This study explored whether and how teachers’ mathematical knowledge for teaching contributes to gains in students’ mathematics achievement. The authors used a linear mixed-model methodology in which first and third graders’ mathematical achievement gains over a year were nested within teachers, who in turn were nested within schools. They found that teachers’ mathematical knowledge was significantly related to student achievement gains in both first and third grades after controlling for key student- and teacher-level covariates. This result, while consonant with findings from the educational production function literature, was obtained via a measure focusing on the specialized mathematical knowledge and skills used in teaching mathematics. This finding provides support for policy initiatives designed to improve students’ mathematics achievement by improving teachers’ mathematical knowledge.

In this article, we discuss efforts to design and empirically test measures of teachers' content knowledge for teaching elementary mathematics. We begin by reviewing the

This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice.This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. Such claims stem originally from the "educational production function literature," or research that examines the contribution of student, teacher, and school resources to student learning. As far back as the Equality of Educational Opportunity study (Coleman, 1966) scholars identified a unique contribution of teacher knowledge to student achievement (in mathematics, see also Hanushek, 1972;Hill, Rowan, & Ball, 2005;Mullens, Murnane, & Willett, 1996;Rowan, Chiang, & Address correspondence to Heather Hill, 6 Appian Way # 445, Cambridge, MA 02138. E-mail: heather hill@harvard.edu , 1997). At the same time, scholars studying teacher knowledge have also identified an impact on instruction (e.g., Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992;Fennema & Franke, 1992;Leinhardt & Smith, 1985;Putnam, Heaton, Prawat, & Remillard, 1992;Sowder, Phillip, Armstrong, & Shappelle, 1998). Taken in combination, there is evidence for the proposition that stronger teacher knowledge yields benefits for classroom instruction and student achievement. MKT AND THE MATHEMATICAL QUALITY OF INSTRUCTION431 MillerYet despite these observations, and despite a wealth of current policy initiatives built on these observations, there is a lack of detailed understanding regarding how teacher knowledge affects classroom instruction and student achievement. Teacher knowledge presumably works through instruction to cause student learning, yet this relationship is not completely understood. Large-scale educational production function studies never peer inside classrooms to compare the practice of higherknowledge and lower-knowledge teachers. Studies in the teacher knowledge and mathematics education literatures have raised strong plausible propositions regarding how knowledge matters for teaching; but with sample sizes of one teacher per study in many cases, and few objective measures of teachers' mathematical knowledge, generalizations have to date been limited.The lack of specifics regarding how knowledge affects instruction leaves critical gaps in our theoretical knowledge, and significant holes i...

Educational reformers increasingly seek to manipulate policies regarding assessment, curriculum, and professional development in order to improve instruction. They assume that manipulating these elements of instructional policy will change teachers' practice, which will then improve student performance. We formalize these ideas into a rudimentary model of the relations among instructional policy, teaching, and learning. We propose that successful instructional policies are themselves instructional in nature: because teachers figure as a key connection between policy and practice, their opportunities to learn about and from policy are a crucial influence both on their practice, and, at least indirectly, on student achievement. Using data from a 1994 survey of California elementary school teachers and 1994 student California Learning Assessment System (CLAS) scores, we examine the influence of assessment, curriculum, and professional development on teacher practice and student achievement. Our results bear out the usefulness of the model: under circumstances that we identify, policy can affect practice, and both can affect student performance.

Value-added models have become popular in research and pay-forperformance plans. While scholars have focused attention on some aspects of their validity (e.g., scoring procedures), others have received less scrutiny. This article focuses on the extent to which value-added scores correspond to other indicators of teacher and teaching quality. The authors compared 24 middle school mathematics teachers' value-added scores, derived from a large (N = 222) district data set, to survey-and observation-based indicators of teacher quality, instruction, and student characteristics. This analysis found teachers' value-added scores correlated not only with their mathematical knowledge and quality of instruction but also with the population of students they teach. Case studies illustrate problems that might arise in using value-added scores in pay-for-performance plans.