Primary teachers’ algebraic thinking: example from lesson study

Previous studies have reported on primary children’s algebraic thinking and generalising in a range of problem settings but there is little evidence of primary teachers’ knowledge of algebraic thinking. In this paper the development in algebraic thinking of one primary teacher who taught a research lesson in a Japanese Lesson Study project involving teachers from three primary schools is presented. The findings suggest the need for professional learning in algebra and reasoning and indicate the value of Lesson Study.

Mathematical knowledge of some entrants to a pre-service education course

Standards for teaching emphasise the need for teachers to have deep content knowledge. To assess the mathematical knowledge of students enrolling in its B.Ed. program, the University of New England has introduced a mathematics diagnostic test. This work is the first stage of an ongoing research project into the numeracy needs of students entering the B.Ed. program. The test is a pen-andpaper test that replaces previous on-line, multiple-choice tests. This paper reports on the test results, discusses some common errors made by students and outlines the future direction of the research.

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.

Connecting with the horizon : developing teachers' appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.

Using everyday experiences in teaching secondary mathematics in Malawi : possibilities and constraints for change

Proponents of socially and culturally oriented mathematics education have argued that teaching approaches which value and connect with the learner's prior knowledge and everyday experience are more likely to promote active, meaningful, relevant and liberatory learning than approaches which rely on transmission and abstract presentation of mathematical content. In Malawi, proposals to reform the outdated secondary mathematics curriculum have been made with the aim of aligning mathematics instruction with the social and political changes in the current Malawian society. Using a case study approach, this study investigated the extent to which everyday experiences could be used as a vehicle for changing the learning and teaching of secondary mathematics in Malawi. The study was collaborative, taking place over a period of five months in severely overcrowded and poorly resourced classes in two schools. It involved three mathematics teachers in a cycle of planning and teaching mathematics lessons based on the use of everyday experiences, and observation of and reflection on these lessons, in order to document the effects of using everyday experiences on student learning and teachers' teaching practices. The data was collected through student questionnaires; classroom observations and fieldnotes; interviews and reflective meetings with teachers; and informal meetings with key education officials in Malawi. Mathematics examination results from students involved in this study and a corresponding group from the previous year were collected. A reflective and critical approach was adopted in the interpretation and discussion of the data. Teachers' participation in this study resulted in heightened awareness of their teaching roles and the value of linking school mathematics with everyday experience. The study also shows that students found mathematics interesting and important to learn despite their lack of success in it. In addition, the study documented a number of constraints to change in mathematics instruction such as teachers' focus on mathematics content and examination requirements, and students' resistance to inquiry learning. It also recorded possibilities and barriers to collaboration both between teachers and researchers, and teachers themselves. The findings of this study are timely since they could serve to inform the reform of the Malawian secondary mathematics curriculum currently being undertaken, which began without a critical examination of the classroom conditions necessary to accommodate a socio-politically relevant mathematics education.

Teaching secondary school mathematics : research and practice for the 21st century

For the last decade or so, educational policy makers and researchers in many countries have been calling for significant changes to the way mathematics is taught in secondary schools. Australian mathematics curriculum documents now promote learning goals that go beyond mastery of a pre-determined body of knowledge and procedures - the traditional emphasis on facts, skills, formulae - to include mathematical reasoning and problem solving, communication, and real world applications. There is also pressure to move away from over-reliance on teacher-centred practices such as exposition and individual seatwork, towards activities that promote learners' involvement in constructing, applying, and evaluating mathematical ideas. Further impetus for reform comes from research recommending that if learners are to develop mathematically powerful forms of thinking and habits of mind, then classrooms should immerse them in the authentic practices of the discipline by supporting a culture of collaboration and sense-making. Teaching Secondary School Mathematics - incorporates recent developments in research and practice and applications to teaching mathematics in Australian secondary schools. Covering such areas as curriculum, pedagogy and assessment; teaching mathematical content; equity and diversity in the classroom; and professional and community engagement, it is an invaluable resource for all practising and pre-service mathematics teachers.

