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The OECD's Programme for International Student Assessment (PISA) measures the achievements of 15-year-olds on a three-yearly cycle. In 2000, reading was the major domain of assessment and mathematics and science were minor domains. In 2003, mathematics was the major domain and reading, science and problem solving were minor domains. In 2006, science will be the major domain. The paper will review the mathematics results from PISA 2000, with the results of additional analyses of the relationship between social background and mathematics achievement across countries, including separate analyses by gender for Australia and some other countries. Results from PISA 2003 will be published on 7 December 2004. The paper will present details of the assessment framework used for testing mathematics achievement in PISA 2003.

Research into the impact and effectiveness of handheld technologies in the teaching and learning of mathematics shows signs of having matured over the past few years. This ?coming of age? will prove useful in the years ahead. Significant developments in the nature and purposes of the technology demand new questions and new approaches from those that would chart a course for others to follow in their effective uptake. This paper addresses some of these new questions and approaches within the context of the tools that drive them. It proposes a research agenda which is mindful of previous ?blindspots? as well as some new imperatives imposed by radical developments in the technology itself.

The notion of community of practice (Lave & Wenger, 1991) has been very influential over recent years. The focus of mathematics education researchers has been mainly on local communities of practice in schools and classrooms. In contrast, this paper provides an insider perspective of MERGA as a community of practice constituted by a group of researchers who together create, share, and apply knowledge. An examination of the 1994 (Bell, Wright, Leeson, & Geake) and 2003 (Bragg, Campbell, Herbert, & Mousley) conference proceedings and the current MERGA Review (Perry, Anthony, & Diezmann, 2004) traces the changing priorities, focuses, styles and values of the MERGA community.

This paper reports a sociocultural analysis of ICT and numeracy practices in a school serving a disadvantaged community. It focuses on a classroom episode involving a rich task that included ICTs and numeracy. The teacher engaged the students in higher-order thinking and established interactive norms of collaboration and shared expertise. The unique features of this site are described, in terms of its context, the activity, the tools and the interactive patterns between participants, in order to understand how such a system might function in other similar contexts.

This paper presents an overview of the research project being reported on in this symposium. I outline the goals and aims of the research, our understanding of the term ?numeracy? and the potential role of technology to support numeracy learning for all students but with a focus on disadvantaged learners in particular. I sketch the research process to date and mention some of the difficulties we are facing.

This paper presents an account of a classroom teacher?s actions with students when using LOGO. The paper focuses on a pedagogical approach aimed at engaging students in transdisciplinary tasks. Using the LOGO environment as a teaching tool, students were engaged in a task where the teacher?s role was to scaffold students with ?nearly-enough? information for them to continue with the task. Problem posing and scaffolding were central to the teaching approaches adopted by the teacher.

This paper discusses the approaches used by some of the teachers in a large project when using ICTs to support numeracy learning. It was found that many of the teachers used a traditional lock-step approach that has been widely criticised in the general mathematics education literature. These approaches are then compared with a more open-ended, studentcentred approach adopted by one of the teachers. It is proposed that the different approaches offer potentially different learning outcomes for the participating students.

This paper arises from a study whose overall purpose was to investigate the relationship between mathematics teachers? conceptions of beginning algebra and their conceptions of their own teaching practices. Drawing from a larger corpus of data collected over a 6-month period, the paper examines the case of a novice teacher?s conceptions of his teaching of beginning algebra, highlighting the tensions afforded by his conceptions of the contextual factors of teaching. The data show that although this teacher greatly emphasised that his knowledge and dispositions were the crucial determinants of his teaching, in the light of his increased knowledge of the contextual factors of teaching, restructuring his teaching meant to him that these contextual factors were also to be considered crucial as he had to fulfil the requirements of his teaching job.

This paper describes data collected from a study that examined links between the use of problem-solving teaching approaches in primary mathematics classrooms and teachers? beliefs about the role of problem solving in learning mathematics. It appears that teachers held diverse views about the role of problem solving in mathematics teaching, that their reported practices were compatible with their beliefs, and that these beliefs and practices were influenced by identified, external constraints. The constraints included the grade level of the class, the school culture and time pressures.

Concerns about social justice issues in mathematics education have a long history stemming from research on gender to issues related to ethnicity and social class. However, almost non-existent in this literature is a theoretical engagement with the concept of social justice itself. This paper further develops a model of social justice elaborated in previous presentations based on writings of feminist theoreticians. It posits social justice as a multidimensional and transcategorical construct. Further, it applies the model to a discussion of social justice in international collaborations in the discipline.

A New Zealand Professional Standard for beginning primary school teachers states that they will ?understand the implications of the Treaty of Waitangi and te reo me ōna tikanga4?. This paper presents how this standard was demonstrated by a group of teachers within their mathematics programs. Dilemmas, supports, perceived effects on children?s attitudes and achievement, and implications for pre-service education are presented. Findings include that teachers felt constrained by time, school planning, school culture, and children?s lack of prior knowledge.

