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…”

“For the majority of relevant studies, however, the classical paper-and-pencil assessment represents the dominating paradigm (e.g. Baumert et al., 2010; Hill et al., 2008; König & Seifert, 2012; Tatto et al., 2012) not least because it enables an efficient and reliable way to measure…”

“Research from psychology (Baldwin and Baldwin 1973; Bandura 1977; Gerrig and Zimbardo 2001) has shown that the behaviour of a modelling person is likely to be imitated if a) it is perceived that the behavior has reinforcing consequences for the modeling person, b) the model is perceived as being positive, popular and…”

“PCK emerged as a particularly valuable construct for science teachers due to its capability to characterize specialized teachers’ professional knowledge and skills and predict effective teaching (Baumert et al., 2010; Hill & Chin, 2018; Kirschner et al., 2016).”

“Research on teaching competence provides insights into teachers' professional knowledge as one of the most important cognitive dispositions that impacts teaching quality and student learning (Baumert et al., 2010; Hill, Rowan, & Ball, 2005).”

“This notion is grounded in empirical evidence that supports the positive relationship between teacher quality and student success in learning (Baumert et al. 2010; Calderhead 1996; Hill et al. 2005; Sanders 2000).”

“For example, teachers’ knowledge influences the mathematics achievement of their students (Baumert et al. 2010; Hill et al. 2005).”

“…approaches have been pursued, ranging from exploring associations of teacher knowledge to the quality of instruction and/or student learning (e.g., Baumert et al., 2010; Hill, Blunk, et al., 2008; Hill, Rowan, & Ball, 2005; Kersting, Givvin, Thompson, Santagata, & Stigler, 2012; Shecthman,…”

“Since these representations play an important role inmathematical understanding (Baumert et al., 2010). as part of SCK, teachers should be familiar with them and translate and form linkages among them (Even, 1993).”

“Therefore, there was a strong rationale for educational researchers and practitioners to formulate programs to assist teachers to improve their CK (Ball, et.al, 2008; Baumert et al., 2010; Houseal, et al., 2014; Duschl, et.al., 2007).”

“2012), which will affect instructional quality and, ultimately, student achievement (Baumert et al. 2010; Hill et al. 2005).”

“In conclusion, the accumulated body of evidence from diverse research endeavors affirms that cognitive activation strategies play a pivotal role in fostering positive outcomes in students’ mathematics achievement (e.g., Baumert et al., 2010 ; Jasmin et al., 2017 ; Keller et al., 2017 ).”

“The discovery of various solutions is part of pedagogical awareness of material, a significant indicator of students' academic achievements in mathematics (Baumert et al., 2010).”

“Since then a number of studies have replicated this finding using different datasets (Baumert et al., 2010; Hill, Charalambous, & Chin, 2015; Kane et al., 2008).”

“Their teachers’ pedagogical ability has been repeatedly identified as the essential factor for impacting students’ learning outcomes and motivational levels (Baumert et al. 2010; Hattie 2009).”

“…for the related student, teacher, and school characteristics described in the literature as influencing student achievement (Akiba et al., 2007; Baumert et al., 2010; Carnoy et al., 2008; Georges et al., 2010; Gillies & Collins, 2008; Goldschmidt & Phelps, 2010; Hattie, 2009; Hungi & Thuku,…”

“The studies by Hill et al. (2005) and Baumert et al. (2010) make a convincing case that the teacher’s mathematics knowledge matters; but there is still the need to identify the pedagogical mechanisms by which this knowledge is transferred to specific teaching moments.”

“Baumert et al. (2010) observed a positive relationship between teachers’ PCK and instructional quality.”

“high level of cognitive activation also show higher achievement gains (Baumert et al., 2010; Fauth, Decristan, Rieser, Klieme, & Büttner, 2014b; Mashburn et al., 2008).”

