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Reported are preservice teachers? intentions to use two types of tasks: those they intend to use most and least frequently in their teaching of mathematics. Preservice teachers? intentions indicated high levels of commitment to using tasks that interest children and low levels of commitment to using tasks that help children memorise frequently used facts. Left unchallenged preservice teachers may form simplistic views about the potential value of different learning tasks. It seems that preservice teachers may benefit from examining the purposes of and contexts in which particular learning tasks are used.

This paper addresses the conference theme of Identities, cultures and learning spaces by exploring how learning spaces might be theorised, how they are created, and how teachers and students negotiate their identities within these spaces. Examples of mathematics classroom practice and teacher learning and development are analysed using concepts derived from sociocultural theories of learning to consider what it might mean to speak of ?technology enriched learning spaces?. The analysis adapts Valsiner?s zone theory to study interactions between teachers, students, technology, and the teaching-learning environment.

The Tao that can be told is not the eternal Tao. (Lau Tsu, Tao Te Ching)

This paper advocates bibliotherapy as a powerful reflective tool in pre-service education. It can provide a new approach to understanding and improving the affective responses of preservice primary teachers. In the study pre-service teachers analysing readings about school students? learning, reflected on and reconstructed their understanding of their own school experiences. The technique of bibliotherapy applied to readings about issues such as mathematics anxiety challenged their assessment of their capacity to learn and teach mathematics. This was a healing process that generated enthusiasm for teaching mathematics.

While the origins of Etienne Wenger?s social theory of learning are located in organisational learning, Graven and Lerman (2003) suggested that a worthwhile challenge would be to relate Wenger?s powerful ideas about learning to the process of becoming a teacher of mathematics. In this paper I attempt to address this challenge by using Wenger?s theoretical discourse to interpret the scenario of Casey, as described in the introductory chapter of this symposium. As such, the role that identity plays in becoming a teacher of mathematics is foregrounded to account for a crucial aspect of what enables learning (or not) in mathematics teacher education programs.

This paper explores the usefulness of marshalling feminist poststructuralist frameworks to think about mathematics teacher education. After describing some of the fundamental aspects of feminist poststructuralism, I use this framework to interrogate a specific vignette from mathematics teacher education ? the case of Casey. My analysis of this case demonstrates the centrality of notions of identities to the feminist poststructuralist project, and the utility of what such an analysis now renders visible.

In this symposium, we argue that a deeper understanding of what impacts on teaching and learning in mathematics education can be gained by foregrounding the concept of identity and exploring its explanatory potential. In this paper we provide on overview of identity and introduce a scenario from mathematics preservice teacher education that is then interpreted from three theoretical perspectives in the papers that follow.

This paper draws on the work of Pierre Bourdieu to understand the resistance of teacher education graduates to their University learning and the alignment with the field of education. This is a perplexing and perennial problem in teacher education. Bourdieu?s theory offers concepts that allow this situation to be theorized through notions of identity and the wider social, cultural arena within which identity can be construed.

This research investigated the use of game playing for mental computation development. As part of a larger case study of a Year 6 class, classroom observation data were used to examine the nature of students? mental computations. Findings indicated that regular playing of a number-based game that was scaffolded by teacher designed learning structures supported students? engagement in mental recall, verbalisation of computation steps, using a range of mental strategies, and experimenting with number combinations.

This paper reports on teachers? perceptions of major curriculum reform in New South Wales at the Higher School Certificate. Quantitative and qualitative data are presented. Measures of teacher self-efficacy and stress related to the innovation, as well as general perceptions of the implementation are reported. Mathematics teachers' views of the curriculum reform are also compared with those of other subject teachers.

This paper explores the usefulness of a framework for investigating teachers? Pedagogical Content Knowledge (PCK). Two primary mathematics teachers completed a questionnaire about mathematics and mathematics teaching, and were interviewed about their responses. These responses were then analysed using the PCK framework. The PCK held by the two teachers was found to differ in many ways, including the connectedness of their knowledge, and the specificity with which they discussed the mathematics involved.

This paper reports some initial results of a project that involved profiling middle school mathematics teachers and their students. Survey data concerning the teachers? confidence in relation to the mathematics topics that they teach, their beliefs about numeracy and effective teaching of mathematics, students? attitudes to mathematics, and their perceptions of the frequency of various events in their mathematics classrooms contribute to a picture of many teachers and their students working in traditional classrooms, believing in the importance of mathematics, but struggling with the conceptual demands of the subject and with finding relevance for the material.

This paper reports on the parental roles of Year 6 and Year 8 mathematically gifted students. A survey is used to evaluate the parents? roles as motivators, resource providers, monitors, mathematics content advisers and mathematics learning advisers. Interview data provides further insights on parents? perspectives on children?s mathematical development from early years to current schooling and future aspirations. The results provide evidence that dispels the myth of the ?pushy parent? and raises implications for parent-school relationships.

Even though school mathematics has been subject to many reforms, the delivery of the high school mathematics curriculum in most schools has changed little since public schooling began. Mathematics is presented as a collection of abstract procedures and, consequently, both student understanding and affect are poor. Year 8 mathematics was traversed through authentic learning experiences, during which students established socio-mathematical norms. The implications from this study are that student-directed learning fosters deeper understanding, and improved affect, compared to traditional methods.

Survey and interview data were used to explore the understandings and use of reform-based teaching approaches of three primary teachers from the same school. The beliefs and practices of the teachers were closely aligned to those recommended in local and international reform-based curriculum documents and to the practices of each other. The data provide insight into contextual factors that facilitate reform-oriented teaching practices within the school setting. Interestingly, it seems that the same structures supporting innovative practice can also foster misunderstandings in teachers? knowledge about reformbased practices.