This paper reports on part of a larger study to compare students? computation and estimation skills. Year 7 students in the Perth Metropolitan area were given a computation test and an estimation test of matched items. The performance on computation was 10 percentage points higher than on estimation. It was clear that students? computations tended to be undertaken mechanically rather than meaningfully. In particular, students were weak at fractions and decimals and showed significant misconceptions. It is recommended that much more time be devoted to estimation as an essential and integrated process in the mathematics classroom.

This paper draws on a study of the mathematics departments of two London secondary schools, and explores some of the ways in which the classroom environment impacts on students? learning of maths. In particular, it looks at the ways in which one school?s efforts to promote equity had some unwanted side effects in terms of limiting the possibilities for students to take responsibility for their own learning. This is set against the very much ?freer? environment in another school, where the opportunities for students went hand in hand with greater inequalities. While resisting a straightforward comparison between the schools, the idea of a ?trade-off? is used to illuminate some of the issues that arose in the two settings. Some suggestions are made for ways to capture some of the best of both schools, but it is argued that there is no panacea, and that ultimately, decisions about how to structure learning must be allowed to reflect the values of the teachers concerned.

This paper describes the first phase of a study that aims to identify the specific English language discourse features that cause problems for English as an additional language (EAL) students in advanced undergraduate mathematics. In this phase observations of lectures and an examination of course notes have been undertaken to try to identify the detailed nature of mathematical discourse features that might cause problems. Building on literature about general features of mathematical discourse, particular examples are identified and analysed, from the point of view of both English and mathematics.

Many rural Indigenous students perform poorly in mathematics as measured by standard tests. This paper discusses contextualisation of mathematics with respect to Indigenous culture and explores the card games the children play out of school. It was found that some of the games were sophisticated and mathematical in terms of strategies needed to succeed. The paper provides descriptions and mathematical analyses of two games which could be used to develop assessment tools to explore the mathematical understanding of Indigenous students and motivating learning activities consistent with existing community mathematics contexts. It draws implications for the use of these games as contextualisation.

This paper reports one aspect of a larger study which looked at strategies used by Grade six students to solve six non-routine mathematical problems. One focus of the study was the relative effectiveness of students? written and verbal communication in revealing their thinking during the problem solving process. The results suggest that students may benefit from instruction on communicating their thinking in writing and that emphasising writing as a guide to students? thinking may disadvantage lower ability students.

This paper reports findings from the case study component of an investigation designed to evaluate the impact of the Count Me In Too early numeracy program in Years 3 and 4 classrooms. Initial anecdotal evidence indicated that the implementation of the program was more difficult than for the corresponding implementation in Kindergarten to Year 2 classrooms. Through the voices of case study teachers, issues surrounding the implementation of the program and potential barriers to professional development emerge and are explored. Interestingly, suggestions as to how some issues are dealt with and how barriers may be overcome are provided by the teachers themselves.

Responses from 42 attendees at a mathematics teachers? conference to a task involving the search for a complete graph of a difficult cubic function were analysed. Neither a specific initial action nor the application of mathematical knowledge guaranteed an immediate global view. However, flexibility of approach, use of the automatic range scaling features, positioning the view shown by the calculator with one?s mental image of the function, and the application of graphing calculator knowledge in conjunction with mathematical knowledge facilitated the teachers? solutions. Focussed experiences involving these should provide the basis of lessons designed to maximise student learning in this topic.

The term affordances is rising in prominence in scholarly literature in mathematics education generally and in technology in mathematics education in particular. A proliferation of different uses and meanings is evident. The roots and use of the term and some of its applications are explored in order to clarify its many meanings. Its potential usefulness for developing a framework for a new research project which aims to enhance mathematics achievement and engagement at the secondary level by using technology to support real world problem solving and lessons of high cognitive demand is investigated.

This paper summarises a report (Brown et al, 2003) to the UK Qualifications and Curriculum Authority of the work of one geometry group. The group was charged with developing and reporting on teaching ideas that focus on the development of geometrical reasoning at the secondary school level. The group was encouraged to explore what is possible both within and beyond the current requirements of the UK National Curriculum and the Key Stage 3 strategy, and to consider the whole ability range.

Integrating everyday and scientific discourses is regarded as essential in developing a deep understanding of specific domains of knowledge. The process of integration, however, may occur in quite different ways. In this paper we analyse two forms of the integration process ? replacement and interweaving ? which provide a heuristic for considering how students might develop facility in mathematical thinking. In an analysis of student talk in a Year 7 classroom we found that replacement and interweaving can facilitate deep understanding. The flexible deployment of both discourse formats is recommended.

How do we make sense of mathematical phenomena? This paper examines the varying discourses of groups of pre-service teachers as they investigated problems using two different approaches: pencil and paper (with equipment available), and spreadsheets. The way this discourse shaped their understanding and generalisation of number patterns, through the differing pedagogical media, is also discussed.