“In other lines of research, teacher knowledge and skills, such as pedagogical content knowledge and content knowledge, are measured directly (see Baumert et al. 2010), but this is not possible to implement in large-scale international studies, unless this is the aim of the study, as was the case…”

“Related studies have found that teachers’ pedagogical content knowledge (Baumert et al., 2010), content knowledge for teaching (Hill et al., 2005), knowledge of student errors and misconceptions (Sadler, Sonnert, Coyle, CookSmith, & Miller, 2013), high-fidelity enactment of standards-based…”

“Tailoring educational experiences to meet the diverse needs of students ensures equitable access to quality education (Baumert et al., 2010).”

“TPACK is rooted in general frameworks, such as the one by Shulman (1986), who proposed three knowledge components for professional teaching (see also Baumert et al., 2010; Hill et al., 2005; Kunter et al., 2013 for empirical applications): a) content knowledge (CK) constitutes teachers’…”

“By contrast, study designs that link directly assessed teacher knowledge to processes of instructional quality and student achievement data potentially allow more substantial and precise correlations (e.g. Baumert et al., 2010; Blömeke et al., 2022).”

“…of knowledge about principles of teaching and learning (Hiebert et al., 2007; Santagata et al., 2018; Seidel & Stürmer, 2014), an aspect of general pedagogical knowledge (Hammerness et al., 2002; Voss et al., 2011) and pedagogical content knowledge (e.g. for mathematics, Baumert et al., 2010).”

“Although there are other important aspects of teaching quality, such as supportive climate or instructional support (Baumert et al., 2010; Hamre et al., 2013; Kunter & Voss, 2013; Lipowsky et al., 2009), classroom management is also a prerequisite for other quality indices (Emmer & Stough, 2001;…”

“In a study involving teachers with significantly different experience, the results of a teaching intervention are difficult to separate from teacher effects (Baumert et al., 2010).”

“[1] also consider selfregulation (SR), beliefs and motivational orientations in their study on teacher competence for mathematics instruction.”

“Ability grouping has been implemented in secondary schools worldwide, and uses students’ academic abilities to assign them to different learning environments (Baumert et al., 2010; Schofield, 2010).”

“This finding is consistent with previous research in this field (Baumert et al., 2010).”

“In the 1980s, Shulman (1987) developed a classification of professional teacher knowledge components, referred to by many empirical researchers when distinguishing between content knowledge (CK), pedagogical content knowledge (PCK), and general pedagogical knowledge (GPK) (see, e.g., König et al., 2016; Baumert et al.…”

“These characteristics include instructional approach (Baumert et al., 2010; Desimone & Long, 2010; Palardy & Rumberger, 2008; Rowan, Correnti, & Miller, 2002), teacher certification (including national board certification) (Clotfelter, Ladd, & Vigdor, 2007; Goldhaber & Anthony, 2007), teacher experience (Clotfelter et…”

“…Practices Scholars have long endeavored to delineate the nature and content of instructional practices that best predict student learning outcomes (Baumert et al., 2010; Darling-Hammond, 2000; Decristan et al., 2015; Hospel & Galand, 2016; Kane & Cantrell, 2010; Marzano et al., 2001; Rjosk et…”

“Assessment tasks can be challenging, activate students’ prior knowledge and engage students in constructive learning processes and therefore make a contribution to cognitively activating instruction (e.g., Baumert et al., 2010; 7 Klieme et al., 2009).”

“Often, implementing a mathematical task in ways that maintain the cognitive demands is one of the more difficult aspects of ambitious teaching (Baumert et al. 2010; Stigler and Hiebert 2004).”

“…(Rowland, Huckstep & Thwaites, 2005); Knowledge for teaching (Davis & Simmt, 2006); conceptualizations focusing on Pedagogical content knowledge (Baumert et al., 2010); Mathematical knowledge for teaching (Ball et al., 2008); and Mathematics teachers’ specialised knowledge (Carrillo, Climent,…”

“Although the terminology used varies [36], the three basic dimensions are (1) classroom management, (2) supportive climate, and (3) cognitive activation.”

“Mit Baumert et al. (2010) und kunter und Voss (2011) ist anzunehmen, dass die hier berichteten effekte über die Inszenierung kognitiv aktivierender und konstruktiv unterstützender Lernumgebungen vermittelt werden.”

