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ОглавлениеHLTA 2: Identifying Higher-Level-Cognitive-Demand Mathematical Tasks
The function of education is to teach one to think intensively and to think critically.
—Martin Luther King Jr.
Developing your team’s understanding of the essential learning standards for the unit helped you answer the first critical question of a PLC, What do we want all students to know and be able to do? The mathematical tasks you and your team choose to use every day during the unit help you answer this first critical question as well.
The mathematical tasks you choose each day and for every unit also partially answer the second critical question of a PLC, How will we know if they know it?
| = Fully addressed with high-leverage team action | |
| = Partially addressed with high-leverage team action |
The What
What is a mathematical task?
NCTM first identified mathematical task in its Professional Teaching Standards (1991, 2008) as “worthwhile mathematical tasks” (p. 24). Melissa Boston and Peg Smith (2009) later provided this succinct definition: “A mathematical task is a single complex problem or a set of problems that focuses students’ attention on a specific mathematical idea” (p. 136).
Mathematical tasks include activities, examples, or problems to complete as a whole class, in small groups, or individually. The tasks provide the rigor (levels of complex reasoning as provided by the conceptual understanding, procedural fluency, and application of the tasks) that students require and thus become an essential aspect of your team’s collaboration and discussion. In short, the tasks are the problems you choose to determine the pathway of student learning and to assess student success along that pathway. As a teacher, you are empowered to decide what and how a student learns through your choice and use of mathematical tasks in class.
The type of instructional tasks you and your team select and use will determine students’ opportunities to develop proficiency in Mathematical Practices and processes and will support the development of conceptual understanding and procedural skills for the essential learning standards. As Glenda Lappan and Diane Briars (1995) state:
There is no decision that 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 the tasks with which the teacher engages students in studying mathematics. (p. 139)
Mathematical Practice 1—“Make sense of problems and persevere in solving them”—establishes the expectation for regularly engaging your students in challenging, higher-level-cognitive-demand mathematical tasks essential for their development. A growing body of research links students’ engagement in higher-level-cognitive-demand mathematical tasks to overall increases in mathematics achievement, not just in the ability to solve problems (Hattie, 2012; Resnick, 2006).
A key collaborative team decision is which tasks to use in a particular lesson to help students attain the daily learning objective. The nature of the tasks with which your students engage provides the common student learning experiences you can draw on to further student learning at various points throughout the unit. Selecting appropriate tasks provides your collaborative team with the opportunity for rich, engaging, and professional discussions regarding expectations about student performance for the unit.
Thus, four critical task questions for your grade-level collaborative team to consider include:
1. What nature of tasks should we use for each essential learning standard of the unit? Will the tasks focus on building student conceptual understanding, procedural fluency, or a combination? Will the tasks involve application of concepts and skills?
2. What are the depth, rigor, order of presentation, and ways of investigating that we should use to ensure students learn the essential learning standards?
3. How does our collaborative team choose the mathematical tasks that best represent each essential learning standard?
4. How does our team ensure the implementation of the tasks as a team in order to avoid wide variances in student learning across the grade level?
Conceptual understanding and procedural fluency are essential aspects for students to become mathematically proficient. In light of this, the tasks you choose to form the unit’s lessons must include a balance of higher- and lower-level-cognitive-demand expectations for students. Your team will also need to decide which mathematical tasks to use for class instruction and which tasks to use for the various assessment instruments given to students during and at the end of a unit.
Higher-level-cognitive-demand lessons or tasks are those that provide “opportunities for students to explain, describe, justify, compare, or assess; to make decisions and choices; to plan and formulate questions; to exhibit creativity; and to work with more than one representation in a meaningful way” (Silver, 2010, p. 2). In contrast, lessons or tasks with only lower-level cognitive demand are “characterized as opportunities for students to demonstrate routine applications of known procedures or to work with a complex assembly of routine subtasks or non-mathematical activities” (Silver, 2010, p. 2).
