Beyond the Common Core. Juli K. Dixon
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.
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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.
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While lower-level-cognitive-demand mathematical