In the mathematics classroom, the teacher, the student and the tasks provide the key structural elements through which the classroom’s social activity is constituted. There has long been a tacit assumption that the completion of mathematical tasks chosen or designed by the teacher will result in the student learning the intended mathematics. This view is persistent despite research that suggests this is not a direct relationship (Margolinas, 2004). Task development, selection and sequencing by teachers represents the initiation of an instructional process that includes task performance (collaboratively by teacher and student) and the interpretation of the consequences of this enactment (again, by teacher and student).

For some time, theories of learning have viewed cognitive activity as not simply occurring in a social context, but as being constituted in and by social interaction (e.g., Hutchins, 1995). From this perspective, the activity that arises as a consequence of a student’s completion of a task is itself a constituent element of the learning process and the artifacts (both conceptual and physical) employed in the completion of the task serve simultaneous purposes as scaffolds for cognition, repositories of distributed cognition and as cognitive products. During the process of task completion, the effectiveness of the task in promoting learning will also be contingent on student intention (with respect to the task) and teacher interpretation (with respect to the students’ activity).

In this symposium, the instructional use of mathematical tasks is explored through two sets of three research reports. The first set examines the use of mathematical tasks to promote domain-specific mathematical reasoning related to key mathematical content. The three reports address: (i) the transition from arithmetic to algebra, as mediated by arithmetic word problems; (ii) “formulas as expressions for functions” serving as a vehicle for facilitating the transition to high school level mathematical reasoning; and (iii) specific tasks scaffolding student development from variational to covariational reasoning. Together, the reported research demonstrates how the role of the selected mathematical tasks was critical in facilitating student development of sophisticated thinking related to variable, algebra, and function.

In the second set of three papers, mathematical tasks are again studied for their capacity to promote the development of forms of reasoning, but the goal in each case is a form of reasoning that transcends conventional mathematical content. Specifically, the three papers report research into: (i) a problem solving situation which involves incidents of specialising, generalising and justification, with the targeted capability the transformation between different semiotic representations; (ii) intention, action and interpretation as these are enacted through task completion, for the purpose of examining the instructional consequences of purposefully amplifying student agency in the classroom; and (iii) the widespread curricular aspiration to develop student facility with both content-specific and higher order thinking skills is investigated through a customised set of “hybrid tasks.”

A discussant will comment on each set of three papers. Both discussants will participate in the concluding general discussion.

In both sets of studies, the mathematical reasoning being promoted is more than procedural expertise. In every one of the six studies the goal is a form of sophisticated mathematical reasoning promoted through the strategic instructional use of purposefully designed mathematical tasks. The research reports present complementary perspectives on the student-task relationship and demonstrate just how diverse are the considerations affecting the instructional deployment of tasks and their role in facilitating student participation in particular types of sophisticated mathematical activity.

Taken in combination, the research reports that constitute this symposium provide powerful illustration of how the strategic use of particular mathematical tasks is critical to the achievement of the most sophisticated curricular goals in mathematics education.

Hutchins, E. (1995). Cognition in the wild. Cambridge, Mass.: MIT Press. Margolinas, C. (2004). Points de vue de l'élève et du professeur: Essai de développement de la théorie des situations didactiques. Université de Provence. http://tel.archives-ouvertes.fr/tel- 00429580/fr/