Providing assistance to teachers in the design of efficacious learning environments is an essential element in promoting teacher change in the teaching of mathematics. The process of teacher change, however, may be demonstrated in quite different ways by different teachers. This paper provides insights into the process of change as perceived by one teacher as she went about employing Collective Argumentation to reconstitute her pedagogy so as to better promote the learning of mathematics in her Year 7 classroom.

In the current educational climate in Australia, there is an imperative on university administrators to maintain student enrolments since funding is explicitly linked to student numbers. The most effective way of achieving this is to retain those students who have already enrolled. Consequently many universities currently seek to identify those students at risk of either failing a course or withdrawing from the university. In this paper we report on an initial study into the use of pre-entry measures to identify at-risk students in the context of a tertiary preparatory course offered entirely in the distance mode. We conclude that such measures are at best rough indicators of at-risk students and that results from such measures should be used in a non-directive manner only.

As part of the moves to reform mathematics teaching in New South Wales, the Years 7-10 Mathematics Syllabus emphasises working mathematically. This paper presents the results of interviews conducted with 39 teachers to examine how they interpret the aims of working mathematically and the extent to which they are implementing working mathematically in their classrooms. The results indicate that while a small number of teachers have embraced the reforms, most have a limited conception of what working mathematically means and have not made any substantial change in their practice. The possible reasons for this situation are discussed.

Numeracy coordinators for schools in the Early Numeracy Research Project led teams of teachers who worked with students in Years Prep to 2. They participated in three years of the research project, investigating ways to improve mathematics learning outcomes for their students. A team of university researchers worked closely with these people, regarding them as co-researchers. Coordinators were supported in their role through professional development, the establishment of a network of coordinators, and through mentoring. Over the course of the project, researchers built a picture of the complexity of the role of the numeracy coordinator in the early years of school (Cheeseman & Clarke, 2005). A year after the end of the research project the same teachers were asked to reflect on their role as numeracy coordinator. The aim of the study was to investigate any changes in the role of numeracy coordinator. Key changes and challenges included responsibility for larger teams spanning more grade levels, reduced budgets and time allocations for the role, and the responsibility for planning and leading professional development sessions for the team.

A common reaction for many experienced teachers when faced with classroom mathematical reforms is ?Why change? What is wrong with the way we teach mathematics?? This paper describes the complex journey three teachers took in attempting to acknowledge, and consolidate the inherent changes required in the mathematical reforms. The journey commenced with their schools? participation in the New Zealand Numeracy Development Projects, a vehicle for mathematical reform, which provided teachers with the impetus to make changes to their personal and pedagogical mathematical content knowledge and practice.

In this paper we present a framework for investigating teachers? mathematical pedagogical content knowledge (PCK). The components of the framework permit identification of strengths and weaknesses in the PCK held by various teachers on different topics. It was applied to teachers? written and interview responses addressing a student?s difficulties with the standard subtraction algorithm. This enabled comparison of the PCK held by individual teachers, together with analysis of the teachers? PCK as a group. Most understood the issues and knew suitable representations, but occasionally lacked key understanding of students? misconceptions and how to help students recognise them.

This paper outlines the history of a famous school mathematics problem ? originally formulated (we believe) by Isaac Newton. It appeared in a U.S. arithmetic text in 1834 and became the source of a controversy that lasted for at least 60 years. We offer two quite different solutions to the problem and provide details of the controversy that emerged. The concept of mathematical elegance, and the difficulty of specifying criteria for elegance, are discussed.

We studied the complex situation of first year university students using computer algebra systems (CAS) for the first time as part of their mathematics subjects. We identified four components of initial experiences with CAS. Existing questionnaires were used to identify two subgroups of students with contrasting approaches to study. When these subgroups were further split by computing experience, their scores on the four components of initial experience with CAS revealed a complex picture that was understandable from an activity theory perspective.

High Stakes examinations used to gain entry to universities have been a formal process in New South Wales since University of Sydney was founded in 1850. This paper reviews mathematics examinations used for the Leaving Certificate from the beginning of World War II in 1939 to the start of the Wyndham Scheme in 1962. Relevant examination papers were analysed to identify changes that took place during this period; furthermore reasons for these changes will be discussed.

This paper reports on fifth graders? proficiency with number line tasks in an interview situation. The results revealed that at least 10% of students were unsuccessful in using a simple number line effectively. Additionally, some students? explanations suggest that they do not appreciate that the number line is a measurement model rather than a counting model. This study concludes with recommendations for explicit instruction, a note of caution for interpreting number line items on numeracy tests, and avenues for future research.

Singaporean students? top-level performance in the TIMSS studies in mathematics has strong connections to the mathematics curriculum in the country. While mathematics curricula tend to be somewhat similar in countries all over the world, there are some subtle differences in the Singaporean mathematics curriculum, which are worth examining. In this paper, I describe some of the important features of the intended, attained and implemented aspects of the mathematics curriculum in Singapore and how they connect to the TIMSS studies. I also briefly comment on the strengths and weaknesses of the curriculum.

The effects of reading comprehension on the mathematical-modelling problem-solving process is yet unresolved. This paper reports on a study conducted with two classes of year four students. It investigated the extent to which a literary organisational structuring strategy: top-level structure, may change students? engagement in mathematical modelling. Mathematical-modelling problems require students to negotiate various texts as they negotiate the problem-solving process to learn mathematical skills. The strategy of top-level structuring aids students to structurally organise textual information, and to elicit and recall the main idea of texts. This research illustrated that top-level structuring can make a difference to young students? mathematical-modelling outcomes to some degree. Further in-depth research on the issues raised here is warranted.

