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Copyright ? 2005 Mathematics Education Research Group of Australasia Inc. Published by MERGA Inc., Sydney, Australia ISBN: 1-920846-09-3 Printed by: Deakin University Press Copies of these proceedings are for sale. Ordering information (subject to availability) can be found on the MERGA web site: www.merga.net.au

Dichotomies such as teacher-centred versus student-centred classrooms, real-world versus abstract tasks, and even teaching versus learning can restrict mathematics educators and educational theorists in general to a fragmented view of the mathematics classroom. Constructing such dichotomies as oppositional offers a set of false choices, sanctifying one alternative, while demonising the other. International research offers insight into possible explanatory frameworks within which such choices are no longer oppositional or even dichotomous, but rather can be seen as complementary. The acceptance of such complementarities is a first step to an integrative theory of classroom practice and learning.

This paper focuses on the New Zealand Numeracy Projects (NZNP), an initiative aimed at reforming mathematics teaching and learning. Evidence shows that students benefited from NZNP, regardless of ethnicity, socio-economic status, gender, or age, but that differential performance and progress were evident for particular subgroups - Asian and European students, high socio-economic status (SES) students, boys, and older students began higher and made greater progress than students in other groups. Evidence suggests that the gaps between some groups (eg, high & low SES) may be narrowing, possibly because of the benefits for low SES schools of being involved in other initiatives. Effect sizes provided a measure of how practically meaningful the benefits of NZNP were. The average effect size for Multiplication/Division, Proportion/Ratio and Addition/Subtraction was 0.40, 0.43, and 0.19, respectively, comparing favourably with those found in the UK (0.17 & 0.18). In a related study, children?s perspectives on their mathematics learning were explored using individual interviews. Overall, children were more positive about the value of explaining their thinking to other people than about knowing the solution strategies used by their peers. Students at City School were particularly enthusiastic about the value of sharing their thinking with others than were those from other NZNP schools. Staff at City School indicated that for some time, they had been developing a collaborative approach to working with their students, encouraging them to discuss their thinking and reflect on their learning in all curriculum areas. The findings indicate that more emphasis needs to be given to the communication of thinking and reasoning in mathematics.

This paper considers the evolution of research in statistics education since the introduction of chance and data into the Australian mathematics curriculum in 1991 and presents selected outcomes of research into students? understanding of the content in the chance and data curriculum, using them to argue for a change in emphasis in the classroom in the teaching of chance and data. These suggestions might also influence current curriculum revisions taking place within Australia and New Zealand. Building on the history of the discipline of statistics and its introduction into the school curriculum, it is argued that topics in the curriculum associated with expectation, such as the mean, generally have preceded those associated with variation, such as the standard deviation. Research however, suggests that children develop an appreciation of variation before expectation, and this knowledge should influence the order of the introduction of associated topics and their juxtaposition in the curriculum and the classroom.

The effects of the increasing international academic contacts and the globalisations of research and curriculum in mathematics education are wide spread and their impacts are experienced differently in many countries around the world. This paper reports on data from three focus groups conducted with mathematics educators from universities in Australia and New Zealand, The Philippines and Colombia. It discusses some tensions raised by these academics in coming to grips with the globalisation of their discipline.

Ensuring student teachers are prepared for the realities and dilemmas of the secondary classroom is an essential goal of pre-service teacher education. This paper describes a pilot study into the effectiveness of using a case-method interview in helping address such issues. Informed by the experiences of a beginning teacher, the case included theoretical links and related tasks. Findings included that student teachers were positive about case method stating it had assisted their personal growth as mathematics teachers.

Interviews were held with five volunteer students in the first cohort to use CAS in end-of-school mathematics examinations in Victoria, to review their experiences and the use they made of CAS. Four students were positive, and one negative. Students were highly reflective about their learning, discussing the need for mathematical understanding to use CAS. Students generally chose to use CAS to do questions quickly, or when by-hand skills were not trusted, or to find equivalent forms matching multiple choice answers in exams.

Investigating students? conceptions of covariation patterns between quantities situated within contextual settings engenders enriched, deep understandings of functional relationships. This paper presents data from a case study of a student (Mary) who solved quadratic contextual problems. Mary?s schemes, constructed from quadratically related quantities and patterns of additive rates, fostered the development of an iterative, summative conceptualisation of quadratics in contrast to the product view. Findings support the use of contextual problems to motivate students to think reflectively and mathematically.

We offer a novel theoretical model of how experienced teachers deal with the enormous complexity of classroom environments through the use of attentional skills, and report on the development of a methodology to explore the empirical basis for this model in a study of experienced mathematics teachers. In particular we address issues of the nature of the knowledge base that enables teachers to operate effectively in the complexity of the classroom, and of the challenges of researching this aspect.

This paper describes a pilot study into the use of a traditional Māori cultural activity as a metaphor and overarching theme for both content and pedagogical knowledge development within a pre-service mathematics education course. Students' feedback on their own learning and their future use of such activities is presented. Findings include that pre-service teachers view the inclusion of cultural activities positively and can identify mathematics in such activities, however they are less confident in describing links to Treaty principles.

Providing teachers with support for the implementation of problem-solving approaches in classrooms is critical if classroom practices are to change. Finding the most appropriate professional development program can be challenging given the constraints of time, cost and personnel. Involving teachers in decisions about their professional learning needs is crucial if they are to embrace such approaches. In this paper, primary and secondary school teachers? responses to surveys about the problem-solving professional development needs of their colleagues, are presented. In general, teachers agreed that considerable support was necessary, particularly in the form of time and resources so that teachers can share teaching ideas, reflect on practice, and develop new understandings about these approaches.

This paper reports a case study of one teacher, Jan, who was part of a whole school case study that was designed to explore the professional development journeys of the teachers in one Intermediate School (year 7 and 8) as they implemented the New Zealand Intermediate Numeracy Project (INP). Jan?s voice is used to highlight individual transformations that she perceived occurred in her teacher knowledge. Her preparedness to acknowledge and challenge her mathematical content knowledge, her pedagogical approaches, her beliefs and her personal ability to implement change over the three year period of the study was a key to her increasing confidence to enact new approaches in her classroom.

The Mathematics and Technology Attitudes Scale (MTAS) is a scale for middle secondary years students that monitors five affective variables relevant to learning mathematics with technology. The subscales measure mathematics confidence, confidence with technology, attitude to learning mathematics with technology and two aspects of engagement in learning mathematics. The paper reports the responses of 350 students from 6 schools to demonstrate the power of the MTAS to provide useful insights for teachers and researchers. ?Attitude to learning mathematics with technology? had a wider range of scores than other variables studied. For boys, this attitude is correlated only with confidence in using technology, but for girls the only relationship found was a negative correlation with mathematics confidence.

This paper reports on the impact of a brief professional learning program for K-8 teachers of mathematics, on teachers? beliefs about effective numeracy teaching strategies and appropriate goals of numeracy teaching, for students with mathematics learning difficulties and for students generally. Evaluation data indicated that the teachers finished the program less inclined to espouse differing beliefs in relation to the two types of students, and that their final beliefs were more in line with the aims of the program.

