assumption that secondary school teachers have strong and flexible knowledge of algebra is unfounded (Ball, 1988). In fact, evidence suggests that secondary school teachers may often have rule-bound knowledge of procedures but lack conceptual, connected understanding of the domain. That some signals existed that high school teachers’ knowledge may not be as robust as had been suspected is not so surprising, since they are educated in the very mathematics classrooms that many seek to improve.

Still, this result signals a more significant problem. First, the changes in and expansion of what is meant by algebra mean that secondary school teachers are increasingly being called on to teach algebraic ideas and connections that they have not themselves studied or have not studied in such ways. Analyses of what the new curricula demand of teachers could make visible the mathematical demands of those curricula and permit investigation of teachers’ current knowledge to teach those materials. Second, the movement of algebraic ideas into the middle school and especially the elementary school curriculum means that teachers who have not in the past taught algebra are now being called on to teach ideas and processes for which they have not in the past been responsible. Prospective elementary school teachers’ knowledge of algebra may be based largely on their own experiences as high school students. The new requirements of elementary school teaching raise important and pressing needs for research on teacher knowledge, teaching, and teacher learning specifically in algebra.

Not only do teachers need knowledge of the mathematical content, however. Equally important (and related) is knowledge of how students think about and develop algebraic ideas and processes. What ideas or procedures are particularly difficult, both in reading and writing mathematical relationships, for many students? As algebra shifts to being a K-12 subject, rather than a pair of high school courses, new questions emerge that warrant investigation: If students learn about variables and equations sooner and engage earlier in algebraic reasoning (Carpenter and Franke, 2001), how will these earlier experiences shape the development of students’ algebraic proficiency over time? How do students of different ages manage and use symbolic notation, both in reading and writing mathematical relationships? What supports the development of meaningful and skilled fluency with mathematical symbols and syntax? In fac-



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