In light of the evidence generated in the classroom, we had to make choices about how to proceed. One option, for example, was to declare that such tasks are too ambitious and to abandon them in favor of assessment tasks similar to those that students are accustomed to completing in class and for homework. Because the goal of the project was to produce tasks and assessments that would enhance instruction and student learning, we decided instead to advance the craft of task development sufficiently to provide students access to what had been previously inaccessible tasks.
Meeting that challenge required looking closely at the students' performances and attempting to determine what was making the tasks so difficult. Two broad themes emerged. First, students are sometimes not given sufficient opportunity to perform, by which we mean some aspects of a task prevent students from showing what they have learned. Opportunity to perform is a primary focus of this chapter. Second, students are sometimes not given sufficient opportunity to learn, by which we mean the students' classroom experiences have not left them well equipped to succeed on certain kinds of tasks. Opportunity-to-learn issues are addressed in Chapter 4.
Of course, opportunity-to-perform and opportunity-to-learn issues are inextricably linked. If students have not had the opportunity to learn, then it will be difficult to identify task characteristics that could prevent students from showing what they know and can do. Nonetheless, it is important to try to separate these issues and to recognize where the responsibility for each lies. Responsibility for opportunity to perform lies with the task and the task developer, and opportunity to learn is primarily the responsibility of administrators, teachers, parents, and students.
In our work, opportunity-to-perform issues emerged as a recasting of the time-honored concept of task validity—whether a task measures what it is intended to measure—because unless students are given sufficient opportunity to perform it is not possible to make valid inferences about what the students know and can do. Thus, when draft versions of a task failed to produce expected results in field trials, we questioned the task's face validity and asked what the task did measure. The challenge was to determine the source of the difficulty and then to revise the task in ways that maintain the important mathematical ideas the task was intended to assess.
This chapter briefly describes the task development process and then illustrates several key concepts that emerged while attempting to construct tasks that provide students with opportunities to perform. In particular, some tasks create cognitive overload by attempting to assess skills, conceptual understanding,