The effect of mathematical games on on-task behaviours in the primary classroom

A challenge for primary classroom teachers is to maintain students’ engagement with learning tasks while catering for their diverse needs, capabilities and interests. Multiple pedagogical approaches are employed to promote on-task behaviours in the mathematics classroom. There is a general assumption by educators that games ignite children’s on-task behaviours, but there is little systemically researched empirical data to support this claim. This paper compares students’ on-task behaviours during non-digital game-playing lessons compared with non-game-playing lessons. Six randomly selected grade 5 and 6 students (9–12 year olds) were observed during ten mathematics lessons. A total of 2,100 observations were recorded via an observational schedule and analysed by comparing the percentage of exhibited behaviours. The study found the children spent 93 % of the class-time exhibiting on-task engagement during the game-playing lessons compared with 72 % during the non-game-playing lessons. The game-playing lessons also promoted greater incidents of student talk related to the mathematical task (34 %) compared with the non-game playing lessons (11 %). These results support the argument that games serve to increase students’ time-on-task in mathematics lessons. Therefore, it is contended that use of games explicitly addressing the mathematical content being taught in a classroom is one way to increase engagement and, in turn, potential for learning.

International multi-perspective analyses of classroom activity

This editorial captures the essence of each of the papers in this journal and synthesises the findings to highlight the complexity of developing practices that enable students to develop mathematical understandings. Papers described and discussed come from sources internal and external to Australasia. They include new perspectives on analysis of classroom practices, analyses of mathematical registers and mathematical language in classroom activity, action research of a teacher developing new pedagogical understandings, and various analyses associated with teacher education (pre-service teacher programs, interactions between pre-service teachers and their mentors, and how limited content knowledge can affect teacher interpretation of classroom activity. The editorial captures many of the complexities highlighted in the papers and points to the enormity of the expertise required to optimise the teaching of mathematics..

Adapting the task : what preservice teachers notice when adapting mathematical tasks

This paper reports on an in-depth study that explores preservice teachers’ pedagogical adaptations to a rich mathematical task. Data were collected from six elementary preservice teachers working in pairs to first solve a mathematics problem and then design adaptations to make the problem more accessible and more challenging for diverse learners. Results indicate that preservice teachers are able to draw upon a range of strategies to vary the mathematical content, the context, and the question asked. However, they also did not notice or attend to how their adaptations changed the mathematical structure of the problem. This study provides insights into what is involved in learning to adapt classroom mathematical tasks as an important pedagogical practice.

Using the Internet in high school mathematics

Whilst the technological, pedagogical and content knowledge (TPACK) model has been increasingly adopted for understanding teachers’ use of technology, there have been many calls for greater discussion about the constituent constructs, their relationship with one another and the central TPACK. This paper analyses qualitatively the TPACK demonstrated by the teacher of a Year 11class who used web-based simulated contexts and interactive web objects in a Mathematics Studies course. The findings indicate aspects of TPK relating to academiclearning time and the transformational mode of the technology were not fully realised in this case study. The implications these has for teacher professional development are discussed.

Content architecture applications in healthcare

Healthcare delivery is undergoing significant transformation in the United States. Many hospitals and clinics are utilizing electronic means for maintaining patient records. Implementation of electronic health records is now a prevalent activity at many healthcare organizations. Despite the use of electronic health records by many healthcare organizations, it is still difficult to obtain meaningful information from electronic data pertaining to healthcare. Intelligent content applications organize the data within a company make all the data in the organization searchable and retrievable for faster access. This paper therefore explores SOA (Service oriented architecture) and intelligent content architecture in an attempt to suggest better structures that enable retrieval of related data from myriad sources within a company. While intelligent content applications are being slowly developed for areas such as electronic publishing, its use in healthcare organizations has been limited. This paper discusses applications of intelligent content architecture for the healthcare domain.

Mathematically-rich interactions in early childhood centres

 This study investigated children’s engagement in mathematically-rich interactions in early childhood centres. While mathematically-rich interactions were found to be occurring with regular frequency, educators sometimes had difficulty identifying mathematical concepts in their planning. These findings may be used to inform the content of pre- and in-service early childhood education courses.

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.