Mathematical-task knowledge for teaching

Abstract

The articles of this issue of the Journal of Mathematics Teacher Education [JMTE] prompted me to think about mathematical tasks and teachers’ knowledge of them. Mathematical tasks are central to the learning of mathematics. For example, they can ‘‘provide the stimulus for students to think about particular concepts and procedures, their connections with other mathematical ideas, and their applications to real-world contexts’’ (National Council of Teachers of Mathematics [NCTM] 1991, p. 24). However, they have no life of their own as a tool for learning. It is the teacher and students who give them life based on how they are interpreted and enacted in the classroom. The teacher is critical in shaping the lived task and directing students’ activities so that students have opportunities to engage meaningfully in mathematics through them. A teacher could turn an open-ended task into a closed one or a closed one into an open one. He or she could treat a task of high cognitive demand as a low level one or vice versa. There are several factors that could influence this; for example, the teacher’s knowledge of content, knowledge of learners, goal for task, instructional orientation, and beliefs about mathematics. In particular, the nature of the teachers’ mathematical-task knowledge for teaching is likely to be the determining factor in their treatment of tasks. Mathematical-task knowledge for teaching deals with the knowledge teachers need in order to (a) select and develop tasks to promote students’ conceptual understanding of mathematics, support their development of mathematical thinking, and capture their interest and curiosity and (b) optimize the learning potential of such tasks. This knowledge includes as follows: