In the past decade, Ball et al. (2008) have worked on two key research projects, namely the Mathematics Teaching and Learning to Teach project and the Learning Mathematics for Teaching project. Through their work over the years, they have developed a model of teacher knowledge which is a practice-based theory of what they call “mathematical knowledge for teaching” (MKfT). As the name implies, this professional knowledge of mathematics is needed by teachers to carry out the work of teaching mathematics (Ball et al., 2005; Ball et al., 2008).
Ball et al. (2008) extended the work of Shulman’s subject matter knowledge (SMK), and pedagogical content knowledge (PCK) into MKfT. Ball’s model of MKfT uses Shulman’s division between SMK and PCK and distinguishes three components of SMK, namely common
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content knowledge (CCK), specialised content knowledge (SCK) and horizon knowledge (HK), as well as three components of PCK: knowledge of content and teaching (KCT), knowledge of content and students (KCS) and knowledge of content and curriculum (KCC), as shown in Figure 3.1.
Figure 3.1 Mathematical Knowledge for Teaching (Ball et al., 2008, p. 403)
This study adopted three of the six domains of (Ball et al., 2008) MKfT: CCK, SCK and KCS, and mapped it with the six strands of (Sapire et al., 2014) error analysis (Figure 3.2). While all of the six strands of mathematical knowledge are critical in the teaching of mathematics, the three domains used in this study were those which the researcher deemed useful in understanding teachers’ engagement with learners’ errors. The literature emphasised that learners’ errors and misconceptions need to be identified, and further argued that learners’
errors and misconceptions actually inform teaching and learning, meaning they are part of the teaching and learning process (Tulis et al., 2016; Siyepu, 2013; Ndlovu et al., 2017). However, the literature does not explicitly explain the type of knowledge that teachers need in order to
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be able to incorporate learners’ errors and misconceptions in the process of teaching and learning. After interrogating Ball et al.’s (2008) strands of MKfT and Sapire et al.’s (2014) strands of analysing errors, the researcher found the three strands, namely CCK, SCK and KCS as critical in this process, thus leading to the new framework design shown in Figure 3.2.
Drawing from Ball et al.’s (2008) framework to explain the types of mathematical knowledge needed for teaching mathematics, this involves identifying the teaching tasks involved and the knowledge needed to teach mathematics effectively. It is based on the premise that teachers need to know mathematics and know how to use mathematics in the work of teaching learners (Ball et al., 2008). Drawing from Ball et al. (2008) and Ball (2017) uncovering the work of teaching, the framework on MKfT encompasses understanding teachers’ use of their teaching ability to improve teaching and learning, the ability to identify the recurrent tasks and problems in teaching mathematics, and the mathematical knowledge, skills, and sensibilities required to manage these tasks. Figure 3.2 provides a diagrammatic representation of the two frameworks combined to form the conceptual framework for this study.
Figure 3.2 Conceptual framework: Enacted from error analysis by Sapire et al. (2014) and mathematical knowledge for teaching by Ball et al. (2008)
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Looking at the combination of the two frameworks for this study in Figure 3.2, the researcher used mapping to link the six criteria of error analysis as purported by Sapire et al. (2014) (procedural understanding, conceptual understanding, awareness of error, diagnostic reasoning, everyday knowledge, and multiple explanations of errors) to three strands of MKfT as posited by Ball et al. (2008), i.e. CCK, SCK and KCS. Ball et al. (2008) focus on explaining teachers’ mathematical knowledge needed for teaching, while Sapire et al. (2014) devise criteria to analyse learners’ errors.
This study explores teachers’ engagement with learners’ errors. The ability to analyse errors requires an individual’s knowledge to teach the subject, in this case mathematics. It is therefore within these parameters that the researcher finds the two conceptual frameworks useful to understand teachers’ engagement with learners’ errors. Teachers’ ability to solve problems that they would expect his/her learners to solve requires their competence with CCK (Ball et al., 2008). However, the ability to analyse learners’ errors also requires teachers’ procedural and conceptual knowledge (Sapire et al., 2014). Therefore, the first key principle needed in the analysis of errors is teachers’ CCK of the mathematics topic, which reveals teachers’
procedural and conceptual knowledge of that topic.
The first block in Figure 3.2 above shows this alignment. In order to understand teachers’
engagement, the researcher explored the extent of teachers’ knowledge to solve the problems given to learners, translated into their ability to explain procedural and conceptual errors made by the learners when teaching and assessing. The second strand, which is the SCK, focuses on teachers’ ability to interpret, diagnose and explain learners’ errors. Possessing such knowledge would allow a teacher to become aware of learners’ errors and to diagnose their reasoning that led to the error in the process of teaching and learning. Therefore, the second principle that grounds the understanding of teachers’ engagement with learners’ errors requires the fundamental knowledge of teachers’ interpretation of errors. The third principle includes KCS, which is also needed in the diagnosis and explanation of errors. The three principles combining three strands of MKfT and strands for analysing errors are essential in exploring teachers’
engagement with learners’ errors.
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In the next section the researcher provides an explanation of the three types of MKfT used in this study, and further discusses the error analysis framework. The researcher brought these two frameworks together to show their connectivity to each other as the lens underpinning this study.
3.5 Mapping teachers’ mathematical knowledge for teaching and error