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ОглавлениеCHAPTER 1
Potential of the 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 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 1.3 (page 16).
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 is similar about the tasks in each column? How do the tasks change as you move up (or down) a column?
The columns of the Benchmark Tasks grid each contain tasks that address related mathematical ideas. In column A, the tasks address division with remainders for students in grade 4. Tasks in column B relate to addition and subtraction of integers in grade 7. The tasks in column C all involve the area of trapezoids in grade 6. However, as you move up or down a column, the tasks provide different opportunities for students’ thinking and reasoning about each mathematical topic. Research shows that attending to the level and type of thinking that a task can elicit from students is equally as important as considering the mathematical ideas in the task, and different types of tasks provide different opportunities for thinking and reasoning (Stein et al., 2009) and impact students’ learning in different ways.
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.
Tasks across the rows of the Benchmark Tasks grid elicit similar types and levels of thinking from students. Tasks in row 1 mainly draw on students’ memorized knowledge or recall of mathematics facts, rules, formulas, or vocabulary. Teachers have described tasks in this row as “You either know it or you don’t”—nothing in the task helps students to learn what it is asking, and there is no procedure they can apply to determine an answer. Students only have access to the task if they are able to recall what it is asking. Taking notes would also belong in row 1 of this grid, as taking notes engages students in writing down or reproducing mathematics rather than doing any mathematical thinking on their own. Tasks such as these are appropriate when the goal for student learning is recall and memorization.
In row 2, students can solve the tasks by applying a procedure, computation, or algorithm. The goal of these tasks is to perform a procedure or computation and arrive at a correct answer. While students may use a conceptually based strategy to solve the task, nothing in the task requires or supports students to make sense of the mathematics or demonstrate their understanding of the mathematics. This denies access to the task if students are not fluent with the procedure or computation used to solve the problem. Successfully completing the task only requires that students perform a procedure and produce an answer.
In row 3, tasks may ask students to engage in problem solving, though they may also ask them to apply specific procedures or use specific representations. The main difference between tasks in row 2 and row 3 is that tasks in row 3 provide opportunities for mathematical connections, reasoning, and sense making. The questions, representations, and contexts in the task support students to develop an understanding of a mathematical concept or procedure or to engage in complex and non-algorithmic thinking. In completing the task, students actually learn mathematics.
Tasks in row 4 contain all of the features of tasks in row 3, with the added feature that the task directions explicitly require students to provide an explanation or justification. In addition to completing the mathematics necessary to solve the task, the task includes a prompt for students to reflect on, explain, or justify some aspect of their work.
Source: Questions adapted from Dixon, Nolan, Adams, Tobias, & Barmoha, 2016, p. 60; Nolan, Dixon, Roy, & Andreasen, 2016, pp. 36, 105.
Figure 1.2: Benchmark Tasks grid.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Figure 1.3: Benchmark Tasks recording sheet.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Students can use prior knowledge to help solve the problems in rows 3 and 4, which allows more access to these types of problems. For example, in problem 2C, if students do not know the formula for a trapezoid, they most likely would be unsuccessful in solving and would not attempt the problem. However, in problems 3C and 4C, students could use their knowledge of other formulas to break up the trapezoid into other shapes, find the area of those shapes, and then find the area of the figure. While this may not be the most efficient strategy, it allows access to the problem and students can find a solution. Then, through discussion with peers, students can connect their solution to others and engage in the lesson. It is interesting that tasks that teachers often consider to be more difficult actually provide more access to students. Additionally, if students are not able to complete the mathematics necessary to solve the task, the explanations and justifications the students provide can assist the teacher in diagnosing gaps in students’ understanding so that the teacher can then address those gaps. (Note: In this book, we use the terms demanding and challenging to mean stimulating and thought provoking, rather than difficult. A difficult task—for example, multidigit long division—may be difficult but not necessarily cognitively challenging.)
How did your responses on the Benchmark Tasks recording sheet compare to these descriptions? Take a moment to consider or discuss any new ideas introduced in this section using your recording sheet from activity 1.2. Then, proceed to the following section, The IQA Potential of the Task rubric, where we present a framework from the IQA Toolkit to assist you and your collaborative team in assessing the potential cognitive demand of mathematical tasks.