The notion of proof is arguably a fundamental concept in mathematics. Mathematics curricula expect students to develop understanding of proof through explaining and justifying their mathematical responses, and communicating these responses in coherent ways. This study reports the findings from a sample of students in Years 5 and 6 in two schools to a question that asked them to prove a mathematical statement of equality. Survey results from 56 students, and twelve follow-up interviews, were analysed using the SOLO model. Implications from the findings that most students could not use a zero statement to justify their responses, and that answers appeared to be related to language-based factors will be discussed.

This report details the responses of middle school teachers and students on a series of survey questions regarding perception of mental computation practice. Teachers were asked to report how frequently they developed mental computation skills with particular number topics and related activities. The students were asked to report how often they used mental computation to help them in the same topics and activities detailed in the teacher survey. From the two sets of questions it was possible to compare and contrast the responses and also observe individual teachers with their classes.

This paper considers the skills needed for data representation. A framework of transnumerative techniques that facilitate data representation is proposed and applied to the responses of 73 year 7 students to two tasks involving association. Students? responses were classified according to their levels of success in representing the association, and the types of techniques used. It was found that while students had techniques for representing data, their choices of graph type were not always suitable, and they overlooked simple techniques such as ordering and grouping data that could have made their representations clearer. Implications for doing and teaching data analysis are discussed.

The Catch the Future project was part of the DEST Literacy and Numeracy Innovative Projects Initiative and sought to improve literacy and numeracy outcomes for children living in low socio-economic circumstances in a region South-east of Melbourne. The intervention strategy that was implemented focused on children in preschool and child-care centres in the year prior to beginning school. Families were invited to take home a disposable camera and to capture the numeracy and literacy activities that their children were engaged in everyday literacy and numeracy activities. Opportunity was provided for them to share these and discuss them with other parents. Similarly, early childhood teachers were invited to document and share with each other their literacy and numeracy practices. This sharing included the presentation of feedback from the parent meeting. This paper will focus on the parents? conceptions of numeracy as reflected in the photographs and the focus group meetings. The photographs represented a rich range of contexts and mathematics that are embedded in the children?s experiences at home and in the community.

The research was carried out over a three-month period in two high schools in the United States. The six focus students who were selected to participate in the study were asked to solve some problems in geometry requiring the use of variables and unknowns. It was found that some of the difficulties that the students had were generic ones that students usually have in algebra but that others were mainly due to a poor understanding of the underlying geometrical or algebraic concepts.

This study explored the thinking exhibited by two teachers as they implemented a mathematical activity designed to introduce young children to investigations. The video data revealed substantial differences in the teachers? distinguishing teaching characteristics. Differences in the quality of their specific teaching characteristics were also identified using a classroom observation instrument. These differences in teaching characteristics suggest that though the teachers ostensibly implemented the same lesson, there was considerable variation in the learning opportunities for each class.

This paper presents the learning pathways of two children who are part of a larger study of the development of multiplicative thinking. The two children changed their approach to the mental solution of two-digit by one-digit multiplication problems over the course of eight mathematics lessons. They began the unit of lessons using repeated addition strategies and ended it using multiplication strategies. The children differed, however, in the range of strategies they used and in the path they took towards more sophisticated understandings.

Changes in society and the workplace necessitate a rethinking of the nature of the mathematical problem-solving experiences we provide our students across the grades. We need to design experiences that develop a broad range of future-oriented mathematical abilities and processes. Mathematical modelling, which has traditionally been reserved for the secondary school, serves as a powerful vehicle for addressing this need. This paper reports on the second year of a three-year longitudinal study where a class of children and their teachers participated in mathematical modelling activities from the 5th grade through to the 7th grade. The paper explores the processes used by small groups of children as they independently constructed their own mathematical models at the end of their 6th grade.

This paper reports on the ways in which one middle grade teacher listened to, supported, and learned from her students? mathematical reasoning within the context of a modelling task in data analysis. The teacher implemented a sequence of tasks that focused on the development of rating systems through selecting, ranking, and aggregating quantities. Analyses of the teacher?s practices suggested that: (a) the teacher?s mathematical background governed her understanding of both the modelling task and the students? reasoning; and (b) the teacher adopted new roles in her interactions with the students, including a focus on listening and observing, asking questions for understanding and clarification, pressing for explanations and justifications, and encouraging reflection on learning at a meta-level. This study illustrates the potential for using modelling tasks that provide a rich source for revealing not only the students? thinking but also the teachers? understanding of the students? thinking. Such tasks provide a learning tool for teachers as they listen to, interpret, and support students? emerging mathematical ideas.

Fourth-year pre-service primary teachers (N=148) engaged in portfolio development during a semester of preparation for a final 10-week school internship. An interpretive research approach was used to analyse data from reflection sheets, classroom observations, lecturer teaching notes, and draft and final portfolio work samples. The findings indicated how the reflective process of portfolio development was supported by development of a mathematics teaching philosophy, submission of draft portfolio items, lecturer and peer mentoring, and formal assessment interviews with school principals and other educators.

