Teachers' Perspectives Using the Instructional Quality Assessment as a Professional Development Tool

Introduction

As one of the most important decisions a teacher will make is choosing the mathematical task their students will engage in (Lappan & Briars, 1995), it is crucial that teachers not only realize the importance of choosing tasks, but also have the knowledge of what tasks can engage students in high quality mathematics. According to the National Council of Teachers of Mathematics (2014), “to ensure that students have the opportunity to engage in high-level thinking, teachers must regularly select and implement tasks that promote reasoning and problem solving” (p. 17). Teachers can impact students’ opportunities to learn by engaging students in high cognitive demand tasks (Smith & Stein, 2018). Cognitive demand refers to the amount of effort a student needs to expend thinking about a problem. Henningsen and Stein (1997) outlined and characterized four different levels of cognitive demand of tasks: memorization tasks, procedures without connections tasks, procedures with connections tasks, and doing mathematics tasks. Memorization and procedures without connections tasks have a low level of cognitive demand while procedures with connections and doing mathematics tasks have a high level of cognitive demand. Memorization tasks involve recalling facts or definitions and do not require computation. An example of a memorization task is stating the Pythagorean theorem. Tasks labeled as procedures without connections involve using a procedure to solve a problem but do not connect the procedure to any other mathematical ideas, such as solving equations for missing variables. Procedures with connections tasks involve using a procedure but connecting it to other mathematical ideas. One such problem is solving a quadratic function, interpreting what the solution means, and relating these values to the graph of the function and the overall given situation. A task labeled doing mathematics does not give an explicit way to solve the problem and may include multiple solution methods, such as figuring out a pattern and coming up with generalized formula. (See Figure 1 for descriptions of the four levels of demand). Many of the descriptors of tasks with a high level of demand align with characteristics of tasks that promote conceptual understanding of mathematics (Doyle, 1983).

Figure 1.

The Task Analysis Guide (Boston & Smith, 2009, p. 122)

While teachers should be able to select tasks appropriately and implement tasks at a high level to support students’ mathematical thinking, they also need to be able to maintain the level of demand during implementation (Henningsen & Stein, 1997). Researchers found that teachers who participated in professional development around high cognitive demand tasks increased both the number of high cognitive demand tasks used and their ability to maintain the level of demand throughout the lesson (Boston & Smith, 2009). Stein and colleagues (1996) found teachers involved in professional development targeted at implementing high cognitive demand tasks had success selecting and setting up high cognitive demand tasks that encouraged students to use multiple solutions and representations, participate in group work, justify answers, and engage in complex mathematical thinking and reasoning.