The IQA Potential of the Task Rubric
The level and type of thinking required by a task is referred to as the level of cognitive demand (Stein et al., 2009). Teachers can use the IQA Potential of the Task rubric (figure 1.4) to rate the level of cognitive demand a task can elicit from students. In activity 1.2, you identified examples and features of tasks at different levels of cognitive demand. The identification of these task features will help you to rate a task using the Potential of the Task rubric, with which you will consider the level of thinking required for students to successfully complete the task as initially written, without any changes, extensions, resources, or other expectations that may arise as students work on the task.
Source: Adapted from Boston, 2017.
Figure 1.4: IQA Potential of the Task rubric.
The IQA Potential of the Task rubric is intended to align with our previous ideas about tasks in rows 1 through 4 of the Benchmark Tasks recording sheet (page 14)—which we hereafter refer to as levels in the IQA Toolkit—and to provide additional detail and support for rating tasks. Look back through your task ratings and rationales for activity 1.2 and consider the following questions.
■ In what ways does the Potential of the Task rubric appear to be consistent with ideas on your recording sheet from activity 1.2? What words or phrases on the rubric do you find helpful?
■ In what ways does the Potential of the Task rubric appear to be inconsistent with or different from ideas on your recording sheet from activity 1.2? In other words, what characteristics of tasks did you identify that are not represented in, different from, or in contrast with the rubric?
Talk about the consistencies and inconsistencies with your collaborative team before moving on to the Application Activities in the following section. While several features of tasks may be important, this framework captures differences in tasks that have been shown to generate differences in students’ mathematical learning (Grouws et al., 2013; Stein & Lane, 1996). The way we categorize tasks according to cognitive demand frames many ideas throughout this book, so it is important to spend the time now to resolve differences with ideas in the rubric and within your collaborative team. These activities will assist you further in using the IQA Potential of the Task rubric (figure 1.4) and assessing your current instructional practices.
Application Activities
The following activities will help you become familiar with the IQA Potential of the Task rubric as you practice rating and adapting mathematical tasks.
Activity 1.3: Rating Mathematical Tasks Using the Potential of the Task Rubric
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 1.3, you may want to print figure 1.5 from this book or the online resources. Note that we have provided grades or grade bands for each task. Because specific mathematics standards may vary from state to state, assume the task is appropriate for the grade level and students for which it is being used.
As you complete the task, consider the following directions.
■ 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 each and every student.
■ Discuss your ratings and ideas with your collaborative team before moving on to the activity 1.3 discussion.
Source: 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.
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 + . Many other scenarios and contexts can support the need for students to make sense of adding equal-sized fractional parts.
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 that prescribe a strategy or direct students how to solve the task will also raise the task level. Beyond procedural tasks, this adaptation also applies to word problems or story problems at all grade levels and for any mathematical procedure. For example, if high school students have just learned the distance formula, and then are given word problems with the directions “Use the distance formula to solve the following problems,” the word problems no longer require thinking and reasoning, but only the application of a previously learned, prescribed procedure. This lowers the cognitive demand of the task.
3. Level 1: Angles—Similar to the idea that level 2 (procedural) tasks are appropriate when the goal for students’ learning is practice or mastery of procedures, level 1 tasks are appropriate when the goal for student learning is recall and memorization. To encourage greater thinking and reasoning, allow students to discover relationships before providing the vocabulary, definitions, properties, postulates, or theorems. For example, before providing the definition and properties of vertical angles, have students measure a variety of vertical angles, conjecture that vertical angles are congruent (which also provides an opportunity to discuss measurement error), and then justify their conjecture using prior knowledge (for example, supplementary angles and the transitive property).
Although the Potential of the Task rubric provides a comprehensive framework for rating and adapting mathematical tasks, certain factors may affect how you rate or select tasks for classroom use. We discuss these issues in the following section.
