The design of tasks in support of teachers' development of coherent mathematical meanings

We examine the role of tasks that have the intended effect of teachers re-conceiving the mathematics they teach as comprising a coherent body of meaningful ideas. We ground our discussion in ideas of trigonometry and modular functions and draw from a professional development research project to illustrate our approach. In this project, many teachers experienced dissonance that was rooted in their commitments to their curricular knowledge of trigonometry. Teachers who built new meanings into a coherent whole were those who coordinated them at a micro level. Teachers who saw implications of their own reasoning for student learning were also successful at expressing that reasoning in natural language. We saw a similar pattern in the case of teachers' creation of meanings for action and process conceptions of mod(f(x),g(x)). Teachers who gained insight into implications of their own activities for student learning were the teachers who reasoned at a micro level in regard to the meaning of mod, who coordinated that meaning with a covariational perspective on the behavior of functions, and who expressed that coordination in natural language. We conclude that a primary feature of tasks that promote teachers' construction of coherent mathematical meanings is that they support an overall effort to have teachers engage in the coordination of meanings in the context of explaining significant ideas and relationships.