In recent years, educators have been calling for an increased focus on complex systems across the school years. One approach to achieving this is through modelling, where children think mathematically about relevant relationships, patterns, and regularities in dealing with authentic problems. This paper reports on the different mathematical models created by three classes of fourth-grade children (9-year-olds) in working a problem that addressed the complex system of team sports. Consideration is given to the problem elements and interactions the children explored, together with the ways in which they operated on and transformed the given data.

This paper, reports on the development of a theoretical framework about statistical thinking and reasoning in relation to data analysis, graphing and graph-sense making. The model developed from a review of the literature is used to construct an assessment instrument that is designed to elicit student prior learning in relation to reasoning about data in an ICT environment. The design of the assessment instrument takes into consideration the different forms of data representations afforded by the graphing software, TinkerPlotsTM Dynamic Data Exploration.

In this paper, enrolment data in year 12 mathematics subjects across Australia for the years 2000-2004 are presented. Based on the Barrington and Brown (2005) year 12 mathematics subject categorisations, the focus is on examining enrolments in ?Intermediate? level subjects, the most likely pre-requisites for tertiary-level mathematics/science-related courses. Enrolment numbers are examined, as too are enrolments expressed as percentages of year 12 cohort sizes. Data for all Australian students, and by gender, are explored and state/territory comparisons are made. The findings have important implications for educational authorities, the tertiary sector, mathematics teachers, and their students.

Traditionally, mathematics has received little attention in prior-to-school settings, however, extensive research in the last 30 years has recognised that mathematical learning is critical to success and achievement in both school and life pursuits. Research has also documented children?s capabilities for complex mathematical thinking and reasoning, and recently curriculum has been developed that addresses young children?s competencies and potentials. This paper reviews current literature to provide a justification for the inclusion of mathematical modelling activities in preparatory settings. Mathematical modelling activities move beyond traditional problem solving to encourage children to develop and explore significant, real world mathematical ideas.

The provision of authentic applications, as distinct from artificial word problems, as vehicles for teaching students to apply their mathematics remains an unfulfilled need. This paper describes the generation and application of a framework that provides principles to inform such a purpose. The framework is grounded in classroom data produced by students solving modelling problems, and its use is illustrated through application to the design of a new problem. The development and application of the framework are on-going.

Three current interpretations of the term ?mathematical modelling? as it is used in mathematics education are described. The modelling cycle appropriate to one of these interpretations forms the basis for research into blockages that emerge in the solution process for problems with real world connections. The development of a framework documenting key elements that enable (or disable) progress during transitions between phases in the modelling process is described, and a selection of elements illustrated. Associated implications for learning and teaching are discussed.

This paper theorises an extension to a framework that structures students? interaction with technology through a series of metaphors: technology as master; technology as servant; technology as partner; and technology as extension of self. These metaphors provide insight into potential relationships between students? intentions, technological engagement and actions. The framework is conceptualized from within a socio-cultural perspective of learning/teaching mathematics and extends the Vygotskian principle of Zone of Proximal Development (ZPD) by elevating computer and graphing calculator technologies beyond that of simple cultural tools to that of quasi-partner or mentor. A component of the framework is used to analyse two episodes of student/student/technology interaction while working on a specific mathematical task. The extension to the framework has the potential to promote more sophisticated uses of technology in mathematics classrooms.

The issue of the research-practice gap ? the discrepancy between what we know about teaching and learning mathematics and what actually happens in mathematics classrooms ? has been recognised as problematic for at least the past two decades. This paper documents a discussion between two researchers, one a teacher and the other a university academic, who have developed a successful professional relationship that has proved to be a productive collaboration both in terms of implementing ideas in the classroom and in terms of the publication and presentation of these ideas. This discussion considers the reason for the success of their partnership as well the implications for this success in narrowing the ?gap?.

This paper reports on part of a larger cross-sectional study of the development of students? quantitative concepts of fractions. In total, 1676 students in Years 4?8 were asked a series of questions designed to elicit their concept images of various fractions. Three questions asked the students to construct a regional model (parts of a circle) for one-half, one-third and one-sixth. The initial analysis of the evoked concept image of one-third revealed an unexpectedly high number of responses shading one-quarter of a circle. Further analysis suggested that what appeared to be one-quarter was frequently intended to be one of three parts, confirming that drawings frequently do not ?speak for themselves?. Area is not always the intended feature of regional models of fractions in students? concept images and it is argued that the ?number of pieces? interpretation is a common response to regional models of fractions.

For some time now the beliefs, attitudes and emotions of preservice primary teachers towards mathematics have been seen as problematic in their development as teachers of mathematics. In response, mathematics educators have changed their preservice courses to attend to these affective qualities. These changes have required mathematics educators to undertake different roles. The focus upon affective reform has given rise to a number of moral and ethical issues, and in this paper these concerns are discussed. The discussion concludes with a call for mathematics educators to give more overt attention to the ethical dilemmas that are inherent in mathematics education courses that have an affective dimension.

Fourteen Year 11 advanced mathematics students participated in individual teaching interviews designed to investigate how they learnt various rate of change concepts. The theoretical framework compared two models of abstraction: the empirical abstraction model of Mitchelmore and White and the nested RBC model of Hershkowitz, Schwarz, and Dreyfus. Examples of learning were found that fitted the nested RBC model, but none that fitted the empirical abstraction model. It was concluded that the nested RBC model is valuable for understanding student learning of the concepts of average and instantaneous rate of change, but that empirical abstraction is likely to be more valuable in understanding how students develop a global concept of rate of change earlier.

