Making Sense of Mathematics for Teaching to Inform Instructional Quality. Juli K. Dixon

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

      Potential of the Task

      [There is] no decision teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of tasks with which the teacher engages students in studying mathematics.

      —Glenda Lappan and Diane Briars

      Why is it important to assess the cognitive potential of instructional tasks? First, the consistent use of high-level instructional tasks has been shown to enhance students’ mathematical learning in elementary (Schoenfeld, 2002), middle (Cai et al., 2013), and high school mathematics classrooms (Grouws et al., 2013). Second, different types of tasks provide different types of opportunities for mathematical thinking and reasoning (Stein et al., 2009). Being aware of both the type of thinking a task can elicit and the type of access a task can give to all students can support you to align tasks with learning goals, and to ensure that students receive opportunities for thinking and reasoning. Finally, research has also shown that the level of the task sets the ceiling for the mathematical thinking, reasoning, and discussion that occurs throughout a lesson, and if a task does not request a representation, explanation, or justification, students typically do not produce or provide these things during a lesson (Boston & Wilhelm, 2015). Therefore, we find it critical for teachers seeking to improve their instructional practice to begin by considering the tasks and problems they are assigning in their classrooms and how these tasks may enable—or inhibit—student thinking.

      What do you look for when selecting tasks? What makes a “good” instructional task?

      In this chapter, you will explore why high-quality tasks are an essential first step in teaching mathematics for understanding. At the conclusion of this chapter, you will be able to answer the following questions.

      ■ How do different types of tasks elicit different opportunities to learn mathematics?

      ■ What types of tasks am I using to engage each and every student in learning mathematics?

Introductory Activities

      Let’s get started by thinking about different types of mathematical tasks. Activities 1.1 and 1.2 ask you and your collaborative team to solve a variety of mathematical tasks and consider the thinking and problem-solving strategies that each task might elicit.

Activity 1.1: Solving a Task

      It is valuable to engage with tasks as learners prior to implementing them as teachers. Be sure to devote attention to this experience. Explore the task on your own before discussing your experience with others.

Engage

      Solve the Leftover Pizza task in figure 1.1. Do not use any procedures or algorithms. Try to solve the task in more than one way, using diagrams or other representations, including in ways students might correctly or incorrectly solve this task.

      Source: Nolan, Dixon, Roy, & Andreasen, 2016.

Figure 1.1: The Leftover Pizza task (grade 6).

      Respond to the following questions.

      ■ What strategies and types of thinking can this task elicit?

      ■ What are the main mathematical ideas that this task addresses?

      ■ How do teachers typically present the mathematical ideas addressed in this task to students? What types of tasks do teachers typically use to present these mathematical ideas to students? What is different about this task?

      ■ How might this task provide access for each and every learner?

      Compare your work and ideas in your collaborative team before moving on to the activity 1.1 discussion.