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I describe the design and iterative implementation of a learning progression for supporting statistical reasoning as students construct data and model chance. From a disciplinary perspective, the learning trajectory is informed by the history of statistics, in which concepts of distribution and variation first arose as accounts of the structure inherent in the variability of measurements. Hence, students were introduced to variability as they repeatedly measured an attribute (most often, length), and then developed statistics as ways of describing ?true? measure and precision. The design of the learning progression was guided by several related principles: (a) posing a series of tasks and situations that students perceived as problematic, thus creating a need for developing mathematical understanding as a means of resolving prospective impasses; (b) creating opportunities for developing representational fluency and meta-representational competence as constituents of conceptual development; (c) introducing statistics as invented measures of the qualities of distribution; and (d) adopting an agentive perspective for orienting student activity, according to which distribution of measures emerged as a result of the collective activity of measurer-agents. Instructional design and assessment design were developed in tandem, so that what we took as evidence for the instructional design was subjected to test as a model of assessment, resulting in revision to each. I conclude with a look at ongoing work to design an assessment system to measure students? understandings of data and statistics, and with some thoughts about prospective synergies between mathematics and science education.

I have long promoted the conjecture that expressing generality lies at the heart of school algebra. Indeed, I have gone further to suggest that ?a lesson without the opportunity for learners to generalise is not a mathematics lesson?. It seems beyond doubt that experiencing and expressing generality is natural to human beings. The pedagogic issue is why there is so much resistance amongst teachers and learners to using this power in mathematics lessons. The notion of generalisation here includes both abstraction from context and generalisation within context. Pondering this question has led us to wonder why generalisation happens sometimes and not others, what can be done to prompt useful mathematical generalisation, and under what sorts of circumstances: in short, what are the conditions for and evidence of imminent or proximal generalisation?

The mathematical problems, tasks, demonstrations, and exercises that teachers and students engage with in classrooms are, in general, specific instantiations of general principles. Indeed, the usual purpose of such examples is to illustrate those principles and thus facilitate their learning. With this in mind, it is clearly important for teachers to be able to choose or design suitable examples, to recognise what is offered (or afforded) by particular examples, and to know how to adapt an already existing example to better suit an intended purpose. Although writers of textbooks and other teaching resources also need these skills, it is ultimately the teacher who puts the examples to work in the classroom. Teachers? choice and use of examples is indicative of their pedagogical content knowledge (PCK)? the complex amalgam of mathematical and pedagogical knowledge fundamental to teaching and learning?and reflects their understanding of the mathematics to be taught and how students can be helped to learn it. This paper examines some of the issues associated with example use and how it is informed by and can inform us about PCK.

This paper examines a practice-based component of a primary teacher education program to gain insight into the type of experiences which assist beginning teachers translate theory based knowledge to their teaching practices. Eighty-six prospective teachers participated in the study. Data were collected from (a) weekly lesson plans; (b) researcher field notes; (c) reflective journals; and (d) interviews with four participants. A theoretical rationale for various aspects of the practice-based component is provided and the implications for teacher education programs are discussed.

This paper explores the number knowledge of 1015 children who began school in 2006 and of a further 3000 children in Grades 1-3. The data show that number knowledge varies considerably when children begin school, and that this variation extends as schooling proceeds. Teachers need to be aware of each child?s current knowledge and ways to customise learning experiences if they are to meet each child?s learning needs.

In the last four years there have been a number of calls for research into many aspects of early childhood mathematics education. As well, there has been an unprecedented increase in Australasian research in this field. How have these two factors matched? That is, are mathematics education researchers studying the aspects of the field that have been identified for further research? This paper provides the beginnings of a discussion around this question by highlighting particular Australasian early childhood mathematics education research endeavours and linking them to recent statements calling for further research in the field.

In recent decades the development of mathematical proficiency has been recognised as a key issue for children and their education. The purpose of this paper is to identify key international perspectives that influence Australian mathematics education in the early years especially those that are in a similar state of technological development to Australia. There are four key trends deserving of discussion: (1) development in the early years, (2) mathematical proficiency in the early years, (3) mathematics policy and curriculum designed for young children, and (4) the existing research evidencebase.