There has been concern for some years about the low mathematics achievement of Maori students in New Zealand. This case study reports on the responses of 18 Maori preservice teachers to an investigative approach to learning mathematics during their compulsory Year 1 mathematics education course, as a possible aid towards helping improve the achievement level of Maori in mathematics.

This paper reports on the espoused views of a group of primary teachers as they discuss issues related to the teaching of school mathematics to Australian Aboriginal students. They believe that their teaching is significantly affected by trying to program and cater for the wide range of abilities, the amount of mathematics content to be covered and the lack of teaching time. They report a lack of teacher education preparation for teaching mathematics across ability groups and the difficulty of inventing appropriate teaching strategies to meet the learning needs of Aboriginal children.

This study examines the views held by students from an inquiry classroom. Individual interviews were used to explore views held by a group of 9-11 year old students towards explanation and justification of their mathematical reasoning. Responses indicated that the students viewed explanations as thinking tools. Challenge, questioning and debate were viewed as opportunities to re-construct mathematical thinking. The study emphasises the need for a safe, power sharing environment to support student construction of positive mathematical attitudes and dispositions.

In the current mathematics education reform efforts teachers are challenged to develop discourse communities where students learn to construct and evaluate arguments used in mathematical reasoning. The challenge for teachers is to know what actions to take to establish such discourse. In this study I investigated the actions a teacher took to establish the discourse of inquiry and challenge within a community of diverse learners. I report on the way in which the teacher shifted back and forth among a range of roles, as a facilitator of the discourse, as a participant in the discourse and as a commentator about the discourse.

This paper reports on the development of a community of practice and its connection with mentoring between mathematics teachers in low socio-economic secondary schools. It follows from an earlier paper (Kensington-Miller, 2005) in which the effectiveness and the difficulties that occur within different mentoring relationships were examined. This study is part of a larger project in which the teachers come together at professional development meetings. Over time a community of practice emerged which had positive implications for the mentoring relationships.

With one-year teacher education programs for elementary and middle-school teachers becoming popular, teacher educators are put on the spot as to how novice teachers should be prepared for teaching mathematics. In this paper I delineate, and hold up to academic debate, epistemological and ontological assumptions that inform my teaching in a newly introduced Graduate Diploma of Education program. From a poststructuralist perspective the question becomes not one of what to leave out, but of how to vary instructional practices to produce competent, inquiry-oriented teachers of mathematics in less time.

A large, national, mathematics data base, the Australian Mathematics Competition [AMC], is used to investigate the appropriateness of curriculum content for different groups of students. The cross-sectional comparisons afforded by the data enable characteristics of mathematical problems found easy and difficult by ?average? and ?able? groups of students at different grade levels to be identified. Gender differences in performance on the papers are also explored.

The traditional long division assumes that users can apply a guess-and-match type mental process of searching for a maximum that is not greater than the dividend. This optimisation procedure frustrates some children because it confuses the concept of division that is embraced is not consistent with their life experiences associated with grouping and sharing. Initial results of our studies suggest that children taught by a new method perform better on a test than those who learn it through the traditional method.

The central focus of the New Zealand Numeracy Development Project (NDP) is to raise student achievement in mathematics by improving the professional capability of teachers. This paper reports on the findings of a contextually responsive evaluation of the NDP professional development guided by Guskeys? critical levels of information and measured through self-assessment.

Partitive quotient construct is instrumental to the development of a deeper understanding of rational number. In this paper we examine this and the need to study the difficulties children experience whilst they attempt to solve fraction problems involving the use of the partitive quotient construct. In order to understand the concept of partitive quotient, it is necessary to examine what partitive quotient is and how it fits within the broader context of numeracy learning. This paper will review key literature sources that examine the meaning and importance of fractions in numeracy, and identify emerging issues that are relevant to the investigation partitive quotient.

This paper reports on the development and refinement of an observation schedule designed to evaluate effective teaching for numeracy and to serve as a starting point for teacher selfreflection. This paper summarises the key findings from the literature as to what constitutes effective numeracy teaching and documents the results of a pilot study designed to test the usefulness of the instrument. The results indicate that the instrument is quite comprehensive in terms of its coverage of effective numeracy teaching indicators and may prove useful to other researchers involved in documenting classroom practice.

A school-based numeracy initiative, conducted in one NSW metropolitan primary school, trialled an innovative approach to improving mathematics for low achievers. The project involved 683 low-achieving students aged from 5 to 12 years, 27 teachers, and three researchers over a 9-month period. A Pattern and Structure Assessment (PASA) interview and a Pattern and Structure Mathematics Awareness Program (PASMAP) focused on improving students? visual memory, the ability to identify and apply patterns, and to seek structure in mathematical ideas and representations. There was a marked improvement in PASA scores particularly in the early grades and substantial improvements found in school-based and system-wide measures of mathematical achievement.

This paper will illustrate the importance of recognising and validating students? informal knowledge as the essential cornerstone for developing mathematical ideas. A single case story is presented to highlight a teacher?s scaffolding strategies which identify and interpret a student?s informal knowledge in relation to school knowledge. From here the case story continues to convey how the teacher builds student understanding as well as her own.

Students, particularly older children, generally do not ask questions or seek academic assistance from teachers in class or their peers. Reasons include concerns about perceptions of incompetence amongst group members, a lack of confidence in helpers to give accurate help or communicate ideas clearly, and self-beliefs that seeking help threatens a student?s independence. Findings are similar for both face-to-face situations and within online learning communities.