Considerations When Rating Tasks
The awareness of different task levels and the ability to rate the level of tasks can equip teachers to be knowledgeable and critical consumers of published and online resources for the mathematics classroom. Published curricular materials often contain tasks at a variety of levels, and small changes to adapt the tasks in ways such as those identified in activity 1.4 can go a long way toward increasing students’ opportunities for thinking and reasoning. Instructional materials featured in online sites for teachers are frequently divided between resources that promote procedural practice and nonmathematical activity (for example, when the main activity is craft based rather than mathematical) and resources that provide ideas for conceptually based lessons, and teachers often have to adapt these resources for use in their own classrooms.
As you begin to assess and rate tasks, there are several issues to consider in order to achieve both successful implementation in your classroom and enhanced thinking and understanding among your students. In this section, we will discuss the practical and conceptual issues that may stem from defining the task itself, considering the implications of higher-level thinking in practice, and aligning tasks with learning goals and standards.
Defining the Task
Sometimes, identifying the task in curricular materials or other resources is not straightforward. In this book, we consider the task to be the mathematical problem or set of problems presented for students to do during a lesson or instructional activity. Tasks in curricular materials or as presented during a lesson may contain several parts. For example, each cell of figure 1.2 (page 13) is considered to be one task. The multiple parts of a task receive one collective rating according to the highest level of cognitive demand of any of the parts. For example, if a task (as presented on a handout or in verbal directions) includes vocabulary recall, a problem-solving activity, and an explanation, we would rate it a level 4. A task that supports students to develop or generalize a procedure and then spend time practicing that procedure would be considered a level 3.
Some curricular materials will clearly identify a mathematical problem or set of problems for students to engage with during the lesson. For other materials, you may need to identify what the task is asking students to do mathematically. Sometimes, the teacher’s manual is necessary to understand exactly what students are being asked to do. This is particularly true for primary grades, in which the teacher often presents the task and directions verbally to align with students’ reading levels. When rating tasks in curricular materials or other resources, consider them as they appear in print. Any directions, manipulatives, representations, or resources indicated by the print materials, including teachers’ manuals, are part of the task.
When using the Potential of the Task rubric (page 15), consider any directions (via textbook, teacher’s edition, handout, whiteboard, or screen) or resources provided to students. Most of the directions will occur before students begin their work on the task. However, the teacher may choose to give students part of a task, allowing them time to explore, and only then provide later parts of the task and additional time for students to continue working or developing their explanations. We would consider any mathematical problems that teachers ask students to do during the lesson as part of the task, even if the task directions are presented in parts throughout the lesson. In later chapters, we explore how a task unfolds throughout a lesson as students work on the task and engage in mathematical discussion, and so it is helpful to consider a lesson as having one main instructional task.
Considering Implications of Higher-Level Thinking
When discussing the activities so far in this chapter, you and your collaborative team may have occasionally determined that the level of the task depends on the grade level or prior knowledge of the students. It is always important to consider how the students’ prior knowledge may impact the cognitive demand of a task. For the activities in the chapter, we instructed you to assume that the task was appropriate for a given group of students. In your school or classroom, you would be familiar with the grade level, standards, and students for a given task. If students have solved a series of very similar patterning tasks, problem-solving tasks, or other level 3 or 4 tasks, subsequent tasks in the series would not elicit the same level and type of thinking as the first. The task would likely become procedural (level 2), with students following a template provided by completing the first few tasks, even though the first task in the series would have been a level 3 or 4. Similarly, if students already know the properties, procedure, or definitions at the heart of a discovery task, there would not be anything for them to discover. For example, the adapted Angles task can help students discover relationships between angles only if students do not already know those relationships.
Additionally, be aware of the implications of the wording of the task and how this can impact the work students produce. While a task may ask students to explain how or show your steps, there is a difference between explaining a procedure and explaining your thinking. For example, if a task requires students to solve fraction division problems such as ¾ ÷ ¼ using the traditional algorithm, asking students to explain might generate responses such as, “I used invert and multiply and computed ¾ × .” While this explanation of how students solved the problem indicates knowledge of an algorithm, it does not indicate conceptual understanding, reasoning, or sense making. Adding a prompt to explain onto a procedural task does not raise the cognitive demand; the task itself must first elicit some thinking, reasoning, problem solving, or understanding for the student to have something worth explaining.
