Cognitive Demand and Technology Use in High School Teachers’ Use Of Mathematical Tasks

The purpose of this study was to investigate and describe how high school mathematics teachers select and implement technology-based tasks and to describe the mathematical thinking required by those tasks. To better understand whether these tasks elicited the kinds of mathematical thinking consistent with the cognitive skills that are increasingly called for in our global society, I used the construct of cognitive demand (Stein, Smith, Henningsen, & Silver, 2000) to characterize the possible thinking that a student might use to solve each task. In this process, I paid special attention to how the use of graphing calculators and computer algebra systems (CAS) influenced the cognitive demand of particular tasks and teachers’ decisions about them. In particular, this study focused on the following research question: In what ways, if any, does the use of instructional technology influence how mathematical tasks are selected, how those tasks are implemented, and the cognitive demand at each stage of the instructional process? Investigation of this question could inform current teaching practice with respect to effective technology use and contribute an additional, technology-based dimension to scholarship on cognitive demand of mathematical tasks.

To organize my thinking about this research question, I used frameworks developed by Stein and her colleagues during work on the QUASAR Project (Stein, Grover, & Henningsen, 1996; Henningsen & Stein, 1997; Stein et al., 2000). First, the Mathematical Tasks Framework provided a way to examine a mathematics curriculum by defining it as a set of individual tasks. During the process of teaching and prior to student learning, each task is defined (and potentially redefined) at three stages: (a) as found in curriculum materials, (b) as set up by the teacher, and (c) as implemented by students. These stages organized my data collection and analysis into discrete modules. In addition to the Mathematical Tasks Framework, I also used the Task Analysis Guide (Stein et al., 2000) to classify the level of cognitive demand of each observed task. This guide divides cognitive demand into four levels, (a) recall tasks, (b) procedures without connections tasks, (c) procedures with connections tasks, and (d) doing mathematics tasks, and provides characteristics of tasks at each level to aid in this classification.

In this qualitative case study, I investigated the practices of three high school mathematics teachers. These teachers were selected based on their use of technology in teaching and their ability to successfully implement high-level tasks during instruction. Data were collected over a two-week period for each teacher focusing on lessons from one class period. Data for each lesson included a videotape recording of the lesson, field notes, and collected student work.

Each teacher also participated in two individual in-depth interviews. The first interview preceded the two-week observation period and focused on teaching philosophy, habits and strategies with respect to selecting tasks for instruction, and technology use during instruction. The second interview followed the two-week observation period. During this interview, each teacher watched selected video segments of her or his instruction and was asked to clarify instructional decisions and interpret student thinking on particular tasks.

Data from the preliminary interview and the two-week observation period were organized by separating the lesson into individual tasks. The Task Analysis Guide was used to assess the level of cognitive demand for each task at each stage of the Mathematics Task Framework. For technology-oriented tasks, further consideration was given to determine whether and how the students’ use of technology affected the cognitive demand of the task. Common themes were identified and confirmed or disconfirmed using data from student work and the final interviews.

Effective teaching observed during this study demonstrated that, although use of graphing calculators or CAS could reduce the cognitive demand for one or more aspects of a task, it enabled students to focus on other aspects of the task requiring a higher level of cognitive demand. For example, allowing students to use CAS to handle tedious symbolic manipulations enabled them to focus on underlying conceptual questions that likely would not have been considered without the availability of technology. Such instances, however, did not occur by accident, but were purposefully orchestrated by the teacher through the selection of appropriate tasks and effective implementation of these tasks.

Two aspects of students’ technology use related to cognitive demand were identified from the data. First, students had to examine the mathematical features of the task to determine whether technology use was appropriate. That is, they had to consider whether the external mathematical representation afforded by the technology was compatible with the mathematical elements and constraints of the task. Second, students had to connect the mathematical context of the task to a technological representation of it. Careful consideration was needed to translate necessary elements of the task into a technological representation and to interpret the resulting output from the technology back into the context of the task. These findings suggested a technology-oriented addendum to the Task Analysis Guide (Stein et al., 2000). This addendum could aid teachers’ selection of appropriate technology-oriented tasks and suggest a focus for discussion of their students’ work on those tasks. 

References 

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.