Using three levels in design of effective teacher-education tasks: The case of promoting conflicts with intuitive understandings in probability

This paper describes a three-level framework (technical, domain, and generic) which enables some generalisability across mathematics topics for effective teacher–education (TE) task design. It argues that TE tasks which encompass these levels increase preservice and inservice teachers’ interest because they transcend the particular mathematical focus and pedagogical activity within the TE tasks and enable translation to classrooms (technical), enhance the success of student learning (domain), and facilitate transfer to other topics (generic). The paper then uses the levels to analyse an effective probability task (based on circular spinners) which involves cognitive conflict between formal and intuitive probability at all three levels, namely, with regard to facilitating non-random results (technical), differences between probabilistic and deterministic mathematics and area and set models (domain), and non-prototypic exemplars and validation in probability experiments (generic). The paper concludes with reference to the power of effective TE tasks in showing how connectivity of mathematics (e.g., fractions and probability) requires similar connectivity in pedagogy.