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…”

“Researchers have recently answered this criticism with the development and use of standardized knowledge tests (Baumert et al. 2010; Hill et al. 2005; Kleickmann et al. 2012; Voss et al. 2011) based on Shulman’s proposed structure of teacher knowledge in content knowledge, pedagogical content knowledge, and generic pe…”

“However, direct empirical support for such a belief is surprisingly scarce (Borasi, 1994; Jacobs et al. , 2007; Baumert et al. , 2010; Heller, 2010).”

“Knowledge of student misconceptions has long been considered a crucial element of a teacher's skill (Baumert et al., 2010; Ergönenç, Neumann, & Fischer, 2014; Keller, Neumann, & Fischer, 2017; Sadler et al., 2013).”

“…relationship we found between PCK and students’ performance is in line with the results of different studies related to mathematics teachers’ PCK (Baumert et al., 2010; Fennema et al., 1996; Kunter et al., 2013), physics teachers’ PCK (Sadler et al., 2013), or primary science teachers’ PCK…”

“For example, Baumert et al. (2010) found that pedagogical content knowledge was a more powerful predictor of instructional quality than teachers’ knowledge of the content.”

“2008; Hill, Schilling, & Ball, 2004), the Diagnostic Mathematics Assessments for Middle School Teachers project (Saderholm, Ronau, Brown, & Collins, 2010), the SimCalc project (Shechtman, Roschelle, Haertel, & Knudsen, 2010), the Knowledge of Algebra for Teaching project (Senk, 2010), and the COACTIV study (Baumert et…”

“It is not surprising then that several studies provide evidence for a positive relationship between teachers’ CK and their students’ learning progress (Abell, 2007; Baumert et al., 2009; Hashweh, 2005).”

“Moreover, while the well-recognized research in this fi eld is often restricted to German-speaking countries (see Baumert et al. 2010; Klieme et al. 2009) and English-speaking countries (Brophy 2006; Good et al. 2009), our fi ndings support previous research in general, but also contribute to…”

“…of teachers focuses on Shulman’s conceptualisation (1987) of pedagogical content knowledge (PCK) as the knowledge base for teachers designing their lessons (Baumert et al., 2010; Cochran, DeRuiter, & King, 1993; Gess-Newsome, 1999; Grossman, 1990; Magnusson, Krajcik, & Borko, 1999; Shulman, 1986).”

“As a result, instructional quality is manifested as a collective and inclusive way of using multiple teaching practices (Baumert et al., 2010), which is adapted and used throughout this study.”

“PCK has been measured through case studies (e.g., Borko & Livingston, 1989; Morrison & Luttenenger, 2015), scenario-based written assessments (e.g., Baumert et al., 2010), and group tasks of examining how to best teach a topic (e.g., Loughran, Mulhall, & Berry, 2004).”

“It has also made progress in providing empirical evidence of the structure in content knowledge, pedagogical content knowledge, and generic pedagogical knowledge by using standardized knowledge tests (Baumert et al., 2010; Blömeke et al., 2009; Döhrmann, Kaiser, & Blömeke, 2012; Hill, Rowan, & Ball, 2005; Voss, Kunter…”

“Although some researchers emphasize the superior validity of open-ended items (Baumert et al., 2010; Schoenfeld, 2007), others observed no difference between open-ended and closed-ended items (Bridgeman & Lewis, 1994).”

“There is substantial evidence that differences in mathematical knowledge for teaching are linked both to the quality of mathematics instruction that teachers enact and the substance of student learning that results (Baumert et al. 2010; Charalambous 2010; Hill et al. 2005; Tchoshanov 2011).”

“The appropriate method, good learning environment, and students’ active involvement create good learning quality through lecturer support (Baumert et al., 2010; Longobardi et al., 2018; Praetorius et al., 2018; Warwas & Helm, 2018), thus, it is expected to affect students’ academic result.”

“Baumert et al. (2010) demonstrated that the effect of teacher pedagogical content knowledge on student learning was also mediated by the teachers’ providing cognitive learning supports to individual students.”

