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Since Shulman’s (1986) seminal work on Pedagogical Content Knowledge (PCK) was released, it has created opportunities for the creation of constructs to scaffold the knowledge and understandings that teachers need in order to be effective. Adapting this work from being a heuristic to an operational structure has seen the development of many models. One such model regarding Mathematical Knowledge for Teaching (MKT) (Hill et al., 2008) has enjoyed a good deal of attention. This paper aims to argue for a reconceptualisation of Hill et al.’s (2008) model to make it as informative as possible for teachers and teaching. When Australia fails to improve or maintain its position in the international testing (Trends in International Mathematics and Science Study [TIMSS] or Programme for International Student Assessment [PISA]) in the subject of mathematics, there is an outcry. Headlines such as “Aussie schools flatline in global education tests” (Australian Broadcasting Corporation, 2012) appear on the national broadcaster’s website as a news item. Recriminations are loud and some media, politicians and academics lament the standard of teaching (Aylmer, 2013; Dinham, Ingvarson & Klienhenz, 2008). In identifying quality of teaching as one of many factors which can contribute to determining mathematical achievement throughout the country, this may in some instances be warranted. However, identifying a problem and providing a solution for that problem are two widely different things. The usual response of developing curriculum and articulating required standards (as has happened in recent years in Australian education) is important, but as Ball, Hill and Bass (2005) state “...little improvement is possible without direct attention to the practice of teaching” (p.14). It may seem almost self-evident that if society requires effective learning, then effective teaching is necessary. The questions which need to be addressed are; what constitutes the professional knowledge required for teaching mathematics effectively and can it be represented in a well-considered heuristic. Shulman’s (1987) Pedagogical Content Knowledge (PCK) In his seminal work, Shulman (1987) specified seven categories of professional knowledge required for teaching. These categories are: content knowledge; general pedagogical knowledge; curricular knowledge; knowledge of learners (their characteristics, cognition, motivation and development); knowledge of educational contexts; knowledge of educational aims, goals and purposes; and pedagogical content knowledge (PCK). The category which caused the greatest excitement amongst researchers was PCK (Abd-ElKhalick, 2006; Ball, Thames & Phelps, 2008). Shulman (1986) argued the existence of PCK, a form of knowledge building upon, but not the same as, subject matter knowledge or knowledge of general principles of pedagogy. Australian Journal of Teacher Education Vol 38. 11. November 2013 55 Rather, the epistemological idea of PCK could be described as a link between the knowledge bases of content and pedagogy with a third necessary domain of context (Figure 1). One could characterise PCK as a practical knowledge of teaching and learning guided through a contextualised knowledge of a particular classroom setting. Figure 1: Shulman’s (1986) domains of pedagogical content knowledge P ark and Oliver (2008) reviewed and analysed literature available on defining PCK and arrived at what they considered to be a comprehensive working definition: PCK is teachers’ understanding and enactment of how to help a group of students understand specific subject matter using multiple instructional strategies, representations, and assessments while working within the contextual, cultural, and social limitations in the learning environment. (p. 264) There is research to support that beginning teachers possess a limited repertoire of PCK (Lee, Brown, Luft and Roehrig, 2007; Nason, Chalmers & Yeh, 2012; Wilson, Floden & Ferrini-Mundy, 2002) and that experience is a major influence on the shaping and development of a teacher’s PCK (Kleickmann et al., 2013; Lee et al., 2007). There is further evidence to support that teaching experience alone is not sufficient and that experience, coupled with thoughtful reflection of instructional practices is required (Kleickmann et al., 2013). This thoughtful reflection may help mitigate the findings of Schneider and Plasman (2007) who warn that PCK might actually decline over time. Gess-Newsome and Lederman (1995) also concluded that teaching experience, although an important factor in the development of PCK, is not as significant in contributing to PCK as a teacher’s opportunity and disposition towards reflection on content knowledge. They state: ...teaching experience alone does not equate with teaching expertise, though the two are often mistakenly confused. Opportunities for a teacher to reflect on classroom practice and implement identified changes, however, greatly influenced teaching “expertise”. If teaching is to be a purposeful act, and if we want teachers to be able to translate integrated understandings of content into classroom practice, the time and opportunity to develop, codify, and implement such beliefs into the classroom must be fostered... (p. 321). Refinements of Pedagogical Content Knowledge (PCK) In 1999 Gess-Newsome proposed two models for PCK, integrative and transformative. The integrative model supposes that the relevant knowledge bases used in Australian Journal of Teacher Education Vol 38. 