The decisions that teachers make in choosing particulars to illustrate the general are important aspects of their pedagogic role. Examples are used by teachers to exemplify concepts and to illustrate procedures. Worked examples frequently form the basis of a teacher?s explanation. This paper focuses on the examples used by experienced teachers and the justification they give for their choice to novice pre-service teachers.

Values are a key component of any teaching situation, yet they are rarely addressed explicitly in mathematics lessons, in comparison with values in science education. In this report, data are presented from a research study into teachers? value preferences and their teaching practice preferences in both mathematics and science. A volunteer group of primary and secondary teachers participated in the questionnaire-based study which revealed some significant differences between both levels of teaching and also the subjects taught.

Studies have been conducted in the broad area of language in mathematics teaching, but the research in this paper investigated the language used by a teacher in her physics and mathematics courses. Several commonalities and differences of this teacher?s talk when teaching the two subjects were identified and are presented here. The style of her talk varied from the casual to the formal depending on which course she was teaching.

In this first year of a three-year study, we report on the abilities of groups of students from years 7-10 to generalise numerical operational strategies involving compensation adjustments in the four arithmetic operations. The Year 7 students, who had all participated in the Numeracy Project, were more successful than those in other year groups in using literal symbols to express generalities. The Year 9 students, about half of whom with Numeracy Project experience, were the least successful despite exposure to formal algebra.

A case study of how affordances are perceived and enacted by a teacher in a technology-rich teaching and learning environment (TRTLE) to maximise learning of functions by senior secondary students is reported. Three conditions were set up by the teacher to optimise students? current and future perceptions of affordances of TRTLE?s. These were an exploratory approach, promotion of multiple solution strategies, and the engagement in mathematical discourse during the learning process.

How do children make initial sense of an investigative situation when approaching it through the pedagogical medium of the spreadsheet? This paper examines the ways groups of ten-year-old children made sense of number investigations explored in a spreadsheet environment, and how their preliminary responses were shaped, and their sub-goals framed, by the features of that setting. It also explores the manner in which this might filter their understanding and conjectures.

Several studies have documented students? approaches to mental computation. Few, however, have explicitly linked strategy use to success or otherwise on mental computation tests. As part of the evaluation of a mental computation program in one primary school, students in Years 3 to 6 undertook pre- and post-tests of mental computation to measure performance and growth over time. Interviews were conducted with twelve of these students, three from each grade. The findings indicated that students who were procedural in their approach were less successful on the pre-test but generally showed growth on the post-test, following a strategy-based intervention. The implications of these findings are discussed.

In response to issues of developing effective teachers of mathematics, this paper investigates the factors, which have contributed to growth related to mathematics teaching and learning for a group of Melbourne primary school teachers. The teachers were surveyed to determine the people or events that had contributed to their development. Major themes identified in the data were the importance of lifelong learning, relationships and reflection upon their classroom experiences. The themes are analysed in order to make recommendations for appropriate professional development for the teachers.

This paper focuses on primary-school students? meta-awareness of their mathematical thinking and communicating of both affective and cognitive factors that enhance their learning. Meta-awareness is emphasised through engaging students in communication about their mathematical reasoning and in reflection on their levels of knowing and confidence to work mathematically. This engages students in a self-assessment and self-regulation of their own learning and valuing of the complexity of linking different mathematical concepts and different knowledge disciplines when working with mathematical contexts. The self awareness of one?s learning is identified as a hidden or subliminal factor that can enhance learning and empower students to engage effectively and confidently when working mathematically.

The value of play has been well established in the early years of schooling, however in the years that follow, a transmission approach where the learner is ?drilled? in mathematical concepts and processes often dominates the curriculum (Von Glasersfeld, 1992a). This paper identifies the need for rethinking the approach to mathematics learning in this phase of development by recognizing the need for pedagogy where concrete materials, sensory-motor experiences and metalanguage are employed through play activities to support learners in the transition to abstract mathematical processing. Both cognitive and affective factors are identified as essential elements of a play-based approach to optimising mathematical understandings in the middle years and beyond.

Thirty-six teachers were nominated as numeracy coordinators in their schools for the Early Numeracy Research Project (ENRP). These people led teams of teachers who worked with students in Years P-2. They participated in three years of the research project, investigating ways to assist the professional growth of their professional learning teams and to improve mathematics learning outcomes for their students. The university research team worked closely with these people as co-researchers. Coordinators were supported in their role through professional development, the establishment of a network of coordinators and through mentoring by members of the university research team. Over the course of the project, researchers listened to coordinators? accounts of their work with school teams and gathered data about their role from time to time. In this way, the research project built a picture of the complexity of the role of the numeracy coordinator in the early years of school, but also the impact of their work on their personal professional growth and that of their teams.

This paper considers the role of content knowledge and pedagogical content knowledge in the teaching of probability to Grade 5 students. Lessons of two teachers were studied to determine the activities and teaching strategies used to bring out ideas associated with chance, and examine how probability understanding is developed in class. The lessons are shown to be rich in deep probabilistic ideas. The complex interplay between these concepts was sometimes handled well by the teachers, whereas on other occasions gaps in content and pedagogical content knowledge had the potential to cause misconceptions for students.

The representation of part-whole relations that are embodied in fractional numbers continues to be a problematic area of learning for many children. In this study I examine this problem with a ten-year old child by analyzing his mappings between the language of fractions, area models and symbols. The visual models in this study were built and modified with computer manipulatives called Javabars. Results of analysis showed that the participant experienced difficulty in mapping symbolic representation of selected fractions to the area analogs.

The use of ICT-based resources in order to provide effective learning environments in which children could explore concepts such as multiplication has received considerable attention. This approach is based on the assumption that teachers who are already teaching and those who are being trained to become teachers, draw on a well-developed knowledge of the multiplicative process, and could exploit the ICT appropriately. The aim of the study reported here was to examine the quality of content and pedagogical content knowledge of multiplication developed by a group of prospective elementary mathematics teachers in the context of an ICT-based software. Analysis of data showed the participants are developing solid understandings but some areas could be strengthened.

The Success in Numeracy Education professional development program was designed to address national goals for numeracy in Catholic primary schools in Victoria. Key components in the program included clinical interviews in Number, the use of growth points and growth point activities, rich assessment tasks in Space and Measurement, the extensive training of Focus Teachers from each school, and regular support from Numeracy Resource Officers. Staff from Australian Catholic University (Melbourne) were invited to evaluate the success of the program in terms of student outcomes and teacher professional growth. The major sources of evaluation data were student assessment interviews, teacher questionnaires and teacher, Focus Teacher, principal and Numeracy Resource Officer focus discussion groups. A picture emerged of great variation across content domains in student understanding, and similar variation between schools in terms of implementation and commitment to the program. Clear advice was offered by focus groups on possible directions for enhancing the program.

Although there has been considerable research into students? learning of length, area, and volume measurement, there is very little research into their relationship. This paper describes a developmental study of how 96 students in Grades 1-4, randomly selected from six public schools across the Sydney metropolitan area, related measurement in these three domains. Two methods of measuring volume (by filling and by packing) were included. The results indicate a clear increase in students? understanding of unit structure of length, area and volume across Grades 1 to 4 with volume (packing) being the most difficult. Volume (filling) was closely aligned to length. These results support the wisdom of highlighting the similarities and differences in the unit structures of length, area, and volume measurement in a teaching context.

