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.
PART 1
Connecting to the T in TQE: Tasks and Task Implementation
In this book, you will analyze and reflect on teaching mathematics at each stage of the TQE process using the Instructional Quality Assessment in Mathematics Toolkit rubrics. Part 1 connects to the T in the TQE process: “Tasks: Select tasks that support identified learning goals” (Dixon, Nolan, & Adams, 2016, p. 4). Implementing tasks that elicit thinking and reasoning can increase all students’ access to high-quality mathematics. Throughout chapters 1 and 2, we highlight features of tasks and instruction in mathematics classrooms that promote access for all learners.
As you explore chapter 1, you will consider the impact of different types of tasks on students’ learning of mathematics. In your work as a mathematics teacher (or with mathematics teachers), you have encountered many mathematical tasks—problems, exercises, homework sets, examples, activities, and so on—some of which have been interesting and challenging, and some of which have been routine and procedural. In this book, we use the term mathematical task to describe a problem or a set of problems that address a similar mathematical idea (Stein et al., 2009). A task can consist of a simple one-step problem, a complex multipart problem, or a series of related problems. Different types of tasks have different potential for engaging students in rigorous mathematics. We introduce the IQA Potential of the Task rubric in chapter 1 to provide a structure for analyzing the level and type of thinking a mathematical task might elicit from students.
Have you ever experienced a mathematics lesson in which you thought the task seemed simple, but surprisingly elicited much greater interest, thinking, and engagement than you anticipated? Conversely, have you experienced a mathematics lesson in which you anticipated the task to be interesting and engaging, but it some-how fell short of eliciting students’ mathematical thinking and reasoning? The IQA Implementation of the Task rubric, introduced in chapter 2, will provide a structure for analyzing how instructional tasks play out during mathematics lessons.
CHAPTER 1