However, selecting a task with higher-level cognitive demand does not ensure that students will engage in rigorous mathematical activity (Jackson et al., 2013). The cognitive demand of a mathematical task is often lowered (perhaps unintentionally) during the implementation phase of the lesson (Stein, Remillard, & Smith, 2007). During the planning phase, your team should discuss how you will respond when students urge you to lower the cognitive demand of the task during the lesson. Avoiding cognitive decline during task implementation is discussed further in chapter 2 (page 71, HLTA 6).
Thus, your teacher team responds to several mathematical task questions before each unit begins:
1. How do we define and differentiate between higher-level-cognitive-demand and lower-level-cognitive-demand tasks for each essential standard of the unit?
2. How do we select common higher-level-cognitive-demand and lower-level-cognitive-demand tasks for each essential standard of the unit?
3. How do we create higher-level-cognitive-demand tasks from lower-level-cognitive-demand tasks for each essential standard of the unit?
4. How do we use and apply higher-level-cognitive-demand tasks for each essential standard during the unit?
5. How will we respond when students urge us to lower the cognitive demand of the task during the implementation phase of the lesson?
Visit go.solution-tree.com/mathematicsatwork to download these questions as a discussion tool.
The How
A critical step in selecting and planning a higher-level-cognitive-demand mathematical task is working the task before giving it to students. Working the task provides insight into the extent to which it will engage students in the intended mathematics concepts, skills, and Mathematical Practices and how students might struggle. Working the task with your team provides information about possible solution strategies or pathways that students might demonstrate.
Defining Higher-Level and Lower-Level-Cognitive-Demand Mathematical Tasks
You choose mathematical tasks for every lesson, every day. Take a moment to describe how you choose the daily tasks and examples that you use in class. Do you make those decisions by yourself, with members of your team, before the unit begins, or the night before you teach the lesson? Where do you locate and choose your mathematical tasks? From the textbook? Online? From your district resources?
And, most importantly, how would you describe the rigor of each task you choose for your students? Rigor is not whether a problem is considered hard. For example, “What is 6 × 7?” might be a hard problem for some, but it is not rigorous. Rigor of a mathematical task is defined in this handbook as the level and the complexity of reasoning required by the student during the task (Kanold, Briars, & Fennell, 2012). A more rigorous version of this same task might be something like, “Provide two different ways to solve 6 × 7 using facts you might know.”
There are several ways to label the demand or rigor of a task; however, for the purposes of this handbook, tasks are classified as either lower-level cognitive demand or higher-level cognitive demand as defined by Smith and Stein (1998) in their Task Analysis Guide and printed in full as appendix B (page 153). Lower-level-cognitive-demand tasks are typically focused on memorization or on performing standard or rote procedures without attention to the properties that support those procedures (Smith & Stein, 2011).
Higher-level-cognitive-demand tasks are tasks for which students do not have a set of predetermined procedures to follow to reach resolution or, if the tasks involve procedures, they require that students provide the justification for why and how the procedures can be performed. Smith and Stein (2011) describe these procedures as “procedures with connections” (p. 16) as opposed to “procedures without connections,” the designation they use for lower-level-cognitive-demand tasks that are not just based on memorization.
Thus, the level of cognitive demand of the mathematical tasks you choose each day can be viewed as either lower- or higher-level cognitive demand as shown in figure 1.6.
Source: Smith & Stein, 2012.
Figure 1.6: Four categories of cognitive demand.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
You may or may not have been fully aware that every task you choose to use with your students each day is either a lower- or higher-level-cognitive-demand task. Lower-level-cognitive-demand tasks take less time in class, and do not require much complex reasoning by students. Their efficiency is appealing. They are much easier to manage in class as a general rule and easily serve direct instruction from the front of the room. The fact that the new state assessments intend to dramatically increase the task rigor compared to current state assessments (Herman & Linn, 2013) is additional motivation for you to increase the cognitive demand of the mathematical tasks you use during instruction and assessment.
The very nature of the mathematical content expectations requires your students to demonstrate understanding, and thus a shift to a balanced task approach during the unit—the use of both higher- and lower-level-cognitive-demand tasks. In most elementary school classrooms, this will require an increase in the use of higher-level-cognitive-demand tasks. Figure 1.7 (page 24) provides six mathematical tasks, one for each grade level from kindergarten through grade 5, along with an identifier for the content standard each supports. Use the discussion tool to examine the mathematical task that most closely relates to the grade-level responsibilities of your collaborative team, and then answer the questions at the end of the tool.