This paper arises from a larger study that aims to investigate how primary school teachers use the Internet for teacher professional development and for teaching mathematics. Through compilation of interview data and classroom observations from one primary teacher, it was found that this teacher uses technology and particularly the Internet as an integral component of her daily mathematics teaching, and regards the Internet as vital to her teaching style. This report provides an examination of the way this teacher integrates the Internet into her mathematics teaching, reasons to use the Internet in teaching mathematics, the benefits for students, and how this approach aligns the intent of the new Queensland mathematics syllabus. The results are discussed in the light of new learning in New Times.

In recent years, there has been a marked increase in the expectations of the mathematical performance of Australian children in their first year of school. Partly, this has been the result of Australasian research emanating from large systemic early childhood numeracy projects. One result of this increase in expectations for the first year of school has been a subsequent change in the expectations held by teachers and parents of children?s mathematical learning in prior-to-school settings and a consequent formalisation of mathematical teaching practices in many of these settings. This paper reports work done with preschool educators as part of the Southern Numeracy Initiative in South Australia in which the educators sought an alternative to the push for further formalisation. The results include the identification of ?powerful ideas? in mathematics, the linking of these to the Developmental Learning Outcomes in the mandated South Australian curriculum documents through pedagogical enquiry questions within a numeracy matrix and how learning stories (narrative assessment) could be used to provide an assessment regime capable of celebrating mathematical learning while remaining compatible with key principles of preschool education. In particular, this paper considers the development and use of the numeracy matrix.

This paper is based on a doctoral thesis about studying calculating instruments and deals with the very familiar primary school notion: the carried-number. We develop this notion in three ways: the history of calculating instruments and their mechanisation; a mathematical study of this notion within the place-value system; and an analysis of experimental data from an investigation of student and teacher understanding. For many students and teachers the notion of carried-number appears to be undeveloped mathematically.

Research suggests that pre-service teachers? beliefs about mathematics and mathematics teaching are a strong indicator of their future teaching practice. In this interview study, 16 pre-service secondary teachers were asked to reflect on their own school experiences and discuss their beliefs about what constitutes a good teacher and good teaching. Results indicate that pre-service teachers enter their teacher training with fixed views about mathematics instruction that are based largely on interpretations of how their own mathematics teachers taught the subject to them.

Sustaining long-term growth from in-depth professional development has become an increasing concern to stakeholders in education. An action research study was undertaken in five New Zealand primary schools exhibiting limited sustained gains through Professional Development in Numeracy. Teachers? lack of in-depth knowledge of numeracy content and pedagogy, lack of mathematics leadership within schools, and limited change in schoolwide practices were identified as contributing factors. The study developed a variety of tools which supported school leaders to embed practices and to provide teachers with opportunities to explore and reflect on ideas encountered within the professional development.

We discuss the nature of multimedia software designed for integrating text and video into professional communications about mathematics teaching, learning, and assessment. Cases that demonstrate teacher professional growth through self-dialogue and peer communication of specific pedagogical insights are exhibited. Critical success factors in implementation of this tool are outlined.

This paper presents some survey data from a mathematical teacher professional development initiative being conducted in a small rural school. The initiative involves an external critical friend working within a school, in mathematics classes, with teachers for week-long sessions. The initiative began with a whole staff professional development day, moving to three week-long visits throughout the year, working one-to-one with staff and their professional partners, concluding with a whole staff reflection day at the end of the project.

To promote equitable student outcomes some Bersteinian scholars advocate pedagogy that preserves the integrity of disciplinary learning, but supports interaction between knowledge fields. This learning, they contend, should be communicated through collective rather than individual endeavour. Collective endeavour can occur within communities of inquiry, which have documented potential to support students? mathematical learning. However, their impact on the mathematics education of very young children has not been explored. This paper uses data collected during an explanatory case study to demonstrate how a community of inquiry comprised of First Grade students and their teacher enables young children?s mathematical learning.

This paper explores an innovative methodology ? self-study through narrative inquiry ? as a way of critically examining pedagogical practice in mathematics teacher education. A unique feature of this self-study was the simultaneous use of narrative inquiry as a research method and a pedagogical tool used with prospective teachers. By juxtaposing my own learning experiences with prospective teachers? learning into accounts of practice I have reconceptualised my approach to mathematics teacher education in terms of establishing an inquiry landscape for co-learning. Three core features of this landscape are outlined to illustrate the potential of self-study to foster critical reflection that impacts on practice. The implications for mathematics teacher educators, and the programs they develop, are threaded throughout the paper.

Blanton and Kaput (2001) encourage teachers, especially in the primary school, to grow ?algebra eyes and ears? (p. 91). This is not an easy task when teachers? vision has for so long been restricted to thinking of arithmetic primarily as computation. This paper looks at how forms of thinking which do not rely on computation can be used with addition and subtraction number sentences. It looks at how these powerful forms of thinking are used by students in Years 5 to 7 in two schools, what distinguishes this thinking from computational thinking, and how consistently it is used by students across different problem types.

When entering university students often find there is a shift in presentation of mathematical ideas, from a primarily procedural or algorithmic school approach to a presentation of concepts through definitions and deductive derivation of other results. For many a course in linear algebra is the first occasion that this shift is encountered, since calculus may approximate to what they have seen at school. This research uses the theory of processes and objects, along with the ideas of embodied or visual, symbolic and formal approaches to mathematics learning to investigate some first year students? understanding of eigenvalues and eigenvectors. We identify some fundamental problems with student understanding of, and hence working with, the definition of eigenvector, as well as with some of the concepts underlying it.

