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Home arrow Mathematics arrow Modeling mathematical ideas: developing strategic competence in elementary and middle school


When deciding on a research goal for their lesson study, teacher teams decide on a content learning goal as well as select several of the core teaching practices that they want to focus on in their research lesson. This allows the teachers to set a personal teaching goal besides the student learning goal. In the Principles to Actions, NCTM (2014) calls for the mathematics education community to focus our attention on the essential core teaching practices that will yield the most effective mathematics teaching and learning for all students (see Text Box 2.1).

Over the years, mathematics education scholars have challenged the profession of teaching to identify a common set of high-leverage practices that yield effective teaching (Ball et al., 2009; Grossman, Hammerness, & McDonald, 2009; Lampert, 2009; McDonald, Kazemi, & Kavanagh, 2013). High-leverage practices have been defined as “those practices at the heart of the work of teaching that are most likely to affect student learning” (Ball & Forzani, 2010, p. 45).

Authors of the Principles to Actions responded to this challenge by defining eight core teaching practices (see Text Box 2.1) drawn from research that includes:

  • (1) Establishing Mathematics Goals to Focus Learning (Hiebert et al., 2007);
  • (2) Implementing Tasks That Promote Reasoning and Problem Solving (Stein et al., 2009); (3) Using and Connecting Mathematical Representations (Lesh, Post, & Behr, 1987); (4) Facilitating Meaningful Mathematical Discourse (Hufferd-Ackles, Fuson, & Sherin, 2004); (5) Posing Purposeful Questions (Chapin & O’Connor, 2007); (6) Building Procedural Fluency from Conceptual Understanding (Hiebert & Grouws, 2007); (7) Supporting Productive Struggle in Learning Mathematics (Dweck, 2008); and (8) Eliciting and Using Evidence of Student Thinking (Sherin & van Es, 2003). Because many of these eight practices are related and support one another, teacher teams often would state several of these practices as the focus of their professional learning goals. In the following chapters, we highlight several lesson vignettes that showcase how teachers implemented some of these eight Mathematics Teaching Practices outlined in the Principles to Actions during our lesson study.

Text Box 2.1 Core Teaching Practices from the Principles to Action, NCTM (2014)

Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.

Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.

Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.

Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

We have used the Japanese Lesson Study Professional Development model (Lewis, 2002; Lewis, Perry, & Hurd, 2004) where we immersed teachers in vertical teams, collaboratively planned, taught the first cycle, observed, debriefed, and taught the second cycle to come to reflect both individually and collectively. In doing so, we reveal the importance of establishing mathematics goals to focus learning, which we referred to as the math agenda during our research lessons. We call it the math agenda because it evokes a sense of immediacy as the word agenda literally means “the things that must be done” in Latin, and it communicates its importance in the planning process.

With that math agenda at the forefront, teachers determine what task can be Implemented so that we promote reasoning and problem solving. Through the use of skillful posing of purposeful questions, we can facilitate meaningful mathematical discourse where students are actively using and connecting mathematical representations as the teacher elicits and uses these representations and strategies as evidence of student thinking to move the math agenda forward. All eight practices are critical when enacting an exemplar problem-based lesson. Depending on the lesson, certain teaching moves may be more accentuated in some lessons than others; however, all eight of these practices ensure a rich math lesson and one should strive for employing all of these practices when planning math lessons.

In one of our past research lessons, the multi-grade team of teachers selected rich tasks to teach at multiple grade levels. The team met to plan their lesson objectives and anticipated student responses. While planning, teachers were asked to focus on these four assessment questions:

  • 1. What does a student at your grade level need to know or be able to do to access this problem?
  • 2. What specific lessons and strategies that you have used in the past will build on this problem?
  • 3. What might be problematic to the students that would require scaffolding in order for them to understand this topic?
  • 4. Develop an assessment task that would be appropriate to your grade level.

As part of the lesson study, teachers studied the mathematics content related to the research lesson, planned the lesson, taught the lesson in different grade levels, and debriefed after each cycle. In the following lesson study case, teachers decided to start the lesson study cycle (see Figure 2.2) in a third-grade classroom to see how younger

The Lesson Study Cycle (Lewis, 2002)

Figure 2.2 The Lesson Study Cycle (Lewis, 2002).

students approached the task. They observed as the third-grade host teacher taught the lesson and debriefed the lesson outcome in terms of the teaching moves and the student learning.

They collaboratively revised the lesson and geared it up for a sixth-grade class and an eighth-grade class. They decided to break up into two teams, observe the lesson, and then come together as a whole group. Teachers then met again for the second cycle of the teaching in the upper-grade classrooms and debriefed the lesson for the final cycle together as a group. Finally, they combined their learning from the research lessons from multiple grades to present at the final conference.

Our typical model for professional develop begins with a summer institute followed by Lesson Study during the academic semesters. There were a number of goals for these follow-up sessions: to provide teachers with continued support in learning and implementing algebraic content; to supply materials and strategies to the participants; to provide opportunities for vertical articulation between and among grades levels to share ideas/resources; and to analyze student learning and work samples.

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