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Discussion, theorising and research in the area of mathematical knowledge for teaching began by asking what level of mathematical qualification teachers need. We have moved on considerably since then, using concepts such as pedagogical content knowledge, and frameworks such as those from Deborah Ball in Michigan or, more recently, Tim Rowlands in Cambridge or Alan Schoenfeld in Berkeley. Some lovely research looking at teachers in action and asking questions about what they bring to their mathematical teaching decisions has pushed our ability to adequately describe the complex environment of the teacher. We are left with the feeling that the practice is still not properly understood. Anne Watson in Oxford has recently written about "mathematical modes of enquiry"-- mathematical practices that teachers emulate in ways appropriate to the classroom. I will discuss this concept through examples, and propose an extension to it that looks not just at what we know mathematically, nor how we behave mathematically, but also at the way we know: how we hold mathematics.

What might it take to support novice teachers to develop the commitment and the capacity to enact ambitious mathematics instruction? In this paper, we describe our experimentation with pedagogies in teacher education to develop novice teachers’ competence in eliciting, responding to, and advancing students’ mathematical thinking. Our efforts centre on the use instructional activities as a tool in supporting the learning and doing of features of ambitious instruction. We explain our use of guided public rehearsal as a pedagogical practice for helping novices practice the interactive contingent nature of classroom teaching in a way that supports their direct interaction with children.

The SIMERR project Mathematical Thinking of Pre-school Children in Rural and Regional Australia: Research and Practice included a review of relevant research literature with the aim of making this accessible to researchers as well as early childhood teachers and educators. This paper introduces the methods used in the project and provides a brief summary of the literature pertaining to the development of mathematical concepts.

This paper describes some key findings from the analysis of video data comprising part of the SIMERR project “Mathematical thinking of pre-school children in rural and regional Australia: Research and practice”. This analysis focused on the identification of young children’s mathematical thinking as it occurred naturalistically in two different pre-school environments. The broad categories of children’s mathematical thinking drawn from the video data were consistent with those identified by early childhood professionals when they were interviewed as part of the larger study.

This paper reports on interview responses to 3 questions seeking preschool practitioners’ perceptions of mathematical thinking in very young children. Generally, the interviewees were found to have a good sense of mathematical concepts relevant to babies and toddlers, and they cited evidence of young children’s mathematical development. It is concluded that this practical knowledge provides a strong foundation for further professional development.

The SiMERR project Mathematical Thinking of Preschool Children in Rural and Regional Australia: Research and Practice concluded its final report with a series of implications for the future. Many of these have the potential to impact on early childhood teacher education and professional development. In this paper, these implications are considered with reference to other recent studies and with particular emphasis on the development of the national Australian Early Years Learning Framework.

This collection of papers outlines the processes used and some results from the SIMERR project Mathematical thinking of preschool children in rural and regional Australia: Research and practice. The project’s focus is on what childcare assistants and teachers think and do in kindergartens and childcare settings (mainly catering for birth to 4 years) in relation so the development of children’s mathematical understandings, language and skills. It was found that many childcare professionals have a good sense of a range of early mathematical concepts but do not think regularly about these when programming and supervising play-based activities. However, many of the children’s play activities were shown to be inherently mathematical. The project report concludes with implications for early childhood professionals and policy makers.

The papers in this symposium focus on crossing perceived divides to improve the quality of the learning process in mathematics. The second and third papers are empirical studies demonstrating how contextual factors can support or prejudice learning, respectively. Psychological, sociocultural and poststructural epistemologies are brought together in the first and final papers to provide an arguably more complete, though more complex portrayal of the learning process in the empirical studies and mathematics generally.

Psychological views of learning have long informed mathematics and numeracy education. Such views support reasoning processes as foundational to the construction of knowledge and its application. In this cognitive sense the learning process is productive, though it can also be productive in the construction of identity (the learner’s sense of self as a legitimate participant in numerate practices). A poststructural axis of analysis recognises the power relationships in all teaching interactions, and the learner’s constitution as active numerate participant with/in them.

Improving quality in the learning process, from a socio-cultural perspective, clearly involves more than adopting best instructional methods. This paper focuses on an attempt to provide a quality inquiry based instructional program for preservice teachers. Volunteered survey responses reveal contextual conditions which facilitate and constrain the bridging of gaps in conceptual understanding, identity and interest in mathematics. The paper may be of interest to teachers and teacher-educators, researchers and policymakers interested in providing quality learning experiences through flexible delivery methods.

Teacher professionalism is a social construct which varies across time and place. In relation to the teaching of mathematics in the middle school, professionalism is a concept under siege; with the advent of national testing some teachers predict dire consequences for student learning. In this small research project several teachers depicted their views of ‘quality’ in relation to the teaching of mathematics and potential threats from the coercive context in which they work.

Teaching mathematics responsibly involves more than the application of disciplinary and pedagogical knowledge; it also involves appreciating and implementing flexible and dynamic interactional patterns that value diversity and students’ idiosyncratic attempts to make sense of learning. New ways-of-being a teacher that support the unpredictable and creative in learning reach into, yet beyond, the psychological and sociocultural for their philosophical ground. An overarching poststructural arm of analysis assumes that power relations inherent in interactional patterns in learning and work constitute or create practice.

In this symposium, we discuss the re-conceptualisation of problem solving in the school mathematics curriculum that is part of the M-ProSE project (Mathematical Problem Solving for Everyone). We propose the use of mathematics practical lessons, similar to practical lessons in science education and the use of an assessment rubric that focuses on key processes of problem solving. We also focus on teacher preparation for a seamless implementation of the problem solving curriculum in school. The following three papers will be presented during the symposium:

In this paper, we discuss the development of a very specific problem solving curriculum in an independent school in Singapore as part of the first phase of our research project. We are using a design research methodology to fine-tune the problem solving curriculum in which we are introducing the mathematics practical, an idea borrowed from science education.

In this paper we elaborate on the ways for assessing problem solving that goes beyond the usual focus on the products of the problem solving process. We designed a ‘practical’ worksheet to guide the students through the problem solving process. The worksheet focuses the solver’s attention on the key stages in problem solving. To assess the students’ problem solving throughout the process, we developed a scoring rubric based on Polya’s model (1954) and Schoenfeld’s framework (1985). Student response to the practical worksheet is discussed.

Rich mathematical tasks are central to the pedagogy that underpins the MITK project. In this paper I present and discuss the characteristics of rich tasks as we have defined them in the project. Activities that have the qualities outlined here can provide the ‘tools’ for teachers to develop their pedagogy so that deep mathematical learning is promoted for all.

Learning is a social activity but mathematics classrooms are often sites for solitary work or at best, parallel work where students may sit in groups but do not interact. For the engagement of learners, working collaboratively in mathematics represents a significant shift in practice. This paper explores the principles of cooperative learning environments ad with the roles that learners need to undertake to ensure group participation and membership. Working in Indigenous environments, there are considerable challenges to implementing a collaborative learning environment despite its synergies with Indigenous ways of knowing.

In this symposium, I argue that the use of home language needs to be viewed as a valuable resource for teachers in Indigenous schools in the Kimberley region of Western Australia. For students to be able to negotiate meaning and mathematical concepts in Kriol can help facilitate their mathematical learning as well as demonstrate an explicit valuing of indigenous cultures and heritage.

This contribution to the Maths in the Kimberley project symposium focuses on teachers' actions around the review of student work. Recognising that teacher actions at each phase of a lesson are connected, it is argued that there are two different but complementary rationales for effective reviews of student work, and two phases at which such reviews occur. The following uses a vignette from a lesson observed as part of the project to elaborate some of these actions.

This symposium presents results from one aspect of a project investigating the use of particular types of tasks in mathematics classes. The first paper provides an overview of the project, and the following three papers elaborate aspects of the three main task types used as the basis of the project.

This paper provides a theoretical background and rationale for a project that involves an investigation of the power of different types of tasks. This background paper focuses on the role of tasks in creating a learning environment and takes into account the effect of the teacher in the process of task implementation. The Task Types and Mathematics Learning (TTML) research project investigates how four types of mathematics tasks contribute to mathematics learning in the middle years of schooling. Through professional development and data collected through observation of teachers and students, interviews, focus groups, and surveys, the project aims to describe the features of successful exemplars of each type, constraints that might be experienced by teachers, and teacher actions that can best support students� learning.

As part of the Task Types for Mathematics Learning Project (TTML), teachers developed and used a range of tasks which focused on tools, models or representations. Data were collected on the ways in which middle years teachers described these tasks, their preferences among particular tasks, the opportunities they saw them as providing, and the constraints they observed during their use. There was general agreement that these tasks form an important part of a balanced mathematics curriculum.

Following professional learning sessions which focused on developing and using tasks in middle years classrooms which began with "real" contexts, teachers trialled such tasks in their classes, and then completed a survey, the results of which are reported here. Teachers were able to articulate the features of such tasks, see potential benefits, and articulate opportunities and constraints in their use. Secondary teachers saw greater constraints in using such tasks than did primary teachers.

After teacher learning sessions on open-ended tasks, teachers trialed such tasks in their classes, and then completed a survey, the results of which are reported here. It seems that the teachers collectively could adequately define open-ended tasks, could give illustrative examples, and could articulate both opportunities and constraints. This knowledge allows teachers to plan to take advantage of opportunities and to minimise the constraints.

The role of the teacher is central to the success of any curriculum innovation. Thus, teachers’ professional development has become an increasingly important subject of discussion in recent education literature. In the design and implementation of the project reported here, teachers’ preparation for the problem-solving curriculum featured prominently. This paper discusses the challenges of selecting a suitable problem and ways of using it productively within a professional development programme that the authors carried out for the teachers involved in the project.

In recent years there has been a noticeable increase in the diversity of backgrounds, abilities and aspirations of students entering first year mathematics courses at The University of Queensland. With the number and diversity of students entering Australian universities increasing, it is important to know what level of mathematical understanding they bring with them. This study investigated University of Queensland first year students’ mathematical abilities.

This paper reports on three of ten themes that emerged from a study of the impacts of a fraction teaching intervention on the mathematics anxiety and fraction competence of eight Year 8 students. Two of the themes that arose from the multiple data sources related to teachers and teaching and a third related to parental support. The students identified practical, hands-on activities and group work as impacting positively on their learning and their confidence in relation to fractions. The influence of parents, either positive or negative was also mentioned by several of the students. The study highlights the connections between mathematics anxiety among middle school students and the nature and extent of support that they perceive from teachers and parents.

An enduring issue for education in Australia is the poor performance of Indigenous students in mathematics. This is more pronounced in remote locations where many of the teachers are new graduates who are enthusiastic but lack experience and are unfamiliar with the complexities of teaching in remote and/or Indigenous contexts. This paper discusses the beliefs and practices of teachers working in a remote, Indigenous region of Australia. It is proposed that the discrepancy between beliefs and practices found in the reconnaissance phase of a design study is due to the teachers realising that they need to implement changed practices to enable students to learn but have little knowledge of what such practices may look like. As such, the discrepancy forms a powerful space for teacher professional development.

This paper presents the questionnaire attitudinal data from a project conducted over two terms to investigate the impact of using two meta-cognitive tools, vee diagrams, and reflective stories on some students’ mathematical understanding, competence in solving problems and mathematics attitudes. Findings have implications for improving students’ mathematics attitudes and thinking and reasoning in mathematics learning and problem solving.

As part of a qualitative longitudinal study on engagement in mathematics during the middle years, a group of twenty Year 6 students were asked to provide their views on mathematics teaching and learning. This paper explores their perspectives of mathematics, firstly drawing on current literature concerning issues of engagement in mathematics and some of the factors influencing students during the middle years of schooling. Students discussed the qualities of a ‘good’ mathematics teacher and these are examined in light of current teacher professional standards.

Numerous recent historical and current developments have occurred that impact on the nature of mathematics and mathematics education, and education in general. In this paper some of these are discussed as they suggest possible new foci for research.

A binomial problem is examined through the use of three methods: a classical approach using the binomial formula, Pascal’s Triangle, and a frequentist approach using a Fathom™ simulation. Using both physical and virtual coins, students explored the distinction between a coin-sequence and a coincombination (where the order is not important) as a foundation for a 5-item True False test task. Pedagogy was based on general principles identified by probability education research as best practice and by Ma (1999), who emphasised deep understanding of concepts through the use of multiple approaches to problem solving.

This paper reports analysis of the concept maps of the understanding of function developed in secondary school constructed by seven experienced secondary mathematics teachers who were part of a larger study. The concept maps were analysed according to (a) key notions related to the definition of function, (b) process or object view of function, and (c) identification of the importance of working within and across representations. The findings suggest a teaching emphasis that might not be supportive of students developing a deep understanding of functions.

Student participation in Collective Argumentation (CA) has been shown to have beneficial effects in promoting student participation in the mathematics of the classroom. However, little is known about the effects of students’ participation in CA on their learning of mathematics beyond that classroom. To provide insights into this issue, students, who had participated in CA classrooms in the past, were asked to respond to a questionnaire designed to elicit their perceptions of the worth of participating in CA. This paper analyses a sample of those responses in terms of the goals of schooling for young Australians.

Developmental assessment processes support the idea of assessment for learning and are based on qualitative judgements of students’ work. Introducing teachers to a developmental model on which to base these judgements aimed to change teachers’ assessment practice. Mathematics teachers in six high schools in New South Wales were provided with professional development over three years that introduced the SOLO model. All teachers reported changes to their practice but these were mainly in the context of teaching rather than assessment. They also identified structural constraints that prevented them from implementing the model.

Statistical literacy is a comparatively new concept in mathematics and while there is some consensus about how it is defined, there has been limited research on how the concept is measured within a school context. This paper, reports on the development and validation of an instrument to measure middle school students’ self-efficacy for statistical literacy. The items were developed from the relevant research literature and then tested on a sample of 366 students. A Rasch measurement methodology was used to create the measure and to provide evidence for its construct validity. The evidence reported in this paper indicates that the proposed instrument has suitable reliability and validity properties.

This paper reports gender differences in the types of interests that middle school students have towards statistical literacy. These differences are detected from the responses of a sample of 366 middle school students to items in a statistical literacy interest inventory. In particular, dominance statistics are calculated in order to detect evidence for gender differences in student responses. Results indicated that girls were more interested in aspects of statistical literacy that related to surveys and boys were more interested in aspects relating to problem solving and also contexts that are associated with sports. The findings are interpreted using Dweck’s achievement goal theory as it pertains to gender, and classroom implications are reviewed.

Although various studies have shown that groups are more productive than individuals in complex mathematical problem solving, not all groups work together cooperatively. This review highlights that addressing organisational and cognitive factors to help scaffold group mathematical problem solving is necessary but not sufficient. Successful group problem solving also needs to incorporate metacognitive factors in order for groups to reflect on the organisational and cognitive factors influencing their group mathematical problem solving.

One finding from a larger investigation of teacher-child interactions that challenged young children to probe their mathematical understandings is reported in this paper. It is the interlinked nature of the conversational exchanges between teacher and child in mathematics lessons in the early years of primary school. These conversations serve to support, extend and challenge children�s thinking. One detailed example of such mathematical conversations, the story of Jordan, is presented to illustrate the phenomenon.

Introducing algebra in high school is a teaching challenge, where appropriate explanations and representations are essential. This study examines teachers’ reactions to a page of explanation about the distributive law from a textbook, and investigates their pedagogical content knowledge about the teaching of algebra. Teachers were aware that students have difficulties with algebra, but the study shows that some of their strategies for teaching it are mathematically unsound. In particular, “fruit salad” approaches are still prevalent.