This paper illustrates the way two teacher-researchers are listening to mathematics education students? voices in a Masters course. Group assignments have their advantages but it is difficult to ensure strong collaboration, high-level analysis and discussion, a good spread of work between group members, and positive social interactions. This research set out to explore one way of attending to these problems in a mathematics education Masters unit. Students submitted (unmarked) individual essays before combining them to create (graded) group assignments. They completed surveys about group work before and after this activity, and some were interviewed. Expecting individual work before group work led to increased levels of engagement, very high quality work, use of skills in analysis and critique, and good levels of student satisfaction.

The essence of the demand for freedom is the need of conditions which will enable an individual to make his own special contribution to a group interest, and to partake of its activities in such ways that social guidance shall be a matter of his own mental attitude, and not a mere authoritative dictation of his acts. (John Dewey)

This paper illustrates the enriched understanding of classroom activity that can occur when the student voice is included in the research design and valid data is generated. It presents interpretations of classroom activity based solely on lesson video, and the same activity interpreted though video stimulated post lesson student interviews. It draws attention to the need to synthesize data from complementary sources.

This paper presents the results of interviews with Year 5 and 6 students about their views of effective teaching practices in mathematics. The students interviewed were part of a largescale study into improving middle years mathematics and science. Their views confirm findings from the literature and other data sources from the project, and provide valuable insights into student perceptions of effective teaching practice in middle years mathematics.

In prior-to-school early childhood settings mathematical play can occur in a natural and unstructured manner. This paper describes the findings of a case study of children in an urban Auckland early childhood centre engaging in socio-dramatic play in the family corner. This data gives rise to the notion that foundational mathematical knowledge can, and does, develop in very young children.

The concept map data from a study of Samoan university students constructing topic concept maps and vee diagrams of problems throughout a semester is presented. Students found that, initially, concept mapping their topic was difficult. However with independent research and multiple critiques, their understanding of the conceptual structure of the topics deepened, becoming integrated and differentiated as evident from the concepts selected, valid propositions and structural complexity of the maps. Students also improved their skills in negotiating meaning, challenging and counter-challenging each others? explanations. Findings imply concept maps can facilitate the effective communication of students? understanding within a social setting.

Data collected from a diagnostics mathematics test taken by some primary student teachers are reported. Student responses were analysed using the Dichotomous Rasch Measurement Model. Error analyses enabled the identification of main misconceptions. Findings showed students performed relatively well with basic computations and visually presented data but struggled with word problems. The more complex and abstract the language used, the more difficult it became, implying that the critical skills of interpreting mathematical concepts, representations, and language and problem solving require explicit remediation. Implications for primary teacher education are provided.

Survey results for anxiety responses and attitude responses to six particular mathematics problems are presented for 43 students from grades 4, 5, and 6. These data are analysed for a relationship between mathematics anxiety and attitude to mathematics. An online survey method is used and is found to be a valuable tool for use in a primary school setting. The six mathematics problems vary in type between traditional levelled tasks in the form of basic mathematical operations and rich tasks. Basic operations are varied amongst three levels of difficulty and rich tasks are varied amongst three degrees of complexity of context. A weak relationship is found between mathematics anxiety and attitude to the six mathematics problems presented to students. Some differences are observed between boys and girls for responses to rich tasks. Also, differences in both attitude and anxiety responses are found due to a variation of problem difficulty for traditional basic operations. Further research is suggested that promises to inform the pedagogies of practicing teachers.

As a teacher educator, I used narrative inquiry to investigate my professional practice in working alongside pre-service primary teachers in mathematics education. One theme that emerged from this research was the exploration of narrative as a powerful means with which to pursue professional development. In this process I encountered, and subsequently changed, previously unknown personal beliefs about learning mathematics. A second theme focused on the value of mathematical investigations, for myself as a mathematical learner and for supporting pre-service teachers to develop their understandings of what it means to learn and teach mathematics.

This paper draws on two studies of mathematics departments in 11-18 comprehensive maintained schools in England to compare and contrast the insights provided and questions raised by differing theoretical perspectives. In one study a mathematics department was viewed as a complex system and analysed accordingly. In the other activity theory was used to describe and analyse features of the departments involved. In both cases the departments involved were considered to be systems and it was the learning of the system rather than of individuals that was of interest. The affordances and limitations of the analytical perspectives are discussed.

