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…”

“, 2021), mediated through instruction including cognitive activation and individual learning support (Baumert et al., 2010).”

“Baumert et al .”

“Cognitive activation refers to teachers providing students with opportunities to solve challenging problems and helping students to develop and apply strategies to solve authentic problems (Baumert et al. 2010; Klieme, Pauli, and Reusser 2009).”

“Teachers play an important role in shaping students’ educational outcomes and experiences (e.g., Ball, Hill, & Bass, 2005; Baumert et al., 2010; Baylor & Ritchie, 2002).”

“Professionswissen von Lehrkräften, kognitiv aktivierender Mathematikunterricht und die Entwicklung von mathematischer Kompetenz (COACTIV): Dokumentation der Erhebungsinstrumente (Materialien aus der Bildungsforschung Nr. 83).”

“In line with the previous research conducted by Myrberg (2007) and Baumert et al. (2010), these studies point to the importance of both content knowledge and pedagogical knowledge.”

“particularly assessment of pedagogical-content knowledge, scores are positively associated with student achievement (Baumert et al., 2010; Hill, Ball, & Schilling, 2008).”

“…possible and necessary for students to actively modify their own learning processes by rewarding intellectual autonomy and creativity (Noftle & Robins, 2007) and encouraging students to fi nd and discuss different solutions to and opinions on a problem or idea (Baumert et al., 2010; Gamoran, 1993).”

“These question-and-answer assessments are the focus of much teacher knowledge assessment research (Baumert et al. 2010).”

“The three dimensions of teaching quality were operationalized according to previous research by Baumert et al. (2010) and Klieme et al. (2009).”

“…(who want to hire effective teachers) because empirical findings show a relationship between pedagogical beliefs and teaching quality (e.g. Baumert et al., 2010; Peterson et al., 1989; Staub and Stern, 2002), while other studies do not confirm these results (e.g., Fang, 1996; Kane,…”

“This is consistent with literature supporting the effects of these sources (Darling-Hammond, 2000; Baumert et al., 2010).”

“Unlike in the field of mathematics education (e.g., Baumert et al., 2010; Hill et al., 2005), the findings in science education regarding the relationship are She et al. at best mixed and underdetermined.”

“…, computational modelling , and conceptual modelling ), level of mathematical argumentation required , and translation processes within mathematics —were deemed most suitable to describe tasks’ potential for cognitive activation with regard to the quality of instruction (Baumert et al., 2010).”

“…subject matter to be taught) and pedagogical content knowledge (knowledge required to teach this subject matter effectively to students; Shulman, 1987); research has demonstrated positive relations of each type of knowledge with student learning (e.g., Baumert et al., 2010; Podhajski et al., 2009).”

“, cognitive activation; [39]) could serve as variables to further explain the relevance of NOS-related learning opportunities for the understanding of NOS.”

“The German project Professional Knowledge of Teachers, Cognitively Activating Instruction and the Development of Students’ Mathematical Competence (COACTIV) analysed the quality of tasks in mathematics instruction (Baumert et al. 2010; Jordan et al. 2008).”

“Thus it is not surprising that several studies provide evidence for a positive relationship between the teacher’s CK and the respective students’ learning progress (Abell, 2007; Baumert et al., 2009; Hashweh, 2005).”

“, 2013), Shulman’s claim was empirically confirmed by several studies in the domain of mathematics, showing that teachers’ PCK positively impacts instructional quality and student learning (Baumert et al., 2010; Kunter et al., 2013). Hence, investing in (pre-service) teachers’ PCK might improve the quality of educatio…”

“Teachers employ cognitive activation strategies to foster students ’ complex thinking abilities, such as critical analysis, problem-solving, and creativity, by actively involving them in the learning trajectory (Baumert et al., 2010).”

“…PCK as one type of content-specific knowledge that is particularly vital for high-quality mathematics instruction and students’ mathematics learning (Baumert et al. 2010; Kersting, Sotelo, and Stigler 2010), compared with other general pedagogical knowledge, such as classroom management skills…”

“Teachers’ mathematical content knowledge MCK and pedagogical content knowledge MPCK are portrayed as having an important relationship with student gains in mathematical knowledge across levels of performance (Baumert et al., 2010), indicating that focusing on mathematical knowledge would be advantageous for all studen…”

“Ausgehend von der Erkenntnis, dass schulisches Lernen in einer Domäne stattfindet (Baumert et al. 2010; Weinert 1994), dürfte die Erfassung fachspezifischer Unterrichtsqualität über die Basisdimensio- K nen hinausgehend für den Lernerfolg von Schülerinnen und Schülern bedeutsam sein.”