This paper presents snippets from a research agenda focused on the interplay of mathematics education and the language, social, cultural, and political issues that affect Mexican American communities in the USA. The lessons learned and issues raised should be of interest to other contexts with non-dominant, marginalized students. The research approach is grounded on a holistic view of education, that includes families / communities, teachers and students. The underlying goal is to develop a better understanding of the appropriate conditions for the participation of all students in mathematics classrooms.

This paper reports on a research methodology for an ongoing PhD thesis, focusing on the author’s experience as he collected data in Ghana. It looks at what the author did during data collection in Ghana and his experiences in the field. It also highlights some of the institutional gaps that appeared to exist between educational institutions in Ghana and Australia, and discusses some implications for planning and execution of mathematics education research methodologies in a developing country like Ghana.

Promoting the concept of density is regarded typically as the domain of science teachers. However, density as the relationship between an object’s mass and volume highlights the importance of foundational measurement concepts located in the mathematics curriculum. In this study, an integrated mathematics and science unit on density was developed for students in the middle years of schooling (Years 4-9) by the research team. Implementation of this unit in Year 5 classrooms is reported here. The lesson sequence appeared to support students’ capacity to use the language of density in meaningful contexts. Implications for teaching are drawn from the data.

This paper reports on strategies 26 Grade 3 students used to solve a range of division word problems in a one-to-one interview. The focus is on the strategies used by the students to solve partitive and quotitive division problems pertaining to four different semantic structures. Of particular interest was the range of strategies used for each form of division. Results suggest that there was little difference between the strategies used for partitive and quotitive division.

This paper reports a design experiment focusing on instruction to support low-attaining 3rd- and 4th-graders’ development of conceptual place value (CPV). CPV is advanced as an instructional domain more suited than conventional instruction in place value, to support learning of multi-digit mental calculation. Distinctive features of CPV are described; a rich description of a realised instruction sequence for CPV drawn from a ten-week, one-on-one instructional cycle of twenty-nine 25-minute lessons is provided; and modifications to the sequence are proposed.

Some studies suggest that teachers make inferential judgements about students’ understanding based on the ways in which they use language. One aspect of language use is that of pronouns. Students’ use of first, second, or third person pronouns indicates the extent of commitment to the truth-value of their utterances. This paper discusses a study that investigated students’ use of pronouns as they explained their understanding of introductory algebraic expressions and equations. The evidence suggested that pronoun use can act as an indicator of the level of students’ conceptual development in this context, which might also be aligned with a theoretical model.

Using a case study approach, this pilot study sought to investigate how teachers' use of particular mathematical tasks in the classroom impacts on low-attaining students. The perspectives of one teacher and two low-attaining students were considered in an attempt to investigate the interplay between task, pedagogy, affective responses and student learning. Four particular teacher actions, potentially beneficial for low-attaining students, were also examined.

This paper originates from a qualitative self-study into the lived experiences of the author, as a beginning teacher of primary school mathematics. The focus of the study was to identify the influences, issues and challenges of a beginning teacher trying to implement reform-oriented teaching practices. One of the important issues to emerge from data collected in both years of the study was the use of a compulsory textbook and the author’s changing perceptions (as teacher and researcher) of its role and value as a teaching and learning tool.

This paper reports on an aspect of a three year longitudinal study which investigated students’ use of technology in individual and collaborative classroom settings. A socio-cultural perspective was adopted to develop the MSPE framework which identifies modes of student technology use and describes the student-student-technology relationships that developed as students worked in individual, small group and whole class settings. Implications are discussed for how the framework might lead to more sophisticated technology rich pedagogies in mathematics classrooms.

This paper presents a theoretical framework for guiding inquiry in an emerging field of research concerned with the learning and development of mathematics teacher educators. Such research is needed for two reasons: to open up the practices of mathematics teacher educators to critical scrutiny in order to better understand the nature of their impact on mathematics teaching in schools, and to systematically inquire into how mathematics teacher educators learn from engaging in teacher education.

Rate of change is an important mathematical concept. Research referring to students’ difficulties with this concept spans more than twenty years. Research suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate of change. This study investigated 23 Victorian Year 10 students’ understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate of change emerged. Four important aspects of the concept were identified and gaps in students’ thinking defined. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.

This paper is an analysis of the challenge of assessing student learning and how that is affected by using descriptions of competencies as a core element when describing the aims of the learning process. Assessment is modelled as a three step process; characterising, identifying and judging, to allow for the following argument: Working with competency descriptions is rightly said to make judging more difficult. This potentially lowers the reliability of the assessment. But competency descriptions also carry a great potential of raising the validity of the assessment by focusing the characterisation part of an assessment process. From a teaching and teachers� perspective, the latter is far more important than the former.

Although the New Zealand Numeracy Development Project has contributed toward achievement successes for many students there still remains a large gap, by OECD standards, between students whose achievement is as expected and students whose achievement is of concern. This gap then extends further to the identified tail of underachievement. For those who are underachieving the cycle could continue throughout their schooling. This paper reports on a new initiative and advocates for a more structured approach for some children, alongside the constructivist and co-constructivist approaches their teachers employ. Of interest is the successful work undertaken with underachieving students by parent volunteers in a primary school.

The transition from school to tertiary study of mathematics is coming under increasing scrutiny in research. This paper reports on some findings from a project analysing the transition from secondary to tertiary education in mathematics. One key variable in this transition is the teacher/lecturer. This paper deals with a small part of the data from the project— analysing secondary teachers’ responses to questions on the differences teachers perceive between school and university and the importance of calculus, a bridging content. The results show that teachers lack a clear understanding of the issues involved in the transition and there is a need for improved communication between the two sectors.

Current reforms in mathematics education strongly advocate the development of mathematical learning communities in which students have opportunities to engage in productive mathematical discourse. In this paper, I address how a teacher used interactional strategies in order to facilitate the participation of students in mathematical discourse. I outline the specific pedagogical strategies the teacher used to shift students’ patterns of participation from passive listeners engaging in non-productive disputational talk to engaging in collaborative interaction and productive mathematical discourse.

This paper employed observational and interview data to explore how Gardner’s Multiple Intelligences points of entry assist teachers in catering for student diversity within primary mathematics classrooms. Drawing upon a sociocultural research approach the study found that Gardner’s multiple points of entry facilitate some ‘valuing and working with difference’ within classrooms.

This study explored successful junior high school principals’ leadership practices for implementing the reformed mathematics curriculum in Taipei. Avolio and Bass’s (2002) full range leadership theory was used to record data through interviews and observations of five Taipei Grade A junior high school principals. Findings revealed that specific leadership practices linked to management by exception-active and contingent reward (transactional leadership), and individualized consideration and idealised influence (transformational) were considered effective for implementing reform measures. Ensuring principals are aware of effective measures may further assist reform agendas.

Being numerate for teaching involves more than personal numeracy; while the construction and communication of disciplinary knowledge will always be important, so to is the creation of a professional self, capable of appreciating and implementing flexible and dynamic interactional patterns that value diversity and students’ idiosyncratic attempts to make sense of experience. New ways-of-being a teacher of numeracy are founded on robust intellectual knowledge, though the learning landscape is constitutive, influencing participation and interactional patterns in the classroom.

This paper provides an account of a recent research study that focused on the implementation of the Queensland, Year 1-10 Mathematics Syllabus (QSA, 2004), at Hillside Primary School in South East Queensland. A context analysis of curriculum change supported by a literature review is provided. This analysis and review enabled the development and validation of a questionnaire used to assess sources of support for curriculum change. The results of this study alerts policy makers, practitioners and researchers to the level of complexity of curriculum change in mathematics and the need to provided both internal and external sources of support during the implementation phase.

