Developing a Practice-Based Theory of Mathematical Knowledge for Teaching

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This project seeks to examine what it is about mathematics that elementary teachers need to know and appreciate, and where and how in teaching such understandings and appreciation are used. Two enduring questions are at the heart of contemporary efforts to improve mathematics instruction:o What mathematical knowledge is entailed by the work of teaching mathematics? o Where and how is mathematical knowledge entailed, and how would it be useful and used, within the work of teaching mathematics? How is mathematical knowledge intertwined with other knowledge and sensibilities in the course of that work?Within the term "knowledge" the project includes content (e.g., topics, theorems, definitions, algorithms) as well as practices and sensibilities. What it might mean, and what it would take, for elementary teachers to teach in ways that are respectful of both students and the mathematics. What could it mean to have "mathematical integrity" as an elementary teacher? The project will also seek to locate the mathematical aspects of elementary teaching -- the tasks, moments, and challenges -- where what a teacher can hear, see, do, and say depends on mathematical knowledge.Rather than asking these questions by examining curriculum frameworks or standards of what teachers are supposed to teach, the project works in and from practice. Our investigations take us first into investigations of the practices of teaching and learning mathematics, as well as into the practices of mathematics itself. Central to this work is a large longitudinal NSF-funded database, documenting an entire year of Ball's third grade public school teaching during 1989-90. By analyzing these detailed records of practice, we seek to develop theory of mathematical knowledge as it is entailed by and used in teaching, to consider instruction over time, examining the work of teachers in developing mathematics and their students across the school year. Products of this project will include detailed analyses of the mathematics entailed in elementary school teaching, as well as grounded theory about the nature and uses of mathematical content, practice, and sensibilities in the day-to-day work of teaching. Written products will target different audiences with interests in this work.