Each of the tasks in figure 1.7 is a higher-level-cognitive-demand mathematical task. What makes a task high cognitive demand? What might a lower-level-cognitive-demand mathematical task look like for the same essential learning standard?
The tasks in figure 1.7 (page 24) represent procedures with connections or problem solving. Notice that the kindergarten task (“Blake has a number of cubes that is 1 more than 15. Jessica has a number of cubes that is 1 less than 17. Who has more cubes? How do you know?”) would not require higher-level cognitive demand for a student in grade 4 to solve. The task’s demand is relative to the students who will engage with the task, and it is connected to a specific essential question and learning objective for the particular unit. However, tasks that are lower-level cognitive demand can still be connected to the same learning standards.
Figure 1.7: Higher-level-cognitive-demand mathematical task discussion tool.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
Use figure 1.8 to work with your collaborative team to adapt the higher-level-cognitive-demand mathematical tasks from figure 1.7 to lower-level-cognitive-demand mathematical tasks.
Figure 1.8: Corresponding lower-level-cognitive-demand mathematical task-creation tool.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
Compare the lower-level-cognitive-demand mathematical tasks you created to the corresponding lower-level-cognitive-demand mathematical tasks in figure 1.9 (page 26). Then, work with your collaborative team to answer the questions in figure 1.9 and compare the two types of tasks for each grade level.
Figure 1.9: Comparing higher- and lower-level-cognitive-demand mathematical tasks.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
While lower-level-cognitive-demand mathematical tasks are crucial for developing procedural fluency, higher-level-cognitive-demand mathematical tasks are essential for improving students’ depth of understanding related to the Common Core–type expectations and, ultimately, student achievement. You and your team should focus on creating tasks of varying cognitive demand in order to meet the expectations of your state assessments.
Identifying the Cognitive Demand of Your Mathematical Tasks
As a first step in understanding the nature of the current cognitive demand level of the tasks you use each day, use figure 1.10.
Figure 1.10: Tool for sorting unit tasks by cognitive demand level.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
What percentage of the current tasks you plan to use fall into the lower-level-cognitive-demand task category? What percentage fall into the higher-level-cognitive-demand task category? Do you have a proper balance in terms of the complexity of student reasoning required by the tasks you present to students throughout the unit?
Consider the grade 4 higher-level-cognitive-demand mathematical task from the Smarter Balanced Assessment Consortium (SBAC) in figure 1.11 (page 28) and the fractions task from the Partnership for Assessment of Readiness for College and Careers (PARCC) in figure 1.12 (page 28).
Source: Smarter Balanced Assessment Consortium, 2013. Used with permission.
Figure 1.11: Smarter Balanced Assessment Consortium practice test grade 4 task.
Source: PARCC, 2013.
Figure 1.12: PARCC practice test grade 4 task.
Using figures 1.11 and 1.12, explain where you would place these mathematical tasks in the cognitive demand table in figure 1.10 (page 27). You can also reference appendix B (page 153) for more detail. If you do believe these are higher-level-cognitive-demand tasks, explain why.
Visit go.solution-tree.com/mathematicsatwork for more higher-level-cognitive-demand mathematical tasks like these from the SBAC and PARCC assessments.
Creating Higher-Level-Cognitive-Demand Tasks
There are several strategies you can use to change a lower-level-cognitive-demand mathematical task to higher-level cognitive demand. A typical strategy that is rarely fruitful is to change the numbers in the problem to greater numbers. For example, with the grade 2 task in figure 1.9 (page 26), it would not be enough simply to change 48 + 25 to 148 + 325. While this might make a task more difficult to complete, it does not necessarily make the task higher-level cognitive demand. You can use the strategies in figure 1.13 to adjust a mathematical task from lower-level cognitive demand to higher-level cognitive demand.
Figure 1.13: Strategies for increasing the cognitive demand of tasks.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