Over time, academia has become a more attractive field for females. Yet males still dominate, particularly among senior academic staff. In this paper we document females’ participation with respect to key academic indicators typically used as criteria for entry to, and promotion within, the tertiary sector. Our focus is on Australian mathematics education journals, and predominantly those published by MERGA. Using these data we examine whether females are (proportionately) as well represented as males or whether these measures of esteem act as subtle structural barriers not yet fully conquered by females in the mathematics education research community.

The effectiveness of educational mathematics games were investigated by testing a game of fractions with Year 8 students. The influences of the game on students’ achievement in fractions were assessed using pre-post quizzes and maths tasks. This paper focused on the achievement differences of students between pre and post quizzes. The objectives were to identify the increases of achievement from pre quiz to post quiz and the relevancy with the game. The results showed that the game enhanced students’ understanding on representations of fractions.

This paper describes an investigation of students’ strategies for solving linear equations. The assessment techniques used for investigating algebra paralleled those used for investigating number in the New Zealand Numeracy Development Projects (NDPs). In this study of 621 Year 7 to Year 10 students oral interviews were used to investigate the strategies that students used to solve equations. Rasch analysis was used to investigate item difficulty and student ability, and then the strategies associated with each question were examined. The data suggest that there is a hierarchy of sophistication of strategies. Many students were unable to solve a lot of the equations as they were restricted to less sophisticated strategies. The most sophisticated strategy of solving equations by performing transformations was understood by very few students.

This case study looks at how a teacher used interactive exercises and exploratory investigations to reinforce concepts that have been learnt in a Year 8 Mathematics class. It highlights the practical problems faced by the teacher in terms of technical issues such as slow download times and logging on problems as well as class management issues. The findings are discussed in the light of the teacher’s mediation with the affordances of the Internet.

This study situates early notions of average within an inquiry classroom to investigate the rich statistical concepts young children draw on to make sense of the questions: Is there a typical height for a student in Year 3? If so, what is it? Based on their deliberations over several lessons, students’ ideas about average evolved through four conceptions - typical as: reasonable, most common (value or interval), not atypical, and representative of the population. Implications for teaching are discussed.

A group of Māori students were identified as high achievers in mathematics in their Intermediate schools. A research project has been following this group of achievers for the last five years. This paper explores how their Māori parents view their relationship and involvement with their child’s school. The voices of Māori parents and some of their offspring testify to a mismatch between the culture of the home and the culture of the school. This paper raises questions about factors that may keep these Māori students involved in mathematics study.

Steady declines in the number of students taking higher-level mathematics courses have been evident across most States and Territories over the last decade. Rather than its potential for positive feedback, mathematics learning is perceived to be ‘hard’, repetitive and demanding of time. No wonder many students question their engagement. This paper draws on the perceptions of teachers and career professionals to focus on three important transitions that students make throughout schooling, and considers the influences on their decision-making.

In this study, 88 Grade 6 students’ responses to three part-whole tasks are analysed in terms of Kieren’s (1988, 1993) four-part model for fractions- measure, quotient, operator, and ratio. Using data from interviews, their strategies were also analysed using traditional part-whole explanations, including identifying the count-and-match misconception which was found to have distinct variations.

A review of literature indicated that there has been a shift in mathematics classroom assessment practice. In the advocated assessment alternatives, mathematics teachers and their students are expected to undergo a cyclical assessment process. This assessment process typically consists of five stages that are planning, designing, implementing, marking, and reporting. As a consequence of this recent demand, traditional mathematics assessment culture is greatly challenged in tandem with the change of roles expected of teachers and students within this assessment process. Hence, this paper is a review reflecting upon the changing cultures of mathematics classroom assessment.

A key test used in Australia to assess the mathematical knowledge of young children uses illustrations of objects such as coins and three-dimensional shapes. This study explored the effects of giving 104 kindergarten children, aged 4-5 years, the questions with either moveable objects or illustrations. It was found that children who were categorized by their teachers as having “higher levels of numeracy” scored well on test questions using either illustrations or objects, while children who were categorized as having “lower levels of numeracy” scored higher with objects than with illustrations. This result could have implications for consideration of test item readability in relation to graphicacy.

This paper reports on the nature and results of a pilot study conducted with the parents of a Grade 1/2 class. The study investigated the mathematical perceptions of a selected number of parents and used an intervention program designed to encourage them to engage in numeracy activities with their child. Preliminary results indicated that the parents were keen to support contemporary classroom practices and were able to describe and evaluate their children’s mathematical understandings. The findings add to the limited research available on the role that parents can play in their child’s numeracy development.

A design-based intervention project was conducted to research the complexities of improving early number-sense learning outcomes for ‘at risk” children in the first year of school. Focusing on the growth of teacher knowledge, a combination of interviews, mentoring sessions, videoed lessons and web-log reflections were used as both learning processes and data collection methods. Preliminary analysis of one teacher’s lesson revealed several key teaching strategies that will enable interpretation of how teacher growth in knowledge translates into effective teaching practice in subsequent lessons.

This paper reports mathematical and real world knowledge application by lower secondary students (aged 13-14 years) in a design-based interdisciplinary project with mathematics, science, and geography as anchor subjects implemented in Singapore . Key findings from two mathematical tasks attempted by 10 case-study groups during the project reveal that whilst students displayed taught knowledge and skill, they lacked in-depth understanding of the use and purpose of scale drawings. There was also limited activation of real-world knowledge during mathematical decision making and little monitoring of accuracy and reasonableness of mathematical results.

This paper reports a case study from a project that worked with teachers using a networked system to obtain feedback from students and collate their responses during mathematics lessons. These data were immediately available in graphical format for teachers to review as a reflective prompt while events of the lesson remained fresh. The study sought to establish the kinds of reflections this approach stimulated and whether teachers found it useful. Overall it appeared this approach prompted higher levels of reflection.

Changes to the relative value of curriculum topics, when using computer algebra systems in secondary school mathematics, have been previously considered in several studies (e.g. Artigue, 2002; Stacey, 2003). This paper extends these findings to an examination of particular topics in undergraduate mathematics, as part of a wider study investigating issues of integrated technology at the tertiary level. This study suggests that issues of curricular value are a critical factor in the successful implementation of integrated technology, and that a re-examination of the relative values of fundamental topics remains a significant challenge for undergraduate mathematics in a rapidly evolving technological environment.

In this study we explored the errors due to analogical reasoning made by students from a sample of 69 Year 12 students in a Junior College in Singapore. The data collected included students’ responses to a test and post-test interview with selected students. The errors seem to stem from their earlier work in mathematics at lower levels. They wrongly transferred structural information from situations that they had met earlier at secondary level.

Research has shown that successful mathematics students use different types of strategies from those struggling with mathematics. Year 4 students were tested using the One Minute Tests of Basic Number Facts (Westwood, 2000) and a paper and pencil Number Screening Test developed by the author and colleagues. Observation of the students during the assessment procedures highlighted the vast difference in the students’ speed and accuracy when recalling basic facts and the types of strategies they used when solving mathematical tasks.

This paper explores the role of a problem-centred pedagogy (Collective Argumentation) in providing teachers with practices and tools to implement a problem-centred approach to teaching mathematics in a senior classroom. The paper focuses on a Year 11 lesson that required students to model and communicate the effects of a disease on a body’s immune system. Using transcripts of teachers’ talk the paper provides teacher perceptions of how students draw on the practices of mathematicians to illuminate and to communicate understanding to others. The study reveals that teachers consider that student capacity to act within a problem-centred curriculum, their sense of agency, is enhanced through participating in the practices of Collective Argumentation.

