This research investigated the relationship between teachers’ espoused professional knowledge, professional knowledge in practice, and student learning, when teaching ‘mathematics for numeracy’ in the New Zealand primary school classroom. The focus was on teaching within the multiplicative and proportional domains, as research at the time the study commenced indicated that these areas of mathematics were problematic for many teachers. The purpose of the research was to identify strengths, weaknesses, and inconsistencies in teachers’ practice; links between espoused views and actions; and to consider the usefulness of a framework for investigating teacher knowledge in practice. This study is intended to inform teacher reflection and professional development, and contribute to improvements in teaching practice and student achievement.
A multiple-case study design, underpinned by an interpretivist paradigm, was used, which included four case-study teachers from two primary schools: School A, was a central city full primary school (Years 1 to 8) and School B, a rural town primary school (Years 1 to 6). The study aligned with a social constructivist perspective on teaching and learning. The data were obtained through four main sources: (1) pre-unit and post-unit student assessment tasks; (2) recorded observations; (3) semi-structured interviews); and (4) teacher questionnaires.
Comparison between students’ initial and final assessment data showed little progress in understanding of multiplication and division, with a more noticeable improvement in fractional understanding. Classroom observations were analysed under three broad categories: content knowledge, pedagogical knowledge, and pedagogical content knowledge, and highlighted important issues relating to the professional knowledge of teachers and the contribution this made to student learning. Results indicated that the mathematical content knowledge of the teachers was stronger than their content knowledge in a pedagogical context. While teaching for conceptual understanding frequently challenged the teachers, they recognised the importance of conceptual understanding prior to procedural learning for their students. They struggled with on-the-spot identification of the next steps of learning for individual students and there was little evidence of focus on questioning that extended students’ thinking that might have assisted in overcoming misconceptions and confusions with concepts. There were times when the teachers’ espoused theories differed from their theory-in-practice, while at times they were similar to each other.
The research concluded that in teaching practice the many facets of PCK, within the broader construct of professional knowledge, were more than topic-specific. Instead, they were person-specific and lesson-specific, with different categories coming to the fore in different proportions, for different reasons, including: lesson structure, context, problem types, the opportunities afforded students for conversation, and use of manipulatives. While not all categories of professional knowledge were evident in every lesson, they combined over a period to underscore the complexities of teaching and ultimately have an effect on student learning.
An outcome of the study was a Wheel of Knowledge designed for teachers, identifying key areas of knowledge to be addressed in mathematics teaching. Alongside this, a more detailed Professional Knowledge Framework was created for researchers, based on categories identified from this research as important in identifying teacher professional knowledge in classroom practice. Both models have the potential to identify areas of teacher professional knowledge required to improve students’ mathematical understanding.