Three selected student tasks from a 2-week study of the statistical concept of distribution in Year 9 class are examined. The tasks considered the exclusion of outliers, analysis of data using a semi-formal framework (GICS) developed for this study, and comparing two distributions. The pedagogy was modelled on current statistics education research best practice, with an emphasis on the cultivation of classroom dialogue where students explain and justify their positions. Fathom? software was used by the students in a computer laboratory, and as a teaching aid in the classroom to support learning.

Eight teacher researchers examined their own practice to analyse their use of questioning in the context of numeracy, in partnership with two researchers. Each teacher researcher devised their own question categories, from which the research team then developed common categories. Teacher researchers found the most helpful way to categorise questions was according to their purposes for asking them, and that only the teacher could reliably determine this. Dichotomies such as open/closed questions, or lower/higher order questions, did not appear to illuminate the complexity that underpins questioning. The teacher researchers discovered that they had asked more questions than they expected, and were surprised that they asked more questions of students working at higher strategy stages. The importance of context was highlighted as the teacher researchers described the many interrelated factors they considered as they formulated questions and presented questions to students.

Research shows that personal experiences have a powerful influence on the views of teaching, learning, and mathematics held by prospective teachers. In this study prospective primary teachers were invited to describe their ideal mathematics classroom in order to explain their views about teaching mathematics. These imagined classrooms provide a valuable insight into their emerging identities as primary mathematics teachers. My analysis of these descriptions addresses the question: What views of the teacher?s role, learners and learning, and mathematics are evident in prospective teachers? visualisation of their ideal primary mathematics classroom?

This paper focuses on notions of function Year 9 students hold as they begin to study functions. As these notions may be fragile, the questions, tasks, and ways of interacting orchestrated by the teacher to elicit depth of understanding, or allow observation of changing notions, are of interest. Extended tasks where students were required to make choices about solution paths provided opportunities for students to develop and consolidate their concept images. Discussion between small groups provided the best evidence of developing and stable conceptions held by students in contrast to written scripts where the strength of these understandings was not evident.

An important aspect of bringing about change in the mathematics classroom is gauging the efficacy of the change in bringing about learning that has application outside the school classroom. The research reported in this paper is situated within an on-going study where over 20 teachers of mathematics in the middle years of schooling are using the practices of Collective Argumentation to bring about change in their classrooms. This paper reports on one aspect of the study that sought to explore whether students who use Collective Argumentation on a regular basis in their classrooms view mathematics as providing a forum where personal understandings can be expressed, re-considered, shared and coauthored when they go about knowing and doing mathematics in a novel context ? an interschool mathematics modelling challenge. The results of the exploration are discussed and situated within the context of the findings of the on-going study.

Several studies have investigated how the formation of informal conjectures, and the dialogue they evoke, might influence young children?s learning trajectories, and enhance their mathematical thinking. In a digital environment, the visual output and its distinctive qualities can lead to interpretation and response of a particular nature. In this paper the notion of visual perturbance is explored, and situated within the data obtained, when tenyear-old children engaged in number investigations in a spreadsheet environment.

Wenger (1998) and Lave and Wenger (1991) developed a social theory of cognition in which learning takes place as a result of one?s legitimate peripheral participation in a community of practice. In this paper, we apply Lave and Wenger?s theory in learning to teach secondary mathematics. We report on clinical interview data concerning the practicum experiences of eight students enrolled in the Graduate Diploma in Education programs at two universities. Factors which influence the pre-service teachers? classroom practice include the pedagogy of the supervising teacher, the academic ability of pupils, and concerns about classroom management.

As part of a larger study exploring teacher behaviours that challenge children to probe their mathematical understandings, children were interviewed about their mathematical thinking and asked to reflect on their learning. Fifty-three interviews were conducted in four schools with 5- to 7-year-old children. The subjects were involved in close conversation with their teachers during the mathematics lesson. Video-stimulated recall was used with a conversational interview to prompt children?s recollections and reflections. Findings indicate that young children in the first years of schooling are able to recall events in their mathematics lessons to reconstruct their thinking and reflect on their mathematical learning.

This paper presents a case study of the journey a teacher/facilitator took to increase her mathematical content knowledge in order to implement reform-oriented teaching approaches in her mathematics classroom, and subsequently supported other teachers to do the same. In the past decade mathematic education reform has been introduced to teachers in curriculum documents and related in-service professional development programmes promoting an inquiry-based approach to the teaching and learning of mathematics to increase student achievement. Recent research findings suggest that the complex mathematical knowledge embedded in these reforms makes it difficult for many teachers to accommodate the reforms in their entirety. This was indeed the case for the teacher in this study.