Students can have difficulty learning to solve equations because different types of thinking are needed. This study investigated scaffolding which allowed students to concentrate on decision making while software carried out other parts of the task. Prototype software developed for this study was trialled by adult students. Many students could solve more equations with the software than in the pre-test and half of these also transferred their learning to pen and paper.

Responses to an item intended to assess teachers’ Pedagogical Content Knowledge (PCK) are discussed. The item examined 92 teachers’ capacity to name the two forms of division, provide a simple representation and story problem for each, and explain which form helps to make sense of dividing a whole number by a decimal. Teachers appeared more familiar with partitive than quotitive division. Apparent understanding of quotition with whole numbers did not always transfer to division with decimals. Some findings from the broader PCK research are also shared.

This paper reports on ongoing research in a cluster of schools in the outer south-eastern suburbs of Melbourne which is utilising Lesson Study as a peer observation model for mathematics teaching. The findings from nine initial Lesson Study sessions undertaken by cluster teachers to develop a Fractions Teaching Program are presented. The results indicate the success of the fractions tasks created. The potential of Lesson Study as an effective model of school-based professional learning for teachers of mathematics became apparent.

Fraction ideas are among the most complex mathematical concepts students encounter in their school years. It is a multifaceted construct that underpinned many numeracy activities and the development of further mathematics. Secure understanding of these ideas is a vital part of being numerate. Despite its importance, there is no known research on students with disabilities learning this concept. This paper reports a pilot study of a doctorate degree to investigate one high school special education teacher and his students with disabilities’ construction of fraction ideas, some of the likely and unlikely difficulties to inform future research.

Details are provided of simple algebraic items which can be used to detect two particular ways students think about the numerical meanings of letters (in contrast to non-numerical thinking). The data from Year 7 students (n=228) and Year 8 students (n=139) on these items is analysed using response patterns to probe student thinking. Less than 10% of the students were correct on these items whilst the prevalence of the two most common error patterns is 20%-30% at each year level. New response patterns are detected, indicating that further improvements to the items will enable students’ thinking to be investigated further.

This study uses number sentences involving two unknown numbers to identify some key junctures between relational thinking on number sentences and an ability to deal with sentences involving literal symbols. In this paper, the focus is on how students were able to make generalisations on sentences involving two unknown numbers, and how these influenced their performance on sentences involving literal symbols. In so doing, it aims to identify some key linkages as students make a transition from arithmetic to algebra.

Applications and mathematical modelling have been a distinctive part of the senior secondary curriculum in Queensland for two decades. Findings from an on-going longitudinal study of this initiative are reported. Twenty-three teachers and curriculum figures from across the state were interviewed and artefacts collected from teachers. A deep understanding of mathematical modelling has alluded some teachers sometimes leading to their not seeing the purpose of gentle nudges from monitoring panels to move forward. A softly, softly approach to implementation has led to some progress on all fronts with approaches described as ranging from minimalist to very rich.

Prior to entering schooling settings, many children exhibit intuitive knowledge of mathematics and many have mastered basic addition combinations. However, often as a result of formal instruction, some children begin to dislike or fear mathematics. In this study, children at a kindergarten in China took a smiley-face survey to determine how their feelings and beliefs about mathematics were affected throughout their kindergarten years. Results suggest that even children in this study have a better number sense and mathematics achievement, they appear to develop mathematics anxiety in Chinese cultural context.

This paper describes the development of an instrument aimed at measuring senior secondary students’ ways of using graphics calculators as part of a larger study investigating the students’ learning styles and their ways of learning and using the graphics calculators. The instrument uses the framework of four metaphors of technology use (Geiger, 2005): Master, Servant, Partner and Extension of Self. Preliminary findings of a pilot study involving 178 senior secondary students from Singapore are discussed.

Mathematical knowledge production is increasingly seen as a multimodal activity, involving complex synchronic and diachronic interactions between semiotic systems. This paper documents a diachronic semiotic analysis of interactions that occurred as two secondary teachers engaged in a mathematising task requiring them to construct graphically an antiderivative, using the context of tramping. The roles of a number of different semiotic resources that were prominent at various times are described, along with suggestions on possible cognitive advantages these offered in terms of reduced cognitive load and increased affordances.

We draw on a 5-year professional development design study to discuss how a group of middle-school mathematics teachers came to view students’ reasoning as an instructional resource. The broadening of the professional development goals to include issues of key importance to the teachers played a key role. The study provides guidance in supporting teachers for whom students’ reasoning holds little relevance to their work at the outset.

This paper explores the question of teacher identity during the practicum experience. Foucault’s insights are applied to explain the ‘making’ of one pre-service teacher. I provide an analysis of the ways in which the realities of the social and material structures in schools play out in everyday activity within schools. The view is towards developing a sensitivity of the impact of regulatory practices on a pre-service teacher’s construction of herself as teacher.

This paper provides an analysis of interview responses from 40 teachers to a student survey question based on a pictograph. Responses comprise the first segment of 30-40 minute interviews exploring teachers’ pedagogical content knowledge (PCK) in statistics at the middle school level. The responses are analysed against a back-drop of previous research on how teachers integrate their understanding to plan for appropriate learning outcomes. Four components of PCK are identified and a wide range of capability observed in the group of teachers interviewed.

This paper reports on the use of a combination of prompted written reflection, online discussion and self-assessment as a vehicle for enhancing learning in an undergraduate primary mathematics subject. At the end of semester, 71 students completed a survey of their perceptions about the influence of the components of the experience on their learning, and their online reflections were analyzed for themes and levels of reflection. The students clearly valued the online discussions with their peers and engaged in quality reflection on a range of substantial issues, however the full role of self-assessment was unclear.

The NSW Counting On program was designed to support the professional learning of teachers in identifying and addressing the learning needs of students in the middle years who have difficulties with the early mathematical concepts and skills. It has a strong theoretical and research base and has undergone major changes which have been regularly evaluated. Counting On 2008 was evaluated and this paper will use the findings of the evaluation report (White, 2009 in press) to examine whether the program was successful in changing student learning outcomes.

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

Associations between resilience or optimism (Seligman, 1995) and the inclination to explore unfamiliar challenging problems in mathematics have been identified (Williams, 2005, 2008). This raised questions about how to build optimism to enhance mathematical performance. In this study, a theoretical framework was formulated to study optimism-building situations (Seligman, 1995; Csikszentmihalyi, 1992). By interrogating ‘Engaged to Learn’ pedagogy (Williams, 2000) through a video-stimulated interview study, situations theoretically expected to be optimism building were identified.

Bibliotherapy as an approach to eliciting and understanding the affective responses of primary school teachers is investigated in this paper. In this pilot study, the author explores bibliotherapy as a reflective tool in teacher professional learning, by analysing affective responses of primary teachers. Year 6 teachers voluntarily wrote responses to readings about school students’ learning, discussed their understanding of their own experiences in the light of the readings and gained some insights into the factors which impacted on them as learners and teachers of mathematics.

Giving student homework as learning activity is a common practice in many schools. How students complete their homework at home remains of interests. In this study, 27 of fourth graders were asked to complete their mathematics homework at home and return to school the next school day. Students were then asked to answer questionnaires after returning their homework to school. We found that parental interventions happened during homework doing process and direct teaching was involved in most of cases.

Two complementary processes involved in mathematical modelling are (1) mathematising a realistic situation, and (2) applying a mathematical technique to a given realistic situation. We present and analyse work from four undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required them to find the antiderivative of a function presented graphically. When determining the height of the antiderivative, they mathematised the situation to develop an elementary mathematical method, and attempted to apply some ideas about definite integration that they had previously learned in class. In the end, however, the participants favoured their more elementary mathematised knowledge over the sophisticated knowledge they tried to apply.