Current mathematics education reform efforts require teachers to learn in the act of teaching. At the forefront of this challenge are questions concerning the content knowledge that teachers bring to their work and how this might develop. In order to characterise how content knowledge is managed to meet the demands of reform, we investigated the processes through which teachers constructed, enacted, and negotiated knowledge within their classrooms. In this paper we report on one aspect of the teachers? experiences, namely ?learning to notice? critical mathematical instances during classroom interactions.

Gender differences in performance in Mathematics have been reported for grade 6 Dutch students both in overall performance, with boys outperforming girls, and for particular item types. Similar gender differences have not been reported for Australian students. Findings reported in this paper suggest that there are types of items which function differently for girls and boys in Australia as well as in the Netherlands.

This paper investigated primary students? (N = 127) knowledge of the properties of Networks, Hierarchies, and Matrices using a set of scenario-based tasks. Firstly, the results revealed differences in students? knowledge of the various properties for each diagram, and differences in their knowledge of some properties across the various diagrams. Secondly and unexpectedly, the performance of older students? was not significantly higher than younger students. Implications for instruction about diagrams and avenues for future research are discussed.

This study focused on how a high school mathematics teacher in the United States interpreted and used the NCTM standards: communication and reasoning and proof in class. The Algebra I class of a grade 10 teacher was observed for one week and after each observation he was interviewed. This teacher valued mental computation, drill and practice, and the use of basic facts and rules. The study showed that he used his own standards based on the wisdom of his experience.

This paper describes the practices of ninety-nine teachers at the beginning of their involvement in a large-scale project investigating the influence of subject cultures on school and teacher change. Data collected from these teachers, using the IMYMS Components of Effective Teaching and Learning mapping procedure, were analysed to investigate similarities and differences in primary and secondary teachers? perceptions of their practice in mathematics. Results of the analysis show different patterns across a range of components of effective teaching and learning.

In the 1890s ?Herbartianism?, a modified form of the education theory of Johann Friedrich Herbart, became the dominant influence on elementary school education on the Continent, in Great Britain, and in the United States. Herbartianism burst on the education scene, meteorlike, and then, after about 20 years, virtually disappeared. In this paper its influence on school mathematics is assessed. It is argued that within Herbartianism lay the seeds of most of the 20th-century?s major mathematics education theories.

This paper addresses 7th-grade children's mathematical modelling at the end of a 3-year program. The children and their teachers participated in a series of mathematical modelling activities from the 5th grade through to the 7th grade. The problems involve authentic situations that need to be interpreted and described in mathematical ways. We examine the mathematical understandings and mathematisation processes that the children used in constructing their models for the final problem (Summer Reading). We report on the nature of the problem factors that the children chose to consider, the operations they applied, the types of transformations they made through these operations, and the representations they used in documenting their models.

Analysis and commentary on the 2002 and 2003 Mathematical Methods (CAS) pilot examinations in Victoria, in particular with respect to common items with the standard course, were reported at the 2002 and 2003 MERGA conferences. This paper extends analysis to student performance on aspects of the 2004 examinations, with consideration of some emerging trends over the three years. Several possible areas for a broader research agenda with respect to implementation of a CAS enabled senior mathematics curriculum are proposed.

Students learning the art of solving equations using formal algebraic procedures are usually presented with examples that require the application of simple arithmetic knowledge. This paper suggests that such contexts do not always encourage students to use arithmetic procedures that are algebraically useful or to see the need for formal algebraic techniques. Excerpts from interviews with students reveal their thinking and some of the strategies they use. Some implications for teaching are suggested.

Mathematics is a language. It has its own vocabulary, symbols, syntax and grammar. It is a powerful way of communicating ideas about the world. Increasingly, students have been encouraged to talk about their mathematics, but the potential for using linguistic features of that talk as indicators of student understanding is only beginning to be explored. This paper presents some evidence that the style of students? responses to questions about groups of similar algebraic items can indicate the level of their understanding.

The integration of Information and Communication Technologies (ICT) into education is recognised as: providing opportunities for developing skills for the 21st century, having the potential to transform pedagogical practices, and playing a role in reforming curricula. Additionally, it is considered an essential tool for developing understanding about mathematical concepts. The challenge for teachers is to use ICT in ways that promote mathematical thinking and concept development. The literature suggests professional development opportunities are necessary for teachers to equip themselves with the skills to integrate ICT effectively into their teaching and learning practices. This report describes part of a study that focuses on the outcomes of ICT professional development completed by Mathematics teachers from a suburban high school. The case study of four teachers demonstrates that the acquisition of ICT skills does not necessarily translate to the adoption of transformative pedagogical practices. Implications of this study include suggestions for designing future professional development programs.

The shortage of skilled teachers has once again intensified investigations into the nature and scope of teachers? work. In this paper, findings are reported from a recently conducted pilot study in which the daily lives ? in and out of working hours - of secondary school mathematics teachers were tracked. Data collection focused on the activities undertaken, their contexts, and the participants? associated affective reactions to them. The efficacy of a key instrument used for data collection, the Experience Sampling Method, and its feasibility for use in a larger scale project, are also examined.

This paper extends the work of an earlier study that theorised four metaphors for the way students made use of calculator and computer technology. Students? responses were obtained on three occasions during a two year teaching program in relation to the way they incorporated technology into their learning of new mathematical concepts and ideas or when solving problems set in both familiar and unfamiliar contexts. These responses were used to develop a finer grained taxonomy of behaviour than proposed in earlier work.

Since our conceptual structures are a major factor in learning, it may be hypothesised that the richer these schemas are the better the learning that will result. This paper reports on a study of students? understanding of derivative, and the thinking they construct. It follows the progress of two students, James and Bob, and describes a snapshot of the richness of their thinking in this area. This is related to a framework of knowing proposed by the authors, and examples of the possible value of schematic extensibility in terms of understanding new ideas.

This paper reports on our research with beginning teachers of secondary mathematics in building a community of practice featuring face to face and online interaction. We analyse bulletin board discussions of our 2003 pre-service 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 sustainability of the community is related to how it was expanded, transformed, and maintained during and after the pre-service program.

The aim of this study was to gain a better understanding of the ways in which children conceptualise fractions as quantities when they are asked to compare the size of fractions. One hundred children aged 11 and 12 were asked to choose the larger of two fractions and to explain their reasoning. The questions progressed from comparing unit fractions to comparing fractions close to one-whole with different denominators. By coding the children?s responses according to the comparative strategy evident in their explanations, a range of strategy categories was identified and exemplified.

Concerns over how I changed my own teaching style during a year of mathematics teaching gave rise to the content of this paper. I will discuss how the influence of the culture of an exam-driven school and the impact of examinations enacted on me, and my teaching practice. In this paper I tell my story using autoethnography. This style of writing invites the reader to share in my experiences and reflections, to enter into my world of mathematics teaching, to draw their own conclusions, and promote further discussion and dialogue.

