Revisiting Theory for the Design of Tasks: Special Considerations for Digital Environments

Abstract

Teachers should and do design tasks for the mathematics classroom, with specific mathematical learning as the objective. Completing the tasks should require students to engage in dialectics of action, formulation and validation (Brousseau in Theory of didactical situations in mathematics : didactique des mathematiques, Dordrecht: Kluwer Academic Publishers, 1997) and to move between the pragmatic/empirical field and the mathematical/systematic field (Noss et al. in Educational Studies in Mathematics, 33(2), 203–233, 1997). In the classroom, students act within a milieu, and where computers are part of this milieu, particular considerations with respect to task design include questions about the mathematics the student does and the mathematics the computer does, and the role of feedback from the computer. Whilst taking into account the role of the computer, the design of tasks can also be guided by theoretical constructs related to obstacles of various kinds; ontogenic, didactical and epistemological (Brousseau in Theory of didactical situations in mathematics : didactique des mathematiques,Dordrecht: Kluwer Academic Publishers, 1997), and, whereas the first two should be avoided, the third should be encouraged. An example of a task taken from empirical research in an ordinary classroom is used to illustrate some of these ideas, also demonstrating how difficult and complex it is for many teachers to design tasks that use computer software in ways that provoke the sort of student activity that would be likely to lead to mathematical learning. Implications for teacher professional development are discussed.