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

have the opportunity to engage in high-level thinking, teachers must regularly select and implement tasks that promote reasoning and problem solving.

      —National Council of Teachers of Mathematics

      In this chapter, you will explore how teachers can implement mathematical tasks during mathematics lessons in ways that support or possibly diminish students’ opportunities to engage in thinking and reasoning. By the end of the chapter, you will be able to answer the following questions.

      ■ What happens when teachers enact high-level tasks in the classroom? How do teachers maintain (or limit) opportunities for thinking and reasoning during the lesson?

      ■ What teacher actions seem to support (or diminish) students’ opportunities for thinking and reasoning as well as provide (or take away) access for students to engage in the task?

      ■ What does student work indicate about students’ engagement in thinking and reasoning during the lesson and their level of access to the lesson?

      ■ What does students’ work indicate about instruction and teaching during the lesson?

Introductory Activities

      Let’s get started by analyzing mathematics lessons and observing how the implementation of a task can affect student learning. In activity 2.1, we ask you and your collaborative team to observe two mathematics lessons that use the same task, but in noticeably different ways. You will then consider the thinking and problem-solving strategies that each teacher’s implementation elicited.

Activity 2.1: Comparing Two Mathematics Lessons

      Before engaging in this activity, consider what opportunities for mathematical learning the Leftover Pizza task could offer students and how you might implement the task in the classroom.

       Engage

      In chapter 1, activity 1.1 (page 7), you considered the level of thinking and reasoning that could potentially be provided by the Leftover Pizza task for students in grade 6. In activity 2.1, you will explore how the task plays out in two different lessons.

      ■ Revisit the Leftover Pizza task (see figure 1.1, page 8).

       Rate this task using the Potential of the Task rubric (see figure 1.4, page 15).

       Be sure to include your rationale—what characteristics or features influenced your rating of the task? Discuss the Potential of the Task level with your collaborative team before moving on.

      ■ Watch the video of the Leftover Pizza lesson version 1.

       Take notes regarding what you notice in the Leftover Pizza lesson version 1. Write down timestamps from the lesson so that you can readily refer to what you noticed when you discuss the lesson within your collaborative team. Throughout the book, we use timestamps to identify particular features of video lessons.

       How would you describe students’ thinking and reasoning during the lesson? Include examples and timestamps.

      ■ Read the transcript of the Leftover Pizza lesson version 2 (see figure 2.1).

       What do you notice about the Leftover Pizza lesson version 2?

       How would you describe students’ thinking and reasoning in the narrative?

       How are the students’ thinking and reasoning the same and how are they different between the two Leftover Pizza lessons?

      Discuss your responses with your collaborative team before moving on to the activity 2.1 discussion.

Leftover Pizza Lesson Version 1: www.SolutionTree.com/Dividing_Fractions_in_Context

Figure 2.1: Leftover Pizza lesson version 2.

       Discuss

      How do your responses compare with those in your collaborative team? What themes emerged during your discussion? In this section, we present ideas for you to consider.

       Rate the Leftover Pizza task using the Potential of the Task rubric.

      The Leftover Pizza task would rate a level 3 on the Potential of the Task rubric. As discussed in activity 1.1, the task provides a context and encourages a model for students to make sense of dividing fractions. The task does not include an explicit prompt to explain.