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