11. November 2013 56 teaching (Shulman’s domains, 1986) are developed separately and that the act of teaching provides opportunity for their integration. Therefore “an expert teacher, then, is one with organised knowledge bases that can be quickly and easily drawn upon while being engaged in the act of teaching” (Silverman & Thompson, 2008, p. 501). The transformative model is a model which recognises the value of synthesised knowledge, the fundamental transformation of knowledge and the creation of new knowledge. According to Silverman and Thompson (2008) the transformative model requires purposefully integrated experiences that allow teachers the opportunity to not only extend their mathematical and pedagogical understandings but also to create connections to create a new knowledge. The transformative model recognises that whilst the knowledge bases of content, pedagogy and context exist they are only useful when transformed into PCK, which by extrapolation cannot occur in a professional learning situation or in pre-service training, but only in the classroom. Is there then an argument to be made that unless teachers are sensitised to the elements that are required for PCK to develop, that they will have issues in creating the required connections whilst in the busy environment of their classroom? Questions need to be asked regarding the efficacy of the current manner in which systems and sectors try to enhance the PCK of teachers and teaching neophytes. To become more responsive when the opportunities for development of PCK in the teaching work place present themselves, teachers need to be given purposeful development opportunities to reflect upon their teaching. This may take the form of having opportunities to observe, analyse and reflect upon other teachers’ teaching (Nilsson, 2008). Furthermore, these other teachers, whether they are classroom based or in a university setting, may in the words of Nilsson (2008) “...need to portray and explicate aspects of their own PCK...” (p. 1296). Developing PCK for teaching mathematics It is well documented that many teachers exhibit weaknesses and lack a deep conceptual understanding of mathematics (Ball, Hill & Bass, 2005; Hill et al., 2008; Ma, 1999;Tsao, 2005). There have also been numerous studies which support the contention that many teachers lack confidence and content knowledge when teaching mathematics (Ambrose, 2004; Evans, 2011; Hill et al., 2008; Kajander, 2005; Norton, 2010; Tsao, 2005). This is supported by Askew (2008) who states that; “...one thing is clear from the research evidence: many prospective and practising primary teachers have, or express, a lack of confidence in their mathematical knowledge” (p. 16). This is a worrying finding, as a positive attitude towards the teaching of mathematics has been shown to have a direct influence on the levels of success that teachers can help students to achieve (Kulm, 1980; Sullivan, 1987). Given this lack of confidence and content knowledge it seems even more important that teachers have some sort of framework upon which to base the development of PCK. Regardless of its pre-eminence, Shulman’s notion of PCK has been challenged (Graeber & Tirosh, 2008) and the concept has been expanded and modified by a number of authors. Ball and Bass (2000) for example, regard the ability to unpack the mathematics from constructs, concepts, analogies, metaphors and images as being an important aspect of PCK. Australian Journal of Teacher Education Vol 38. 11. November 2013 57 Mathematical Knowledge for Teaching (MKT), Hill, Ball and Schilling (2008) Ball, Thames and Phelps (2008) argue that although the term PCK is very widely used, it actually lacks clarity of definition, and its potential has not been fully realised, a view shared by Schneider and Plasman (2011). Their refinements of the concept of PCK and its attempt to reframe the study of teaching knowledge are predicated around placing the emphasis on the use of “knowledge in and for teaching rather than on teachers themselves.” (p. 394). In elaborating on Shulman’s construct of PCK, several research teams (Hill, Ball & Schilling, 2008; Ball, Thames & Phelps, 2008; Delaney, Ball, Hill, Schilling & Zopf, 2008) use a construct (Figure 2) which maps their domains of content knowledge for teaching onto two of Shulman’s (1986) initial categories for PCK, those of subject matter knowledge and pedagogical content knowledge. Hill et al.’s (2008) constru

What Teachers Need to Know to Teach Mathematics: An argument for a reconceptualised model