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 a level 3 or 4. Similarly, if students already know the properties, procedure, or definitions at the heart of a discovery task, there would not be anything for them to discover. For example, the adapted Angles task can help students discover relationships between angles only if students do not already know those relationships.
Additionally, be aware of the implications of the wording of the task and how this can impact the work students produce. While a task may ask students to explain how or show your steps, there is a difference between explaining a procedure and explaining your thinking. For example, if a task requires students to solve fraction division problems such as ¾ ÷ ¼ using the traditional algorithm, asking students to explain might generate responses such as, “I used invert and multiply and computed ¾ ×
Aligning With Learning Goals
While we categorize procedural tasks at a level 2, note that there are many important mathematical procedures that students should be able to apply fluently and with automation after having established an appropriate level of conceptual understanding (NCTM, 2014). There are also appropriate occasions when you would use or assign a level 2 task to provide students the opportunity to practice or demonstrate their ability to perform a procedure, or a level 1 task when the goal is memorizing rules, properties, or definitions. Task levels (as well as mathematical content) should align with the goals for students’