Following assessment by their teachers, about 40% of Grade 1 and Grade 2 participants in the Early Numeracy Research Project were identified as vulnerable in aspects of number learning. This paper explores the domains and combinations of domains in which these children were vulnerable. The findings highlight the diversity of the instructional needs of this group and the challenge teachers face in meeting their diverse needs.

Two methods for teaching drug calculation were compared?one using traditional formula-based teaching methods, the other building on students? existing mathematical problem-solving skills. On the basis of quantitative measures, the formula-based approach appeared more effective. However, students? interview responses revealed that the learning experiences of the two student groups were quite different. The findings are supported by other evidence that alternative teaching methods may be more effective in increasing students? confidence, and achieving better long-term recall and transfer of skills.

For some time it has been known that student use and knowledge of the constituent parts of equations, have not mirrored those of mathematicians. This paper describes some understandings of particular parts of the object ?equation? displayed by lower secondary school students and seeks to analyse them in terms of properties of those parts of equations, including the equals sign. We find that students display a number of different conceptions of what an equation is, and this appears to be connected with their perspective on the role of operators and transitivity. This data has assisted with construction of a framework for understanding the mathematical equation object.

This paper reports on our research with pre-service secondary mathematics teachers in building a community of practice featuring face to face and online interaction. We analyse bulletin board discussions involving our 2003 student cohort in terms of Wenger?s (1998) three defining features of a community of practice: mutual engagement, joint enterprise, and a shared repertoire of resources. The emergent design of the community has contributed to its sustainability in allowing pre-service teachers to define their own professional goals and values.

The importance of building home, school and community partnerships is increasingly acknowledged since family and community involvement in education is thought to be associated with children?s success at school. This paper reports on aspects of an Australian Government commissioned research project that analysed educational partnerships aiming to enhance children?s numeracy education. Snapshots of two school case studies are presented to highlight features of effective partnerships and the kinds of numeracy learning they supported.

There has been concern for some time about how mathematically anxious preservice primary teachers can best be helped to deal with their fears and negative feelings about mathematics so that they can move on to focus on how best to teach children mathematics. This study reports on the effectiveness of a course that was especially established to meet the needs of such students in a preservice primary teacher education program.

The main aim of the present study, using first year university students as subjects, was to discover whether there are any differences in the success rates of students between problems in which they just translate the word problem into an algebraic equation and problems in which they are asked to solve the equation as well. Although the success rates were not very different, a large proportion of the correct solutions to the equation were obtained without using an algebraic equation as part of the solution process. A secondary aim of the study, motivated by the Student/Professor problem, was to investigate students? errors when the relationship between two variables was given as a quotient instead of a product. Because this study raised further questions, it was repeated with some modifications to the questions. This provided more detailed information about students? understanding of variables and showed a significant difference between the proportions correct depending on the order in which the questions were presented.

Most research on both the assessment and teaching of decimal fractions has dealt with the decimal fractions as static units, not ones involved in operations that involve compensation. This study examined 1356 students? ability to add, subtract, multiply and divide numbers that included a decimal portion using the appropriate compensation for each operation. A fifteen-item test was given to 12-year-old students that assessed both their ability to use compensation and decomposition with whole numbers and with numbers that included a decimal fraction. Results showed that if students were able to use the appropriate compensation with whole numbers, between 47% and 64% were also able to use compensation to solve problems dealing with numbers that included a decimal fraction. This indicated that these students understood the part-whole nature of decimal fractions. It is suggested that while compensation has not been used to teach decimal fractions in our schools, it provides an additional way of enabling students to operate with decimals in ways similar to that used by people outside of school.

This paper reports on a study, which seeks to understand mathematics teacher development in low socio-economic schools. It initially aimed to evaluate the effectiveness of four strategies: professional development meetings; peer observations; mentoring teachers in their own classrooms; and providing readable literature. However, it quickly became apparent that implementing these strategies was problematic, and consequently the focus shifted to investigate what was happening and to find an explanation. There existed a contradiction between teachers? espoused desire for exactly these types of professional development and their lack of participation when it was made available and supported. The study indicated that many of the problems with implementing professional development were not with individual teachers but systemic throughout the schools. The obstacles related to teachers? time commitments, energy demands, and their working environment.

Australia?s Teachers: Australia?s Future (DEST, 2003) highlights the increasing demands on the education system to train, inspire and retain outstanding teachers of mathematics. Such teachers, it is stated adopt ?innovative approaches? to teaching while developing in students ?the capacity to be innovative? (p. 6). However, the document itself is far from innovative in its views of how this ideal is likely to be realised. In this paper I adopt a poststructuralist view that the prospective teacher?s capacity to act in innovative ways, though based on knowledge, skills and attitude, is (im)mobilised through how s/he is, and has been, positioned in teaching/learning interactions and relationships in teacher education and schools. Discursive relationships shape professional and mathematical identities and abilities though they are not mentioned in the DEST (2003) document.