As part of an ongoing project, we have developed a model of planning and teaching that is designed to assist teachers to help students overcome barriers they might experience in learning mathematics. The following is a discussion of one aspect of the model that we term ?enabling prompts?. These refer to the directions, invitations, or questions that a teacher offers when interacting one-on-one with students experiencing difficulties. We argue that teachers should plan to pose subsidiary questions in the first instance, rather than, for example, offering further explanations. We outline our overall planning and teaching model, we present some examples of enabling prompts used by our project teachers, and we propose some considerations for teachers when structuring their own enabling prompts.

Students enter Australian high schools with a wide variety of knowledge, background experiences, attitudes to school and discourses, their ?funds of knowledge?. Much of this comes from their ?first space? ? what they bring from the home and community. Some of it comes from their ?second space? ? what they bring from prior school experiences and what they experience in the current school environment. This second space is profoundly shaped by the experience of formal, textbook-oriented approaches to high school mathematics. This theoretical paper looks at the experience of students entering their first year of high school mathematics in terms of first and second space, and casts forward to the creation of a ?third space? ? one which values and builds on students? funds of knowledge.

I am researching equitable and socially just teaching practices when using technology for the mathematical learning of disadvantaged and marginalised students in junior secondary school. Using data gathered from teacher interviews and a meeting of teachers, I present a case study of one teachers? practice. The case suggests that there are some equity considerations for the use of an integrated project approach to teaching mathematics and that whole class problem solving with technology can provide access to mathematical ideas when students have limited access or skills with technology.

Scaffolding has become increasingly popular as it provides teachers with an appealing alternative to traditional classroom techniques of teaching. Recent research identified a number of different ways that scaffolding can be used in the classroom to improve students? numeracy levels in primary schools. However, despite the importance of scaffolding, pre-service teachers experience difficulties in understanding the complex techniques of scaffolding and often fail to make connections between theoretical explanations and their practical use. This paper examines current perceptions of scaffolding by a cohort of pre-service teachers, both in its conceptual framework and its practical implications to teaching in the classroom, and to teaching numeracy in particular. The results indicated that the participants appreciated the importance of scaffolding as an alternative to the traditional forms of educational instruction. However, they continue to demonstrate a limited appreciation of the more complex and theoretical aspects of scaffolding.

Recent reform efforts in New Zealand prescribe classroom grouping as a key pedagogical strategy, yet current understandings of how and for whom grouping makes a positive difference is by no means conclusive. This paper unpacks what the literature says about grouping and how it influences students? active participation in a classroom community. The research evidence reveals that quality grouping arrangements are built on knowledge of the different purposes of, and roles within, the particular social arrangements established, and demand constant monitoring for inclusiveness and effectiveness for the classroom community.

This paper reports on a model developed to support 6 Year 1 teachers as they developed understandings of the new Patterns and Algebra strand in the revised Queensland Syllabus. Traditional Professional Development models are grounded in notions of teacher growth and change. This particular model was grounded in theories of learning, particularly those grounded in the socio-constructivist perspective. Teachers worked in pairs developing and implementing learning experiences for three differing aspects of the Patterns and Algebra strand. The results indicate that not only did the model offer positive professional learning experiences for the six teachers but also assisted them in becoming experts in their own right.

This paper examines the use of Blanton, Westbrook & Carter?s (2005) extension of Valsiner?s (1987) zone theory in interpreting one teacher?s Zone of Proximal Development (ZPD) with regard to professional development (PD) from teaching experiments. It shows that if what the researcher sets in the PD is outside the teachers ZPD, little teacher development takes place. It also shows that there may be an inherent conflict between teaching experiments and professional development that is hard to overcome.

As part of a teacher profiling instrument, 42 middle school teachers were presented with a mathematics problem dealing with fractions and wholes and asked to suggest solutions that would be given by their students. Further they were asked how they would address inappropriate responses in the classroom. The students in their classes were presented with the same question as part of a larger survey of mathematical concepts important in the middle years. This study compares the expectations of teachers and their suggested remedial actions with their years of teaching, their previous mathematics study, and the performance of students. Results suggest explicit questioning of teachers is an effective way to explore teacher knowledge for teaching mathematics.

Year 5 teachers from eight Brisbane schools took part in a professional development project to encourage them to use spreadsheet technology with their classes. Twelve of the teachers participated in follow-up research to investigate teachers? attitudes and perceptions relating to spreadsheets, implementation methods and uses, and factors that may enhance or hinder acceptance and usage. Results showed that teachers who were more confident of their ability to use spreadsheets were more likely to perceive them as useful in the teaching of Chance and Data, and more likely to use activities that developed higher order thinking. In using spreadsheet technology with their classes, teachers had to overcome a range of personal, technological and school-related problems.

The nature of creative and insightful thinking of Year 8 students in mathematics classes was studied through simultaneous examination of post-lesson video stimulated interviews and lesson video. One case is used to illustrate these findings. The concept of Space to Think emerged as a space manoeuvred by each of the five (out of eighty-six) students from Australia and the USA who creatively developed new knowledge. This Space to Think illuminates pedagogical moves that provide opportunities for creative thinking.

This paper presents results from a study of the non-routine mathematical problem solving employed by 17 preservice teachers. Analysis of task-based interviews led to the identification of five cognitive phases: engagement, transformation-formulation, implementation, evaluation, and internalisation. Corresponding metacognitive behaviours were associated with each of these cognitive phases. A five-phase model for problemsolving, which incorporates multiple pathways, is described. Since various pathways between the categories are possible, the model accommodates the range of metacognitive approaches used by students.