“For math competence, a number of studies controlling for school track differences or focusing only on the Gymnasium documented the importance of students’ shared perception of efficient classroommanagement as well as high levels of cognitive activation (e.g., Klieme et al., 2001; Klieme and Rakoczy, 2003; Lipowsky et…”

“Given that teacher education is a crucial period for obtaining a profound deep and conceptual understanding of fractions (Borko et al. 1992; Ma 1999; Newton 2008; Toluk-Ucar 2009; Zhou, Peverly & Xin 2006), for effective teaching to take place, there is a need to focus on the content knowledge…”

“Finally, a number of studies indicate that teacher pedagogic and content knowledge are linked to better instruction (Hill et al., 2008) and student mathematics achievement (Baumert et al., 2010; Kunter et al., 2013).”

“…oriented toward developing scientific understanding and competence in order to prepare students for university, nonacademic track education is aimed at providing students with a general, rather practical education that prepares them for the working world (e.g., Baumert et al., 2010; Blömeke, 2016).”

“…both CK and PCK are considered critical professional development resources for teachers, each requiring special attention during teacher training and classroom teaching practice (Baumert et al. 2010), teachers’ professional knowledge is still difficult to define and articulate (Berry et al. 2008).”

“…posteriores escolhas das estratégias a seguir, recursos a utilizar e feedback a fornecer – a escolha dessas estratégias, recursos e feedback (sua adequação e forma) sustenta-se na especificidade do conhecimento matemático do professor, que molda o seu conhecimento didático (BAUMERT et al., 2010).”

“…researchers (Ball, Lubienski, & Mewborn, 2001; Hill, Schilling, & Ball, 2004) have conducted studies to understand teachers ’ knowledge for teaching mathematics, only a few recent studies (Baumert et al., 2010) have attempted to measure teachers ’ knowledge of data analysis (Bradshaw et al., 2014).”

“Contrary to some empirical approaches in measuring subject-didactic knowledge in Germany, done largely by educational scientists12 (e.g. Baumert et al., 2010; Krauss et al., 2017; Kunter et al., 2013), we postulate a much wider understanding of subject-didactic knowledge (SDK) based on the outline…”

“Research has shown that CK alone is not sufficient to ensure adaptive learning and the learning success of students (e.g., Abell 2007; Baumert et al. 2010; Förtsch et al. 2018).”

“…a teacher’s professional knowledge is related to student achievement (Metzler & Woessmann, 2012; Selling, Gargia, & Ball, 2016), a claim that has been supported by the findings of Baumert et al. (2010), who showed a substantial positive impact of pedagogical content knowledge on student learning.”

“, [30,31]), and corroborative research shows that teacher education programs with a content focus improve the PCK of pre-service teachers who have had few formal opportunities to develop PCK [28,32].”

“One of the next great challenges for teacher research will be to determine how this knowledge can best be conveyed to both preservice and inservice teachers” (Baumert et al., 2010, S. 168).”

“[12] Many studies have found that teachers’ MKT is essential to the improvement of teaching and learning mathematics [13,14] and has been linked with improvements in students’ achievement.[12,15] However, despite such obvious importance, research studies have shown evidence of inadequate knowledge of mathematics for t…”

“In addition, teachers’ lack of content and pedagogical knowledge of fractions has also been pointed out as a critical reason for failure to provide students opportunities for diverse learning experiences with fractions (Baumert et al., 2010; Hill, Rowan & Ball, 2005).”

“The CK instrument consists of 59 items on French grammar and vocabulary that are based on the CK definitions of Shulman (1987), Baumert et al. (2010), and Krauss et al. (2008).”

“This finding builds on previous research that suggests cognitive activation provides students with increased support from their teachers (Förtsch et al., 2017; Hanze & Berger, 2007; Krapp, 2002) and fosters greater interest in learning (Baumert et al., 2010; Fauth et al., 2019; Klieme et al., 2009).”

“Researchers have provided evidence of PCK being an important factor in relation to pupils’ achievement and also teachers’ quality of instruction (see e.g. Baumert et al. 2010; Voss, Kunter, and Baumert 2011).”

“For example, such instruments exist in the mathematics research community regarding the quantification of teachers’ content knowledge within specific topics (Hill et al ., 2005; Baumert et al ., 2010).”

“Transforming content knowledge for student understanding requires teachers to use their PCK (Halim & Meerah, 2002), indicating PCK is the greatest single contributor to explaining student progress (Baumert et al., 2010).”

“Ensuring support for teachers struggling with the mathematical ideas is important since teacher knowledge is closely linked to student achievement in mathematics (Baumert et al., 2010).”

“Recent large-scale classroom-based research has linked specialized mathematical knowledge by teachers to student achievement 19 , and the mathematical quality of classroom lessons has also been studied by the Learning Mathematics for Teaching Project and connected to teacher capacity 20 .”

“By incorporating these strategies, teachers can create a more dynamic and interactive learning environment that stimulates students ’ interest and curiosity in mathematics (Baumert et al., 2009).”