Task
[There is] no decision teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of tasks with which the teacher engages students in studying mathematics.
—Glenda Lappan and Diane Briars
Why is it important to assess the cognitive potential of instructional tasks? First, the consistent use of high-level instructional tasks has been shown to enhance students’ mathematical learning in elementary (Schoenfeld, 2002), middle (Cai et al., 2013), and high school mathematics classrooms (Grouws et al., 2013). Second, different types of tasks provide different types of opportunities for mathematical thinking and reasoning (Stein et al., 2009). Being aware of both the type of thinking a task can elicit and the type of access a task can give to all students can support you to align tasks with learning goals, and to ensure that students receive opportunities for thinking and reasoning. Finally, research has also shown that the level of the task sets the ceiling for the mathematical thinking, reasoning, and discussion that occurs throughout a lesson, and if a task does not request a representation, explanation, or justification, students typically do not produce or provide these things during a lesson (Boston & Wilhelm, 2015). Therefore, we find it critical for teachers seeking to improve their instructional practice to begin by considering the tasks and problems they are assigning in their classrooms and how these tasks may enable—or inhibit—student thinking.
What do you look for when selecting tasks? What makes a “good” instructional task?
In this chapter, you will explore why high-quality tasks are an essential first step in teaching mathematics for understanding. At the conclusion of this chapter, you will be able to answer the following questions.
■ How do different types of tasks elicit different opportunities to learn mathematics?
■ What types of tasks am I using to engage each and every student in learning mathematics?
Introductory Activities
Let’s get started by thinking about different types of mathematical tasks. Activities 1.1 and 1.2 ask you and your collaborative team to solve a variety of mathematical tasks and consider the thinking and problem-solving strategies that each task might elicit.
Activity 1.1: Solving a Task
It is valuable to engage with tasks as learners prior to implementing them as teachers. Be sure to devote attention to this experience. Explore the task on your own before discussing your experience with others.
Engage
Solve the Leftover Pizza task in figure 1.1. Do not use any procedures or algorithms. Try to solve the task in more than one way, using diagrams or other representations, including in ways students might correctly or incorrectly solve this task.
Source: Nolan, Dixon, Roy, & Andreasen, 2016.
Figure 1.1: The Leftover Pizza task (grade 6).
Respond to the following questions.
■ What strategies and types of thinking can this task elicit?
■ What are the main mathematical ideas that this task addresses?
■ How do teachers typically present the mathematical ideas addressed in this task to students? What types of tasks do teachers typically use to present these mathematical ideas to students? What is different about this task?
■ How might this task provide access for each and every learner?
Compare your work and ideas in your collaborative team before moving on to the activity 1.1 discussion.
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.
What strategies and types of thinking can this task elicit?
The Leftover Pizza task is set in a context that is conceptually helpful for understanding the division of fractions. By thinking through the action in the problem, students can make sense of a situation that requires the division of fractions and solve the problem without needing to know a set procedure for dividing fractions. The context encourages the use of a drawing or manipulatives. Students are likely to draw circles or rectangles to model the pizzas, divide the pizzas into thirds or sixths, and create groups of ⅔ of a pizza. Students can also use pattern blocks to model the problem nicely, using the yellow hexagon as the whole, the blue rhombus as ⅓, and the green triangle as ⅙.
Students often determine that they can create seven whole servings of ⅔ of a pizza. The remaining piece of pizza elicits a dilemma and a common misconception in interpreting fraction division—the remaining piece is ⅙ of a pizza, but ¼ of a serving. Students often wrestle with determining if the answer is 7¼ or 7⅙ servings.
The task could be solved by applying a procedure for dividing fractions, but this would first require the student to make sense of the situation and realize (a) the need to divide 4⅚ by ⅔ and (b) what the answer of 7¼ means in the context of the problem. The ¼ refers to one of four parts of a serving of pizza, rather than ¼ of a whole pizza. The ⅙ refers to the part of the whole pizza remaining, rather than a part of the serving size.
What are the main mathematical ideas that this task addresses?
The Leftover Pizza task engages students in interpreting a contextual situation, dividing fractions, and interpreting the meaning of the quotient. While the main mathematics underlying the task is division of fractions, the task also provides opportunities for using diagrams or manipulatives, modeling a contextual situation, and making sense of the action in the problem and of the result. In this way, the task aligns with national standards, such as from the Common Core State Standards (CCSS) for mathematics: “Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem” (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010; 6.NS.A.1). The task also aligns with standards from the National Council of Teachers of Mathematics (NCTM, 2000): “Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers” (p. 214).
How do teachers typically present the mathematical ideas addressed in this task to students? What types of tasks do teachers typically use to present these mathematical ideas to students? What is different about this task?
Educators often present fraction division as a rote procedure, modeling the process for students in example problems and accompanying this modeling with hints, such as “Remember to invert and multiply,” or “keep-change-flip.” Sometimes the examples are set in a context, but often students are provided a procedural solution to the examples and not encouraged to draw or model the situation or to make sense of the result. For example, students might be given the problem 4⅚ ÷ ⅔ along with several similar problems (for example, “Complete classwork examples 1–20”) that could be solved by applying the same procedure to each
