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

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СКАЧАТЬ 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 specifically calls for. The directions tell students exactly what operation to use; there is little ambiguity about what to do, removing students’ opportunities for thinking and sense making. The main mathematical activity left for students is dividing the first number in the problem by the second number in the problem. Even though this set of problems is set in a context (for example, “word problems” or “story problems”), this task is a level 2. Each problem follows a very similar format, and students can apply a procedure given to them in the directions of the task (division) and obtain an answer without considering the action in the problem or making sense of the situation. While the context of each problem supports an understanding of the equal groups or measurement model of division, the directions instruct students to use division at the outset of the problem, thus minimizing their opportunity to think through an appropriate model and operation themselves. Note that removing the directions and varying the format of each situation would make the task a level 3, as students would then need to make sense of the situations and select an appropriate model. Then, also including the prompt “Explain how you know your strategy and solution make sense” would raise the task to a level 4. While this task rates a level 2, it still provides good access to students because it is situated within contexts that allow students to solve the problems in multiple ways, including drawing pictures or models of each situation.

      Level 1 refers to tasks that elicit recall and memorization. We rated the following task at level 1.

      ■ Properties of Multiplication task: The potential of the task is limited to engaging students in recalling memorized knowledge of the properties of multiplication. Nothing in the task helps the students learn about the properties; they are simply asked to name the property displayed in each example. If students do not know each property, they are not able to access this task. It is difficult to modify tasks that are level 1 to increase access for more students without altering the mathematical goal of the task.

      Activity 1.4 provides an additional opportunity to use the Potential of the Task rubric to rate and adapt the levels of tasks.

Activity 1.4: Using the IQA Potential of the Task Rubric to Rate and Adapt Tasks

      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.

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