A range of assessment tasks was developed for use in one-to-one interviews in December 2005 with 323 Grade 6 students in Victoria. In this paper, we summarise briefly the research literature on fractions, describe the process of development of assessment tasks, share data on student achievement on these tasks, and suggest implications for curriculum and classroom practice. Particular emphasis in the discussion is given to students? judgements and strategies in comparing fractions. A particular feature of this report is that one-to-one interview assessment data were collected from a larger number of students than is typically the case in these kinds of studies. Recommendations arising from these data include the importance of teachers understanding and presenting a wider range of sub-constructs of fractions to students in both teaching and assessment than is currently the case, using a greater variety of models, and taking available opportunities to use the interview tasks with their own students.

Recent mathematics education reform calls for pedagogical practice that is responsive to students? personal articulations of mathematics ideas. In such initiatives, listening to students is fundamental to advancing students? thinking. Our study explored the relationship between teachers? orientation towards listening and teachers? content knowledge. We investigated how four teachers listened to and made sense of students? ideas, and the influence of content knowledge on their capacity to listen. The study revealed that the depth of teachers? content knowledge ? both subject matter knowledge and pedagogical content knowledge ? mediated their enactment of effective listening practices.

This study compared the thinking of five high performing and five low performing primary students on a set of graphically-oriented numeracy items. Generally, their thinking differed in four ways. First, high performers drew on existing knowledge and skills, which low performers appeared to lack. Second, high performers used multiple cues to complete tasks, whereas low performers worked from a single cue or overlooked cues. Third, high performers used simple solution procedures correctly; in contrast, low performers used more mentally demanding procedures with limited success. Finally, high performers were more knowledgeable about everyday graphics than low performers.

This paper reports how high school students from two different schools used patterns and generalisations while working on some selected problems. The results show that the initial identification of a pattern was crucial in determining the type of symbolic generalisation, which for successful students? seemed to proceed through four sequential stages.

This paper reports on part of a larger study and examines the changing nature of mathematics teaching and tasks. Two Year 4 classes were compared after mathematical modelling tasks were undertaken with and without top-level structuring. The results indicate that mathematical-modelling and top-level structuring tasks can advance mathematical literacy. Where students are guided through information organisation and mathematising through quality teaching, they can make sense of the mathematical world. Also evident was the vital role of the teacher in creating a positive learning environment through facilitating discourse and literacy development in mathematics students. Recommendations for teaching are given. Indications evidenced here warrant further investigation.

This paper reports on one aspect of a larger study into student understanding of scale. Thirteen students from Years 7 and 8 were interviewed, using a diagnostic assessment designed for the purpose, to identify how they went about locating numbers on, and reading numbers from scales. A range of student strategies were identified, most of which can be classed as informal knowledge. These strategies can be sorted into a progression that relates to the level of number thinking involved.

Aspects of students? arithmetic knowledge are described via two case studies of responses to tasks during a videotaped assessment interview. Tasks include reading numerals, locating numbers, saying number word sequences by ones and tens, number word after or before a given number, incrementing and decrementing by ten, addition in the context of dot strips of tens and ones, and addition and subtraction involving bare numbers. On many tasks the students had significant difficulties and responded differently from each other. The paper demonstrates the idiosyncratic nature of arithmetical knowledge, and the significance of context in students? multidigit thinking.

This paper examines one approach to promoting creative and flexible use of mathematical ideas within an interdisciplinary context in the primary curriculum, namely, through modelling. Three classes of fifth-grade children worked on a modelling problem (Australia?s settlement) situated within the curriculum domains of science and studies of society and environment. Reported here are the cycles of development displayed by one group of children as they worked the problem, together with the range of models created across the classes. Children developed mathematisation processes that extended beyond their regular curriculum, including identifying and prioritising key problem elements, exploring relationships among elements, quantifying qualitative data, ranking and aggregating data, and creating and working with weighted scores.

When students? responses to a test of introductory algebra items were Rasch modelled, three distinct ?ability? clusters occurred. The question then arose as to the mathematical thinking that could characterise each of these groups. Data from the test revealed that the tendency to conjoin terms inappropriately occurred with different frequencies in each of the three groups. Interview data and error analyses provided further insight into the students? thinking that resulted in these types of errors. Implications for classroom practice are considered.

