Strengths and weaknesses of the mathematical knowledge for teaching model

The MKT model has been praised and is referred to as “probably the most influential reconceptualization of teachers’ PCK within mathematics education” (Depaepe, Verschaffel & Kelchtermans, 2013, p. 13). In a systemic review on how PCK was conceptualized in empirical studies in mathematics education, Depaepe et al. (2013) mentioned three distinct merits of the MKT model: First, it was developed out of empirical evidence on the knowledge needed by teachers to teach mathematics. Second, the MKT construct adapted Shulman’s heuristic into a reliable and valid tool to measure teachers’ mathematics knowledge about teaching, and lastly, “MKT provides empirical evidence for a positive relation between teachers’ PCK and student learning outcomes” (Depaepe et al., 2013, pg. 14).

As with all theories, a theory is never an entirety (Foucault & Deleuze, 1977) - there are always conceptual discrepancies and room for refinement and the MKT construct is not exempted from these conceptual discrepancies. Four problems of the MKT model were highlighted by Ball et al. (2008). The first problem stems from the strength of the model and how it addresses the “messiness and variability of teaching and learning” but the framing of the theory is informed by mathematics teaching practice, the classification of knowledge domains does not take into account that some “situations can be managed using different kinds of knowledge.” (Ball, et al., 2008, p. 403). The second and third problems emerged from the first problem on the strength of the theory. The second problem concerns the static nature of the knowledge categories (Ball, et al., 2008). In constructing the MKT model, Ball and her colleagues (2008) explicitly set out to measure how mathematical knowledge plays out in the context in which it is used, namely the classroom, but they did not examine and make the thinking behind the mathematics knowledge explicit. As a result, the “features of pedagogical thinking” that shape the use of the mathematics knowledge “remain tacit and unexamined.” (Ball, et al., 2008, p. 403). The third problem is also referred to as the ‘boundary problem’. The boundaries between knowledge domains are blurry, which “affects the precision (or lack thereof) of our definitions” as it is difficult in some cases to distinguish between CCK and SCK from KCS (Ball et al., 2008). A final weakness identified by Ball et al. (2008), is the lack in understanding whether

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the formulation of these knowledge domains are specific to a particular culture or are culture independent. Furthermore, they also need to understand how different teaching styles/habits might have an impact on the formation of their knowledge domains.

A key aspect that has been overlooked in the development of the MKT model is the lack of understanding how teachers’ conceptualise of the nature of mathematics, and what it means to understand mathematics influences on their own knowledge formation. In other words, the question can be asked what impact teachers’ beliefs of the nature of mathematics has on their knowledge formation. Teachers’ conceptualisations of what mathematics is vary greatly as depicted in the illustration, in Figure 7.

Figure 7: Illustration of continuum of teachers' views of mathematics

As illustrated in Figure 7, at the one end of the continuum mathematics is viewed as a body of facts and procedures and knowing mathematics is seen as the ‘mastery of facts and procedures (Schoenfeld, 1992). On the other end, mathematics is viewed as the “science of patterns” which is parallel to natural science, because it emphasises the empirical seeking of patterns (Schoenfeld, 1992, p. 28). Teachers whose mathematics knowledge for teaching is influenced by an epistemology of mathematics as the “science of patterns”, develop not just mathematical concepts but also mathematical thinking processes i.e. problem-solving, reasoning and proof, communication, connections and representations and exploring of patterns (National Council for Teachers of Mathematics, 1989, 2000). The student teachers whom I studied in this study had different beliefs about problem-solving, for example.

"Mathematics is viewed as a body of facts and procedures

Mathematics is viewed as the "science of patterns"

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Ball et al.’s (2008), MTK framework does not explicitly describe and analyse mathematics content knowledge as consisting of both mathematical concepts and factual knowledge or processes (procedural knowledge). Schoenfeld (2016, p. 4), argued that scientists and mathematicians use a combination of knowledge of “concepts and tools, and practices and habits of mind” to inquire into what makes things work. In addition, scientists and mathematicians refer to this combined knowledge as the “content” of the discipline or disciplinary knowledge (ibid, p. 4). Mathematics content knowledge should, thus, also be described as consisting of both concepts and processes for it is the ‘cornerstone of teachers’ knowledge which affects both what they teach and how they teach it (Harel, 2008c). By implication, the other knowledge domains in Ball et al.’s (2008) MKT framework should also be understood in terms of the mathematical thinking processes as proposed by Foster, Wake and Swan (2014). Foster et al. (2014) concurred that the MKT model does not explicitly describe and analyse teachers’ content knowledge as consisting of both concepts and processes. They therefore, saw the need to adapt the framework to place mathematical thinking processes more “prominently within the consciousness of the mathematics education community” (Foster et al., 2014, p. 98).

2.3.3 Frameworks combining mathematical content knowledge and mathematical