2: Designing and Using Tasks to Teach Mathematical Knowledge for Teachingⁱ

Research over the last two decades clearly shows that teachers make a difference in student achievement (National Mathematics Advisory Panel, 2008). In the whirlwind of problems that plague mathematics education in the U.S.—from weak student achievement, to a repetitive and unfocused curriculum, to persistent racial and income-related gaps in learning opportunities—one important and promising finding is that instruction matters and that teachers are key to effective instruction.

Teachers’ skill in making mathematics accessible to and learnable by all students depends on a wide range of resources, including their own mathematical knowledge. Knowing mathematics for teaching can enable sensitive analysis of a student’s difficulty, skilled use of diagrams or models, clear explanations, well-posed questions, and strategic selection and use of tasks and examples. However, the mathematical knowledge that supports effective teaching involves more than sheer adeptness with the school curriculum. It requires learning more and different mathematics. Furthermore, such knowledge is not necessarily the product of conventional mathematics study (Begle, 1979; Monk 1994; National Mathematics Advisory Panel, 2008) because such study is not usually aimed at the specialized knowledge of mathematics needed for instruction.