This paper considers task design related to the geometrical understanding of a class of nine- and ten- year old children. The research was interested in identifying the principles underpinning these tasks. A first iteration involved a nationally-recognised sorting task (National Numeracy Strategy, DfEE, 2000) and a second principled task was designed to delve further into the children?s understanding of the inclusive nature of the definitions of quadrilaterals. We consider to what extent van Hiele?s levels of geometrical understanding can be used at the classroom level and raise the issue of appropriate tasks for children to engage with in order to challenge and stimulate their understanding of geometric definitions.

This study investigated the change in student mental computation strategies for addition and subtraction following eight half-hour lessons over an eight-week period The principal researcher provided the teacher with a theoretical background for mental computation and support materials for the development of the instructional program Twenty-one Year 2 students participated in pre- and post-testing using individual interviews to identify the students? mental computational methods The results indicated that students who employed inefficient methods such as counting moved to more sophisticated strategies such as wholistic compensation Other students who already employed some sophisticated strategies increased their repertoire.

In this paper, I present the initial plans for a beginning collaborative research partnership (Grundy, 1998) involving myself, two graduate students, and nine middle school mathematics teachers who work with students aged ten to 14 This paper provides information about the guiding literature (on discourse, teacher beliefs, and practitioner research) and the project plans I hope to get critical feedback from this international audience of mathematics educators so that I can improve the plan prior to intense data collection during the upcoming academic year.

Students? concept image of rate of change may be incomplete or erroneous. This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students? existing schema of informal understandings of rate of change to more formal mathematical representations. Students developed numerous ?models of? rate of change in a motion context which then transferred to serve as a ?model for? rate of change in other contexts.

A sample of 572 children from 70 schools was assessed for number knowledge at the beginning of schooling and at the end of each year for the first five years of school. During the five years children?s mathematical understanding developed at different rates and many moved position relative to their peers. While just under two thirds of the students who began with their number understanding below the median were still below the median at the end of Grade Four (the fifth year of school), this means about a third moved from the lower group to the upper half of the class. Nearly 13% had moved into the upper quartile, demonstrating that mathematical behaviour on entry to school was not necessarily the strong predictor of future performance as has been shown in other studies.

This paper describes the phases in a journey one teacher and her students travelled during a year long teaching experiment designed to reform classroom communication norms. A study group environment supported the teacher to examine reflectively the discourse patterns enacted in the classroom. Data indicated that the communication norms constituted in the classroom significantly influenced the discourse context and student engagement in mathematical practices. Student autonomy and collective responsibility increased within the enacted inquiry and argument contexts as the teacher positioned herself as a facilitator.

This paper reports on mentoring relationships that developed as a professional development strategy for mathematics teachers in secondary schools in low socio-economic areas. It follows from an earlier paper (Kensington-Miller, 2004) in which four different professional development strategies, one of which was mentoring, were evaluated for their effectiveness in understanding mathematics teacher development in low socio-economic schools. The paper then theorises the effectiveness and the difficulties that occur within different mentoring relationships. The relationships are described using a continuum model ranging from judgmental to developmental.

This paper provides a critique of the notion of sustainability situating the analysis in the context of recent developments in mathematics education in New Zealand. It examines how the term sustainability is used internationally in the literature on curriculum reform and nationally with reference to the Numeracy Development Project. I argue that the complexity and multi-dimensionality of the term is underestimated and it appears to have become an inappropriate slogan for the next stage of numeracy development in New Zealand.

The written responses of 88 secondary students to a task involving the interpretation of two data sets were examined to study the interplay between students? mathematical/statistical knowledge and context knowledge when interpreting data. Findings indicate that most students relied on rudimentary mathematical/statistical procedures although contextual aspects of the task did figure into the work of some students. Characteristic patterns of students? reasoning are reported.

This study investigated the language issues for senior Pasifika mathematics students in one school in the Manukau region. Using the experiences of the Pasifika teacher/researchers, a test was constructed to investigate likely language difficulties. Some interviews were conducted about the test Initial perceptions that vocabulary was the main problem were not borne out in the first test, although the interviews indicated that the language issues were interwoven. There was some evidence for differences between Tongan and Samoan students, and between those who knew their Pasifika languages well and those who did not.

Teacher educators play a part in trying to shape the nature of classroom mathematics teaching by their work with pre-service student teachers. There is usually an established curriculum that is underpinned by some particular philosophical or political agenda. Is, however, the intended curriculum what is perceived by the students to be occurring in their academic mathematics teacher education learning environments. Here the perceptions of primary student teachers of their mathematics education lecturers? practice are considered using constructivism as a referent.

This investigation examines the role of the home supervisor (a parent) in the development of children?s numeracy in a distance education learning environment. The case study explores the relationship between home and school partnerships as young students make sense of mathematics outside of a ?more traditional? classroom context. The learning materials provided by the school were more likely to be used in flexible and dynamic ways if the supervisor initiated the (re)construction of learning activities. Although the home supervisor had the most dramatic influence on the teaching and learning processes being implemented to support young student?s numeracy development, it is argued that new advances in technology may provide opportunities for classroom teachers to become more involved in the learning process.

In the context of the current education reform in Papua New Guinea which strongly encourages the use of both Indigenous knowledge-based systems and local vernaculars in teaching various school subjects in elementary and primary schools, the research reported here is the first of an on-going longitudinal research project carried out to determine the long-term effects of teaching early arithmetic strategies using the K?te counting system. The sample consisted of 52 elementary school children, aged between 8 and 10 years attending Elementary 1 class from 3 different elementary schools. The elementary school children?s performances on 7 different numerical tasks were obtained via an individual-task-based interview Contrary to the negative view held by many parents in Papua New Guinea, the results show that the use of counting and arithmetic strategies embedded in the children?s own traditional counting systems is an added advantage in their learning of formal English arithmetic strategies normally taught in schools.

Chris and Leesa are two Indigenous people who have come to mathematics education through their involvement in the struggle for Indigenous rights; Chris from being an Indigenous researcher with a PhD in Applied Mathematics and Leesa from a degree in Commerce and experience publishing material on Indigenous culture. Tom and Annette are two non-Indigenous people who have come to Indigenous culture through working in schools in Indigenous communities; both are mathematics educators and researchers. This paper attempts to describe the profound effect of Chris and Leesa?s perspectives that mathematics education devalues Indigenous culture (see Matthews, 2003) on Tom and Annette?s Indigenous mathematics-education projects at QUT. It indicates the potential of partnerships to empower Indigenous research. Note: The paper is written in heteroglossic form: Chris and Leesa?s Indigenous perspectives are written in normal font; Tom and Annette?s reactions and reflections are written in italics.

Professional development programs usually aim to support teachers as they explore innovative approaches to making their teaching more effective. The experience of the Early Numeracy Research Project is discussed, indicating that change in teacher beliefs and classroom practice can be an outcome of a major professional development project. It is posited that seeing teachers as co-researchers and reflective professionals contributed to teacher professional growth in this project.

