The entire Illustrative Mathematics team spends a lot of time reading about teaching and learning. Most recently, we have been reading—some of us rereading—and reflecting on the 5 Practices for Orchestrating Productive Mathematics Discussions by Mary Kay Stein and Margaret Schwan Smith. Members of the team were asked to reflect on the following two questions to share with the Illustrative Mathematics community:
- What idea stood out to you when reading the 5 Practices for Orchestrating Productive Mathematics Discussions?
- Why do you feel this idea is important?
Quote: Introduction: “The key is to maintain the right balance. Too much focus on accountability can undermine students’ authority and sense making and, unwittingly, encourage increased reliance on teacher direction. Students quickly get the message—often from subtle cues—that “knowing mathematics” means using only those strategies that have been validated by the teacher or textbook; correspondingly, they learn not to use or trust their own reasoning. Too much focus on student authorship, on the other hand, leads to classroom discussions that are free-for-alls.” (p. 2)
Importance: The idea of balance is so important when thinking about the different teacher moves in the classroom and what they communicate to students about learning mathematics. We need to give students permission to make mistakes, and have ownership of their reasoning, while also being mindful of the discussion focus. As students own their learning, it prepares them to independently problem solve, maintain authority in their learning and avoid relying on the teacher as the source of direction in the math classroom.
Quote: Introduction: It is the teachers’ responsibility to move students collectively toward, and hold them accountable for, the development of a set of ideas and processes that are central to the discipline—those that are widely accepted as worthwhile and important in mathematics as well as necessary for students’ future learning of mathematics in school. (p. 2)
Importance: Productive mathematical discussions encourage students to play an active role in the learning process. In such classrooms, the teacher’s role may have to shift from directly transmitting information to guiding the discussion and synthesizing to achieve the learning goal set for the lesson.
Vice President, Portfolio Management
Quote: Chapter 1: “Many teachers are daunted by an approach to pedagogy that builds on student thinking. Some are worried about content coverage, asking, ‘How can I be assured that students will learn what I am responsible for teaching if I don’t march through the material and tell them everything they need to know?’” (p. 7)
Importance: This resonated with me from a teacher’s perspective, and made me realize that teachers face many challenges and feelings of being overwhelmed by the pressure of covering everything in their curriculum. Teachers have a constant sense of responsibility and want to do what is best, but it can be daunting. Teachers need, and welcome, tools that are realistic and relevant to what they are doing in the classroom each day.
Algebra 1 Lead
Quote: Chapter 4 (re: planning the questions to ask in response to students’ work): “Because questions are bound to the context in which they are asked, it is critical to pose a question that makes a connection with issues that are currently being addressed. Developing questions only ‘in the moment’ is very challenging for a teacher who is juggling the needs of a classroom full of learners who need different types and levels of assistance. When teachers feel overwhelmed […] it is easy for them to revert to just telling students what to do when an alternative course of action does not immediately come to mind.” (p. 36)
Importance: The process of making mathematical ideas visible and advancing students’ thinking takes considerable thought. The better a teacher could plan for possible responses and their connections to the mathematical aims, the better prepared he or she would be to move students’ understanding toward those aims. The demands on a teacher’s attention during class time are numerous. Planning the questions in advance gives the teacher a greater bandwidth to attend to students’ thinking and make on-the-spot decisions about unanticipated responses.
PD Content Developer
Quote: Chapter 4: “By first anticipating the wide range of things that a student might do (and identifying which of those might be mathematically useful in achieving the lesson’s goals), a teacher is in a better position to recognize and understand what students actually do. Teachers who have engaged in this kind of anticipation and prediction can then use their understanding of student work to make instructional decisions that will advance the mathematical understanding of the class as a whole.” (p. 42)
Importance: It is important to think about what it means to advance a student’s mathematical understanding. We often think of moving thinking in terms of going from least efficient to most efficient, but even a student using the most efficient strategy can be moved forward in their thinking by understanding, or better understanding, the concrete underpinnings of their strategy.
Algebra 2 Lead
Quote: Chapter 5: “Selecting is the process of determining which ideas (what) and students (who) the teacher will focus on during the discussion. This is a crucial decision, since it determines what ideas students will have the opportunity to grapple with and ultimately to learn. Selecting can be thought of as the act of purposefully determining what mathematics students will have access to—beyond what they were able to consider individually or in small groups—in building their mathematical understanding.” (p. 44)
Importance: When considering the work of teachers, this is an example of the challenging and intellectual rewarding part that is all too often misunderstood. When we put student voices and ideas front and center, we are not abdicating our responsibilities as architects of our students’ learning experience. Instead, we are ensuring our students see new tools and ideas in an order that helps them build a robust understanding.
Lesson Writing Team Lead
Quote: Chapter 5: “Selecting is critical because it gives the teacher control over what the whole class will discuss, ensuring that the mathematics that is at the heart of the lesson actually gets on the table.” (p. 44)
Importance: In traditional, direct instruction classes, teachers take control over the delivery of mathematics and give up control when discussing the lesson. Calling only on students who voluntarily raise their hand or accepting shouted answers leaves the discussion open to many directions and does not provide any insight into the variety of understanding in the classroom. In a problem-based class, students take control of doing the math during their time for exploration, and the teacher can then control the flow of responses during the discussion by asking selected students to share their approaches in a sequential way.
High School Professional Development Lead
Quote: Chapter 5 (Determining the Direction of the Discussion: Selecting, Sequencing, and Connecting Students’ Responses): “Selecting is critical because it gives the teacher control over what the whole class will discuss, ensuring that the mathematics that is at the heart of the lesson actually gets on the table. We have come to think of the question, ‘Who wants to present next?’ as either the bravest or most naïve invitation that can be issued in the classroom. By asking for volunteers to present, teachers relinquish control over the conversation and leave themselves—and their students—at the mercy of the student whom they have placed at center stage.” (p. 44)
Importance: Students recognize that we aren’t just interested in hearing from the students who are “first and fast,” but from all students. They bravely learn to share their ideas when they are selected because they know that our discussion isn’t built around volunteers who raise their hand to share. Everyone has something to contribute and will be called on to do so when it connects to the mathematicals goals.
- Click here to learn more about professional development of the 5 Practices for your classroom or school.