Making Sense of Mathematics for Teaching to Inform Instructional Quality. Juli K. Dixon
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 problem. This set of problems (considered as one “task” according to our definition of a task as a set of problems that address the same mathematical idea) encourages students to apply a previously learned procedure, but does not support them to think or reason about division of fractions. The task directions and number of problems suggest that the focus of the task is on performing or practicing a procedure to produce an answer. The expected solution would look similar to:
While 4⅚ ÷ ⅔ and the Leftover Pizza task both require the same mathematical operation and perhaps address similar content standards (dividing fractions), they provide much different opportunities for students’ thinking and reasoning. The Leftover Pizza task engages students in interpreting, modeling, and making sense of a context that requires the division of fractions, the process of dividing fractions, and the meaning of the quotient. In this way, the Leftover Pizza task elicits the types of mathematical thinking identified in the Process Standards of NCTM’s (2000) Principles and Standards for School Mathematics, such as representations and connections. It also engages students in the Mathematical Practices called for by the Common Core’s Standards for Mathematical Practice, such as Mathematical Practice 1, “Make sense of problems and persevere in solving them,” and Mathematical Practice 4, “Model with mathematics” (NGA & CCSSO, 2010).
How might this task provide access for each and every learner?
This task has many features that allow access for each and every student. It can be described as having a low threshold and a high ceiling (McClure, 2011). There are numerous ways to solve the pizza task, ensuring multiple entry points for all learners. Students can use fraction tiles, pattern blocks, or other manipulatives to model the situation and utilize those models when communicating how they solved the problem to their peers and teachers. The pizza task allows students access because they are able to model the serving sizes and determine that they can make at least seven servings. Then, using the models or through discussion with peers or the teacher, students can get to the ¼ of a serving that remains. By allowing students entry into the problem, teachers provide them with something to discuss with the class to further their understanding. If a student was just given the problem 4⅚ divided by ⅔ and did not have the means to perform the operation or solve the problem, he or she would not have access or the ability to solve the problem and would then be left out of the conversation regarding the solution. Multiple entry points and solution methods, as well as a meaningful context, allow students to interact with the task on at least some level so that when there is a discussion, all students have ideas to bring to the table and then can use those ideas to make sense of the mathematics and build a better understanding of the division problem.
In activity 1.2, you will continue to explore how different types of tasks provide different opportunities for students’ thinking.
Activity 1.2: Considering Different Types of 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 1.2, you may want to print figure 1.2 (page 13) and figure 1.3 (page 14) from this book or from the online resources (see go.SolutionTree.com/mathematics). Look over the tasks in figure 1.2 and use figure 1.3 to record your responses to the following questions.
■ What is similar about the tasks in each column? How do the tasks change as you move up (or down) a column?
■ What is similar about the tasks across each row? Identify phrases that characterize the nature of tasks in each row of the grid, and write these phrases on your recording sheet.
Compare your work and ideas in your collaborative team before moving on to the activity 1.2 discussion. Keep your recording sheet to use as your rubric in activity