In this paper we contemplate the potential dangers of binary thinking in school mathematics. From a poststructuralist perspective, we suggest that binary thinking insinuates itself into classroom practices and relationships and supports or suppresses students? participation and sense of themselves as competent, numerate persons. As well, binary thinking is conservative in that it blinds educators and researchers, and the students themselves, to the long-term effects of discursive practices that can sometimes result in a ?dumbing down? of the curriculum and make authentic participation on the students? part a pretence.

Students? attributional beliefs about the causes of their success and failure can have a substantial impact on their motivation in mathematics. In this paper I present data from a group of Maori and Pacific Island students who were chosen to participate in a summer scholarship offered by the Department of Mathematics. Students? enjoyment of being part of a bigger group and their appreciation of the well-structured support was evident. The analysis shows that for these students the encouragement from external sources was a major motivational factor. The weak link on their motivation is the intrinsic motivation.

Authentic artefacts (including brochures, menus, bus timetables and photographs) provide opportunities for students to develop skills in knowing when and how to use mathematical knowledge for representing and solving problems in both practical and realistic situations. This paper examines the influence artefacts can have on Grade 5 children?s capacity to make sense of scenarios by applying personal knowledge and experiences to the problem context. In this investigation the children routinely modified the problem when they felt that the context was problematic or not realistic. Moreover, the participants revised or extended the problem in ways that made the task considerably more sophisticated and more closely aligned to their perception of authenticity.

Twelve students across Grades 2 to 6 were interviewed individually using a range of tasks, where the mathematical focus was a conceptual understanding of fractions. Careful listening established that despite giving a correct answer and appearing to have conceptual understanding, further probing sometimes revealed that the child had only a faulty procedural understanding. Similarly, success on one task did not guarantee success on a different but related task. Conversely, a task involving a continuous quantity enabled a child to move between discrete and continuous interpretations of fractional parts. This study supported the claimed advantages of one-to-one interviews over pen and paper tests, but also highlighted the importance of careful listening and the need for multiple tasks in the one mathematical domain in eliciting understanding.

Lesson planning usually involves the generation of a hypothetical learning trajectory. This paper illustrates a teaching strategy that is one focus of a major research project. Alternative learning trajectories with different entry level prompts were used to enable students to access the concepts and procedures necessary for their joining the main learning trajectory. The strategy is being trialled in primary classrooms that have a large proportion of lower SES students, with the aim of maximising success in mathematics for all students.

This paper provides an overview of The Numeracy Research in NSW Primary Schools Project. An interview-based Numeracy Assessment Instrument K-6 and a Numeracy Achievement Scale monitored students? numeracy growth using Rasch modelling. Students from Trialling schools demonstrated greater than expected numeracy growth compared with their counterparts in Reference schools. Factors found to be ?making a difference? in numeracy achievement in the 45 Case Study and the 10 Trialling schools highlighted the leadership and support of key group teachers and the principal; consistency and continuity of teaching practices and whole school planning, a focus on the language of mathematics and application of practical resources.

Orthodox mathematics education emphasises conscious deliberative modes of knowing and learning, and neglects the constructive workings of the unconscious which are crucial for fluent and effortless mathematical know-how. This know-how, in part, results from a mindful attentiveness that is primarily unselfconscious, ethical and aesthetic. Effortless mastery is a mode of learning that leads to this know-how, and, in order to attain it, effortlessness must be a central focus of mathematics learning from the start. Effortless mastery also provides new insights into the phenomenon of ?maths anxiety?.

The orthodox mode of social justice in mathematics education is code-justice, and, in its disregard for the ethical realm of qualitative discrimination, it boils down to proceduralism. An alternative mode, ethics-justice, gives primacy to the ethical, and shares features in common with the jazz metaphor. From the perspective of ethics-justice, but not necessarily from the perspective of code-justice, streaming can be seen as an injustice. Recent changes in the social world, and in the self-constitution of the individual, have led to the need for ethics-justice to be given primacy. Further, humanistic mathematics has become a justice?and not just a philosophical or pedagogical?issue, and needs to be given renewed emphasis.

This paper reports the results of a factor analysis of data from a survey of teachers? beliefs and practices relating to the compulsory numeracy tests conducted in Years 3, 5 and 7. The resulting six factors related to giving feedback to students, using the tests for diagnosis of pupils and content, changes in teachers? practice, comparing results with other schools, test validity, and preparing pupils for the tests. Analysis of factor scores showed the significance of professional development in teachers? practices associated with the tests. School location and size also had an effect on teachers? beliefs and practices associated with the tests.

This paper examines Year 7 students use and learning of ratio concepts while engaged in the technology practice of designing, constructing and evaluating simple machines, that used cogs and pulleys. It was found that most students made considerable progress in accounting for ratio concepts in their constructions and some constructed sophisticated machines and provided explicit and quantitative descriptions involving ratio reasoning. The findings have implications for the study of mathematics in integrated and contextual settings.

