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 mastery and does not promote understanding and sense making. For example, one can know and perform the procedure for dividing multidigit numbers but not be able to explain why you “bring the number down” or know when division applies to a contextual situation. For these reasons, more conceptual tasks (levels 3 and 4) align with goals and standards when students are beginning to develop an understanding of the mathematical topic, and procedural tasks may align better with goals and standards at the end of students’ learning trajectory of a particular mathematical topic, once they have developed their understanding.
To increase the cognitive demand of a procedural task, use a context or representation that supports students to make sense of the operation and provides the need to develop a new strategy. To increase access, allow students to use multiple strategies or manipulatives to engage with the task. Consider the following task from Making Sense of Mathematics for Teaching Grades 3–5 (Dixon, Nolan, Adams, Tobias, & Barmoha, 2016):
Brandon is sharing four cookies equally between himself and his four friends. Brandon wants to start by giving each person the largest intact piece of cookie possible so each person receives the same size piece of cookie to start. How might Brandon divide the cookies? (p. 73)
While fifth-grade students might easily determine that four cookies shared among five people is ⅘ of a cookie per person, requiring the largest intact piece of cookie to be shared equally first provides a context for adding fractions with unlike denominators, such as ½ + ¼ +
Another approach to increasing the cognitive demand of a task is to ask students to develop a new procedure based on prior knowledge before teaching the procedure to students. In this case, knowledge of equivalent fractions and adding fractions with like denominators is all students need to figure out how to add fractions with unlike denominators. Similarly, removing structure or directions
