Mathematical Knowledge for Teaching (MKT) framework

Three content-related domains of knowledge, according to Shulman’s (1986) typology, have been at the center of mathematics education research after he referred to these dimensions of knowledge as the ‘missing paradigm’ in research on teachers. Pedagogical content knowledge (PCK) as the most influential teacher knowledge base has drawn the most interest in research

4 _{Shulman (1987) expanded teachers’ knowledge base to seven knowledge domains; content knowledge,}

pedagogical content knowledge, curriculum knowledge, knowledge of learners and their characteristics, general pedagogical knowledge, knowledge of educational purpose and values, knowledge of educational contexts ranging from the working of group etc.

24

“….because it identifies the distinctive bodies of knowledge for teaching. It represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interest and abilities of learners and presented for instruction” (Shulman, 1987, pg. 8). The notion of pedagogical content knowledge was and still is a topic of interest, not only in research in mathematics education but spanning across subject disciplines such as physics, chemistry, biology, social sciences, English, and so forth (Ball, Thames & Phelps, 2008). However, amidst all this interest, the notion of pedagogical content knowledge has remained ill defined. Ball et al., (2008) assert that definitions of pedagogical content knowledge in studies across subject disciplines have been superficial and lacked clarity. Furthermore, in the instances where there were clearer definitions of it, the boundaries between pedagogical content knowledge and other domains of knowledge remained blurred (Ball et al., 2008). In addressing the lack of clarity in defining what pedagogical content knowledge is and identifying the ‘blurriness’ between knowledge domains, Ball5 (2005, 2008) and her colleagues refined the notion of pedagogical content knowledge and introduced the term, mathematical content knowledge for teaching (MKT) to describe the professional knowledge needed for teaching mathematics.

The conceptual framework of MKT is an artifact of the two projects conducted by Ball and her colleagues over the past two decades (Ball, 1990; Ball 1993a; Hill, Rowan & Ball, 2005; Ball et al., 2008). The aim of these two research projects, the Mathematics Teaching and Learning to Teach Project and the Learning Mathematics for Teaching Project was to identify the mathematical knowledge required to teach. This was achieved by analysing teaching practices in grade K-8 classrooms in the US, and, based on these analyses, “a set of testable hypotheses about the nature of mathematical knowledge for teaching” were developed (Ball et al., 2008, pg. 390). These testable hypotheses lead to the refinement of the notion of PC. As a result, the MKT-model consists of two overarching knowledge categories, subject matter knowledge and PCK, mapped onto Shulman’s ‘subject matter knowledge and pedagogical content knowledge’. Each overarching knowledge category consists of three sub-domains of knowledge. Subject matter knowledge highlights the importance of knowing mathematics and consists of common content knowledge (CCK), specialized content knowledge (SCK) and horizon content knowledge (HCK). Pedagogical content knowledge highlights the importance of learning mathematics and the knowledge needed by teachers to promote this learning and, therefore, consists of knowledge of content and students (KCS), knowledge of content and teaching

25

(KCT) and knowledge of content and curriculum (KCC). The sub-domains of knowledge are illustrated in Figure 6.

SUBJECT MATTER KNOWLEDGE PEDAGOGICAL CONTENT KNOWLEDGE

Common content knowledge (CCK) Specialized content knowledge (SCK) Knowledge of content and students (KCS)

Knowledge of content and curriculum (KCC) Horizon content knowledge (HCK) Knowledge of content and teaching (KCT)

Figure 6: Mathematical Knowledge for Teaching Model (Source: Ball et al., 2008, p. 403)

The first sub-domain in the category, subject matter knowledge, is common content knowledge (CCK). This knowledge domain is referred to as ‘common’ because the knowledge and skills are not just used in teaching - it is used in other settings (e.g. medicine, banking, architecture, engineering etc.) and thus not unique to teaching (Ball et al., 2008). Moreover, the term ‘common’ does not “suggest that everyone has this knowledge” because a person with CCK is able to recognise an error, calculate correctly and successfully solve a mathematical problem (Ball et al., 2008). Ball et al. (2008) cautioned that although they refer to common content knowledge as knowledge known in common with other adults educated in mathematics, they found that it is not always clear what they mean. As a result, the boundaries between common content knowledge and specialized content knowledge is blurry because in some instances it is difficult to discern common from specialized content knowledge (Ball et al., 2008).

The second sub-domain, specialized content knowledge (SCK), is the sub-knowledge domain that drew the most interest from Ball et al. (2008, pg. 400) for it is the “mathematics knowledge

26

and skills unique to teaching”. The teaching of mathematics requires of teachers to have a unique understanding of mathematics and reasoning, meaning teachers should know more mathematics than what is being taught to the learners (Ball, et al., 2008) and have to model mathematical behaviour to the ‘apprentices’ (Brown, Collins & Duguid, 1989) in their care. Furthermore, teaching of mathematics requires “significant mathematical knowledge, skill, habits of mind and insight” (Ball et al. (2008, p. 398). Specialized content knowledge consists of various skills and knowledge. Skills such as ‘unpacking’ (analysing features of) mathematics, seeing whether there are patterns in the errors learners make, and understanding different conceptual models of basic operations constitute specialized content knowledge. Horizon content knowledge (HCK), or otherwise referred to as horizon knowledge, refers to the ability to grasp the interrelatedness of mathematical topics across the curriculum, as well as the interconnectedness of mathematical concepts in general. Teachers with HCK are not only able to understand connections and relationships between topics and concepts but they are able to see connections between representations.

The first sub-domain of pedagogical content knowledge, knowledge of students and content (KCS) is also a sub-domain in the typology of Shulman (1986). Knowledge of students and content is a blending of “knowing about students and knowing about content” (Ball et al., 2008, p. 401). KCS is evidenced through the teachers’ ability to predict the possible errors learners will make in a mathematics task, what learners’ conceptions and misconceptions are when engaged with specific concept learning and which mathematical activities will be difficult, easy, interesting, confusing or motivating. Furthermore, it is evidenced by being able to interpret learners’ thinking, based on their justifications (Ball et al., 2008). In summarizing what knowledge of students and content entails, Ball et al. (2008, pg. 401) say that a teacher’s “knowledge of students and content is an amalgam, involving a particular mathematical idea or procedure and familiarity of what students often think or do.”

The last sub-domain of knowledge, knowledge of content and teaching (KCT), referring to knowledge about best practices - the ability to decide which teaching approach will yield the best learning outcome. KCT comprises (1) designing the instructional task by sequencing the content, (2) thinking about which examples would lead to deeper conceptual understanding of the content at hand, and (3) assessing the pros and cons of different representations when teaching a particular concept. In addition, KCS also includes initiating and leading classroom discussions by, deciding when to pause for more clarification, which questions to pursue to

27

gain a deeper understanding of learners’ thinking, deciding when to ask a new question, or deciding when to assign a new task to enhance learning. Each of the instruction designing tasks involves the blending of mathematical content knowledge and “pedagogical issues that affect student learning” (Ball et al., 2008, pg. 401).