The purpose of this study was to explore the mathematics teaching practices of graduates of a pre-service primary education program designed to develop teachers? capacities to implement non-traditional mathematics curricula. As a complementary component of a large survey study of graduate teachers, eight graduates were interviewed to examine their mathematics teaching practices and influences upon their practices. The teachers were implementing personally developed, constructivist-oriented curricula, while also acting as curriculum leaders. They indicated awareness of how aspects of their pre-service education provided them with the knowledge, skills, and confidence to enact their beliefs about effective mathematics teaching.

This paper explores the number learning in 2006 of over 7000 children in the Ballarat Diocese for the purpose of identifying any issues that may inform the development of a Diocesan professional learning plan. The data for each grade level were examined to find if there were any apparent learning, teaching, or curriculum issues. The study found that there was a spread of knowledge within each grade level, and that there were groups of students who may be vulnerable. In particular, it was found that notable numbers of students beginning Grade 6 were not yet able to read, write, order, and interpret four-digit numbers nor use reasoning-based strategies for calculations in addition and subtraction, and multiplication and division. These findings need to inform the professional learning plan.

This paper reports on the initial phase of a research study that is investigating how and why secondary school mathematics teachers use digital technologies to help their students learn. Case studies of a beginning teacher and an experienced teacher, both of whom are regarded as effective users of technology, aim to identify critical factors that support or hinder innovative teaching and learning. The findings are analysed with the aid of Valsiner?s (1997) zone theory to study interactions between teachers? knowledge and beliefs, their professional contexts, and their formal and informal professional development experiences.

This paper describes a project that supported a group of secondary mathematics teachers in implementing the new Queensland Mathematics Years 1-10 Syllabus. The purpose of this paper is to evaluate the effectiveness of the professional development model that was used to assist teachers move towards an investigative approach to ?working mathematically?. The model integrates a zone-theoretical approach to understanding teacher learning into a framework for designing professional development of mathematics teachers. The effectiveness of the model is evaluated via case studies of teachers? professional learning throughout the project and examination of the impacts on their teaching and assessment practices.

In writing this paper we draw considerably on the work of Jo Boaler and Leone Burton. Boaler?s studies of Railside have been particularly poignant in alerting the mathematics education community to a number of key features of successful classrooms, and how such features can turn around the successes for students who traditionally perform poorly in school mathematics. This is supplemented by the more recent work of Leone Burton who worked extensively with research mathematicians in order to understand their communities and ways of working. Collectively these two seminal works provide valuable insights into potential ways to move the field of school mathematics forward. In times where there is international recognition of the plight of school mathematics, there is a need for new teaching practices that overcome the hiatus of contemporary school mathematics.

Mental strategies are a desired focus for computational instruction in schools and have been the focus of many syllabus documents and research papers. Teachers though, have been slow to adopt such changes in their classroom planning. A possible block to adoption of this approach is their lack of knowledge about possible computation strategies and a lack of a clear organisation of a school program for this end. This paper discusses a framework for the categorisation of mental computation strategies that can support teachers to make the pedagogical shift to use of mental strategies by providing a framework for the development of school and classroom programs and provide a common language for teachers and students to discuss strategies in use.

This paper examines student experiences in VCE Further Mathematics. In a survey conducted in 2006, 866 year 12 graduates who had studied Further Mathematics the previous year were asked about their experiences of Further Mathematics classes and their views on the subject and the teacher. The students who did Further Mathematics as their only mathematics subject were less confident about doing well, had a less positive view of the classroom as a learning environment and more negative attitudes towards their mathematics teachers, compared to students who studied both Further Mathematics and Mathematical Methods. The practice of allowing Mathematical Methods students also to study Further Mathematics may contribute to higher results in Further Mathematics for these students, but it may inhibit the capacity for teachers and schools to cater properly to the needs of those for whom the subject was initially designed.

This paper reports on insights into students? understanding of the concept of rate of change, provided by examining the gestures made, by 25 Year 10 students, in videorecorded interviews. Detailed analysis, of both the sound and images, illuminates the meaning of rate-related gestures. Findings indicate that students often use the symbols and metaphors of gesture to complement, supplement, or even contradict verbal descriptions. Many students demonstrated, by the combination of their words and gestures, a sound qualitative understanding of constant rate, with a few attempting to quantify rate. The interpretation of gestures may provide teachers with a better understanding of the progress in their students? thinking.