This paper focuses on the growth of understanding about ratios and fractions by a group of four pre-service teachers engaged in an on-line knowledge-building community. Pirie and Kieren?s (1994) model for the growth of mathematical understanding was used as the framework for analysing their growth in understanding. This analysis revealed that the pre-service teachers made considerable advances in their understanding of ratio and fraction instructional representations and mathematical constructs embedded in ratios and fractions.

Students are documented as struggling in high school with using formal proofs. In particular, students believe that algebra is expected by teachers but is a component of responses not easily understood by themselves. This research documented the use of algebra by 35 students from a range of high schools throughout New Zealand when responding to two justification question in an externally set end-of-high-school examination. Although the numbers of students are small, some differences according to gender and socioeconomic background and question format suggest that further research needs to be done.

Twenty-six children from Years 5 to 8 were given a task-based interview on fraction knowledge. The verbal responses of the children to three tasks out of the 35 offered are discussed in depth here. The use of measurement and spatial language was evident in correct and incorrect answers. What was more critical were misconceptions about understandings of measurement principles. The rational number domain with its infinite number of points between any two fractions can be seen as more like the measurement domain where continuous properties are measured, than the whole number domain where discrete items are counted. This analogy may be useful for children?s conceptual understanding of fractions.

The concept of mathematical understanding is central to curriculum development, classroom interaction, and the training of mathematics teachers. In this paper, some models of the growth of understanding that the literature presents are outlined. Some results of a study that documented four primary teachers? mental models of, and beliefs about, different forms of understanding are reported. It is proposed that linear models may restrict ways that these teachers plan lessons. Questions for further research are raised.

This paper reports one aspect of a larger study which looked at the strategies used by Grade 6 students to solve six non-routine mathematical problems. This paper summarises the findings in relation to the students? success at various stages of the problem-solving process and the influence of metacognitive thinking on this success. Results suggest that most difficulties arise from a lack of metacognitive activity during the execution and verification stages of the problem-solving process.

In this paper, we analyse grade 6 students? reasoning in some tasks, which involved comparing two simple expressions. The tasks allowed students to move from procedural to structural understanding of simple expressions. Analysis of students? responses revealed various patterns of thinking and different ways of communicating the reasons. Students were found to justify their responses using language or symbols. The students moved from using predominantly language to using symbols predominantly, as the expressions became more complex.

This paper is a report of an investigation of differences in the professional development needs and preferences of teachers of primary mathematics across Queensland. A state-wide survey was employed to investigate teachers? prior PD experiences, their current PD needs, their dispositions towards engaging in PD and their preferences for the timing, duration and location of PD events The survey data were analysed in terms of geographical location and school size. The majority of primary teachers who responded reported that they had experienced little or no professional development (PD) in mathematics in the previous two years However they indicated that were eager to participate in such events. Differences in PD preferences were detected among teachers in remote, rural, provincial city and capital city regions and across small, medium and large schools. The issue of PD has become more important since the recent introduction of a new syllabus for Years 1 to 10 mathematics in Queensland and pressures to improve performance in benchmark numeracy tests, and the major education systems will need to design programs of PD to cater for the varying needs and preferences of teachers across the state.

Females remain underrepresented in tertiary Mathematics Research has indicated that this is, in part, due to the way that mathematics is traditionally taught. This paper reports the results of a participant observation intervention in the teaching of proportional thinking to Year 6 girls. It was found that by taking account of their preferred learning styles with the provision of specific scaffolding and the use of integrated and authentic tasks most females enjoyed the tasks and developed a stronger disposition to study mathematics.

Lesson study describes a process that supports teacher collaboration and the implementation of reform pedagogy This study investigated whether teachers? practices changed while participating in a lesson study and sought to identify aspects of a lesson study that advanced this growth. Analysis indicated that the discourse patterns of teachers who participated in a lesson study changed when they personalised an insight, suggesting that critical reflection is a catalyst for teachers? professional growth.

Twenty preservice teachers participated in a collaborative action research project in which they taught mathematics lessons about shapes in 12 classrooms of a local primary school. This paper reports on the development of teachers as they establish substantive communication about shapes in two upper primary classrooms. The paper highlights common themes from analysis of the other students? reports. Videotape or audiotape transcripts of the lessons and student work samples were available as well as my own classroom observation notes. The preservice teachers? reflections were critical in their development of questioning skills. Their desire to modify approaches and their growing knowledge of concepts built their self-confidence as teachers.

This paper reports a study monitoring the development of 53 preschool children?s patterning skills in two matched preschools, in one of which a 6-month intervention promoting patterning was implemented. The development of an interview-based assessment comprising identification, repetition, visual recall, extension, transformation, and representation of patterns is described, and a preliminary analysis of the interview data is presented. Children participating in the intervention program showed much greater improvement over the course of the 6 months than the non-intervention children.

This paper reports on part of a larger study that investigated the effect of a single context on students? mental computational performance and strategy choices across a range of school ages in the Perth metropolitan area. Money was the context and students? prior experience with money was also considered. Students were given the same mental computation test twice ? once with the items in context and once without. Despite many curriculum documents espousing the importance of teaching and learning mathematics in context, this study found that neither experience with nor use of the context of money had any effect at all, except in Year 3.

Pre-service primary teachers in their first mathematics pedagogy subject at the University of Western Sydney completed three surveys: an achievement test of the mathematics they would be expected to teach; a survey of their beliefs about mathematics, mathematics teaching and mathematics learning; and a survey of their attitudes towards mathematics. This paper reports the data from the achievement test and the beliefs survey, and investigates the relationships between these.

Historical conceptions of projectile motion have varied from the Aristotelean through impetus theory to Newtonian mechanics; but its standard mathematical treatment is only possible within the Newtonian framework. This paper reports a study suggesting that many Australian Year 12 mathematics students do not conceptualise projectile motion within that framework, but rather use a variety of Aristotelean, impetus and Newtonian conceptions. The implications for the teaching of projectile motion are briefly discussed.

This study describes errors and misconceptions of pre-service primary teachers, at course entry, across the mathematics curriculum. A Rasch analysis of a multiple-choice mathematics test uncovered patterns of errors in a cohort of 426 students at the beginning of their teacher education course. These errors were of varying sophistication. A map of an individual?s errors is also presented and we discuss how teacher educators and students can confront subject matter knowledge misconceptions using the diagnostic capability of the test.

Helping young students develop mathematical literacy involves promoting their development of mathematical knowledge, but also their ability to adapt and apply mathematical knowledge to a range of tasks. Extensive research has been directed towards the development of large-scale programs that have successfully promoted young children?s acquisition of early mathematical knowledge. Less research has explored the application of mathematic knowledge to tasks which cross disciplinary boundaries. This paper argues for further research of this type, to support the development of approaches that promote comprehensive early mathematical literacy.

Commencing and graduating preservice teachers completed written surveys and interviews. Results from an item in the written questionnaire indicated preservice teachers intend to provide children with good examples and help them replicate them in both literacy and numeracy lessons. Follow-up audiotaped interviews with 17 of the 186 graduating preservice teachers surveyed found preservice teachers were as likely to use the strategy at the end of their courses.