This research examines the relationship between the mathematical background of approximately 300 first year B. Ed (primary) students entering the education faculty and their achievement in the first year mathematics foundations unit at Queensland University of Technology (QUT). Students? mathematical backgrounds were divided into five categories according to level of achievement at high school ranging from success at advanced level mathematics to having done no recent mathematics at all. The performance of each group was compared for overall achievement in the Mathematics Foundations unit. In addition the results in the Foundations Unit were correlated with the results in the Mathematics Curriculum unit and two other core units taken by all students. The paper draws some implications regarding the selection of students for the course and their mathematical needs.

Many researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. This paper examines the persistence of whole number schemes among 14 year-old students who appear to have successfully mastered routine algorithms for working with fractions. Uncovering whole number thinking among such students is therefore difficult, and is illustrated through the use of several probing interview tasks, revealing quite different forms of whole number thinking. These forms of thinking can give correct answers also making it difficult for teachers to identify incorrect thinking about fractions. Representations of fractions using number lines can assist in identifying and correcting such thinking.

This paper describes the early experiences of secondary mathematics teachers in four rural schools in NSW as they began to explore the changing nature of assessment and the implications it has for their practice. Two important findings emerged from their involvement and experiences. First, teachers reported that using a theoretical framework to guide decisions acted as a catalyst to their rethinking of how and what to assess. Second, teachers used assessment information as a trigger for improving their teaching practices to enhance student learning.

This paper reports findings on students? approaches to learning from a larger study which investigated the effects of a university calculus assessment program on students? attitudes, perceptions, learning approaches and associated learning quality. Data on sixteen students? learning approaches were collected through the use of questionnaires, interviews, assessment submissions and anecdotal evidence. The data suggest students? use of deep learning approaches was facilitated by the nature of assessment tasks and students? perceptions of assessment requirements, and had a positive effect on learning quality.

This paper is based on the results of a 4-year study investigating primary pupils? real-world problem solving and modelling strategies. In order to foster and highlight cooperative mathematical modelling processes, different real-world related Fermi problems were given to three grade 3 and 4 classes. Interpretative analyses of all group work episodes from these classrooms suggest that while most groups did not develop and implement a solution plan, in most cases multiple modelling cycles led to highly appropriate solutions.

Encouraging students to develop effective use of Computer Algebra Systems (CAS) is not trivial. This paper reports on a group of undergraduate students who, despite carefully planned lectures and CAS availability for all learning and assessment tasks, failed to capitalize on its affordances. If students are to work within the technical constraints, and develop effective use of CAS, teachers need to provide assistance with technical difficulties, actively demonstrate CAS? value and unambiguously reward its strategic use in assessment.

We describe the work of two groups of six 8 year-old children as they plan Father Christmas?s epic journey on December 24th. We trace how the children draw upon formal and personal knowledge to make connections with a range of external representations. We conclude that the context, despite its distracting potential, is critical in supporting engagement, while the representations act in mutual support of the construction of a utility for directed number (Ainley & Pratt, 2002).

This study investigated views held about mathematics education by parents of a small inner-city primary school. Similarities in responses indicated consistency of ideas. Respondents demonstrated a broad understanding of mathematics and disagreed with a number of stereotypical statements about mathematics and mathematics education. They were keen to increase their knowledge and understanding of the mathematics curriculum and teaching practices. They showed high levels of interest in assisting their children with mathematics and provided a wealth of ideas for supporting learning.

Forty-eight students across Grades 3 to 6 were interviewed individually using a range of tasks, where the mathematical focus was decimal knowledge and understanding. Students who may be categorised as ?apparent experts? on a decimal comparison test were found to differ considerably in their ability to perform ordering and benchmarking tasks. Those students whose explanations when comparing decimals reflected a greater place value knowledge and who were not following a rule which ultimately treats decimals as whole numbers, appeared to have a more conceptual understanding of the decimal numeration system and were able to apply this understanding to more difficult (or novel) tasks.

This paper reports on an action research study in which a composite class of Years 6-8 students were involved in a collaborative learning training programme. The intervention was designed to promote the students? engagement in verbal interactions. Categories of talk were analysed before and after the training sessions to determine both the amount and the cognitive level of talk. Students showed a gain in the mean amount of cognitive talk and higher cognitive talk after the intervention.

A prior socio-cultural inquiry into the professional socialisation experience of immigrant teachers of secondary mathematics in Victoria had focussed on their responsive approaches to perceived value differences (Seah, 2003b). By adopting a different angle of a similar lens of inquiry complementarily, through questioning the nature of the value differences perceived, the immigrant teachers? socialisation experiences are contextualised in terms of interactions between cultural values. This paper outlines some of these value differences, and highlights values related to the Victorian mathematics educational culture.

Jane was a mathematically talented female who chose not to study mathematics in Year-13 at secondary school. Jane had many ?open doors? available to her and opted to continue with an arts course. She preferred classics to calculus. Her reasons included that calculus was boring, not enjoyable, and had no perceived future use. Jane stated she ?hated maths?. This article aims to determine the factors that caused this dislike and influenced her decision to choose an alternative path away from mathematics.

