Peter Sullivan

Converting mathematics tasks to learning opportunities: An important aspect of knowledge for mathematics teaching

As part of a research and professional development project that focused on the opportunities and constraints provided by different kinds of mathematical tasks, a group of 67 primary and 40 secondary practising teachers of mathematics were asked to complete a survey focusing on their use of tasks. In this article, we discuss responses to one particular item which sought teachers’ ideas on taking a fraction comparison task (which is larger: 2/3 or 201/301?) and converting it into a mathematics lesson in the middle years of schooling. Drawing upon a number of components of ‘mathematical knowledge for teaching’ as a framework, we attempt to examine those aspects of mathematical knowledge which are involved in making such a conversion. Our recommendation following this analysis is that greater emphasis is necessary in professional development settings on taking a potentially useful task and converting it into a worthwhile mathematics learning experience for students. Knowing the relevant mathematics also seems necessary even if not sufficient to make this conversion.

Thinking Teaching: Seeing Mathematics Teachers as Active Decision Makers

Many teachers are now incorporating a broader range of strategies into their teaching, including problem solving, investigations and open-ended questions. Among other things, such teaching requires teachers to talk less but to make more decisions. The acknowledgment of this complexity and the centrality of active decision making have implications for teacher education and teacher development, and for the strategies and resources used.

The Mathematics Teacher and Curriculum Development

The teacher is the key to worthwhile mathematical experiences for children. In this chapter, we offer an appreciation of the crucial role played by teachers in any meaningful curriculum; a recognition that teachers need to be supplied with appropriate resources (text or student materials, teacher support material, relevant technology, and an appropriate physical environment); and an acknowledgment that teachers need time: Time to plan, time to meet together, time to assimilate new content and pedagogy into their repertoire, and sufficient hours timetabled for mathematics. The views of teachers as either irrelevant or merely agents of change are examined, and these are contrasted with a view of the teacher as curriculum maker. Some factors which constrain the teacher’s role in mathematics curriculum development are considered, and a range of curriculum projects and approaches to curriculum policy are discussed. During the discussion, reference will be made to experiences in a variety of countries and contexts, illustrative of the points we are making. We also report in greater detail on three specific examples of curriculum development (in Papua New Guinea, a DutchlU.S. joint initiative, and Australia), and reflect on the implications of these examples for those seeking to maximise the role of the teacher in curriculum development.

The contexts of mathematics tasks and the context of the classroom: are we including all students?

Mathematics teachers are encouraged to use realistic contexts in order to make mathematics more meaningful and accessible for all students. However, the focus group research reported in this article shows that decisions on the suitability of contexts are complex and multidimensional. Similarly, the way the task contexts are presented, and the way the tasks are incorporated into classroom routines have potential to alienate some groups of students. We suggest that teachers and researchers should be sensitive to difficulties that students might experience as a result of both the task and classroom contexts, and take specific steps to avoid or overcome the difficulties.

Using a schematic model to represent influences on, and relationships between, teachers’ problem-solving beliefs and practices

Schematic models have been used extensively in educational research to represent relationships between variables diagrammatically, including the interrelationships between factors associated with teachers’ beliefs and practices. A review of such models informed the development of a new model that was used to plan an investigation into primary school teachers’ problem-solving beliefs and practices. On the basis of the findings from the research, the model was revised to include the important variable of prior mathematics learning, as well as a repositioning of the influence of teaching experiences in classrooms.

Making the pedagogic relay inclusive for indigenous Australian students in mathematics classrooms

Many students are unsuccessful in the study of school mathematics, not because of some innate ability, but because of pedagogical practices. Bernstein (1996) has argued that pedagogy serves as a mechanism for cultural reproduction, so that for those students whose cultures are different from that represented in and through pedagogy, the task of constructing school mathematics is made more difficult. The paper explores the ways in which a teacher changes the pedagogic relay in order to be more inclusive of her students. Her practice is informed by understanding the ways in which pedagogy is a subtle tool for marginalization in mathematics.

FACTORS INHIBITING CHANGE - A CASE STUDY OF A BEGINNING PRIMARY TEACHER

Classrooms are crowded with people and activity, with many things happening at once. The class meets with the teacher for extended periods with events affecting subsequent actions. Many of the constraints have only limited potential for manipulation by teachers. Some popular models of change do not recognise the complexity of the classroom. Mathematics educators and curriculum developers need to acknowledge the difficulty of translating theory into classroom practices which are feasible, and to identify mechanisms for overcoming inhibiting factors. To investigate the constraints on teachers, a number of primary pre-service education students were studied both in their final year and in their first year of teaching. The report on one teacher is presented here. Data are presented on her background, attitudes and beliefs about mathematics teaching, and on her practice teaching experience. Similar information was collected during the beginning year, including beliefs and classroom practice. The observed teaching appeared to emphasise narrow and instrumental goals, which contradicted her stated beliefs and intentions. Some constraining factors were identified.

Quality mathematics teaching: Describing some key components

Survey responses of 125 teacher educators and experienced teachers to fixed-format and open-response items on aspects of mathematics teaching are presented. A qualitative analysis of responses revealed six major categories. These were Communicating, Problem Solving, Building Understanding, Engaging, Nurturing and Organising for Learning. Within each of these categories, the frequency of use of particular phrases and descriptors indicate general beliefs about the important characteristics of quality mathematics teaching. There was a great diversity in the language used to describe particular components. A model is proposed which suggests a way in which the categories are linked.

Issues and Directions in Australian Teacher Education

In most states graduating teachers are required to have 4 years of post school education. For secondary teachers this usually consists of 3 years of discipline study (e.g. a Bachelor of Arts) and 1 year of education studies (e.g. Graduate Diploma of Education). Such education courses in Victoria, for example, commonly consist of the equivalent of about 15 hours per week over 26 weeks of lectures and tutorials, and a further 9 weeks of full time supervised experience in schools.

Students' Responses to Content Specific Open-Ended Mathematical Tasks

The results reported here are from one component of a larger project that explores the classroom potential of content specific open-ended tasks. We propose a framework for discussing such tasks, and justify our focus both on open-endedness and on content. In particular, we examine responses of 1200 students to comparable closed and open-ended tasks and explore the effect of using specific contexts for such tasks. We propose a method for comparing the degree of difficulty of open-ended and closed tasks and conclude that both open-ended and closed tasks can contribute to effective classroom programs.