## Scientifically-Based Research

Recent research on how to best prepare teachers directs us to focus on specialized content knowledge related to teaching within the broad category which Shulman (1986) identified as pedagogical content knowledge (Ma, 1999; Ball, 2000). This work suggests that we must improve “… not just what mathematics teachers know, but how they know it and what they are able to mobilize mathematically in the course of teaching” (Ball, 2000, p. 243). Ball and her colleagues have set out to identify and measure a mathematical knowledge for teaching based on the mathematical work of teaching. “Teaching mathematics requires an appreciation of mathematical reasoning, understanding the meaning of mathematical ideas and procedures, and knowing how ideas and procedures connect” (Hill and Ball, 2004). They assert that teachers must understand the mathematical definitions, representations, examples and notations that are most powerful in supporting children’s understanding; they must hear the mathematical thinking of children, and guide and extend that thinking; they need to recognize the nature of children’s alternate conceptions, appraise novel student methods for solving problems, and help students to create counterexamples and arguments.

Cohen and Hill (2001) found that professional development is most likely to positively impact teachers’ practice when it focuses not only on content specific to the curriculum, but also on the mathematical concepts underlying the curriculum, student thinking about the mathematics content and on the teaching of that content. The Learning Mathematics for Teaching (LMT) project at the University of Michigan investigates the mathematical knowledge needed for teaching, how such knowledge develops and has developed assessment items that reflect the mathematics tasks teachers face in classrooms.

When Hill and Ball (2004) evaluated California’s Mathematics Professional Development Institutes using these LMT measures, they found that teachers’ opportunities for mathematical engagement, such as analyzing solutions and methods, exploring representations, communicating mathematically and making arguments, positively affected posttest scores on their instrument. Based on this work, they assert that “the more teachers engage with mathematics in ways that afford them opportunities to explore and link alternative representations, to provide and interpret explanations, and to delve into meanings and connections among ideas, the more flexible and developed their knowledge will be. It may even be that the extent to which teachers encounter mathematics as a domain in which reasoning and representation are central, and not simply as one comprising rules and routines, the more they learn” (p. 346).

Our aim is to provide teachers with opportunities to explore, link, interpret and connect mathematical ideas. This research has informed and will continue to inform the design and delivery of this project’s activities, and we will use the LMT measures as our primary means of evaluating the development of mathematical knowledge for teaching in our program participants.