This paper reports on one small component of a much larger study that explored the perspectives of students towards mathematics learning. Students were asked ?What do you think maths is all about?? Some students responded in terms of mathematical content. Others commented on learning in general, or on problem-solving in particular. Some students talked about the usefulness of mathematics for everyday life. An overwhelming number of students answered the question by talking about the importance of mathematics for the future.

In this paper we draw on data from a larger study where we have been exploring the ways in which teachers in diverse settings use ICTs (or not) to enhance students? numeracy learning. Drawing on the data from a survey implemented in 6 schools that were representative of considerable diversity within the Australian education system, we report the ways in which various programs are used, the levels of ICT usage across grade levels and teachers? levels of skill and confidence with the use of ICTs. We find that there are differences between our schools in these areas that may be constitutive of aspects of a digital divide in mathematical learning.

This paper describes part of a study which involved 29 third year pre-service student teachers answering Maths Anxiety and Maths Self-efficacy questionnaires and questions regarding previous and current mathematical experiences. Results indicate that previous depth of mathematics learning is not a factor in the level of Maths Anxiety and neither is the level of success in current mathematics teacher education courses, and that Maths Anxiety is highly correlated with Maths Self-efficacy.

The argument in this paper is that identity as a mathematical thinker develops through self-directed learning within a supportive community of practice. This paper discusses how identity as a mathematical problem solver and investigator develops through selfregulation. This development is illustrated by considering students undertaking a mathematics and technology subject in a primary teacher education degree. It shows how students set goals, plan, organise, self-evaluate, record keep and structure their learning environment to achieve self-regulation. The role of the tutorial group and technology is also important in establishing their identity as a mathematical problem solver and investigator.

Following a concern for the engagement of boys in a NSW Year 6 classroom, action research was undertaken to explore the effect of the use of computer technology and the change in role of the teacher from a giver of knowledge to a facilitator, on the motivation and engagement of the boys in the class. Data were obtained from a number of sources, including a Motivation Scale, focus group discussions, observations, student reflective logs and a teacher diary. Initial results indicated that while boys responded enthusiastically to the challenge of manipulating data using the Tinkerplots software and developing their own questions and research topics, there was no indication that they preferred to use technology.

This paper reported the influence of the Numeracy Development Projects (NDP) in revision of the number strand of the New Zealand mathematics curriculum. It documented how four types of teaching and learning research frameworks were synthesised to provide evidence for validity of the number framework and associated pedagogies. Examination of students? responses across number domains showed consistency and suggested a general growth path. Strategy stage norms to set expected levels of achievement were described and challenges posed for mathematics teaching and learning.

Everyone has beliefs about how learning should take place and what the best practices are to enable this to happen. Although it is believed that students? beliefs about ?best practice? will mirror those of their teachers, and change as they change teachers (Kershner & Pointon, 2000; Kloosterman, Raymond, & Emenaker, 1996) the importance of listening to the ?students voice? is becoming recognised (McCullum, Hargreaves & Gipp, 2000). This paper reports on one aspect of a larger study that explored Pasifika student achievement in mathematics at Year 7. What is it that these students consider ?best practice? in learning mathematics? Do their beliefs truly mirror those of their teacher?

This paper reports on a study of five, Year 2 students? strategy choice, flexibility and accuracy when answering 20 addition and subtraction mental computation questions. All five students were identified as being inaccurate. However, two students employed a range of calculation strategies while the other three students remained inflexible in their strategy choice, choosing low order strategies. Individual interviews were conducted to identify these aspects of calculation. Two conceptual flowcharts developed by Heirdsfield (2001b) were utilised to identify factors and relationships between factors that impact on mental computation. Use of these flowcharts provides an avenue for identification of the breakdown in the structures thereby providing an understanding of where the child is operating and how they may be moved forward.

The development of place value understanding is an essential foundation concept that enables students to have a strong number sense. However, place value is a common problem area experienced by students and pre-service teachers alike. This paper reports the results of an initial investigation of pre-service teachers? understanding of the symmetry of the place value system. The results suggest that pre-service teachers? mathematical content knowledge with respect to the symmetry of the place value system is weakest for fractions and when they are asked to generalise their understanding to a base a system. Overall more of the pre-service teachers had a pre- or uni-structural understanding of the symmetry of the place value system.

Student misconceptions of projectile motion in the physics classroom are well documented, but their effect on the teaching and learning of the mathematics of motion under gravity has not been investigated in the mathematics classroom. An experimental unit was designed that was intended to confront and eliminate misconceptions in senior mathematics secondary school students studying projectile motion as an application of calculus to the physical world. The approach was found to be effective, but limited by the teacher's own misconceptions. It is also shown that teachers can reinforce student misconceptions of motion because they cannot understand why students have difficulty understanding it.

Derived initially from the observation of children's methods of counting, Mathematics Recovery and Count Me in Too, the New Zealand Numeracy Projects have, as a starting point for the training of teachers, the understanding and use of a strategy framework that traces children's development in number reasoning. Research indicated the usefulness of teachers interviewing students in their own class, and viewing video clips of strategic reasoning across of a wide range of ages of student. This paper outlines how interviews and the video clips are incorporated into the initial stages of teacher Professional Development in learning to use the strategy number framework.