In this paper I make a case that the process of becoming a teacher of mathematics can be enhanced when teacher educators adopt a theory/practice/reflection cycle of inquiry that occurs within a community of inquirers. An account of prospective teachers? narrative inquiries undertaken in the final year of their B Ed (Primary) course is presented to illustrate the nature of learning within a community of inquirers. Through the narrative analysis of data focused on the presentation of posters created at the end of a year-long subject, I explore the metaphors that portrayed prospective teachers? collective experience as practical inquirers. The choice of metaphors suggest that prospective teachers had begun to develop an inquiry stance as part of their forming identities as future classroom teachers.

The Mathematics Education Research Journal is a highly regarded international research journal that allows contributors to disseminate their research findings to a wide audience. The peer review process that contributes to its quality depends very much on the reviewers who for the most part are from the ranks of MERGA membership. The reviewing process is sometimes questioned by contributors so an attempt to clarify some of the critical issues involved in the process is timely. The purposes, criticisms and the criteria used are explored in relation to twenty papers reviewed during the years 2000-2003.

This paper provides an overview of the major results of a large-scale longitudinal study of students? misconceptions of decimal notation, drawing them together and presenting refined results. Best estimates of the prevalence of various misconceptions about decimal numbers from both cross-sectional and longitudinal perspectives are provided, as well as some estimates of persistence. Strengths, limitations and suggestions for improvements to the Decimal Comparison Test as well as major implications for teaching are discussed.

Grouping according to perceptions of ability is widely used in the teaching of mathematics in many countries. These practices may be viewed as operating within discursive complexes concerned with the mathematically able child. This paper uses Foucault?s theories of discourse to argue that such a child is discursively produced, and that the differentiating pedagogies that characterise mathematics teaching in many New Zealand primary schools are supported within such discourse. It investigates the dispositif surrounding the ?mathematically able child?, and considers implications of dominant discursive accounts of mathematical ability for young learners.

This paper confronts the issue of pre-service secondary teaching identities. Rather than theorising learning to teach as an outcome of belief change, an aftermath of being there in the classroom, or as a function of pedagogical experience, it makes a strategic engagement with social practice theory. Drawing on Lave?s work, one student is traced through the process of becoming a secondary teacher, within and between three unique contexts, each of which represents different and competing relations of knowledge, dependency, commitment and negotiation. In the telling, the student moves through legitimate periphery participation towards approximating full membership of the secondary teaching community of practice.

This paper reports on an investigation of the nature and extent of teacher?s knowledge of mathematical patterning and how this knowledge is applied in their daily planning and day-to-day interactions with children. Two case studies indicated that teachers acknowledged the salient role of mathematical patterning and were satisfied with their understanding of patterning processes. However observation of these teachers? classes suggests that there were limited worthwhile patterning opportunities for children.

This report considers the difference in performance of students in Grades 3, 5, 7, and 9 over a two-year interval, 2000 to 2002, on a survey of concepts in the chance and data curriculum. Two types of comparisons occur. First, longitudinal change within students is measured and this is compared for students in schools where an instructional intervention occurred and those in schools with no intervention. Second, in Grades 5, 7, and 9 in 2002 performance is compared with that in the same grades in the same schools in 2000. Reasons for differences are suggested based on the educational experiences that the 2002 students experienced in the intervening time. Students answered questions concerning basic chance and data understandings, and variation in chance, data and graphing, and sampling. Scores on these four subscales as well as the overall survey are used to make comparisons.

This report considers beliefs about chance in relation to understanding of random processes and luck. Changes in chance beliefs over two- and four-year periods were measured for 265 students initially in Grades 3 and 6. Students were asked three questions on repeated occasions, based on the meaning of the word random, their beliefs about luck, and their beliefs about how luck affects winning a lottery. Change was also measured for performance on three questions related to chance measurement. The association of chance belief and chance measurement was found to be weak. Educational implications are considered.

The New South Wales Pedagogy Model defines three dimensions: Intellectual Quality, Quality Learning Environment, and Significance. The elements of Background Knowledge and Connectedness from the Significance dimension are particularly pertinent to mathematics. The former refers to teaching so that new knowledge is built on existing knowledge, while the latter refers to applying results in ways that have meaning beyond the classroom. This paper argues that mathematics teaching of both early ?empirical? and later ?invented? mathematics too often has tenuous links to previous knowledge and at best provides superficial applications to real life. It is argued that quality teaching at both levels, while having different emphases, should employ a similar approach to Background Knowledge and Connectedness, namely teaching for abstract-general concepts. However, what constitutes meaningful learning varies with the individual, and invented mathematics may be inappropriate for a large number of students.

This paper concerns algebraic errors committed by secondary school Thai and Japanese students. The first author created the Equation Test in the two languages, and the study showed that both groups of students made errors concerning the equalising property and in understanding the verbal problems. However the Thai and Japanese students also committed different types of errors, and the paper relates these errors to the style of problems found in their respective textbooks.

