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

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.

      Source: Questions adapted from Dixon, Nolan, Adams, Tobias, & Barmoha, 2016, p. 84; Nolan, Dixon, Roy, & Andreasen, 2016, pp. 128, 133; Nolan, Dixon, Safi, & Haciomeroglu, 2016, p. 46.

Figure 1.5: Tasks for activity 1.3.

      Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.

       Discuss

       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 all students.

      Level 4 refers to tasks that promote meaning, sense making, connections between representations, or problem solving and explicitly require explanations or justifications. We rated the following two tasks at level 4.

      ■ Water Fountain task: This task provides an opportunity for students to make connections between representations. It provides a context, a graph, and two forms of a quadratic function. The task asks students to consider which of the symbolic representations would be most useful for answering different questions about the water fountain. It explicitly prompts them to explain their choices. Students have access to the task because they are able to visualize the representation of the graph and relate the functions to the graph.

      ■ Science Quiz task: This task requires students to determine how to compare the data. Students must make sense of the type of data, the distribution of scores, and what these both suggest about appropriate representations to model and compare the data. Students are solving a genuine problem and developing an explanation for why their choices make sense. The task explicitly prompts students to explain how they know which class did better on the quiz. This task allows access because there are multiple ways to compare the sets of data and make an argument using mean, median, mode, range, dot plots, and box plots. Students could argue for either class depending on what measure of center they choose or which type of graph they create and are not confined to using one particular procedure.

      Level 3 refers to tasks that promote meaning, sense making, connections between representations, or problem solving but do not explicitly require explanations or justifications. We rated the following three tasks at level 3.

      ■ Shapes Pattern task: Identifying patterns and forming conjectures provide opportunities for thinking and reasoning as well as recognizing and using structure. While the task asks students to make conjectures, the task does not prompt students to provide mathematical evidence for those conjectures. This task can engage students in thinking about important mathematics at grade 2 (for example, multiples or division with remainders). This task allows access because most students can identify a pattern and then engage in a conversation with peers around how to justify and generalize the pattern. Including a prompt such as “How do you know?” or “Determine whether your conjecture is always true” would increase the task to a level 4.

      ■ Swimming Pool Deck task: This task would be rated as a level 3 because it provides a context and opportunity for students to make sense of area, but it is not a level 4 because it does not ask students to form a generalization or justify their solutions. The shape is nonstandard, and students cannot just apply an area formula and obtain an answer. There are multiple ways to find the area of the deck and the task suggests no specific pathway to the students. The task provides access because students who cannot recall the formula for finding the area of a trapezoid can use other area formulas by decomposing the shape into other shapes for which they know the formulas.

      ■ Fraction Pizza task: The task has the potential to engage students in complex thinking and creating meaning about the relative size of fractions. The task provides a context in which students can compare the relative size of unit fractions and fractions one part away from a whole. While students could use a procedure to compare fractions (for example, common denominators), the task provides a context to support students to reason about the relative size of fractions close to 0 and close to 1, even if they did not know the procedure. The opportunities inherent in this task to solve it in different ways increase its access to more students. Note that the task requires no explanation, hence it is not a level 4.

      Level 2 refers to tasks that require procedures, computation, or algorithms without connection to meaning and understanding. We rated the following two tasks at level 2.

      ■ Multistep Equations task: The potential of the task is limited to students performing a procedure or series of procedures to solve multistep equations. The number of problems in the set suggests that the task requires students to apply procedures quickly and efficiently. Solving equations is an important and useful algebra skill, and this particular task provides the opportunity for students to practice and demonstrate their ability to perform previously learned procedures for solving equations. For this reason, the task is a level 2. The task does not support students to develop an understanding of the underlying mathematical concepts (for example, the property of equality). The prompt to “show your work” does not require students to explain their thinking and reasoning, but simply to show the steps in the procedure. This task does not allow access if students do not already have a set procedure for how to solve multistep equations.

      ■ Division Story Problems task: The potential of the task is limited to engaging students in a procedure that the task