students actually learn mathematics.
Tasks in row 4 contain all of the features of tasks in row 3, with the added feature that the task directions explicitly require students to provide an explanation or justification. In addition to completing the mathematics necessary to solve the task, the task includes a prompt for students to reflect on, explain, or justify some aspect of their work.
Source: Questions adapted from Dixon, Nolan, Adams, Tobias, & Barmoha, 2016, p. 60; Nolan, Dixon, Roy, & Andreasen, 2016, pp. 36, 105.
Figure 1.2: Benchmark Tasks grid.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Figure 1.3: Benchmark Tasks recording sheet.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Students can use prior knowledge to help solve the problems in rows 3 and 4, which allows more access to these types of problems. For example, in problem 2C, if students do not know the formula for a trapezoid, they most likely would be unsuccessful in solving and would not attempt the problem. However, in problems 3C and 4C, students could use their knowledge of other formulas to break up the trapezoid into other shapes, find the area of those shapes, and then find the area of the figure. While this may not be the most efficient strategy, it allows access to the problem and students can find a solution. Then, through discussion with peers, students can connect their solution to others and engage in the lesson. It is interesting that tasks that teachers often consider to be more difficult actually provide more access to students. Additionally, if students are not able to complete the mathematics necessary to solve the task, the explanations and justifications the students provide can assist the teacher in diagnosing gaps in students’ understanding so that the teacher can then address those gaps. (Note: In this book, we use the terms demanding and challenging to mean stimulating and thought provoking, rather than difficult. A difficult task—for example, multidigit long division—may be difficult but not necessarily cognitively challenging.)
How did your responses on the Benchmark Tasks recording sheet compare to these descriptions? Take a moment to consider or discuss any new ideas introduced in this section using your recording sheet from activity 1.2. Then, proceed to the following section, The IQA Potential of the Task rubric, where we present a framework from the IQA Toolkit to assist you and your collaborative team in assessing the potential cognitive demand of mathematical tasks.
The IQA Potential of the Task Rubric
The level and type of thinking required by a task is referred to as the level of cognitive demand (Stein et al., 2009). Teachers can use the IQA Potential of the Task rubric (figure 1.4) to rate the level of cognitive demand a task can elicit from students. In activity 1.2, you identified examples and features of tasks at different levels of cognitive demand. The identification of these task features will help you to rate a task using the Potential of the Task rubric, with which you will consider the level of thinking required for students to successfully complete the task as initially written, without any changes, extensions, resources, or other expectations that may arise as students work on the task.
Source: Adapted from Boston, 2017.
Figure 1.4: IQA Potential of the Task rubric.
The IQA Potential of the Task rubric is intended to align with our previous ideas about tasks in rows 1 through 4 of the Benchmark Tasks recording sheet (page 14)—which we hereafter refer to as levels in the IQA Toolkit—and to provide additional detail and support for rating tasks. Look back through your task ratings and rationales for activity 1.2 and consider the following questions.
■ In what ways does the Potential of the Task rubric appear to be consistent with ideas on your recording sheet from activity 1.2? What words or phrases on the rubric do you find helpful?
■ In what ways does the Potential of the Task rubric appear to be inconsistent with or different from ideas on your recording sheet from activity 1.2? In other words, what characteristics of tasks did you identify that are not represented in, different from, or in contrast with the rubric?
Talk about the consistencies and inconsistencies with your collaborative team before moving on to the Application Activities in the following section. While several features of tasks may be important, this framework captures differences in tasks that have been shown to generate differences in students’ mathematical learning (Grouws et al., 2013; Stein & Lane, 1996). The way we categorize tasks according to cognitive demand frames many ideas throughout this book, so it is important to spend the time now to resolve differences with ideas in the rubric and within your collaborative team. These activities will assist you further in using the IQA Potential of the Task rubric (figure 1.4) and assessing your current instructional practices.
Application Activities
The following activities will help you become familiar with the IQA Potential of the Task rubric as you practice rating and adapting mathematical tasks.
Activity 1.3: Rating Mathematical Tasks Using the Potential of the Task Rubric
It is valuable to engage with tasks as learners to make sense of what those tasks have to offer students. Be sure to devote attention to this experience. Explore the tasks on your own before engaging in the activity.
Engage
For activity 1.3, you may want to print figure 1.5 from this book or the online resources. Note that we have provided grades or grade bands for each task. Because specific mathematics standards may vary from state to state, assume the task is appropriate for the grade level and students for which it is being used.
As you complete the task, consider the following directions.
■ Rate each task in figure 1.5 from level 1 to level 4 using the Potential of the Task rubric and provide a reason for the level you selected. Determine the ways each task provides access to each and every student.
■ Discuss your ratings and ideas with your collaborative team before moving on to the activity 1.3 discussion.
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