Many older people bemoan the numeracy skills of a nation?s youth and cast youth in deficit models. This public display has created new forms of training and policy that seek to redress perceived innumeracy. This paper reports on a small component of a larger project investigating the numeracy practices of contemporary work. Using a case study approach with a tool known as stimulated recall, young people across 19 sites were each workshadowed over a period of at least 3 days. The outcomes of these cases show that young people approach tasks very differently from older people and that technology has shifted the emphasis on what and how numeracy activities are undertaken.

Many students have traditionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. This study investigated the value of introducing students, after standard methods, to a Vedic method of multiplication of numbers that is very visual in its application. We wondered whether applying the method to quadratic expressions would assist student understanding, not only of the processes but also the concepts of expansion and factorisation. We found that there was some evidence this was the case, with some students preferring to use the new method. Thus results suggest it may be a useful adjunct to traditional approaches.

Many proponents of the use of technology in mathematics education argue that to be effective, such usage should be fully integrated, but exactly what is meant by integrated technology is seldom defined. A number of tertiary mathematics teachers were surveyed to investigate the degree to which technology is used in their courses, and the information received was used to identify several aspects critical to an examination of integrated technology. The paper then proposes a model for helping to describe and compare the degree of integration of technology in different courses.

This project assesses the underlying mathematical knowledge of 53 children from two Sydney pre-schools, one year prior to formal schooling, and monitors their on-going mathematical development. The ability to generate and abstract simple mathematical patterns is explored through a range of problem-solving tasks. Data comprise structured videotaped individual interviews, children?s representations, observational and anecdotal records. A case study of one pre-school analyses an early mathematics intervention program highlighting mathematical patterns, structures and relationships. The study analyses teacher and community perceptions; the effectiveness of this intervention on children?s mathematics learning, the professional development of pre-school staff and their programs.

An integral part of all initial teacher education programs is the school-based practicum where preservice teachers get an opportunity to develop their teaching skills and knowledge in a classroom. Many have suggested that these experiences are very powerful in shaping prospective teachers views of teaching as they are perceived as being ?real? as opposed to the ?artificial? environment of the tertiary courses. This can mean that the practicum experiences can legitimate or negate the learning that occurs within tertiary courses. In particular, preservice teachers can make significant positive changes in their affective responses to mathematics, but the longevity and stability of these changes can be severely challenged through their school-based practicum experiences.

Teaching young adolescents today requires not only an understanding of content and pedagogical knowledge, but an understanding of students? developmental stages and an approach which prioritises the students? preferred learning environment. From a case study of a Year 9 mathematics classroom, this report will identify and explain how certain characteristics and qualities of a student-directed program and a range of classroom conditions influenced student attitudes to the learning of mathematics. The focus will be student perceptions of the learning of mathematics along with an examination of students? preferred learning conditions and the influence of the classroom environment on mathematical learning.

The benefits of a holistic approach to education has been acknowledged by educators and, in view of the historical background of mathematics as a discipline, it may enhance performance in mathematics if more emphasis is placed on the wonder and mystery formally associated with the subject than is currently the case. In a tentative trial in this regard, two secondary pre-service students were asked to respond to a series of questions and images related to the spirit of mathematics and their own views of spirituality. From two different backgrounds, these responses identified concepts of order and relationships as key factors in their views of mathematics and its spirit.

Researchers have emphasized the importance of developing relational thinking in regard to early equation solving. An assessment instrument was designed to assist and facilitate the process of improving the development of students? strategies. The purpose of the instrument was to assist teachers to analyse and classify students? reasoning by examining students? responses to a variety of tasks involving number sentences. The assessment instrument focuses on concepts of equality, identifying strategies employed, as well as the ability of students to generalise number behaviour. This assessment instrument was trialled with 51 Year 5/6 and 26 Year 8 students. The paper focuses on a limited number of test items within the assessment instrument. In particular, it analyses the different types of strategies utilised and the proportions of students who employed those strategies. We seek advice and direction from participants regarding further possible development of the assessment instrument as well as the identification of tests items that would provide useful information to improve the practice of teachers.

The diversity of mathematics achievement in junior secondary mathematics and the problems of engaging students have been well documented. Many researchers including regular presenters at MERGA conferences have explored the misconceptions and mathematical thinking strategies of students who are at risk of not achieving the numeracy benchmark standards. Howard and colleagues have reported the successful outcomes of a structured intervention program conducted in the middle years with students in schools in New South Wales. However teachers in junior secondary mathematics classrooms struggle to adopt reform practices and implement intervention strategies or programs that adequately address the needs of at risk students. During this round table discussion, the findings from a numeracy intervention project in which teachers from one school participated in action research will be presented. The teachers gained new insights and understanding of their students? mathematical thinking and made small steps in changing classroom practices and learning activities that enabled improvements for some students in their classes. Division thinking and the relationship of division concepts to fractions emerged as one issue for further research that participants of the round table discussion may want to pursue in particular. Through this round table I would like to invite other researchers to discuss their experiences of working with secondary school teachers on intervention with a view to collaborating in the development of a large-scale numeracy intervention project for students in junior secondary mathematics.