Student's pedagogical knowledge: a source of pedagogical content knowledge

This paper presents the results of interviews with Year 5 and 6 students about their views of effective teaching practices in mathematics. The students interviewed were part of a largescale study into improving middle years mathematics and science. Their views confirm findings from the literature and other data sources from the project, and provide valuable insights into student perceptions of effective teaching practice in middle years mathematics.

Who owns pedagogy? (Re)locating pedagogy and pedagogical content knowledge

This paper draws upon a study in Victoria, Australia which investigated pedagogical understandings and practices specific to three domains: Science, Thinking and ICT. Teachers working in these three domains and key stakeholders from the profession participated in focus group and individual interviews. This study explored the pedagogical knowledge particular to each of these three domains, and common pedagogical knowledges across the domains. Participants in each domain claimed the distinctive nature of their domain and therefore the distinctive nature of their pedagogical content knowledge (PCK). A final common thread was the identification of teacher strategies as a substantial part of pedagogy. The key issues arising from this study are the implications of ascribing pedagogy to particular disciplines and the future possibilities for a curriculum reform agenda within a regressive pedagogical climate.

Deepening the mathematical knowledge of secondary mathematics teachers who lack tertiary mathematics qualifications

A professional learning program for unqualified practising secondary mathematics teachers regarding senior secondary mathematics teaching is described in this paper. The VCE (Victorian Certificate of Education) mathematics professional learning program for senior secondary mathematics was designed for practising secondary teachers of mathematics who had no experience of teaching advanced senior secondary mathematics and who had not completed the recommended qualifications. Professional learning episodes, artefacts and reflections of three teachers who participated in the program are analysed to identify the development of these teachers' mathematical and pedagogical content knowledge (PCK). The PCK framework developed by Chick et al. was used to analyse teachers' knowledge, and the cases of teachers' knowledge presented in the paper illustrate the entwining of knowledge of mathematics and knowledge of teaching and learning The findings indicate that a program designed for senior secondary mathematics can enable practising teachers to deepen and broaden their understanding of junior secondary mathematical pedagogy.

First year pre-service teachers' mathematical content knowledge : methods of solution to a ratio question

In this article, pre-service teachers' mathematics content knowledge is explored through the analysis of two items about ratio from a Mathematical Competency, Skills and Knowledge Test. Pre-service teachers' thinking strategies, common errors and misconceptions in their responses are presented and discussed. Of particular interest was the range and nature of common incorrect responses for one whole-whole ratio question. Results suggested pre-service teachers had difficulty interpreting a worded multi-step, ratio (scale) question, with errors relating to ratio and/or conversion of measurement knowledge. These difficulties reveal underdeveloped knowledge of mathematical structure and mathematical connections as well as an inability to deconstruct key components of a mathematical problem. Most pre-service teachers also lacked knowledge of standard procedures and methods of solutions.

Future teachers' professional knowledge on argumentation and proof : a case study from universities in three countries

In this paper, qualitative results of a case study about the professional knowledge in the area of argumentation and proof of future teachers from universities in three countries are described. Based on results of open questionnaires, data about the competencies these future teachers have in the areas of mathematical knowledge and knowledge of mathematics pedagogy are presented. The study shows that the majority of the future teachers at the participating universities situated in Germany, Hong Kong and Australia, were not able to execute formal proofs, requiring only lower secondary mathematical content, in an adequate and mathematically correct way. In contrast, in all samples there was evidence of at least average competencies of pedagogical content reflection about formal and pre-formal proving in mathematics teaching. However, it appears that possessing a mathematical background as mandated for teaching and having a high affinity with proving in mathematics teaching at the lower secondary level are not a sufficient preparation for teaching proof.

Opportunities to promote mathematical content knowledge for primary teaching

Understanding the development of pre-service teachers’ mathematical content knowledge (MCK) is important for improving primary mathematics’ teacher education. This paper reports on a case study, Rose , and her opportunities to develop MCK during the four years of her program. Program opportunities to promote MCK when planning and practicing primary teaching included: coursework experiences and responding to assessment requirements. Discussion includes the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. By fourth-year, Rose demonstrated development of different categories of MCK when practicing her teaching because of her program experiences.

Developing primary pre-service teachers' mathematical content knowledge during practicum teaching

While it is recognised that a teachers' mathematical content knowledge (MCK) is crucial for teaching, less is known about when different categories of MCK develop during teacher education. This paper reports on two primary pre-service teachers, whose MCK was investigated during their practicum experiences in first, second and fourth years of a four-year Bachelor of Education program. The results identify when and under what conditions pre-service teachers' developed different categories of their MCK during practicum. Factors that assisted pre-service teachers to develop their MCK included program structure providing breadth and depth of experiences; sustained engagement for learning MCK; and quality of pre-service teachers' learning experiences.

Elements of a model of effective teachers

Much research in teacher education has concentrated on individual elements of effective teaching such as the best way to teach content. There has been less emphasis on understanding the complex process of effective teaching in its entirety. Teacher educators are in the business of creating effective teachers and as such need a clear, evidence based model of an effective teacher. We believe that current models of effective teachers are limited because they fail to give sufficient emphasis to many important aspects of effective teachers and fail to integrate these components into a coherent whole and so provide a language for discussion of and a conceptual framework for developing teacher education.

This paper discusses the elements needed for a model of an effective teacher. This model emphasises not only the domains of effective teaching which receive most of the attention in teacher education and evaluation, namely content knowledge, pedagogical knowledge and, more recently, pedagogical content knowledge but also takes into account the teacher's personal knowledge and knowledge of context. We suggest that it is not just this knowledge that teachers have in these domains but the way this knowledge overlaps and interacts both within the teacher and with the teacher's physical, social, intellectual and emotional environment.

An examination of the effective teacher challenges not only teacher educators to rethink the way we educate both preservice and inservice teachers; but also the way we assess, judge and reward teachers.

Generic versus subject specific pedagogy - should mathematics be in the centre?

Research into pedagogy and school change is a high priority in Australia and many other countries. This paper, which includes some preliminary findings from the Improving Middle Years Mathematics and Science: The role of subject cultures in school and teacher change (IMYMS) project, argues that, while there are key features that are common to quality learning environments across all subject areas, generic formulations of pedagogy fail to take account of the extent to which the disciplines being taught shape pedagogy or the contribution of Pedagogical Content Knowledge (PCK) to effective teaching - i.e. that there really is a need to put "mathematics in the centre ".

Cultural diversity as an educational advantage

This paper is based on a longitudinal project that looked into the management of cultural diversity in secondary schools. It discusses how Arab-Australian students and their families understand and construct their own social and educational experiences in relation to wider social discourses. The paper outlines key pedagogical initiatives and community-school partnership initiatives that have been collaboratively developed to effect positive change in the multifaceted schooling experience.