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

and sense making. For example, one can know and perform the procedure for dividing multidigit numbers but not be able to explain why you “bring the number down” or know when division applies to a contextual situation. For these reasons, more conceptual tasks (levels 3 and 4) align with goals and standards when students are beginning to develop an understanding of the mathematical topic, and procedural tasks may align better with goals and standards at the end of students’ learning trajectory of a particular mathematical topic, once they have developed their understanding.

      To increase the cognitive demand of a procedural task, use a context or representation that supports students to make sense of the operation and provides the need to develop a new strategy. To increase access, allow students to use multiple strategies or manipulatives to engage with the task. Consider the following task from Making Sense of Mathematics for Teaching Grades 3–5 (Dixon, Nolan, Adams, Tobias, & Barmoha, 2016):

      Brandon is sharing four cookies equally between himself and his four friends. Brandon wants to start by giving each person the largest intact piece of cookie possible so each person receives the same size piece of cookie to start. How might Brandon divide the cookies? (p. 73)

      While fifth-grade students might easily determine that four cookies shared among five people is ⅘ of a cookie per person, requiring the largest intact piece of cookie to be shared equally first provides a context for adding fractions with unlike denominators, such as ½ + ¼ + . Many other scenarios and contexts can support the need for students to make sense of adding equal-sized fractional parts.

      Another approach to increasing the cognitive demand of a task is to ask students to develop a new procedure based on prior knowledge before teaching the procedure to students. In this case, knowledge of equivalent fractions and adding fractions with like denominators is all students need to figure out how to add fractions with unlike denominators. Similarly, removing structure or directions that prescribe a strategy or direct students how to solve the task will also raise the task level. Beyond procedural tasks, this adaptation also applies to word problems or story problems at all grade levels and for any mathematical procedure. For example, if high school students have just learned the distance formula, and then are given word problems with the directions “Use the distance formula to solve the following problems,” the word problems no longer require thinking and reasoning, but only the application of a previously learned, prescribed procedure. This lowers the cognitive demand of the task.

      3. Level 1: Angles—Similar to the idea that level 2 (procedural) tasks are appropriate when the goal for students’ learning is practice or mastery of procedures, level 1 tasks are appropriate when the goal for student learning is recall and memorization. To encourage greater thinking and reasoning, allow students to discover relationships before providing the vocabulary, definitions, properties, postulates, or theorems. For example, before providing the definition and properties of vertical angles, have students measure a variety of vertical angles, conjecture that vertical angles are congruent (which also provides an opportunity to discuss measurement error), and then justify their conjecture using prior knowledge (for example, supplementary angles and the transitive property).

      Although the Potential of the Task rubric provides a comprehensive framework for rating and adapting mathematical tasks, certain factors may affect how you rate or select tasks for classroom use. We discuss these issues in the following section.

Considerations When Rating Tasks

      The awareness of different task levels and the ability to rate the level of tasks can equip teachers to be knowledgeable and critical consumers of published and online resources for the mathematics classroom. Published curricular materials often contain tasks at a variety of levels, and small changes to adapt the tasks in ways such as those identified in activity 1.4 can go a long way toward increasing students’ opportunities for thinking and reasoning. Instructional materials featured in online sites for teachers are frequently divided between resources that promote procedural practice and nonmathematical activity (for example, when the main activity is craft based rather than mathematical) and resources that provide ideas for conceptually based lessons, and teachers often have to adapt these resources for use in their own classrooms.

      As you begin to assess and rate tasks, there are several issues to consider in order to achieve both successful implementation in your classroom and enhanced thinking and understanding among your students. In this section, we will discuss the practical and conceptual issues that may stem from defining the task itself, considering the implications of higher-level thinking in practice, and aligning tasks with learning goals and standards.

Defining the Task

      Sometimes, identifying the task in curricular materials or other resources is not straightforward. In this book, we consider the task to be the mathematical problem or set of problems presented for students to do during a lesson or instructional activity. Tasks in curricular materials or as presented during a lesson may contain several parts. For example, each cell of figure 1.2 (page 13) is considered to be one task. The multiple parts of a task receive one collective rating according to the highest level of cognitive demand of any of the parts. For example, if a task (as presented on a handout or in verbal directions) includes vocabulary recall, a problem-solving activity, and an explanation, we would rate it a level 4. A task that supports students to develop or generalize a procedure and then spend time practicing that procedure would be considered a level 3.

      Some curricular materials will clearly identify a mathematical problem or set of problems for students to engage with during the lesson. For other materials, you may need to identify what the task is asking students to do mathematically. Sometimes, the teacher’s manual is necessary to understand exactly what students are being asked to do. This is particularly true for primary grades, in which the teacher often presents the task and directions verbally to align with students’ reading levels. When rating tasks in curricular materials or other resources, consider them as they appear in print. Any directions, manipulatives, representations, or resources indicated by the print materials, including teachers’ manuals, are part of the task.

      When using the Potential of the Task rubric (page 15), consider any directions (via textbook, teacher’s edition, handout, whiteboard, or screen) or resources provided to students. Most of the directions will occur before students begin their work on the task. However, the teacher may choose to give students part of a task, allowing them time to explore, and only then provide later parts of the task and additional time for students to continue working or developing their explanations. We would consider any mathematical problems that teachers ask students to do during the lesson as part of the task, even if the task directions are presented in parts throughout the lesson. In later chapters, we explore how a task unfolds throughout a lesson as students work on the task and engage in mathematical discussion, and so it is helpful to consider a lesson as having one main instructional task.

Considering Implications of Higher-Level Thinking

      When discussing the activities so far in this chapter, you and your collaborative team may have occasionally determined that the level of the task depends on the grade level or prior knowledge of the students. It is always important to consider how the students’ prior knowledge may impact the cognitive demand of a task. For the activities in the chapter, we instructed you to assume that the task was appropriate for a given group of students. In your school or classroom, you would be familiar with the grade level, standards, and students for a given task. If students have solved a series of very similar patterning tasks, problem-solving tasks, or other level 3 or 4 tasks, subsequent tasks in the series would not elicit the same level and type of thinking as the first. The task would likely become procedural (level 2), with students following a template provided by completing the first few tasks, even though the first task in the series would have been