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Introduction

Effective teaching practice can be learned.

—National Research Council

Improving instructional quality is an important aspect of teaching mathematics effectively, as instructional quality in mathematics has been associated with student achievement (Darling-Hammond, 2000). Our vision of quality mathematics instruction aligns with the effective mathematics teaching practices described by the National Council of Teachers of Mathematics (NCTM, 2014) in Principles to Actions: Ensuring Mathematical Success for All. According to NCTM (2014), “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” (p. 17). In this book, we provide you with a process and toolkit for improving instructional quality that are specific to mathematics teaching and learning, that identify aspects of instruction that impact students’ learning, and that provide data to influence the planning and teaching of future lessons.

The Importance of Making Sense of Mathematics for Teaching

Previous books in the Making Sense of Mathematics for Teaching series have focused on developing a deep understanding of important mathematical content at different grade bands. They provide opportunities for mathematics teachers in grades K–2, 3–5, 6–8, and high school to engage deeply with the mathematics they teach. A central premise in each of these books is that you learn about the mathematics you teach by doing the mathematics you teach (Nolan, Dixon, Roy, & Andreasen, 2016).

In this book, we focus on the opportunities for thinking and reasoning embedded in mathematical tasks at all grade levels and present an instructional assessment framework to determine whether those opportunities are actualized during instruction. Just as the previous books in the Making Sense of Mathematics for Teaching series promote mathematical learning by asking readers to do mathematics, this book engages teachers in reflecting on mathematics instruction through the activities within each chapter. Throughout the book, we ask readers to solve mathematical tasks. In each of those instances, readers should take time to solve the tasks and discuss their solution strategies within their collaborative teams. This will provide greater insight when they later analyze the task or the lesson featuring the task.

About This Book

Making Sense of Mathematics for Teaching to Inform Instructional Quality provides teachers and teacher leaders with the opportunity to reflect on the quality of mathematics instruction at any grade level. As with previous books in the Making Sense of Mathematics for Teaching series, this book highlights the use of tasks, questions, and evidence (referred to as the TQE process; see figure Part 1 engages teachers in considering the quality of tasks and task implementation (connecting to the T in the TQE process). Part 2 supports teachers as they explore the quality and impact of their questions and other discourse actions (the Q in the TQE process). Part 3 guides educators to examine the evidence of their students’ thinking and participation in the classroom community (the E in the TQE process).

Source: Dixon, Nolan, & Adams, 2016, p. 4.

Figure I.1: The TQE process.

Throughout the book, we present a set of rubrics—the Instructional Quality Assessment (IQA) Mathematics Toolkit (appendix A, page 127, offers reproducible versions)—as a framework to focus reflections, conversations, feedback, and the planning and teaching of mathematics. The IQA rubrics provide a set of instructional practices and detailed descriptions of levels of quality within those practices. The Mathematical Tasks Framework and Levels of Cognitive Demand (Stein, Smith, Henningsen, & Silver, 2009) served as the foundation of the IQA. The set of IQA rubrics follow the progression of a mathematical task throughout a lesson as teachers engage students with the task, pose questions, orchestrate mathematical discussions, and collect evidence of students’ learning. The IQA rubrics, and levels of cognitive demand within each rubric, form the core of this book and provide a way for teachers to focus on their instructional practice over time. Using the IQA rubrics, you will be able to identify instructional practices that support students’ learning, areas for growth and improvement, and pathways for promoting that growth and improvement.

We encourage you to use Making Sense of Mathematics for Teaching to Inform Instructional Quality within a collaborative teacher team, which we define as at least two people with similar goals for improving the quality of mathematics instruction. In each chapter we encourage you to engage in activities, discuss your ideas about those activities together, read our analysis of the ideas in the activities, relate those ideas to the IQA rubrics, and apply the ideas and rubrics to your own mathematics classrooms.

Figure I.2: Play button icon.

Figure I.3: Task icon.

Throughout this book, we will use different icons to call your attention to various tasks to think about or perform. The play button icon, shown in figure I.2, indicates that an online video depicting a lesson is available for you to watch. You can find the videos either by scanning the adjacent QR code or by following the provided URL. (For a full list of videos and figures used in this book, see appendix E, page 143.)

The task icon, in figure I.3, highlights academic tasks to perform or problems featured in the videos. The tasks and lessons throughout the book represent tasks from a range of grade levels in elementary, middle, and high school. Regardless of the grade level or levels you teach, we have discovered that all teachers find value in exploring tasks and lessons across a variety of grade bands in order to illuminate the features of quality instruction and students’ thinking.

Throughout the book, we ask you, as educators and collaborative teams, to focus on instructional quality by observing teaching. Sometimes we show this teaching through provided videos; at other times, we ask you to observe the teaching of members of your collaborative team. You may choose to accomplish this through live observations, by video recording the lessons, or by a combination of the two—whichever fits best within your individual contexts. Throughout the chapters, the IQA rubrics provide a helpful tool for teacher peers to both provide and receive feedback on the quality of your instructional practice—a highly important outcome allowing for more targeted and content-specific feedback than what is more frequently received from administrators, who may or may not have expertise in teaching mathematics (Darling-Hammond, 2014/2015).

Each chapter begins with introductory activities that engage you and your collaborative team with the key ideas in the chapter before we formally introduce the related IQA rubric or rubrics. Once you have an understanding of the rubric, we provide application activities to allow you and your team to practice using it and further reflect on your instruction. We encourage you to engage in the activities and discuss ideas with your collaborative team before moving on to the discussion following each activity. In each chapter, we provide resources that you may want to view or print as you complete the activities. These activities, materials, and videos are key to supporting your journey as you begin to reflect on mathematics instruction. To connect to practice, each chapter closes with a transition activity that applies the ideas in the chapter to the mathematics classroom and is then revisited in subsequent chapters. We close the book by providing you with the opportunity to use the entire IQA Toolkit to reflect on instruction and consider how to use IQA data to improve instruction.

We challenge you to reflect deeply as you explore one of the most influential characteristics related to student achievement—the quality of instruction.