There is a common belief amongst teachers, particularly in the middle school grades, that students must have a firm foundation in basic skills before they can benefit from more challenging instruction Views about mathematics pedagogy were explored through interviews with fifteen experienced middle school teachers. They were passionate about mathematics but were caught in a "back to basics" dilemma, fuelled by their perceptions of declines in student knowledge and strong beliefs that basic skills can only be mastered through direct instruction, rote learning and repetition and must precede more complex learning. These findings have implications for middle school mathematics pedagogy.

(The research team and authors involved in this project also included: Helen Goldblatt, Margaret Hyslop, Louisa Linterman, Sandra Morgan, Chanda Pinsent, Julie Roberts, Brenda Sherley, Sandi Tait-McCutcheon, Marama Taiwhati and Leyton Walker.) To gain a representative snapshot of mathematics teaching following intensive numeracy professional development in New Zealand, twelve primary school classrooms were observed simultaneously within a geographical region. Researchers identified ‘mathematical events’ embedded within a typical school day, noting how teachers recognised and sought opportunities to make connections to mathematics throughout. Following data collection, the team analysed and interpreted findings - both individually and collectively. This resulted in the examination of ‘effective’ teaching practice as demonstrated by teachers who showed commitment to the learning of their students, focused on conceptual development, and made connections to prior learning and related mathematical ideas.

This paper describes research on the classroom practices of seven teachers who taught a lesson on multi-digit multiplication using array-based materials. Students� understanding of multi-digit multiplication just prior to the lesson is contrasted with their performance several weeks after the lesson. Differences among the teachers in the ways they taught the lesson are examined in relation to students� subsequent understanding of multiplication. An important issue was the match (or mismatch) between the demands of the lesson and students� understanding of multiplication prior to the lesson. For example, Teacher A carefully scaffolded her students from single-digit to two-digit by two-digit multiplication problems using the dotty arrays and her students made substantial progress in their understanding of multiplication. Teacher C�s students did not seem to have a solid understanding of place value or multiplication with small quantities, and did not learn how to use dotty arrays for multi-digit multiplication. Teacher G�s students already understood the partitioning processes needed to solve two-digit by two-digit problems prior to the lesson and probably did not benefit much from the lesson on using dotty arrays.

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here is as part of the first named author's recent PhD thesis where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the learning and teaching of some linear algebra concepts. This paper highlights some of the overall findings of this research and suggests applications for learning and teaching in undergraduate mathematics classrooms.

(Poster) The study is a collaborative effort between a teacher educator and three preschool practitioners. It adopted a narrative approach and yielded a ‘practical’ experience bearing direct relevant as well as significance value to preschool teacher educator and practitioners in the teaching and learning of math for young children. Examples of how children learn mathematical concepts through the use of computer software and hands-on activities will be showed in the poster.

Large-scale assessments such as NAPLAN (Australia) provide a wealth of data about the mathematical understanding and achievement of students. However, measurement is only the first step in the implementation of systems aimed at achieving improvement. This paper presents a framework for analysis of NAPLAN data from the perspective informed by the systems approach. Rather than ask how well the children perform, the exploratory analysis presented here focuses on identifying the obstacles to children’s success.

Approximately 200 students from two focus schools were asked to draw ‘what maths or doing maths means to you’. This task was used to augment information from a larger study of students’ beliefs about mathematics, collected from questionnaires containing likert-type questions and open-ended questions requiring written responses; 848 year 5 and 6 students, aged between 9 and 11, from 17 New Zealand primary schools completed the questionnaire. The concern was that not all students of this age are able to express their beliefs through a method reliant on their literacy skills. The results showed marked differences between the tasks.

The study aims to develop and validate Mathematical Thinking Assessment (MaTA) framework. It involved a total of 7 classes of 16 years old students and their mathematics teachers from Malaysia. The MaTA is implemented in the school context to assess students’ mathematical thinking: the performance assessment is administered to elicit students’ thinking processes during problem-solving; the Metacognition Rating Scale is used to specify students’ awareness while working on problems; the Mathematical Dispositions Rating Scale is used to indicate students’ disposition towards learning of mathematics whereas the Mathematical Thinking Scoring Rubric is used to score and grade students’ mathematical thinking according to the domains defined in this study.

(Poster) The poster illustrates two programs in numeracy assessment: the International Competitions and Assessments for Schools (ICAS) and the Early Literacy and Numeracy Assessment (ELNA). ICAS is a 40-item assessment of mathematics skills and content for Year 3-12 students which allows horizontal and vertical equating of student performance. ELNA is an online diagnostic computer-adaptive assessment program that assesses literacy and numeracy skills and knowledge for Pre-school-Year 2 students. Strands tested in numeracy are number, space and measurement, and patterns. ELNA enables individualised assessment and eliminates the ‘ceiling effect’. ICAS and ELNA provide an interesting research area in mathematics education.

School-based instructional leadership of professional development initiatives is important to embedding high quality instructional practices, so that students’ mathematics achievement is enhanced. By looking beyond designated leadership positions it may be possible to explicate alternative enactments of leadership in primary schools as they are played out in the school setting, with a specific focus on setting up school structures that support mathematics teaching. This paper uses a backward mapping strategy to explore ways of conceptualising instructional leadership in primary mathematics, as well as identifying school support structures that enable teachers to advance towards a vision of high quality mathematics instruction.

Each year a new cohort of 240 students undertakes the challenge of second year mathematics. The student feedback from one year is often a reversal of that received about the course run in previous years. The 2009 class was surveyed on their attitudes to mathematics at both the beginning and the end of the course. This paper reports changes in teaching methods implemented in response to the first survey, changes in student perceptions about mathematics and on the student feedback compared to previous years. From this data, some conclusions are made about how students’ best learn.

The aim of this study was to gain a better understanding of the misconceptions and ways in which pre-service teachers responded to a mathematical skills and knowledge test during their first year of the Bachelor of Education. Two hundred and ninety seven pre-service teachers’ results were collected. This paper will explore the answer of one question identified as difficult for nearly 90% of the cohort.

(Poster) This poster discusses a project about feet, undertaken by the class of children ranging in 4 to 5 years in a kindergarten of Hong Kong. Throughout this project, the children studied parts of leg, measured and compared their feet, found out the relation between the size of their feet and shoes, and experimented with making their own shoes. The article documents the children’s work through photographs, and provides the teachers’ reflections.

The importance of a consideration of large numbers in primary and early secondary school should not be underestimated. A study of history of India (and of the Maya) reveals that contemplation of large numbers provided the impetus for a construction of place value number system. While today’s students do not have to create a number system they do need to understand its structure in order to develop number sense and operations and to progress to algebra. This can perhaps be done more quickly by contemplating on large numbers. This study examined and categorised Year 9 students’ responses to a questionnaire on large numbers. The results suggest that many students show competence in naming and using large numbers and some are in the process of thinking beyond their curriculum level.

This paper reports how a group of prospective secondary school teachers in Singapore solved a quadratic generalising task. The findings show that the teachers were capable of deriving multiple equivalent expressions for the task. Three approaches were used to work out the rule: numerical, figural and pragmatic, with the last two being more predominant. It was also found that the same expression could be obtained from different visual representations, and conversely, the same visual representation could also produce different expressions.

Within a larger study of a Year 9 class investigating sample size students examined explicitly the legitimacy of the data generated by a Fathom die simulation. Instrumental genesis - the process where students adopt a tool as their own - is an established analytical framework used with computer algebra systems (CAS), and this framework has potential application in electronic probability simulations. In addition to a examination of students adopting the simulation's procedures and features, any reservations students have regarding the legitimacy of the data generated may act as a potential impediment to learning. More recently instrumental genesis has been extended to acknowledge the role of the teacher and the classroom environment through a process defined as instrumental orchestration.

