complications and decisions that have to be made when you bring student ideas to the forefront of the classroom experience that many teachers have little experience making. I'm currently trying to figure out how to better support teachers who have very little experience orchestrating classroom discussions with being more prepared when they try the routines with their students.
David, Thank you for sharing the New Visions for Public Schools video. Having access to resources supporting mathematics teaching strengthens the profession. I really like the idea of teachers rehearsing instructional routines. I use video resources from NCTM's Principles to Actions toolkit as examples for orchestrating classroom discussions.