**2 Theoretical Background and Relevant Research**

**2.2 Beliefs of Teaching Knowledge**

Over the last few decades, a growing body of research has been conducted to elucidate teachers’ mathematical knowledge needed for teaching (Hill et al., 2008). However, most of the extant studies have focused on the ways teachers’ knowledge and beliefs influence student performance (Hill et al., 2005; Rockoff, Jacob, Kane, & Steiger, 2011) and instructional practice (Ball, 1990; Ben-Peretz, 2011; Fennema &

Franke, 1992; Mapolelo & Akinsola, 2015; Wilkins, 2008). In particular, only a small portion of these studies have focused on teachers’ views and understanding of the knowledge needed for teaching mathematics (Hatisaru, 2018; Mosvold & Fauskanger, 2013).

Research on teachers’ beliefs about teaching knowledge, while limited, has yielded some valuable findings (Ferguson & Brownlee, 2018; Fives & Buehl, 2008; Hofer, 2002; Mosvold & Fauskanger, 2013;

Sinatra & Kardash, 2004). Fives and Buehl (2008), for instance, described the concept of personal epistemology in the context of studies about teaching knowledge. The authors employed qualitative and quantitative methods to examine pre-service teachers’ and practicing teachers’ beliefs about teaching knowledge and teaching ability. While

the authors provided valuable insights for developing a framework to conceptualize teachers’ beliefs about teacher knowledge, they called for further investigations using longitudinal and cross-sectional methodologies to explore this topic further, as “such studies would indicate whether these beliefs are developmental in nature and change as one experiences the profession” (Fives & Buehl, 2008, p. 172).

Drawing upon the work of Fives and Buehl (2008) and Philipp’s (2007) concept of belief, Mosvold and Fauskanger (2013) explored the epistemic beliefs that teachers have about the knowledge needed to teach mathematical definitions. The researchers gathered pertinent data via focus-group interviews involving 15 pre-service and in-service teachers in Norway, which was subjected to content and inductive analysis. Their findings revealed that, while some teachers believed that knowledge of definitions is an integral part of their mathematical knowledge for teaching, others opined that the mathematical definitions are important for higher grades but are not necessary for lower-grade students. The participating teachers were, however, aware of the cultural differences in accepted mathematical definitions.

In a subsequent study, Mosvold and Fauskanger (2014) focused specifically on the domain of mathematical horizon content knowledge.

In this context, the authors discussed the beliefs pre-service and practicing teachers have about the knowledge at the mathematical horizon for teaching. A significant finding that emerged from this study was that teachers did not seem to emphasize HCK in their education and practice. When discussing aspects of broader content, participants tended to focus mainly on whether a particular mathematical content was directly related to the curriculum for a specific grade level. This investigation illustrates difficulties encountered when investigating teachers’ beliefs about mathematical knowledge for teaching, as this is a complex phenomenon that some teachers might find difficult to articulate.

In other studies, focus was primarily given to specific characteristics of teaching knowledge. For instance, Leikin and Zazkis (2010) interviewed secondary school teachers about their usage of advanced mathematical knowledge acquired during undergraduate studies at colleges or universities. The authors adopted a qualitative approach based on grounded theory (Strauss & Corbin, 1990), aiming to identify common themes in the teachers’ data. Their findings indicate that most teachers acknowledge the relevance of advanced mathematical knowledge but have difficulties in generating specific problems or recalling situations in which advanced mathematics knowledge can be useful (Leikin & Zazkis, 2010). In particular, only a few participants were able to provide content-specific examples for the purposes and advantages of their advanced mathematical knowledge for student learning, such as personal confidence, and the ability to make connections and respond to students’ questions (Leikin & Zazkis, 2010).

Based on these findings, the authors called for a more articulate relationship between advanced mathematical knowledge and mathematical knowledge for teaching.

While the research briefly reviewed in the preceding sections is relevant to the understanding of how teaching knowledge influences the quality of teaching, how teaching knowledge functions in the teaching-learning process of pre-service teachers during teacher training remains to be established. Extant studies on this topic suggest that teaching knowledge should be examined through the lens of pre-service teachers’

perceptions of knowledge domains (Kilic, 2015), their self-perceptions of the tasks of teaching (O’Meara, Prendergast, Cantley, Harbison, &

O’Hara, 2019), and their views on and understanding of the certainty of teaching knowledge (Ferguson & Brownlee, 2018). Empirical evidence shows that teachers’ beliefs have a strong influence on the way they approach students’ specificities and learning needs (Givvin, Stipek, Salmon, & MacGyvers, 2001), comprehend mathematical knowledge (Cady & Rearden, 2007), and develop their identity as a teachers (Ponte, 2011). Thus, there is a need to examine whether pre-service teachers

truly understand the teaching tasks and the knowledge needed to carry out these tasks during mathematics instruction.

Influenced by the works of Mosvold and Fauskanger (2013) and Fives, Lacatena, and Gerard (2015), the present study focuses on the pre-service teachers’ understanding of the knowledge defined as relevant to the practice of mathematics teaching. Thus, its aim is to elucidate how pre-service teachers develop their understanding of the knowledge necessary to teach mathematics throughout teacher education.

Having presented the theoretical background of this study, it is now important to clarify the concept of belief and the reasons for adopting understanding as the focus of this study.

**2.3 Clarification of the Terms: Belief and Understanding **