Aligning With Learning Goals
While we categorize procedural tasks at a level 2, note that there are many important mathematical procedures that students should be able to apply fluently and with automation after having established an appropriate level of conceptual understanding (NCTM, 2014). There are also appropriate occasions when you would use or assign a level 2 task to provide students the opportunity to practice or demonstrate their ability to perform a procedure, or a level 1 task when the goal is memorizing rules, properties, or definitions. Task levels (as well as mathematical content) should align with the goals for students’ learning.
Task levels should also align with students’ learning progression for a particular mathematical idea at a particular grade level. For example, rigorous state standards suggest providing opportunities consistent with level 3 or 4 tasks while students are in the process of learning to make sense of multiplication in grades 2 through 4, as expressed in the grade 4 Common Core standard:
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (NGA & CCSSO, 2010; 4.NBT.B.5)
By grade 5, we expect students to demonstrate mastery of the standard algorithm for multidigit multiplication, which would be supported by engaging in tasks at level 2 that provide practice and promote automation and mastery.
Note that in the progression of learning multiplication from grades 2 to 5, opportunities for students to understand, unpack, and develop strategies and procedures, using problems, contexts, and representations that make sense, precede the memorization of multiplication facts and mastery of standard algorithms. Too often, students’ learning of a mathematical topic or procedure begins with the teacher telling or showing students everything they need to know, modeling procedures, or providing definitions, while students’ mathematical activity is limited to practicing procedures (level 2) or taking notes and memorizing (level 1). Typically, teachers wait until students have mastered procedures or memorized the appropriate facts or properties before providing opportunities for real-world applications or problem solving. After all, how would students know how to solve problems if they were not shown how to solve them first? On the contrary, students can develop mathematical procedures given contextual problems prior to any direct instruction or modeling (Carpenter, Fennema, Franke, Levi, & Empson, 2014). Students can discover many mathematical properties through investigation, such as the congruent angles formed when parallel lines are cut by a transversal (figure 1.6, page 22), the formula for area of a trapezoid (figure 1.2, page 13), or the rules for adding integers (figure 1.2, page 13). The Making Sense of Mathematics for Teaching series provides many suggestions for supporting students’ learning of specific mathematical topics at each grade band.
Summary
It is important to rate the level of instructional tasks to be aware of the type of thinking and access a task can provide for each and every student. A task at level 1 or 2 does not provide much space for discussion, as the focus is on the correctness of memorized knowledge or rote procedures. Additionally, a task at level 1 or 2 often does not provide access for students unless they know the mathematics to be recalled or the specific procedure requested. A level 3 or 4 task is often necessary to support quality mathematical discourse and teacher questioning, as we will discuss in upcoming chapters. We provide additional support for rating tasks in appendix D (page 141).
Even though reform efforts call for mathematics learning for each and every student (NCTM, 2014), learners who struggle in mathematics or who have special education placements often have less access to demanding mathematics (Weiss, Pasley, Smith, Banilower, & Heck, 2003). To successfully include all learners in the mathematics classroom, we need to design instruction that is accessible to all.
Chapter 1 Transition Activity: Moving From Tasks to Implementation
Before moving on to chapter 2, engage in the transition activity with your collaborative team. The transition activity will enable you to build on ideas about tasks from chapter 1 to begin to explore implementation in chapter 2.
• Select a chapter, unit, or any set of two to three consecutive lessons in the mathematics curriculum materials you use in your school or classroom. Rate the tasks that appear in a set of lessons over two to three days of instruction.
What opportunities would students have to engage in thinking and reasoning?
What is the balance of levels across the lessons?
• Identify a task at level 3 or 4 to use as the main instructional task to teach a mathematics lesson. Indicate what features make the task a level 3 or 4.
What thinking, reasoning, or sense making would the task potentially elicit from students?
What products or processes would serve as evidence that students actually engaged in this thinking, reasoning, or sense making?
How does the task provide access for all students?
• Implement the task in your class. Collect sets of student work (at least four samples). Select samples that show a variety of strategies, thinking, and reasoning.
• Analyze students’ responses. Did students actually engage in or produce the level and type of thinking you identified when considering the potential of the task?
Save the sets of student work, your ratings, and notes or any written reflections from the transition activity, as you will refer to them in chapter 2.