In New Zealand by the end of 2009 most primary schools will have participated in Phase 1 professional development of the Numeracy Development Project (NDP). The project is now in its 9th year after the pilot ‘Count Me In Too’. Having self-managing schools in numeracy is the focus of Phase 2 of the project. Therefore the direction of support in the future will be towards maintaining quality teaching in classrooms. As schools are working on sustainability of good numeracy teaching and learning, the role of Lead Teacher is becoming more critical. They have as part of their role the continuous improvement of teacher practice. They also have a greater responsibility in improving, monitoring and reporting progress of students. Maintaining the momentum can be a difficult task and requires support from principals and facilitators in order to carry out Lead Teacher multi-layered responsibilities. Further to Deb Gibbs and Marilyn Holmes roundtable presentation at MERGA in 2006, facilitators have continued to explore issues around the role of Lead Teachers. Three years later, have changes for the better taken place or has the status quo remained? This small study focuses on how well the Lead teachers make use of the time allowed and looks at answers to the following questions: 1.Has the perceived value of the Lead Teacher role increased over time? 2.Has the support given to Lead Teachers to carry out their role effectively changed? Discussion around this study will highlight implications for lead teachers, principals and facilitators in numeracy.

The beginning of any lesson is an important event and may take on many forms and functions. In this presentation we reveal how three mathematics teachers began their year 9 lessons. Using data from the New Zealand component of the Learner’s Perspective Study (LPS) we examine the lesson event characterised as the ‘starter’. Within all three classrooms, the starter activities typically involved students in solving teacher-provided problems. However, closer analysis of video and interview data from each classroom sequences of ten lessons reveals differences in the intent, the nature, and the enactment of student/teacher activity. We explore how particular learning opportunities and modes of participation, both in the moment and later in the lesson, are interwoven with the starter activity.

The structure of two tier testing is such that the first tier consists of a multiple-choice question and the second tier requires justifications for choices of answers made in the first tier. This study aims to evaluate two-tier tasks in ‘proportion’ in terms of students’ capacity to write and select justifications and to examine the effect of different two-tier formats on student performance. Twenty students each from Y7 and Y8 participated in the study in Melbourne in March 2008. The students took eight similar tests with each test having eight two-tier tasks. Eight students were interviewed individually after the testing. Analysis of students’ responses revealed that i) Y7 and Y8 students were able to select and write justifications to two-tier tasks, ii) Y7 and Y8 students’ success in writing or selecting justifications varied on ‘marked answer’ and ‘select answer’ formats, and iii) Y7 and Y8 students’ justifications gave some information about their misconceptions in proportional reasoning. Implications for teachers looking for alternative assessment tasks tracing students’ reasoning behind their correct and incorrect answers are discussed.

A range of models of teacher professional learning have been utilised within Australian schools, with varying success. The views of 125 university-based ‘Critical Friends’ in the Australian Schools Innovation in Science, Technology and Mathematics (ASISTM) project were sought on the relative effectiveness of 15 models of professional development in relation to their capacity to impact on teacher professional learning and student learning respectively. The findings are shared, along with recommendations to those considering these models for future professional development programs.

(Poster) Year 9 students compared explicitly the fairness of three dice: a standard factory-made die, a die students fabricated using Sculpey (TM) modelling clay, and a virtual die in a Fathom (TM) simulation. During class discussion students developed a fairness measure - a single statistic quantifying the fairness of the die - calculated as the difference between the observed and the expected frequency in 30 rolls of the die. The students pooled the fairness measures and constructed poster-size dot plots, and the dot plots were then used in whole-class discussion to compare the fairness of the three dice.

This short presentation provides some preliminary results of a one-on-one numeracy intervention program developed as part of the Commonwealth Government Literacy and Numeracy Pilot Projects in low SES schools. The intervention was modelled on a reading recover program (Clay, 1993), using a lesson structure based on research into how the brain learns mathematics (Sousa, 2007). Students identified as having low numeracy levels in years 1, 4 and 8 at several Catholic schools in the West and South of the Archdiocese of Canberra and Goulburn were withdrawn for thirteen weeks and given on-on-one intervention by a specially trained teacher.

A student's mathematics achievement can be promoted or obstructed by their beliefs about intelligence in general (mindset), and about their own mathematical capability in particular (mathematics self-efficacy). Ways of investigating the effects of a mindset intervention and a mathematics self-efficacy intervention, and how these effects might be associated with changes in students' mathematics achievement, will be discussed. An exploratory study with Year 4-5 students and their teachers, drawing on design research methodology and including quantitative and qualitative methods, will be described. A case will be argued for extending our understanding of New Zealand students' self-beliefs in relation to mathematics.

Streaming (or ability grouping) for mathematics is a hotly debated and contentious issue. In this paper, data are presented from 44 Victorian secondary teachers who responded to an online survey. The aims of the study were to explore the extent to which streaming is implemented in Victorian post-primary schools, and to examine teachers views on the policies adopted in their schools. The findings indicated that streaming is widespread, even at grade 7, and that most teachers supported the policies in place. In supporting their views, various limitations to the streaming practices were also identified.

This paper reports the effect of task structure on two Y7 students’ performance in a pre-post testing situation. The tests consisted equal number of ‘tasks with confidence level’ and ‘tasks with safety-net’ structures. Two cases were analysed: i) one girl’s performance in eight weeks gap, and ii) one boy’s performance in two weeks gap. The boy’s performance was consistent and showed little effect from task structure. The girl’s performance seemed significantly affected showing progress in safety-net tasks which could be attributed to teacher’s feedback. Task structures uncovered different patterns of girl’s reasoning between the tests which was confirmed in interview.

The over emphasis of examination results has created phenomenon where teachers teach-to-test and students learn by rote such as memorizing all the facts without really know how to apply the knowledge and skills in the real life context. In order to reduce the focus on examination results, the Blueprint of Education Development suggested that students’ assessment should be measure through school-based grading system which is holistic in nature. In Malaysia context, there is yet available a well-established school-based mathematics assessment framework to be used by the school teachers. Hence, this study aims to develop and validate a framework for Mathematical Thinking Assessment (MaTA). However, prior to this, we see the importance of seeking teachers’ perceptions about school- based assessment. Hence, this paper discusses the interview analysis of 18 mathematics teachers from six secondary schools on the issues and possible challenges faced for the implementation of school-based assessment for mathematics. The results show that the lack of exposure, availability of assessment guidelines, preference to current traditional assessment, time constraint, students’ mathematical ability and English language proficiency; are the main concerns of the participating teachers in this study.

Student difficulties in learning algebra can arise from the diverse meanings assigned to variables. We propose teaching different aspects of variables (unknown, generalised number and function) in parallel with each other using real contexts and call this a multifaceted variable approach. We are investigating whether learning about variables using a multifaceted variable approach before moving on to symbol manipulations can reduce student misconceptions regarding variables and improve their algebra learning outcomes. This paper reports results from student interviews administered after the algebra teaching intervention. Results indicate that students of the experimental classes showed fewer misconceptions regarding variables than comparison classes and they were able to recognise that variables can have multiple values.

Thai education is moving toward student-centred learning. However, Thai students have had little experience with cooperative learning strategies. This qualitative study reports on Thai students’ perceptions about their engagement in cooperative learning in mathematics classrooms. The study found that Grade 4 students utilised four different but complementary processes (peer tutoring, peer relationships, peer assessment and group role) to facilitate their cooperative group work learning. These findings indicated that the alternative teacher preparation workshops need to provide additional workshops to improve the implementation of peer relationships, peer assessment and group role.

This paper proposes a new technique for data collection in classroom studies. The approach adopted is to regard teachers as a co-researcher and give them the decisions of what and when to collect data by using hand held video cameras in the classroom. Coupled with the data collection is an emphasis on the teachers following through a process that enables them to self critique their data and consider whether they have changed their practice. Pitfalls and successes in our first attempts at using this technique are documented.

