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

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.

       Engage

      For activity 1.4, you may want to print figure 1.6 (page 22) from this book or the online resources. The tasks in figure 1.6 are examples of tasks at levels 1 through 3.

      ■ Provide a rationale for each task level using the IQA Potential of the Task rubric.

      ■ Consider how to adapt each task to increase the cognitive demand. Use ideas from the Potential of the Task rubric to make small changes to each task to provide greater opportunities for students to provide their thinking and reasoning while still addressing the same mathematical content.

      ■ Before moving on to the activity 1.4 discussion, discuss your rationales and task adaptations with your collaborative team. Include in your discussions how your task adaptations might increase the potential for access to the task by more learners. Compare your ideas with the rationales and suggestions for adaptations in appendix B (page 137).

       Discuss

      How do your responses compare with those in your collaborative team? What themes emerged during your discussion? In this section, we present ideas for you to consider.

       Provide a rationale for each task level using the IQA Potential of the Task rubric.

      Rationales for the levels of tasks in figure 1.6 appear in appendix B (page 137). Were your rationales consistent with the ones provided? Were there any ratings you questioned? In the following description of adaptations, we provide additional detail regarding the rating of each task.

       Consider how to adapt each task to increase the cognitive demand.

      In this activity, we suggested using ideas from the Potential of the Task rubric to make small changes to adapt the tasks in figure 1.6 to provide greater opportunities for students’ thinking and reasoning, while still addressing the same mathematical content. Here, we describe three types of changes that would increase the tasks to a level 3 or 4.

      1. Level 3: Number Pairs That Make 10—The Number Pairs task has the potential to engage kindergarten students in creating meaning for how to generate sums to ten. It is important for students at this age to know how to compose and decompose numbers, especially with tens. Because we would not yet expect kindergartners to have memorized the number facts that sum to ten, this task allows students to explore different ways to make ten. There are many ways students could think through the problem and model their ideas and strategies, which is an important feature in providing access to all students. The Number Pairs task does not rate a level 4 because there is no explicit prompt for an explanation or justification. This task would provide greater opportunities for thinking and reasoning than traditional tasks that ask students only for answers such as:

      Source: Level 3 question adapted from Dixon, Nolan, Adams, Brooks, & Howse, 2016, p. 65.

Figure 1.6: Tasks for activity 1.4.

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

      5 + 5 = ____ 8 + 2 = ____ 3 + 7 = ____

      Note, however, that with older students who have the number facts memorized, we would characterize the Number Pairs task as level 1 (memorization).

      In general, level 3 tasks provide opportunities for thinking, reasoning, and sense making. Added prompts for students to explain their thinking, compare strategies, reflect on their strategy choice, or justify their conjectures or generalizations are examples of how to raise the task to a level 4. Asking students to find all possible solutions, and to explain how they know they have found them all, can engage students in analyzing patterns and making generalizations. Alternatively, asking students for two different ways to solve the problem or to find more than one solution, and prompting students to explain, compare, or relate the different solutions, also increases the cognitive demand. In the Number Pairs task, asking students to explain why more than one number pair works would more deeply engage students in decomposing and recomposing numbers and explaining their reasoning. Finally, requiring students to create a representation and explain something about the representation can also increase cognitive demand.

      2. Level 2: Adding Fractions With Unlike Denominators—The Adding Fractions task is a typical procedural task. There are numerous procedures for every grade level and mathematical topic that we could substitute in place of “adding fractions with unlike denominators” (for example, multiplying or dividing multidigit numbers, cross-multiplying, applying the Pythagorean theorem, or factoring). Such tasks provide opportunities for students to practice or demonstrate a previously learned procedure. While practice or mastery of certain mathematical procedures is often useful and even necessary, as teachers we want to be aware that engaging in procedural tasks only promotes practice and rote