“Teacher’s pedagogical content knowledge, mediated by individual instructing, is known to have a positive effect on students’ learning (Baumert et al. 2010).”

“Classroom management depends on teachers’ general pedagogical knowledge (see e.g., Baumert et al. 2010).”

“However, these few known studies—COACTIV (Baumert et al., 2010), TEDS-Instruct/ Validate (König et al.”

“Based on research about instructional quality and teacher effectiveness (e.g. Baumert et al., 2010; Kunter et al., 2013), we selected 32 items referring to 10 distinguishable aspects of desirable instructional practices which themselves are well compatible with the three overarching ‘basic…”

“…of Shulman’s ideas about content knowledge and pedagogical content knowledge, some researchers have found that these two are clearly related (e.g., Baumert et al. 2010; Hill 2010), particularly when the measures used focus on the same topic—for example, knowledge of operations with fractions and…”

“CK also has a positive influence mediated by PCK (Baumert et al., 2010).”

“…research focusing on continuous linear relationships between teaching quality and student outcomes, we hypothesize that students taught by teachers who prioritize cognitive activating behaviors will themselves achieve more positive cognitive gains (Baumert et al. 2010; Klieme et al. 2001).”

“In line with this, pedagogical content knowledge has been found to predict both quality of instruction and student achievement (Baumert et al., 2010; Hill, Rowan, & Ball, 2005; Sadler, Sonnert, Coyle, CookSmith, & Miller, 2013).”

“Similarly, in a one-year study of grade 10 students and teachers in Germany, Baumert et al. (2010) found relations between their own assessment of teachers’ pedagogical content knowledge and both instructional practice and student learning gains.”

“preservice teacher education, science teacher education, teacher knowledge learning progress of students (Abell, 2007; Baumert et al., 2010).”

“Baumert et al. (2010) identified differences between non-academic and academic tracks only with regard to the cognitive and curricular level of tasks and respectful treatment of the students.”

“Only Baumert et al. (2010) examined non-mathematical aspects of instruction.”

“…Benachteiligung der Kinder auf niedrigeren Schulstufen an, nicht nur weil die Ausbildung für diese Lehrämter weniger Kompetenzen vermittelt (Baumert et al. 2010), sondern auch weil die Studierenden die vorhandenen Lerngelegenheiten aufgrund ihrer unvorteilhaften Eingangsmerkmale womöglich…”

“almost independently from each other (Baumert et al., 2010).”

“…body of research has highlighted the importance of teachers’ understanding of the subject matter in teaching and student learning (e.g., Copur-Gencturk, 2015; Baumert et al., 2010; Borko et al., 1992; Campbell et al., 2014; Hill et al., 2008; Hill, Rowan, & Ball, 2005; Tchoshanov, 2011).”

“…teaching, the Professional Competence of Teachers, Cognitively Activating Instruction, and Development of Students’ Mathematical Literacy (COACTIV) study showed that student achievement was positively influenced by PCK, mediated by the level of cognitive activation in class (Baumert et al., 2010).”

“…attempted tomeasure a number of factors that influence teachers’ enactment of high-quality mathematics instruction, including teachers’ knowledge, conceptions, and experience (Baumert et al., 2010; Copur-Gencturk, 2015; Hill et al., 2008; Stipek, Givvin, Salmon, & MacGyvers, 2001; Wilhelm, 2014).”

“…researchers in the field of science education still use this work to organise research efforts or to design and carry out research studies (e.g. Baumert et al., 2010; van Driel, Beijaard, & Verloop, 2001; Hashweh, 2005; Hill, Ball, & Schilling, 2008; Lee & Luft, 2008; Loughran, Mulhall, &…”

“Moreover, some studies have used the word ‘generic’ instead of, or interchangeably with, ‘general’ (Baumert et al., 2010; Tröbst et al., 2019).”

“mentioned studies such as COACTIV (Baumert et al., 2010), we analysed – with respect to quality of instruction – teachers’ actions in supporting students’ cognitive activation in classroom lessons.”

“that with low CK, there can be little TSPCK or as stated by Baumert et al. (2010) ‘PCK is inconceivable without a substantial level of CK’ (p.”

