Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress

“…affect students’ learning gains and motivational development (Bransford, Darling-Hammond, & LePage, 2005; Bransford, Derry, Berliner, & Hammerness, 2005; Grossman & McDonald, 2008; Grossman & Schoenfeld, 2005; Hiebert, Morris, Berk, & Jansen, 2007; Munby, Russell, & Martin, 2001; Reynolds, 1989).”

“In contrast to Hill and colleagues (Hill et al., 2004; Hill et al., 2007), who conceptualized mathematical knowledge for teaching as an amalgam of the mathematical everyday knowledge that all educated adults should have, a purely mathematical understanding of topics typically taught at school, and mathematical knowled…”

“, cognitive activation, classroom management, and individual learning support) have previously been shown to predict student outcomes (Baumert et al., 2010; Dubberke et al., 2008).”

“The findings for PCK are consistent with the particularly low performance of Germany’s nonacademictrack students on the Third International Mathematics and Science Study and PISA assessments—assuming that teachers’ PCK is a crucial prerequisite for student learning (Baumert et al., 2010; Hill et al., 2005).”

“Challenging problems and questions—rather than repetitive tasks and narrow questions— should be used to stimulate cognitive conflicts, engage students in higher-level thinking processes, and support metacognition (Baumert et al. 2010; Lipowsky et al. 2009).”

“Our tests focused on knowledge and conceptual understanding of a specific teaching unit (short term effects), whereas other studies have focused on one-year effects on broader constructs, such as mathematics literacy (Baumert et al., 2010; Dubberke et al., 2008) or the understanding of the nature of science (Fraser &…”

“Professional development with mathematics teachers, for example, focuses on mathematics tasks and problems, which serve as a vehicle for exploring students’ mathematical thinking and problem-solving and how teachers can guide these processes (Baumert et al., 2010).”

“This basic idea forms the foundation of several recent studies on different aspects of professional competence such as teachers’ knowledge (Baumert et al., 2010; Förtsch, Werner, von Kotzebue, & Neuhaus, 2016; Keller, Neumann, & Fischer, 2016), professional beliefs (Kleickmann, Vehmeyer, & Mö ler, 2010; Kunter et al.”

“In line with Shulman’s ideas, some approaches have focused on providing empirical evidence for the proposed structure of teacher knowledge (Baumert et al., 2010; Hill, Rowan, & Ball, 2005; Voss et al., 2011).”

“Direct measures of teachers’ domain-specific knowledge—that is, their content knowledge and pedagogical content knowledge—have now been developed (Baumert et al., 2010; De Jong, Van Driel, & Verloop, 2005; Hill, Schilling, & Ball, 2004; Krauss et al., 2008; Schmidt et al., 2007).”

“Moreover, studies that rigorously investigate the relationship between the different kinds of teacher knowledge and student gains in understanding of science are rare (Baumert et al., 2010).”

“…as well as students’ engagement with the learning content (e.g., Hiebert et al. 1997) by exploring and building on students’ prior knowledge, by stimulating cognitive conflict, by activating higher-order thinking processes, and by fostering students’ metacognition (e.g., Baumert et al. 2010).”

“Specifically, we included scales related to three commonly agreed upon basic dimensions of teaching quality, namely, cognitive quality, effective classroom management, and supportive climate (e.g. Baumert et al., 2010; Klieme, Pauli, & Reusser, 2009; Pianta & Hamre, 2009).”

“Studies using external observer ratings, such as Baumert et al. (2010) and Lipowsky et al.”

“Baumert et al., 2010). More data on MPCK would be urgently needed then. Furthermore, Hsieh, Lin and Wang (in this issue) would prefer an assessment of teacher knowledge as conceptualized by Niss (2003) who emphasized the fundamental characteristics of mathematical thinking across the content”

“For instance, Baumert et al. (2010) showed in a longitudinal study that the objectively measured PCK of teachers can explain class variance in mathematics achievement reasonably well (R 2 = 39%).”

“…teaching on learning teacher competencies ! quality teaching ! student learning; the COACTIV project has shown, for a representative sample of German secondary mathematics teachers, that subject-related teacher competencies have a strong influence on students’ performance (see Baumert et al. 2010).”

“In this manner, Voss, Kunter and Baumert’s (2011) model is similar to those of Slavin (1984) and Carroll (1963) where students and teachers are considered interconnected and interdependent in the teaching and learning process. According to Voss, Kunter and Baumert (2011), their model is domain-general and necessary fo…”

“student achievement have rarely been examined: we identified one study where a significant medium (according to Cohen, 1988; see this reference also in the following for categorization of effect sizes) effect was found for German lower-secondary mathematics teachers (Baumert et al., 2010).”

“Research in the domain of content knowledge and pedagogical content knowledge has shown that deep theoretical and conceptual knowledge is a decisive condition of successful teaching in terms of student learning (Ball & Cohen, 1999; Baumert et al., 2010); the same holds for a teacher’s ability to observe and interpret…”

“At t2, math ematics achievement was assessed by a test developed to cover the standard content stipulated in the federal states' curricula for grade 10 students (Baumert et al., 2010).”

“Clearly, the relevant subject-specific knowledge differs considerably depending on the characteristics of the subject in question (Baumert et al., 2009; Wilson, Shulman, & Richert, 1987).”

“1Humboldt University of Berlin, Germany 2Michigan State University, East Lansing, MI, USA 3National Taiwan Normal University, Taipei, Taiwan Corresponding Author: Johannes König, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany Email: johannes.koenig@staff.hu-berlin.de The growing body of resea…”

“Empirical research has illuminated the many ways in which this knowledge influences their practice (Baumert et al., 2010; Campbell et al., 2014; Carpenter et al., 1989; Franke et al., 2001; Hill et al., 2008; Kersting et al., 2012; Sadler et al., 2013) and measures of teachers’ knowledge have been…”

“On the other hand, Baumert and Kunter‘s [20,25] model is explicitly designed as a generic model, which lends itself for specification and concretization, which has proven itself in studies on action competence of teachers and which can show empirical evidence [36].”