Since finishing a Masters in Mathematics Education, I have taught mathematics concepts in industry and in a secondary classroom, and have frequently come across able students who have difficulty learning and achieving to their academic potential in mathematics. These difficulties seem to stem from anxiety that the students experience when doing mathematics. This is case study of Year 10 students (14-15 years old) who are in the top achievement class of an Otago secondary school. Students were chosen for this class by the school at the beginning of Year 9 because they demonstrated excellence in one or more fields, not necessarily mathematics. Through mainly qualitative methods (classroom observations, questionnaires, and individual interviews), the following is being investigated: ? the mathematical identities of students in the class; ? the identification of maths anxiety and potential maths anxiety in students; ? interventions to improve the mathematics experience of anxious and potentially anxious students.

When contemplating the division of two common fractions, the ?invert and multiply? algorithm, does not develop naturally from using manipulatives. (Borko; Eisenhart; Brown; Underhill; Jones & Agard. 1992) suggest it is for this reason, that it is unlikely that children will invent their own ?invert and multiply? algorithm. Before a student can be expected to ?invent? this algorithm, knowledge of whole number division and basic fraction concepts, including the notion of equivalent fractions is essential. (Sharp, 1998). The purpose of this round table discussion is to take cognisance of the suggestions attested to by Borko et al. and Sharp and examine the teaching approaches adopted by classroom teachers, as they relate to the process of division with fractional numbers. To highlight this, six Year 7 and Year 8 teachers were asked to solve and then describe their mathematical approaches and processes used to calculate 2/3 ? 1/2; illustrate the meaning of the operation and describe the means by which they would explain their respective methods for solving problems involving the division of fractions with their students. The opportunity to examine the use of mathematical equipment that may be used to support and illustrate the mathematical process of division with fractions, will also be examined with the view to generating a conceptual representation of the meaning of the division of fractions. References Borko., H; Eisenhart., M; Brown, C.A., Underhill, R., Jones, D., & Agard, P. (1992). Learning to teach hard mathematics: do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education 23 194-222. Sharp, J. (1998). A constructed algorithm for the division of fractions. In: The Teaching and Learning of Algorithms in School Mathematics. Morrow, L.J. & Kenny, M.J. (editors) Reston, VA: National Council of Teachers of Mathematics.

New Zealand, along with many other countries, has been investigating ways of raising children?s achievements in mathematics by improving teachers? professional knowledge, skills and confidence. The Numeracy Development Project (NDP) in New Zealand grew out of The New South Wales Department of Education and Training initiative ?Count Me In Too? in 2000, and was further developed by ?research evidence about mathematics education, effective teaching, teacher learning, effective facilitation practice, and educational change? (Ministry of Education, 2004). Pivotal to the success of the project are facilitators, principals and lead teachers who work with classroom teachers to effect changes in teaching practice. Schools participating in NDP can expect continuous and focussed support throughout their year long professional development. Sustaining the numeracy momentum within a school is a difficult task and can be further complicated by the employment of untrained numeracy teachers or provisionally registered teachers with varying levels of understanding. During their time in the project, lead teachers have mainly an administrative responsibility. However, in subsequent years, it is magnified to include a complex, multi-layered facilitation role. This small study by Deborah Gibbs and Marilyn Holmes highlights two challenging aspects: (a) The needs of untrained numeracy teachers as they try to come to grips with the numeracy project as well as the school?s mathematics programme; and (b) the complexity of the lead teacher?s responsibility in mentoring new untrained staff. Discussion generated around this study will highlight the implications for facilitators in numeracy and pre-service educators. References Ministry of Education. (2004). The Numeracy Story continued. What is the evidence telling us? Wellington: Learning Media.

This round table discussion will focus on the efficacy, pedagogies and practice of nonspecialist teachers of mathematics in the middle years of schooling in a regional Victorian setting. Case study data, collected through questionnaires and interviews of 26 junior secondary teachers with no mathematics methods in their training, from six, rural, coeducational Victorian Government schools, will be presented. Current middle years? reform and initiatives such as Victorian Essential Learning Standards (VELS) provide guidelines for effective teaching strategies to engage young adolescents generally. However, in order to provide expanded learning opportunities in mathematics, teachers require professional knowledge of their subject. Being able to present an understanding to student of big ideas, the main branches and concepts of mathematics, and providing a sense of their interconnections aids students to engage in mathematics. This research project was undertaken to investigate whether non-specialist mathematics teachers? pedagogical practices in secondary schools were influenced by their limited training and mathematical pedagogical knowledge. The findings indicate that lack of method training and knowledge of pedagogical content impacts on both the teachers and the students? mathematical engagement at junior secondary level. Through this round table I invite other researchers to discuss their experiences of working with middle years mathematics teachers to seek their input in the possible development of further work in investigating the challenges of non-specialist mathematics teachers in junior secondary schools and overall student engagement in mathematics.

Researchers and educational policymakers have given encouragement to the use of electronic technology in the teaching and learning of school mathematics and in the assessment of senior mathematics and statistics courses. If used appropriately (judiciously) is hand-held technology able to offer secondary mathematics and statistics teachers and their students a significantly richer mathematics learning experience? Little research into hand-held technology and statistics has been done and during this round table discussion, the researcher describes the use of hand-held technology [graphical calculators] in three, N.C.E.A. Level 3, statistics classrooms in a large co-educational urban secondary school and the impact that they had on teacher pedagogy, student learning and understanding of statistical concepts will be presented. The roundtable is to discuss aspects of this study and to invite other researchers to share their experiences in working with secondary school teachers in statistics. The feedback provided in this round table is to assist and inform the researcher about planning, implementing and appropriate methodology for a more in-depth study in hand-held technology use and statistical literacy and thinking.