A controlled randomized experiment was conducted to examine the hypothesis that by using real-life cover stories from learners' real-life to explain fractions, the new concepts could be more readily assimilated into existing knowledge held in long-term memory compared to the more traditional geometric contexts. Grade 5 students (n=32) from a Sydney public school participated in this study. A 2x2 ANOVA with repeated measures was used to analyse the results. The result supported the hypothesis.

This paper explores the impact of the project on the teaching of mathematics in senior secondary classrooms. The Secondary Numeracy Project provides professional development for secondary mathematics teachers. Schools opting into the project participate in school based in-service development of their mathematics teaching team. The specific focus of the Secondary Numeracy Project is on enhancing the teaching of mathematics to students in their first two years of secondary school. The impact on pedagogy in the senior secondary school mathematic resulting from the project, as reported by mathematics teachers who participated in the project, will be discussed.

(Poster) This poster outlines a study that aimed to investigate the potential of Computer Algebra Systems (CAS) to enhance the processes associated with mathematical modelling and application tasks. Data for the study was drawn from a one year study of five different secondary school classrooms. Analysis of the data revealed that there were significant differences in the uptake of CAS technology and of the use of applications of mathematics between schools. Implementation of technology rich approaches to mathematical modelling varied from classroms where students engaged in complex rich tasks to those where use of applications of mathematics was limited. This poster also highlights the affordances and constraints experienced by teachers in implementing new approaches to learning/teaching mathematical modelling through the use of technology.

The teaching and learning of basic facts is a topic of perennial interest and significance to teachers, teacher educators, and the mathematics education community. This study builds on to a prior study of basic facts teaching and learning in New Zealand by examining in detail the practice of one teacher as she reflects on her teaching and learning programme. From this study the authors seek to advance knowledge and lead to the provision of advice for teachers and researchers.

This presentation provides some preliminary results of the use of videos as a vehicle for reflection and planning with pre-service secondary teachers. A lesson was developed to illustrate the elements of a framework that we term ‘mathematically significant classrooms’, consisting of tasks (Watson & Mason, 2007), norms (Yackel & Cobb, 1996) and conversations (Chapman, 1993). Students’ responses to the framework, including their use of it in planning lessons during professional experience, will be discussed. The framework provides a robust framework through which teachers can reflect on the intellectual quality of a mathematics classroom (Gore, Griffiths & Ladwig, 2004).

This study explores how Collective Argumentation has given the students in a middle-school mathematics classroom a framework which allows them participate in mathematical discussions to develop the skills and desire to think, reason and work mathematically, where personal understandings can be expressed, re-considered, shared and co-authored. This has led to students sharing the authority and promoted student engagement

In Singapore primary schools, an approach of representing relationships between quantities using blocks, called the model method, is used to solve word problems. While very useful for difficult problems at upper primary grades, this method is taught at primary two for solving simple one- or two-step problems. A research study was designed to find out whether beginning primary three pupils would prefer to use the method or other methods for simple word problems. The pupils’ ability to handle the component parts of the method and the difficulties encountered were also investigated. This presentation will describe the findings of this study.

With the release of the New Zealand Curriculum document (2007) we, as two pre-service teacher educators, formalised our ongoing conversations about the nature of mathematics. A literature review is being undertaken as part of these conversations. This reveals a wide variety of explicitly made and implicit conceptions of mathematics. Of particular interest is Barton's description of NUC (near-universal conventional) mathematics and the suggestion that 'mathematics could have taken many forms, the forms and preferences of NUC-mathematics were not inevitable; they are the result of a particular historical trajectory that includes many social influences, including language' (2008, p. 24). Implications for mathematics in the NZC, and our work as pre-service teacher educators are being considered.

This project involved a teacher examining her use of an interactive mathematics program with a class of 13-14 year old boys to promote engagement in mathematics. Students identified a variety of mathematical concepts implicit in the learning experience, and described ways in which it presented opportunities to encounter, apply or develop these ideas. It provides an example of how the affordances of such software can be accessed by teachers developing new approaches to teaching through in-depth professional development. The teacher’s heightened awareness allowed her to capitalise on the novel context to promote increased opportunities for learning and participation in mathematics.

"Captivate", a product from the Adobe(TM) suite, is a video screen capture software that records students’ actions on a computer as a video recording. The software operates in parallel and behind the primary software - a typical example is Microsoft(TM) Excel. Students are aware that Captivate is running, but they soon ignore its presence and this allows a natural recording of students’ use. The software includes a audio recorder; if students work in pairs and are encouraged to discuss their work the audio recording adds immensely to the richness of the data collected. The video recordings are easily exported to widely-used file formats such as AVI and MPEG, and this allows the video-clip to be used independently of Captivate. The files may then be analysed, using for example, N-Vivo, and edited, reproduced, transmitted, and displayed as needed. Practising teachers may find application of the software as an instructional tool. Teachers could create short instructional video for students that they can replay on-demand in computer laboratories. This has the potential to improve teaching practice by liberating the teacher from some repetitive classroom tasks. The round-table seeks to discuss other researchers’ use of screen capture technology as a data source. The presenter will demonstrate recordings made during class and small group interviews.

Current shifts in teaching and learning practices in mathematics classrooms challenge teachers to develop mathematical communities which offer all participants opportunities to engage in mathematical inquiry, explanations, justification and generalisations. The complexities and challenges to achieve this are many, particularly with the diverse learners in our classrooms. We want to explain a framework we have developed to explore ways teachers can constitute classroom norms which support such dialogue. Teachers participating in the research study will describe how they have adapted and used the framework in their numeracy classrooms and also how they have used it as a reflective tool to plan next possible growth areas. In this session we seek feedback on where adjustments need to be made to meet the explicit needs of Maori and Pasifika students in particular. Whilst knowledge of how to develop productive communication and participation patterns with diverse learners in New Zealand is the focus of our study we hope that experiences from Australia and other countries will add to the discussion.

The University of Waikato Numeracy advisers will start the round table discussion by presenting the findings from a case note and a small research project involving teachers with significant numbers of Maori students in their classrooms and Maori students from schools that are working in depth in numeracy across the Waikato region. The international evidence cited in Wendy Nielsen, Cynthia Nichol, and Jenipher Owuor (2008) positively supports the enhancement culturally responsive pedagogy has on student’s connection with the learning process. In New Zealand, Best Evidence Synthesis research highlights the importance of relationship building with Maori students to increase engagement and raise achievement (Alton -Lee 2003). Aspects that might be considered in this discussion include the underpinning principles of Te Ao Maori, classroom strategies that seem to improve achievement for Maori students, Ka Hikitia, the Ministry of Education, Maori Education Strategy, and the work of Dr Russell Bishop et al.

The general acceptance afforded the national image of Australians as gamblers have given gambling a legitimacy rare in other countries. Concerns with the social effects of this have led many State governments to implement programs to counteract negative social effects. The Queensland Treasury has allocated funds for the development of teaching resources for this purpose including the development of the Unit presented here. In 2006 the author constructed a Unit of work for Queensland Senior Secondary (Years 11 and 12) Mathematics classes entitled ‘The Mathematics of Responsible Gambling’ as a consultancy to the Queensland State Government. Towards the end of 2007 the ‘Secondary Mathematics Teaching Resources Kit’ was distributed to all secondary government schools. This paper describes the activities of the Unit, their relationship to the Queensland Syllabus objectives, the research upon which the Unit is based, and the current research into the effectiveness of its implementation which began in November 2008 and will continue in Semester 1 of 2009.

This study examined the college mathematics placement exam results of 1,277 students learning from nine secondary mathematics curricula and two Advanced Placement (AP) mathematics programs in 25 different high schools in the United States. The results suggest that students learning from several traditional mathematics programs and AP Calculus significantly outperformed students learning from the reform-oriented mathematics program, Core-Plus Mathematics Project, on algebra manipulation and calculus readiness questions. Prior mathematics achievement, course completion, and gender also significantly influenced mathematics placement scores.