In both the United States and Europe, concerns have been raised about whether preservice and in-service training succeeds in equipping teachers with the professional knowledge they need to deliver consistently high-quality instruction. This article investigates the significance of teachers' content knowledge and pedagogical content knowledge for high-quality instruction and student progress in secondary-level mathematics. It reports findings from a 1-year study conducted in Germany with a representative sample of Grade 10 classes and their mathematics teachers. Teachers' pedagogical content knowledge was theoretically and empirically distinguishable from their content knowledge. Multilevel structural equation models revealed a substantial positive effect of pedagogical content knowledge on students' learning gains that was mediated by the provision of cognitive activation and individual learning support. American Educational Research Journal March 2010, Vol. 47, No. 1, pp. 133-180 DOI: 10.3102/0002831209345157 Ó 2010 AERA. http://aerj.aera.netat Max Planck Ins on July 27, 2011 http://aerj.aera.net Downloaded from KEYWORDS: teacher knowledge, teacher education, mathematics, instruction, cognitive activation, hierarchical modeling with latent variables S ince Lee Shulman's presidential address at the 1985 American Educational Research Association meeting-in which Shulman went beyond the generic perspective of educational psychology, emphasizing the importance of domain-specific processes of learning and instruction-educational research JÜ RGEN BAUMERT is a co-director at Max Planck Institute for Human Development, Center for Educational Research, Lentzeallee 94, 14195 Berlin, Germany; e-mail: sekbaumert@mpib-berlin.mpg.de. His research interests include research in teaching and learning, cultural comparisons, large-scale assessment, and cognitive and motivational development in adolescence. MAREIKE KUNTER is a research scientist at Max Planck Institute for Human Development, e-mail: kunter@mpib-berlin.mpg.de. Her research interests include teacher research, motivational processes in the classroom, and assessment of instructional processes. WERNER BLUM is a professor of mathematics education at University of Kassel, e-mail: blum@mathematik.uni-kassel.de. His research interests include empirical research on instructional quality in mathematics, national and international comparison studies in mathematics, approaches to application, modeling, and proofs in mathematics instruction. MARTIN BRUNNER is an associate professor at University of Luxembourg, e-mail: martin.brunner@uni.lu. His research interests include research on cognitive abilities, achievement, and achievement motivation by means of modern measurement models. THAMAR VOSS is a predoctoral research fellow at Max Planck Institute for Human Development, e-mail: voss@mpib-berlin.mpg.de. Her research interests include research on instruction and learning, teacher research, and teacher beliefs. ALEXANDER JORDAN is an academic staff member at University of Biel...

Many psychological constructs are conceived to be hierarchically structured and thus to operate at various levels of generality. Alternative confirmatory factor analytic (CFA) models can be used to study various aspects of this proposition: (a) The one-factor model focuses on the top of the hierarchy and contains only a general construct, (b) the first-order factor model focuses on the intermediate level of the hierarchy and contains only specific constructs, and both (c) the higher order factor model and (d) the nested-factor model consider the hierarchy in its entirety and contain both general and specific constructs (e.g., bifactor model). This tutorial considers these CFA models in depth, addressing their psychometric properties, interpretation of general and specific constructs, and implications for model-based score reliabilities. The authors illustrate their arguments with normative data obtained for the Wechsler Adult Intelligence Scale and conclude with recommendations on which CFA model is most appropriate for which research and diagnostic purposes.

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