Pages: 13

The Teaching for Robust Understanding (TRU) Framework identifies five key dimensions of equitable and ambitious instruction. When things go well along these five dimensions, students become knowledgeable and resourceful thinkers and problem-solvers and develop positive mathematical identities. We present the TRU targets, the main focus of this book. The outer rings of the targets highlight the properties of tasks that are OK, but that could be modified to offer more for students to engage with. The middle rings show that a task is rich but still might be enhanced. If a task lands near the center of the targets, students can learn a great deal from it. This book is about using the targets to identify opportunities to make tasks and activities richer. Here we preview the general idea, taking a typical task and modifying it to provide much richer opportunities for students to engage fully and equitably with the mathematics. This sets the stage for the use of the targets in general.

Pages: 15

This chapter takes readers through the task improvement process. It shows how we can start with a typical textbook or curricular task, and use the TRU targets to improve it. With a math target, we explore ways in which the mathematics in the task could be enriched and modify the task accordingly. With a cognitive demand target, we explore ways in which students can dig into the task and engage in productive sense making. This too may result in task modifications. Similarly, with an equitable access target we try to make sure the task is meaningfully accessible to all students; with an agency, ownership, and identity target we look for ways the students can relate to the task and make it their own; and with a formative assessment target, we increase the likelihood that there will be rich and powerful conversations about the task.

Pages: 20

This chapter provides a “deep dive” into the first dimension of the TRU Framework, the Mathematics, and the mathematics targets. We begin by discussing the importance of the mathematics in the classroom environment and provide some history that helps understand where we are today. We then introduce the mathematics targets, showing how they can be used to make the mathematics in a task much richer. Finally, we take three tasks, one each at the elementary, middle, and secondary grade levels. For each task we work through the targets, showing how the targets can be used to make the task much more rich mathematically.

Pages: 20

This chapter provides a “deep dive” into the second dimension of the TRU Framework, Cognitive Demand, and the Cognitive Demand targets. We begin by discussing the concept of cognitive demand, which is related to “productive struggle.” If the work they are assigned is too easy or too rote, students are not challenged and don’t learn very much; but if the work isn’t within reach, they can’t engage in sense-making. The goal is to hit the “sweet spot” where students are working productively to put ideas together for themselves. We then introduce the cognitive demand targets, showing how they can be used to raise questions about the tasks and activities we use and make tasks meaningfully accessible and challenging to more students. Finally, we take three tasks, one each at the elementary, middle, and secondary grade levels. For each task we work through the targets, showing how the targets can be used to make the task more accessible and amenable to sense-making.

Pages: 24

This chapter provides a “deep dive” into the third dimension of the TRU Framework, Equitable Access to Mathematics, and the Equitable Access targets. We begin by discussing the concept of equitable access. Access to what? Every student must be able to engage productively with meaningful and powerful mathematics. But the issues are subtle. Who participates in classroom activities and in what ways? Do all students have opportunities to engage with important content and contribute to collective discussions? How is diversity valued, and how are students’ widely divergent backgrounds and experiences leveraged to create rich learning opportunities for all students? How are classroom tasks and activities constructed, so that as many students as possible can contribute meaningfully? We then introduce the equitable access targets, showing how they can be used to raise questions about the tasks and activities we use, and make tasks more accessible and meaningful to more students. We then take three tasks, one each at the elementary, middle, and secondary grade levels. For each task we work through the targets, showing how the targets can be used to open up the task so that more students can engage with it productively.

Pages: 24

This chapter provides a “deep dive” into the fourth dimension of the TRU Framework, Agency, Ownership, and Identity (AOI), and the AOI targets. A student’s mathematical agency involves participating actively in mathematical conversations, taking intellectual risks, and sharing ideas. It’s a willingness to take initiative in mathematics. A student’s ownership of mathematics comes from that students working through ideas, individually or collectively, and making those ideas their own – the sense that “I (or we) figured this out; it makes sense to me; it’s not simply what ‘they’ told me is true.” A student’s mathematical identity refers to who they see themself to be in relation to mathematics. A student with a positive mathematics identity is likely to think, “I like math; it makes sense; and I can figure things out.” Math is part of themself, in and out of school. We introduce the AOI targets, showing how they can be used to raise questions about the tasks and activities we use and provide increased opportunities for students to understand and make progress on tasks and to contribute to classroom conversations in ways that enable them to see themselves as productive mathematical thinkers. We then take three tasks, one each at the elementary, middle, and secondary grade levels. For each task we work through the targets, showing how the targets can be used to provide more opportunities for students to engage with it in ways that support their agency, ownership, and identity.

Pages: 23

This chapter provides a “deep dive” into the fifth dimension of the TRU Framework, Formative Assessment, and the Formative Assessment targets. The classic definition of formative assessment is “all those activities undertaken by teachers, and by their students in assessing themselves, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged. Such assessment becomes “formative assessment” when the evidence is actually used to adapt the teaching work to meet the needs of students”. We discuss the challenges of anticipating student thinking, crafting tasks that get students to reveal their thinking, and crafting a learning environment that responds to their evolving understandings productively. We introduce the Formative Assessment targets, showing how they can be used to raise questions about the tasks and activities we use, to help reveal student thinking and move it forward. We then take three tasks, one each at the elementary, middle, and secondary grade levels. For each task we work through the targets, showing how the targets can be used to provide more opportunities for students to reveal their thinking and to engage with the task in ways that enhance their learning.