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...

This study is a comparative study of the relationship between corneal structure, morphology, and function in a range of mammalian species. Xray scattering patterns were gathered at regular spatial intervals over the excised cornea (and in most cases also the scleral rim) of humans, marmosets, horses, cows, pigs, rabbits, and mice. All patterns were analyzed to produce quantitative information regarding the predominant orientation of fibrillar collagen throughout the tissue. The predominant direction of corneal collagen varies between mammals. This variation is not related to the size, shape, or thickness of the cornea or the frequency with which the animal blinks. A relationship does, however, appear to exist between corneal collagen arrangement and visual acuity. An excess of collagen directed toward one or both sets of opposing rectus muscles is a feature of animals that have an intermediate to high level of visual acuity. There is a significant variation in the arrangement of corneal collagen between different mammalian species. This finding may be related to differences in the frequency of action and the forces generated by the various extraocular muscles during eye movement and image fixation. Anat Rec, 290:1542Rec, 290: -1550Rec, 290: , 2007Rec, 290: . 2007 Wiley-Liss, Inc.

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Challenges to EBP utilization and fidelity should be monitored as EBPs contribute to the delivery of high-quality care. Collaborations between universities and rural agencies may support an agency's abilities to adopt EBPs, train staff, and systematically assess impact.