“Drawing on prior research such as that of De Corte, Verschaffel, and Masui (2004), Baumert et al. (2010), and Vosniadou, Ioannides, Dimitrakopoulou, and Papademetriou (2001), Dignath and Veenman (2021) describe environments as constructivist when teachers prompt students to associate new…”

“(p. 239) Baumert et al. (2010) show that teachers’ pedagogical content knowledge influenced the cognitive level of tasks with significant impact on pupil attainment in mathematics – this is something that the present study does not account for.”

“Assessment tasks can point to aspects of teacher content knowledge that need to be acquired as part of professional preparation (Ball et al., 2008; Baumert et al., 2010; Döhr-mann, Kaiser, & Blömeke, 2014; Phelps et al., 2020).”

“Professional knowledge has been shown to be highly predictive of instructional quality as well as student learning outcomes (Baumert et al., 2010; Kunter et al., 2011, 2013).”

“2017) and mathematics (Baumert et al. 2010) education to explore the relationship between PCK and practice.”

“Baumert et al. (2010) found that ameasure ofPCKis a significant predictor of studentmathematics learninggains in Germansecondaryschools; furthermore, they found that empirically theycoulddifferentiate teachers’ knowledge of subject matter from PCK, with the latter having a stronger weight in predicting achievement. It…”

“…of teacher professional competences according to Blömeke et al. (2015) focusing on GPK. Teachers’ professional knowledge is considered to be an essential prerequisite for planning challenging, varied and motivating learning opportunities, as well as for student outcomes (Baumert et al., 2010).”

“…certification in the subjects they are teaching (i.e., teaching out-of-field), they often lack the requisite pedagogical content knowledge (PCK) and engage in lower quality instruction (Baumert et al., 2010; Hashweh, 1987; Hobbs, 2013; Ingersoll, 1999; Jones & Carter, 2007; Sanders et al., 1993).”

“In the literature, many authors (such as Baumert et al. (2010); Akanni and Oduaran (2018); Bruffaerts et al. (2018); Chalapati et al. (2018); Keller et al. (2017); Mortenson (2006)) relate lecturers’ pedagogical competencies to students’ motivation for learning, claiming that this is the key to…”

“A number of authors have therefore pointed to the need to narrow down the scope of studies in order to analyse the situation in relation to more limited fields of mathematics (Baumert et al., 2010; Depaepe et al., 2015; Hill, Sleep, Lewis, & Ball, 2007).”

“The distinction between pedagogical and subject-matter knowledge is in line with findings that confirm the distinguishability of these two types of knowledge (Baumert et al., 2010; Kleickmann et al., 2013; Krauss et al., 2008).”

“This support is important because SMK is the foundation for the development of PCK and thus influences their instruction (van Driel et al., 2014).”

“There is evidence that a teacher’s pedagogical math knowledge accounts for some of the variance in students’ mathematical achievement (Baumert et al. 2010; Hill et al. 2008).”

“Selected studies have investigated the role of teacher behavior in the mathematics classroom discussing teacher knowledge, teacher beliefs, and to a lesser extent teacher ‘instructional goals’ (e.g., Sowder, 2007; Philipp, 2007; Baumert et al., 2010; Roesken, 2011).”

“[10,15–17].”

“However, there is no detailed understanding of how CK, PCK, and PK impact teachers’ actions in physics (Cauet et al., 2015; Kulgemeyer & Riese, 2018; Keller et al., 2016), even though promising results come from other disciplines like mathematics (e.g. Baumert et al., 2010).”

“Some studies also have found that cognitive activation strategies enhance mathematics achievement in academic assessments (Baumert et al., 2010; Lipowsky et al., 2009; Yi & Lee, 2017).”

“Naturally, teachers’ professional competence has a demonstrable effect on student ability to learn (Baumert et al., 2010).”

“More recently, Baumert et al. (2010) used a measure of German high school teachers’ pedagogical and common content knowledge, and they found both were significant predictors of the quality of instruction and student performance.”