Perspectives on pedagogical content knowledge in the senior secondary mathematics classroom

Pedagogical Content Knowledge (PCK) is an essential and complex facet of mathematics teacher knowledge that impacts on teaching and learning. Drawing upon Shulman's (1986) early conceptualisation and subsequent research (e.g., Ball, Thames, & Phelps, 2008; Chick, Baker, Pham, & Cheng, 2006; Magnusson, Krajcik, & Borko, 1999; Rowland, Huckstep, & Thwaites, 2005), PCK may be defined as the intricate blend of subject matter knowledge and other aspects of teacher knowledge (e.g., knowledge of curriculum and assessment). This study focused on PCK within a senior secondary mathematics teaching and learning context, by examining the ways in which mathematics teachers convey advanced mathematics ideas to students. The research investigation was designed to explore PCK, as a complex social phenomenon, in a multi-dimensional way. A qualitative research approach in the form of a collective case study, was used to generate and analyse data showing evidence of PCK from multiple sources and perspectives. Three senior secondary mathematics teachers and their Year 11/12 Mathematics Methods students, from two different schools in Northern Tasmania, took part in the study. Data generation methods included observation and video recording of 18 lessons (six per class), post-lesson interviews with the three teachers, as well as focus-group interviews and short written reflections from participating students. These methods allowed the researcher to obtain evidence of the teachers' enacted PCK through: observation, supplementary insights from the teachers' own perspective, and the students' interpretation of the PCK demonstrated by their teachers. Data were analysed for evidence of elements of PCK including those defined in the literature relating to mathematics teacher knowledge (e.g., Chick et al., 2006; Rowland et al., 2005). The results were qualitatively described, depicting the different elements of PCK in action in the classroom and offering insight into this knowledge from the perspectives of the teachers and their students. Findings suggest that multiple and interconnected aspects of PCK were enacted by the teachers in ways that focused on the teaching and learning of mathematics procedures with an emphasis on solving standard text-book exercises. Elements of PCK including knowledge of examples, teacher demonstration, knowledge of student errors, and anticipation of complexity were particularly evident in the data. The teachers' justification for their own instructional choices and actions, evident in their post-lesson interview responses, enhanced the depth and quality of evidence of PCK. Their comments reflected a perception of the constraints of the context of the Mathematics Methods syllabus, particularly in relation to the high stakes external examination. The teachers made pragmatic decisions about what to teach and how to teach it, particularly when unexpected situations arose where teachers had to call upon their own mathematical content knowledge in-the-moment. In general, the teachers avoided addressing the deeper conceptual underpinnings of mathematical ideas in favour of solving standard text-book questions. The students noticed and appreciated aspects of their teacher's PCK, particularly those relating to explicating the steps involved in completing these questions. This study contributes to the research into mathematics teachers' knowledge by exploring some of the complexities and tensions of PCK within the context of the senior secondary mathematics classroom, particularly in relation to the ways in which perceived contextual constraints impact upon this knowledge and its growth.