The purpose of this paper is to show empirically that ?the awareness of assumptions? is an effective teaching principle. For this purpose, lessons were designed and taught by the author with emphasis on ?the awareness of assumptions? for 9th Grade students in Kanagawa Prefecture. Lessons were analysed based on videotape-records, students? descriptions in the solution process and written impressions. For this group of students it was confirmed that ?the awareness of assumptions? was effective as a teaching principle.

Concerns about the importance of variation in statistics education and a lack of research in this topic led to a preliminary study which explored pre-service teachers? ideas in this area. The teachers completed a written questionnaire. Responses were categorised in relation to a framework that identifies levels of statistical thinking. Although the pre-service teachers have had more real-life experiences involving probability and have been involved in the study of probability concepts at secondary school level, they still demonstrated the same misconceptions as younger students. Additionally, while more students showed competence with the birth problem, they were less competent on the die toss question. This could be due to task format or contextual issues. The paper concludes by suggesting some implications for further research.

This paper reports on the first stage of the development of a methodology to evaluate mathematics textbooks. The methodology depends on the development of a set of mathematics curriculum principles relevant to the context in which the textbook is used, in this case the state of Queensland and its new years 1 to 10 syllabus. Based on these principles, as well as syllabus content statements and published research, sets of specific curriculum goals are being developed for key topic areas in the curriculum and these goals form the basis for the evaluation of textbooks. The paper reports on the development of the goals and sets of indicators for the topic of ratio and proportion and on a small trial application of the methodology.

In an environment where cross-curricular or interdisciplinary activities are being recommended for classrooms, the question of the assessment of multi-faceted objectives becomes important. Should multiple tasks be employed for multiple objectives or can a single task be devised in a manner such that more than one rubric can be used to assess the multiple objectives? As a first step in the direction of using a single task for more than one objective, a task from statistics is employed to explore the possibility of assessing the two ideas of Expectation and Variation using two rubrics on the same task. Advantages, disadvantages, and educational implications are explored.

This paper demonstrates the use of relative risk, a statistic which is widely used in other areas but currently under-utilised in education. Relative risk analysis provides a language for comparing educational outcomes as well as statistical tests of significance. We illustrate this statistic with data on students? understanding of decimal notation. In particular, we determine differences in how misconceptions operate at different ages, by analysing the relative risk of primary and secondary students persisting with particular misconceptions, and becoming experts.

While the use of technology at university level is an accepted part of some courses such as statistics and applied mathematics, there is still considerable resistance toward its use in pure mathematics courses in some universities. Regardless of whether one is sceptical or favourably inclined about the inclusion of technology, there are concerns relating to its implementation and instrumentation that are worth considering. As part of this research 502 stage one pure mathematics students at The University of Auckland volunteered to express their views on the use of TI-89 calculators in their course. The results for these questionnaires are discussed in this paper.

Teaching via an integrated approach is regarded as beneficial to students because connections are drawn between content areas so as to help them develop greater understanding of concepts inherent across disciplines. This paper presents the results of a case study that was designed to explore the knowledge, beliefs and practices of primary school teachers who integrate mathematics and music. The study extends our knowledge concerning teachers? beliefs and practices and broadens our understanding of issues surrounding subject integration.

In England and Wales, National Strategies promote pedagogies that emphasise interactive whole class teaching, although this is not defined precisely. In recent years major investment has been made in Interactive White Boards (IWB) and, whilst they do not determine pedagogy, as cultural tools they tend to support and encourage whole class teaching. This paper discusses the nature of interactive teaching and suggests that deep rather than surface features of interaction must be addressed if learning is to be improved.

This paper reports on data from 183 nine-to eleven-year old children attending six Waikato schools. An analysis is provided of children?s responses to a question about their teacher?s role in helping them learn mathematics. Four major roles were identified, including mentor, classroom manager, transmitter of information, and arbiter of decisions. The implications for students? mathematics learning of viewing their teacher in each of these roles are then discussed. The importance for teachers of recognising the ways they are perceived by their students is also considered.

465 students, comprising students in Year 9 classes in Thailand, Year 10 students in Brunei Darussalam, and second-year university students in the United States, attempted to solve the same quadratic equations. Most of the school students and many of the university students were confused about the concept of a variable and the meaning of ?solution to a quadratic equation?. Most of the students in the three subsamples acquired neither instrumental nor relational understanding of elementary quadratic equations.

In this paper we describe a program to enhance pre-service teachers? mathematical and pedagogical knowledge through a partnership with middle years teachers and learners in a primary school. We wanted to investigate how pre-service teachers? experience of teaching mathematics could enhance both their knowledge of mathematics and the development of generative practices. The findings have inspired us to develop similar partnerships with other primary schools.

Much primary mathematics teachers? work consists of designing, selecting, assigning, and monitoring learning tasks. This paper uses postmodern perspectives to examine task-oriented pedagogies of mathematics, demonstrating how the discourse supporting such pedagogies conflicts both with the discourse embodied in international declarations which advance children?s right to participation in decisions affecting their lives, and also with participatory principles espoused within the discourse of ?good governance?. The paper raises issues about children?s autonomy, entitlement to control their learning environment, and spontaneous determination of their own educational journeying and considers alternative discourses of participant-determined mathematical learning.

This paper examines teacher actions that support young children to consider repeating pattens as co-variational (functional) relationships, to use this understanding to predict uncountable steps in the relationships, to express these relationships in general terms, and use repeating patterns to introduce proportional thinking. A teaching experiment was conducted in two classrooms, comprising of a total of 45 children whose average age was 9 years and 6 months. This experiment focused on exploring teacher actions (including the use of concrete materials, recording of data, and questions asked) that supported young children?s development of co-variational reasoning. The results indicated that explicit instruction assisted children to find patterns across the table as well as down the table, to find the relationships between the number of tiles and an uncountable number of repeats. Also the results indicate that young children are capable of not only thinking about the relationship between two data sets, but also of expressing this relationship in a very abstract form.

Contextualisation of mathematics for Indigenous children has received an increased degree of attention over the past two decades. This paper offers an Indigenous perspective to contextualised activities undertaken at one Indigenous learning space, an inner-city kindergarten and pre-school centre (referred to throughout this paper as The Centre), situated in Brisbane, Queensland. It discusses these activities from the perspectives of cultural safety and empowerment.

This study uses Rasch modelling to link student outcomes over the decade since the introduction of chance and data into the curriculum of an Australian state in 1993. Although improvement is observed over time for intact groups of students, and between grade levels in a given year, improvement across cohorts for given grades over time is not observed. The distribution of the items used in the 2003 survey across the statistical literacy variable supports earlier models of the hierarchical nature of statistical thinking obtained from a larger pool of items.

The theory of Teaching for Abstraction was used to construct a unit for teaching percentages in Grade 6. The unit focused on percentage as a multiplicative relationship and put more emphasis on why percentages are used than on how they are calculated. Analysis of the data collected showed that many students did learn to apply percentages meaningfully, although the level of achievement was not as high as expected. It was found that the Teaching for Abstraction approach provides many challenges for students and teachers, but is worthy of further research.