“…artifact approach has been used by researchers to assess instructional quality because of its feasibility, cost-effectiveness, and less disruptive collection of multiple snapshots of teachers’ instruction (e.g., Baumert et al., 2010; Boston, 2012; Clare & Aschbacher, 2001; Joyce et al., 2018).”

“PCK emerged as a particularly important construct for science teachers because it has the potential to characterise expert teachers’ professional knowledge and skills and predict effective teaching [5,6,12].”

“For this reason, other authors have preferred to develop quantitative tools to measure PCK on biology (Schmelzing et al., 2013) or mathematics topics (Baumert et al., 2010).”

“PCK could be considered as a measure of teacher quality, with teachers having higher levels of PCK being more likely to teach classes showing gains in content knowledge (Baumert et al., 2010; Hill et al., 2004, 2005).”

“Pedagogical content knowledge (hereafter, PCK) has been a particularly compelling construct among educational researchers (e.g., Ball et al., 2008; Baumert et al., 2010; Charalambous & Hill, 2012; Hill & Charalambous, 2012; Monte-Sano & Budano, 2012; Shulman, 1986), who have widely agreed that it is important in teach…”

“This is supported by studies that found no direct relationship between CK and students’ achievement [104–106] or found that PCK is the less relevant predictor for students’ achievement [74].”

“For in-service teachers, teachers’ professional knowledge may become the main focus as it has been proved to be a key predictor of instructional quality, teachers’ professional knowledge may become the main focus (Baumert et al., 2010).”

“…challenging tasks to their students, refer to students’ prior knowledge (Hiebert & Grouws, 2007), encourage students to find multiple solution ways (Baumert et al., 2010), to generate own solution pathways, or to evaluate own and other’s solutions or ideas (Stefanou, Perencevich, DiCintio, &…”

“These studies used classroom-level data from both the primary (Baumert et al., 2010; Fauth, Decristan, Rieser, Klieme, & Büttner, 2014; Kunter et al., 2013) and secondary school level (Klieme, Pauli, & Reusser, 2009) and found significant associations between classroom management and student…”

“Studies have ranged from investigations of what teachers (and preservice teachers) know (or lack) (e.g., Ball, 1990; Baumert et al., 2010; Hill, 2007; Rowland et al., 2005; Thompson, 1984); what teachers learn from interventions, or other opportunities to learn mathematics (e.”

“Other studies, drawing on Shulman’s (1986) framework, have shown that teachers’ pedagogical content knowledge in mathematics better predicts student learning than pure, or basic, content knowledge (Baumert et al., 2010; Campbell et al., 2014).”

“…able to show that pedagogical content knowledge could be distinguished empirically from content knowledge and that pedagogical content knowledge had a substantial positive effect on student learning that was mediated by cognitive activation and individual learning resources (Baumert et al., 2010).”

“[13] showed that PCK was decisive in influencing both the quality of instruction and students’ learning gains and that content knowledge alone is insufficient for effective teaching of secondary school mathematics.”

“( 2010 ) studied the relationship between instruction and pedagogical content knowledge, which includes knowledge of students.”

“Because MCK can be regarded as an important precondition for applying MPCK successfully in class on a cognitively demanding and flexible level (Baumert et al., 2010), efforts to enhance the mathematics knowledge of U.S. primary teachers seem to be meaningful.”

“Teachers’ design and implementation of tasks and activities is considered a fundamental aspect of teachers’ pedagogical knowledge and has been found to be predictive of student knowledge gains in mathematics (Baumert et al., 2010).”

“[38]) and previous findings on the importance of content-related knowledge facets for subject-specific instructional quality [33,34], PCK can be considered as a pivotal knowledge facet of teachers’ diagnostic competences when the diagnostic focus is on subject-specific instructional aspects.”

“…has also tried to address these problems by following up 9th graders in the PISA 2003 International Assessment Outcomes and Educational Production sample with a curriculum-based student test in the 10th grade and by testing teachers on their subject matter teaching knowledge (Baumert et al., 2010).”