Content preparation of teachers is a current topic of interest to mathematics educators across the nation. Many experts now agree that reforming teacher preparation in postsecondary institutions is central to sustaining and deepening efforts to provide quality mathematics education for all students (National Science Foundation, 1996; National Research Council, 1996). Traditionally, recommendations about the mathematical content preparation of teachers have taken the form of a list of courses or a list of topics. Yet, research has shown that the number of mathematics courses taken by teachers does not correlate significantly with their effectiveness as measured by student learning (Begle, 1979; Monk, 1994). In other words, what the teachers take from their mathematics courses does not necessarily serve them well in their classroom practice. Recent recommendations about the mathematical preparation of teachers, such as the Conference Board of the Mathematical Sciences' document *Mathematical Education of Teachers* (in preparation), have attempted to move beyond lists of content, stressing the need for knowledge about mathematical connections, communication, modeling, or use of technology, for example. In practice, however, these recommendations are often realized as courses or sequences of courses organized by mathematical content and taught via a narrow repertoire of approaches, which alone will not be effective for prospective teachers nor for the students they will be teaching.

Despite the plethora of recommendations over the past forty years, questions about what mathematics content teachers need to know, how and where they should come to know the material, what they need to know about the nature and practice of mathematics, and how their knowledge of mathematics relates to teaching practice, though fundamental, are largely unresolved in current research. There is a tremendous need to address these questions systematically, to develop new ways of thinking about the role of content knowledge in teacher preparation, and to identify focused questions for future research involving members of the mathematics education community—mathematicians, scientists, mathematics educators, teacher educators, K-12 teachers, and administrators.

Questions about teachers' knowledge of content are of interest across a range of communities that are concerned with teacher preparation. Some feel strongly

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Knowing and Learning Mathematics for Teaching
Introduction
Content preparation of teachers is a current topic of interest to mathematics educators across the nation. Many experts now agree that reforming teacher preparation in postsecondary institutions is central to sustaining and deepening efforts to provide quality mathematics education for all students (National Science Foundation, 1996; National Research Council, 1996). Traditionally, recommendations about the mathematical content preparation of teachers have taken the form of a list of courses or a list of topics. Yet, research has shown that the number of mathematics courses taken by teachers does not correlate significantly with their effectiveness as measured by student learning (Begle, 1979; Monk, 1994). In other words, what the teachers take from their mathematics courses does not necessarily serve them well in their classroom practice. Recent recommendations about the mathematical preparation of teachers, such as the Conference Board of the Mathematical Sciences' document Mathematical Education of Teachers (in preparation), have attempted to move beyond lists of content, stressing the need for knowledge about mathematical connections, communication, modeling, or use of technology, for example. In practice, however, these recommendations are often realized as courses or sequences of courses organized by mathematical content and taught via a narrow repertoire of approaches, which alone will not be effective for prospective teachers nor for the students they will be teaching.
Despite the plethora of recommendations over the past forty years, questions about what mathematics content teachers need to know, how and where they should come to know the material, what they need to know about the nature and practice of mathematics, and how their knowledge of mathematics relates to teaching practice, though fundamental, are largely unresolved in current research. There is a tremendous need to address these questions systematically, to develop new ways of thinking about the role of content knowledge in teacher preparation, and to identify focused questions for future research involving members of the mathematics education community—mathematicians, scientists, mathematics educators, teacher educators, K-12 teachers, and administrators.
Questions about teachers' knowledge of content are of interest across a range of communities that are concerned with teacher preparation. Some feel strongly

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Knowing and Learning Mathematics for Teaching
that prospective teachers need deep and substantial knowledge of mathematics content; however, there is varying opinion about the level, breadth, and formality of this content. Others are concerned that prospective teachers have content knowledge that will be useful to them in classroom settings with diverse student populations. To some, there is the perception that these two perspectives are mutually exclusive, while others look for a balance. Recently developed standards-based K-5 curriculum materials impose substantial new mathematical demands on K-5 teachers, and developers worry that teacher preparation courses will not adequately prepare teachers to employ these mathematically challenging materials in the classroom or to manage the mathematical discussions that emerge in classrooms when they are used. Thus, one of the two major questions addressed at the workshop was: What is the mathematical knowledge teachers need to know to teach well?
At the same time, professional developers in mathematics education (Ball & Cohen, 1999; Schifter, Bastable, & Russell, 1999; Shulman, 1992; Stein, Smith, Henningson, & Silver, 2000), particularly at the inservice level, are building experience and expertise with professional development materials and tools, including the use of videos, case studies, teacher reflections on practice, mathematicians' commentaries, analyses of student work, and the use of curriculum materials by teachers. While many of these efforts have shown promise with inservice teachers, there is very little experience in the mathematics education or mathematics communities with incorporating such practice-based approaches to teacher development into preservice content courses. Thus, a second question of the workshop was: How can teachers develop the mathematical knowledge they need to teach well?
REFERENCES
Ball, D. L., & Cohen, D. K. ( 1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In G. Sykes & L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3-32). San Francisco: Jossey Bass.
Begle, E. G. ( 1979). Critical variables in mathematics education: Findings from a survey of the empirical literature. Washington, DC: Mathematical Association of America.
Conference Board of the Mathematical Sciences, (in preparation). Mathematical education of teachers. Draft report available on-line: http://www.maa.org/cbms.
Monk, D. H. ( 1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125-145.
National Research Council. ( 1996). From analysis to action: Report of a convocation. Washington, DC: National Academy Press.
National Science Foundation. ( 1996). Shaping the future. Washington, DC: Author.
Schifter, D., Bastable, V., & Russell, S. J. (with Yaffee, L., Lester, J. B., & Cohen, S.) ( 1999). Developing mathematical ideas, number and operations part 2: Making meaning for operations casebook. Parsippany, NJ: Dale Seymour.
Shulman, J. ( 1992). Case methods in teacher education. New York: Teachers College Press.
Stein, M. K., Smith, M. S., Henningson, M. A., & Silver, E. A. ( 1999). Implementing standards-based mathematics instruction: A casebook for professional development . New York: Teachers College Press.