Indicator C.2.3. Implement Effective Instruction
Well-prepared beginning teachers of mathematics use a core set of pedagogical practices that are effective for developing students’ meaningful learning of mathematics.
Teachers must not only understand the mathematics they are expected to teach (Ball, Thames, & Phelps, 2008) and understand how students learn that mathematics (Fuson, Kalchman, & Bransford, 2005), they must be skilled in using content-focused instructional pedagogies to advance the mathematics learning of each and every student (Forzani, 2014). Well-prepared beginning teachers of mathematics have begun to develop skillful use of a core set of effective teaching practices, such as those described in Principles to Actions (NCTM, 2014a) and listed in Table 2.2 below.
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.
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.
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.
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.
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.
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.
Note. Reprinted from Principles to Actions: Ensuring Mathematical Success for All (p. 10) by the National Council of Teachers of Mathematics, 2014. Reston, VA: NCTM. Copyright 2014 by National Council of Teachers of Mathematics. Used with permission.
Well-prepared beginners enter classrooms with commitment to, and initial skills for, enacting effective mathematics instruction. They can identify mathematics learning goals for lessons, articulate how those goals relate to the selected tasks, and use the goals to guide their instructional decisions throughout a lesson. They can distinguish between high-level and low-level tasks on the basis of the cognitive demand for their students. They are growing in their abilities to implement high-level tasks that promote reasoning and problem solving with students and to engage students in meaningful mathematical discourse on those tasks, without lowering the cognitive demand or taking over the thinking and reasoning of students (Stein, Grover, & Henningsen, 1996; Stigler & Hiebert, 2004). For example, well-prepared beginners provide opportunities for students to work together on mathematical tasks and then engage students in whole-class discussions to share, compare, and analyze student strategies and solutions. They endeavor to position students as authors of ideas—students who discuss, explain, and justify their reasoning using varied representations and tools. They also know that the purpose for whole-class discussions is not show-and-tell but rather an intentional discussion of selected and sequenced student approaches and use of mathematical representations and tools to move students through a trajectory of sophistication toward the intended mathematics learning goal of the lesson. This skillful orchestration of student interactions in whole-class discussions is complex and takes years for teachers to fully develop. Furthermore, although well-prepared beginners can see key mathematical ideas within students’ representations, the accomplished teacher can weave a mathematical idea across many representations (e.g., proportional relationships shown with discrete objects, tables, tape diagrams, double number lines) in ways that help students see connections among representations and see affordances of different representations. Well-prepared beginners have clear visions of effective mathematics instruction centered on mathematical discourse and sense making; they begin with strong foundations of initial capabilities upon which to build that vision into their classrooms.
Effective teaching requires attending to students’ mathematical thinking and reasoning during instruction. Well-prepared beginners commit themselves to noticing, eliciting, and using student thinking to assess student progress in understanding the mathematics and to adjust instruction in ways that further support and advance learning toward the intended learning goals. For example, well-prepared beginners assess students’ understanding and reasoning at multiple points throughout a lesson. This assessment might include posing purposeful questions to gather information or probe students’ understandings while they work individually or in small groups or asking students to respond to a prompt on their whiteboards during a lesson.