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.

Growth of pre-service teachers’ knowledge and teaching ideas about decimals and fractions : The case of Vivi

This paper discussed and analysed the growth of one pre-service teachers’ knowledge about decimals and fractions during a teaching experiment. Evidence of her progress is based on responses to written test and interview questions. This case shows with probing questions and appropriate teaching ideas, it is possible for a pre-service teacher with initially weak and fragmented knowledge about decimals and fractions to develop a meaningful knowledge about decimals and fractions. The stronger conceptual base provided by use of a concrete representation of decimals enabled Vivi to move away from reliance on memorised facts and rules and towards conceptually based explanations of ideas.

Using the results of diagnostic testing to promote mathematical pedagogical knowledge

Teachers require a range of knowledge bases, including both content knowledge and pedagogical knowledge (Shulman, 1986). In recent times there have been calls from a variety of sources for teacher preparation courses to improve the mathematical knowledge of teachers, particularly primary teachers. These calls have been underlined by the recent formation of bodies such as the Institutes of Teachers in Victoria and NSW, as well as the development of teaching standards by professional bodies including the Australian Association of Mathematics Teachers. Rather than simply adopt a "back-to-basics" approach, work is required that uses the results of educational research to design courses that help pre-service students to understand how and why errors are made (by themselves and by children in their own classrooms). Diagnostic testing of preservice students is the first step in the process. However, it is not enough to simply test students and to remediate their misconceptions. Instead, the aim is to use the results of the testing to improve students' pedagogical knowledge as well as their subject content knowledge. This paper outlines one approach to the use of diagnostic testing with pre-service students and how the results can be used to assist in the development of pedagogical knowledge.

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.

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.

A multiplication curriculum : a short outline

Teachers not only know how to multiply, they also know how to teach children multiplication. But at any one time, most teachers are only concerned with a small section of the whole large process of teaching multiplication. It is easy to lose sight of the wood, because of the close attention being given to individual trees. What is the larger picture? How do children learn to multiply, asks the author? The author discusses the progressive stages of ideas and processes that are involved in learning to multiply. He also provides questions to assist teachers with identifying how far students have progressed in their understanding of multiplication.

Is proof dead in the age of computers?

Some knowledge of what it means to construct a proof is an extremely important part of mathematics. All mathematics teachers and students should have some exposure to the ideas of proof and proving. This paper deals with the issue of creating proofs in mathematics problems.

Effect of social background on the development of probabilistic concepts

This research explores how the social backgrounds of a group of students contributed to their intuitive knowledge in probabilistic reasoning, and influenced their processing of the associated mathematics. A group of Year 11 students who came from families for whom the phenomenon of track gambling formed an important part of their cultural background was identified. Another group consisting of students in the same mathematics course (Year 11 Maths in Society) but from families for whom the phenomenon of gambling in any form was totally absent from their social backgrounds was identified. Twenty students were selected from each group. The research employed a qualitative methodology in which a phenomenographic approach was used to investigate the qualitatively different ways in which individuals within the two groups thought about concepts involving probabilistic reasoning, and processed the related mathematical skills and concepts. The cognitive processes involved in the applications of probabilistic and related mathematical concepts in a variety of both gambling and non-gambling situations were studied in order to determine whether this culturally based knowledge could be viewed as a type of ‘ethnomathematics.’ Data were obtained through individual structured interviews which enabled patterns of reasoning to be compared and contrasted. Analyses of these data enabled intuitive mathematical understandings possessed by the gamblers not only to be identified, but also to be linked with their social backgrounds. Also differences between how individuals in the two groups processed probabilistic and associated mathematical knowledge were determined. This research complements and extends existing knowledge and theories related to culturally-based mathematical knowledge. Implications for further research, for classroom teaching, and for curriculum development in the study of probability in senior secondary mathematics classes are discussed.

Knowledge Construction: individual or social?

For many years, battles raged between those who saw knowledge as perception of a given reality and those who saw it as being constructed through rational activity. More recently, epistemological debates have focused on that activity being essentially individual or social. Initially, the International Group for the Psychology of Mathematics Education (PME) was heavily influenced by the idea that mathematical knowledge is constructed by individuals, and particularly by von Glasersfeld’s “radical” constructivism. The social constructivist thesis that mathematics is a social construction challenged this dominant notion, but Steve Lerman critiqued the “shared consciousness” interpretation of social constructivism and articulated the sociocultural idea of the centrality of social interactions. His influence led a strong turn to the social, with a focus on intersubjectivity in mathematical knowledge and mathematics learning. Steve not only challenged individual people’s ideas but also drew PME into a position where sociocultural and other poststructural theories are in regular use.

Introduction: why pedagogy of school mathematics is important for us?

This textbook is designed to empower pre-service teachers to teach mathematics in schools in China. In this chapter, we briefly described the history of mathematics teaching in the West and in China and the history of mathematics education becoming a science. Then we reviewed some hot research issues associated with mathematics education that occurred from the mid-1950s. We selected four research in literature as examples to illustrate the research methods that PSTs could try in future. We hoped to help PSTs understand that having solid mathematical knowledge is not enough to be an excellent teacher. They need to learn professional knowledge and deal with teaching issues basing on research.

The octagonal cardboard frisby-hovering stability

This article explores the mathematics behind frisbys for the primary school classroom.

Exploring contributions to scholarship in e-learning : weighing up the evidence

Recent views on the scholarship of teaching and learning have conceptualised the work of teachers as a form of inquiry, drawing on scholarly discourse about teaching and learning and communicating new understandings back to the scholarly community for peer review and evaluation. Knowledge about teaching may be based on a variety of forms of evidence, including research, evaluation, reflection, review, and the discussion or development of theoretical perspectives, ideas and concepts. This raises questions about the quality and forms of evidence about teaching which contribute to scholarship, whether these are the same in relation e-teaching and learning as they are for teaching in other contexts, and the implications of different forms of evidence for the relationship between research and teaching. In this paper we examine articles from three recent issues of three journals (two of them relating to e-learning and one to higher education in general), in order to draw some preliminary conclusions about the kind of contributions to discourse about e-learning which may be regarded as valuable in advancing the scholarship of teaching and learning.