In this study bibliotherapy encouraged pre-service teachers to reflect on their school experiences and their views of themselves as learners of mathematics while studying school students? experiences. Pre-service teachers? responses to readings on students? learning problems allowed them to reconstruct their experiences, changing their self-image of themselves as students, and their assessment of their capacity to learn and teach mathematics. Their reflections emphasised the lasting influence of individual teachers, and showed awareness of alternatives to the teaching approaches they experienced.

The Numeracy Project suggests a language model based on teachers listening and questioning students? justifications and explanations. This paper presents the analysis of the discourse in one classroom lesson and compares it to the language features identified in the Numeracy Project. The interview and lesson analysed here display features that appear to be consistent with the aims of the project. There is also evidence to suggest that students used some characteristics of this discourse with one another when working on problems together. The discourse in a second classroom provides a contrast. The Numeracy Project places little emphasis on using correct mathematical terms and presenting complete evidence of the forms that guide more advanced mathematics. It is suggested that educators become aware of these weaknesses and place an emphasis on these aspects of mathematical discourse in addition to those aspects promoted by the Numeracy Project to increase the likelihood of the students developing mathematical language and thinking.

The importance of conceptual understanding in mathematics has been well documented (Hiebert & Carpenter, 1992). This paper reports selected results from a research study that used a conjecture-driven teaching experiment (Confrey & Lachance, 1999) to enhance Year 3 students? conceptual understanding of multiplication. The teaching experiment employed children?s literature as a mediational tool for students to explore and engage in multiplication activities and dialogue. The SOLO taxonomy (Biggs & Collis, 1989) was used to both frame the novel teaching and learning activities, as well as assess the level of students? conceptual understanding of multiplication.

Seeking to represent the views of participants, as opposed to those the researcher, can require the latter to suspend their views. To achieve this in a way that prioritises the researched over the researcher, the method of stimulated recall can offer potential. This paper discusses the method used for collecting young people?s ways of working in situ across a number of workplace settings.

The first set of examinations, for the original cohort of 78 students from three volunteer schools, were conducted in November 2002 and some initial analysis of student performance, in particular with respect to items common with the standard course, was reported at the 2003 MERINO conference. This paper further investigates student performance on this aspect of examinations for 270 students of the expanded pilot program in November 2003.

The use of textbooks in mathematics classrooms has the potential to displace a teacher?s ability to shape in their learners an identity of participation. In such settings, issues of inclusion and exclusion in learning communities are revealed. Thus, some young people are shaping an identity of non-participation maintained by the practice of relying on textbooks to teach mathematics. This paper draws on research in progress to argue that the practice of using textbooks influences identities and the forms of participation in mathematics learning communities.

When students move from school to university they have to overcome many problems, academic, social and personal. One of the important factors influencing this transition is the students? own expectations. In this study we have undertaken an investigation of mathematics students in the UK and in Portugal. We concentrated on exploring how students felt their university experience would compare with school, in a number of areas, covering aspects of their own work and study regimes, the anticipated conditions of learning and their interactions with mathematics. The work was part of a wider investigation of the transition from school to university mathematics carried out at the University of Southampton in the UK, and Universidade do Minho (Braga) in Portugal, made possible by a grant from the British Council Treaty of Windsor programme. We conclude that in a number of important areas the students? expectations are not realised, in significant ways. It is also clear that in some areas there are differences between the UK and Portugal, and this suggests that the issues should be explored in relation to other countries as well.

Concerns about students' difficulties in statistical reasoning and a lack of research led to a study which explored form five (14 to 16 year olds) students? ideas in this area. The students were interviewed using open ended questions which related to sampling and variation, graphical representations and probability. This paper presents and discusses the ways in which students made sense of sampling and variability questions. While some students demonstrated sound reasoning, the findings revealed that many of the students used strategies based on prior experiences. The use of non-statistical responses (beliefs, everyday and school experiences) were considerably more common than that discussed in literature. The paper concludes by suggesting some implications for teachers and researchers.

Despite increasing girl retention in advanced high school mathematics programs, female college entrants disproportionately avoid math-related IT majors (National Academy of Engineering, 2002; National Research Council, 2001). Girls on Track (Berenson, 2004; Berenson, Vouk, & Robinson, 2002; Howe & Berenson, 2003), a longitudinal multi-institutional intervention program, works to increase middle-grade girls? interest in math-related careers. Project data shows that parental influence is a key factor in the girls? success in school mathematics and in their career choice. We identify four major roles of parents: ? Providing learning infrastructure such as books, software and tutoring ? Being a role model and providing access to other community role models ? Managing time and tasks ? Emotional support and encouragement

In Korea, the gifted education appears to gain much popularity lately that it has become a ?hot issue? in the education circle. Unusual enthusiasm of Korean parents for education led them to take their children to a reputable centre for gifted education programs, which enormously increases the demand for gifted education. Gifted education programs were primarily operated by gifted education centres run by universities. Now many elementary and secondary schools have started launching gifted educational programs because government?s policies encourage gifted education. At this juncture where gifted education has been rampantly expanding without thorough planning, it is necessary to review the gifted education form a reflective perspective. Gifted education is mainly implemented following three: selection, education and evaluation for reselection. In this paper, the issues of selection and evaluation for reselection will be touched upon from a critical point of view. The aim of this paper is to investigate mainly problems for the selection and reselection of mathematically gifted students. Inappropriate and appropriate problems and examples of student answers will be presented.

This paper presents a case study of a student who developed quadratic schemes by solving a contextual problem during a teaching experiment. Mary reflectively abstracted patterns from graphical transformations, critical points, rates of change and equations representing variations of a base context whilst interacting and negotiating meanings with the researcher. A multi-representational software assisted Mary in verifying/justifying her conjectures. Findings include Mary?s schemes to characterize quadratic covariations. The student-researcher interactions fostered the development and consolidation of Mary?s quadratics conceptions.

This paper explores the problem-solving frameworks of 621 Secondary 2 students (13 to 14 years old). These students were asked to complete in writing the statement ?When I am given a mathematics problem to solve, this is what I do...?. The data were coded under the four phases of problem solving which were arrived at from a preliminary analysis of the data. The four phases: Understand / Represent the Problem (U), Find a Way to Solve the Problem (F), Solve the Problem (S), and Check the Solution (C) were noted. Secondary 2 students were found to rely on individual problem-solving frameworks to guide them when solving problems. The frameworks of different mathematics-anxiety students were similar, brief and specific in nature.

This paper reports the upshot of a poster contest ?Mathematics-Language of the Universe? conducted for prospective teachers on 10.11.2002 in ?Tuticorin, India?. 20 student teachers (12 females& 8 males) from 7 colleges of education in ?India? participated in the poster contest. I designed the poster contest to meet the objective of out bringing the concept orientation of student teachers (STs) towards mathematics and to provide an opportunity to organise their thinking about mathematics in a creative way. The contest delineates the specific objectives of exposing the significance of mathematics, utility value of mathematics in daily life and social values of mathematics by way of eliciting the prospective math teacher?s ideas. Philosophy in Mathematics, aesthetic beauty of Mathematics, Mathematics and art of living, Mathematics and religious tolerance are some of the themes selected by the STs. Role of Mathematics in the development of human science, Mathematical symbols and shapes used in daily life, Mathematics and aeronautics are some other concepts used by the contestants. 4 STs depicted how Mathematics promotes other subjects by choosing different personification of Mathematics such as the rising sun, tree root, the sun with planets and peacock with feathers. One contestant tried linking mathematics with the structure of chemical molecules. This contest served as a window to analyse the beliefs of future teachers about Mathematics as a subject.

