At the school level, 'tasks' in mathematics classrooms are generally equivalent to 'mathematical tasks' and so tasks can be seen as the mediating tools for teaching and learning mathematics. In secondary school mathematics teaching, it is particularly important to understand that tasks on their own cannot scaffold the shifts in perception and cognition required to understand abstract mathematical ideas. Zaslavsky and Sullivan have coordinated the diversity of approaches to tasks for mathematics teacher education, some of the goals being pedagogic, some being about development of professional habits of mind. In Stylianides session the task sequence includes a range of tasks in which prospective teachers primary or immediate engagement shifts several times between mathematics and pedagogy, thereby promoting discussions on the interplay of mathematics and pedagogy. The design of the task sequences shapes their immediate experience in the session, and also mediates interplay between past and present experience, school and university experience, and mathematics and pedagogy.