“In an investigation of teacher knowledge involving experienced Grade 10 mathematics teachers in Germany, Baumert et al. (2010) empirically distinguished teachers’ knowledge of mathematics content and pedagogy in an open-ended assessment.”

“Pianta and Hamre 2009) that can be regarded as crucial for students’ learning (Baumert et al. 2010; Klieme et al. 2009; Lipowsky et al. 2009).”

“Baumert et al. (2010) found evidence for the hypothesis that CK forms the prerequisite for PCK in the domain of mathematics.”

“Various studies—mainly from mathematics education—have provided evidence that these three dimensions are empirically separable and significantly influence student progress (e.g., Baumert et al., 2010; Kunter et al., 2013; Praetorius et al., 2018).”

“Even though the concept of instructional quality is understood differently by different researchers in the field of educational effectiveness research, there is agreement that it is a multidimensional construct (Baumert et al. 2010; Creemers and Kyriakides 2008).”

“Interest in these areas has been driven by current federal and state guidelines for mathematics instruction (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010; National Mathematics Advisory Panel, 2008), by prior research that has found consistent links between tea…”

“On the other hand, it is widely known in the tracking literature that teachers in low tracks provide lower-level instruction (Baumert et al., 2010; Oakes, 1985) and have lower expectations of their students (Brophy, 1983), which in turn influences students’ own perceptions of their abilities (Good, 1987).”

“Therefore, we can conclude in line with Baumert et al. (2010) that CK is a necessary, although not sufficient condition for PCK.”

“Positive associations between teacher content knowledge and student outcomes were also found in studies based in Germany (Baumert et al. 2010) and Peru (Metzler and Woessman 2012), and in a comparative study using Programme for the International Assessment of Adult Competencies (PIAAC) data…”

“So far, important educational research studies have been conducted showing that teachers’ knowledge as part of their expertise and professional competence is related to effective teaching and the quality of instruction (e.g. Baumert et al. 2010; Hill et al. 2008).”

“…in Europe (i.e., cognitive activation, supportive climate, and classroom management; Klieme, Pauli, & Reusser, 2009; for a similar model, see Baumert et al., 2010) and in the United States (i.e., instructional support, emotional support, and classroom organization (Pianta et al., 2008;…”

“Theories of instructional quality in school classrooms underline the impact of three dimensions of instructional quality: classroom management, supportive climate, and cognitive activation (e.g., Baumert et al., 2010; Klieme, Pauli, & Reusser, 2009).”

“The studies that actually test teacher knowledge mainly focus on teachers of mathematics (Ball, Thames, & Phelps, 2008; Baumert et al., 2010; Hill et al., 2005; Schmidt et al., 2007; Tatto et al., 2012), thus enriching our understanding regarding mathematics as a core school subject worldwide (Organisation for Economi…”

“1999), there is agreement with respect to two critical aspects of PCK: knowledge of students’ understanding and knowledge of instructional strategies (Park and Oliver 2008; Baumert et al. 2010; Lee and Luft 2008; van Driel et al. 1998).”

“classroom discussions, and other pedagogical strategies (Baumert et al., 2010; Osborne et al., 2015).”

“Most of the existing studies have focused on elementary school or secondary school teachers (Baumert et al., 2010; Blömeke & Delaney, 2012; Hill, Rowan, & Ball, 2005).”

“…knowledge considered most important for school mathematics teaching, but we know much less about the knowledge most important for teaching secondary school mathematics than for primary mathematics (see, e.g., Baumert et al., 2010; Hill, Sleep, Lewis, & Ball, 2007; Schmidt et al., 2007).”

“By doing so, cognitive activation was understood from a content-specific perspective with a focus and an emphasis on cognitively activating and mathematically demanding tasks (Baumert et al., 2010; Klieme et al., 2009; Lipowsky et al., 2009).”

“With only a few exceptions discussed below, most studies examining teacher knowledge kept the curriculum in the background—acknowledging its role, but not exploring its interaction with teacher knowledge in shaping instruction (e.g. Hill et al. 2008, Baumert et al. 2010).”

“Instead, we care whether the construct itself, as measured by our instrument, is of importance to teaching and learning, similar to the approach Baumert et al. (2010) take with teachers’ pedagogical content knowledge.”

“As CK is regarded as a prerequisite for PCK development by many researchers (e.g., Ball, Hill, & Bass, 2005; Ma, 1999), the result of COACTIV could reflect a transfer effect of CK on PCK (Baumert et al., 2010).”

“Teachers ’ professional knowledge may become the focus point because it has been shown to be an important predictor for instructional quality (Baumert et al., 2010).”

“Furthermore, the results challenge the assumption that content knowledge, pedagogical content knowledge and general pedagogical knowledge are of the same nature (Bl€ omeke, Kaiser & Lehmann, 2010; Baumert et al., 2010; Shulman, 1987; Schoenfeld & Kilpatrick, 2008).”

“Establishing links between teachers’ knowledge and students’ achievement eluded researchers for decades, but a few recent studies (e.g., Baumert et al., 2010; Hill, Rowan, & Ball, 2005) have done so using measures inspired by the knowledge categories Shulman (1986) discussed as necessary for…”

“This measurement issue also limits meaningful comparisons that could shed light on how teaching effectiveness relates to educational systems and teacher education in multilevel settings (Baumert et al., 2010; Scherer and Gustafsson, 2015).”