In a study of 744 Western Australian Year 8 and 9 students, three different types of misperception were identified when students performed linear transformations. To examine their particular perceptual problems, nine persistent misperceivers (those who misperceived in the same way on three tests) were given one-on-one remedial instruction. Case studies of two different types of persistent misperceiver are presented here, one involving angles, the other involving reflections, to demonstrate to maths teachers what may be happening in their classrooms unbeknown to them.

The acquisition of the ?language of zero? is important in the developmental progression of place value understanding. This short communication paper reveals the ?language of zero? that children and teachers use in Christchurch primary school classrooms and considers educational implications of research findings from a larger study. Key international literature regarding place value development and acquisition of number are referenced. The paper is intended to invite feedback to inform subsequent studies regarding the acquisition of place value language.

While there is more research on kids ages eight and up, less is known about young children reasoning with tables. A series of design experiments (Droujkova, In preparation, In press) led to a table reasoning model (TRM) describing children, ages three to eight, working with qualitative and numeric tables. TRM includes four components of reasoning: ? Sameness/variation: what and how entities are varied or kept the same. ? Action/object: ways children conceptualise entities as objects and actions. ? Local/global: reasoning about neighbour cell patterns vs. thinking of rows and columns. ? Coordinates/function: focus on the position of entries, or on functions forming entries. Competent work with tables requires fluent use of all components in multiple contexts. Moving toward such competency, young children exhibit a variety of reasoning patterns, often surprisingly different from adult approaches (Brizuela & Lara-Roth, 2002). For example, a beginner typically fills every cell with the same picture with no variation. Children view the 2*3 cell in the multiplication table as a product (action) of two numbers (objects), or as a doubling (action) of the number three (object). During missing cell value tasks, children search clues in row and column interactions, exhibiting global reasoning, or look at local neighbour cell patterns. In ?jigsaw? tasks of putting cut-up tables back together, children can focus on coordinates of each cell within table structures, and/or on the function used to create the cell entry. TRM helps to analyse ways children reason about tables. It is a robust tool for designing learning activities developing table reasoning.

The purpose of the roundtable is to discuss with interested people the challenges for mathematics education faced by Pacific Island Nations. The form of the roundtable will be oral presentations and discussion rather than papers to reflect the oral style traditionally used in Pacific Island countries. The session will start with people working in these countries speaking about the problems in their particular localities with a focus on cultural, political, geographic, and economic influences on mathematics education. This will be followed by a more general discussion about the possible ways that some of these challenges might be addressed. Particular issues that are expected to be discussed include: ? the ways that members of MERGA might possibly help the Island nations; ? the possible role of research; ? the types of research that might be appropriate, and ? the involvement of non-indigenous researchers. Although research will be a major focus of discussion, other development activities are also likely to be discussed.

High school students across Australia traditionally take at least one mathematics subject in their senior years of study. These subjects are seen as necessary for success at university and an important part of a students? well rounded education. Similarly statements are often released relating the importance of mathematics in the changing technologically based world. Yet more and more universities have removed the mathematics entry requirements from their courses and few universities list numeracy as an attribute of one of their graduates. Currently in Queensland, the majority of students planning to study at university still choose to study mathematics in their senior years, although there have been significant decreases in the number of students studying calculus inclusive mathematics courses. But will students still opt to study mathematics at all if universities do not include it as a prerequisite? This round table discussion will present preliminary results of research into the current status of mathematics entry requirements in Australian universities and the academic numeracy demands of courses for which senior mathematics is not an entry requirement. We aim to investigate, through discussion, how to determine what mathematics is required for university study, to answer the question ?What are the numeracy demands of university study??

Numeracy has been a major focus of recent professional development initiatives for primary school teachers in many countries. Research has shown that a variety of factors contribute to a school?s ability to sustain gains after completion of professional development programmes. An exploratory action research study undertaken with five inner-city primary schools focused on identifying and providing a variety of tools to sustain effective numeracy practices. Teacher surveys, professional discussions with staff, and classroom observations informed the intervention. Numeracy facilitators/researchers worked with principals and Lead Teachers in mathematics to develop increased knowledge and understanding of issues surrounding numeracy teaching and learning. A central theme was the use of reflective practices which promoted the use of critical peers, observation, and videoing of mathematics lessons. Major issues found to be inhibiting sustainability included: leadership and staff changes within schools, a lack of school wide numeracy policies, and teachers? lack of in-depth knowledge of numeracy content and pedagogy. Interim findings suggest that teachers require further support and opportunities to experiment with, and reflect on, new ideas encountered within the professional development. Offering external support, creating a forum to address the issues, and promoting professional learning communities within the schools, enabled the schools to refocus on effective numeracy practices. Some reluctance by teachers to deprivatise or critically reflect on their practice was apparent. Round table participants are invited to contribute to a discussion of factors which may contribute to the sustainability of effective numeracy practices, to inform further research.

We have already entered an era of greater scrutiny of the nature and impact of our research from practitioners and we are about to receive active evaluations of our research productivity from the government and other funding agencies. First, we need to understand DEST policy developments and implications. Next, we need to understand the processes used by funding agencies in awarding grants, especially the ARC. Then we need to consider the types of projects that will have an impact on the field of education, and especially mathematics education. This roundtable will allow participants to share understandings of policy directions, especially the anticipated research assessments and the implications for our activities, it will review processes for applying for competitive grants, and it will allow discussion of implications for future mathematics education research. There will also be discussion of the MERGA research policy and consideration of whether variations or additions are needed to anticipate the forthcoming evaluative environments.

The aim of this round table discussion is to examine the merits of a project-based teaching approach for teaching statistics. My proposed research involves the lecturer using a project-based teaching approach that covers all of the prescribed content concepts in a statistics paper. This project would be introduced initially, and then every concept would be taught with respect to its relevance in the project analysis. The taught content would include the method of sampling, data analysis and the drawing of conclusions. Data measuring the level of reasoning in statistics along with student confidences in using statistics would be measured for students taught a similar content in two different ways. It will be hypothesised that students will display significantly greater ability to reason statistically with more confidence than those that have been taught using a traditional topic order approach with a multitude of examples. Yesilay (2000) suggested that students tend to learn more by doing a project than in any regular coursework. Murtonen & Lehtinen (2003) identified linking theory with practice as one of five difficulties experienced by education and sociology students in quantitative methods courses. It is hoped that the use of a project will enable the student to learn more effectively as it will provide a link from theory to a real world problem. Discussions will revolve around the use of a project-based teaching approach in either the assessment or delivery of a paper in statistics. Specific questions addressed will be: How have you used projects in the teaching of statistics? How effective did you find the use of these projects?