This chapter aims to provide insights into students’ perspectives about the meanings and purposes of mathematical tasks, and to understand how appropriate task design might help minimize any gaps between teacher intentions and student mathematical activity. Throughout the chapter, we explore accounts of how students understand the meaning and purpose of the mathematical activity they undertake, as well as how task design might take account of what we know about these perspectives. For instance, we discuss research that indicates ways in which the perceptions of students may differ from the intentions of teachers and task designers, and attempt to articulate the nature of those differences to raise both theoretical and methodological challenges concerning how an observer can appreciate the student’s point of view. We also discuss ways in which task design that takes account of students’ responses might reduce the discrepancies between the intentions of designers and/or teachers and students’ perceptions of their activity and achievements.