This paper presents case study data from an exploratory study investigating six preschoolers? patterning skills using three learning modes: concrete materials, screen-based technological tools, and combined modes. Children using dynamic interactive software and virtual manipulatives to solve pattern-eliciting tasks engaged in more ?experimental? representations and created more patterns and transformations than children using concrete materials. However, there were no qualitative differences observed between children's understanding of simple repetition. This research highlights new ways of mathematics learning that can be enhanced through explicit techniques afforded by technology.

During initial attempts at filmmaking by my Year 7 class my focus was on the technology. However, I observed many positive learner behaviours inherent in the filmmaking process. Could these positive learner behaviours be harnessed, through filmmaking, to improve learner outcomes in mathematics? Two trials were conducted comparing a mathematical mini-documentary making revision strategy with pen-and-paper revision. It was concluded that mini-documentary makers retained information at least as effectively as, if not better than, students who used pen-and-paper strategies. This implies that mathematics educators can be confident of positive effects on students? knowledge retention through student filmmaking in mathematics.

In this paper we discuss the considerations and challenges in designing instructional tasks that support both students? mathematical engagement and their developing mathematical competence. We draw on Dewey?s work and take the perspective that cultivating students? content-related interests should be an instructional goal in their own right rather than solely serving the instrumental purpose of supporting students? conceptual understanding. We reflect on our learning from two classroom design experiments to offer illustrations of issues related to supporting students? interests. We offer these illustrations, not as exemplary cases, but instead, as points of reflection and discussion. In this paper, we focus specifically on instructional tasks by presenting a retrospective analysis on the role of tasks in supporting students? interests and access to important content ideas.

Strong relationships established between schools and communities can improve the mathematical learning outcomes for Aboriginal students. The 2005-2006 Building Community Capacity project sought to identify key aspects of meaningful engagement between schools and communities focusing on the development and implementation of contextualised, relevant and connected mathematics curriculum and appropriate teaching and learning strategies to enhance Aboriginal students? mathematics outcomes. Using case study methodology within two school sites in New South Wales, the paper identifies critical elements of community engagement and provides underlying principles, which other communities might consider in their own community capacity building.

A literature-based instrument gathered 147 final-year preservice teachers? perceptions of their mentors? practices related to primary mathematics teaching based on five factors for mentoring (i.e., personal attributes, system requirements, pedagogical knowledge, modelling, and feedback). Results indicated acceptable Cronbach alpha scores for each factor: 0.91, 0.77, 0.95, 0.90, and 0.86, respectively. Furthermore, less than 45% of mentors were perceived to provide specific practices associated with mentoring system requirements. This paper discusses possibilities for using the survey instrument including benchmarking mentees? perceptions of their mentoring for developing their mathematics teaching and as a reference point for delivering professional development for mentors.

Student transition from arithmetic to algebraic reasoning has been acknowledged as an essential but problematical process. Previous research has highlighted the difficulties in shifting students towards relational thinking when solving equivalence problems. This paper reports on an investigation into students? use of relational thinking to solve equivalence problems after they have been in classrooms where specific focus has been on developing flexible, efficient computational strategies. The results reveal that most students used computational strategies to solve the equivalence problems rather than relational strategies. Many of the common errors, students made reflected a lack of understanding of the equal sign.

In the current reform of mathematics classrooms teachers are required to develop discourse communities in which all students have equitable opportunities to engage in productive discourse. The challenge is for teachers to engage students in the mathematics talk across a range of classroom situations. In this paper I address how a teacher used interactional strategies to scaffold participation of her diverse students in small group interactions. I report on the actions the teacher took to shift the patterns of discourse from a disputational form to one in which the students collectively constructed group explanations and justification.

This qualitative multiple case study involved eight Year 6 and 7 students and six classes and investigated their capacity to recognise, apply, and question the use of mathematical ideas embedded in a range of contexts. It also considered the extent to which students? capacity to connect mathematical knowledge to other contexts could motivate them to learn mathematics. In particular, it investigated the effect of the Mathematical Search strategy in achieving these ends. It found that student thinking about mathematics and their attitudes towards it could be enhanced by targeting mathematical connections through the use of the Mathematical Search.

This paper presents a case study of a secondary school mathematics student in New Zealand. Stories about this student relating to the context of mathematics form his mathematical identities and are told by his parents, his teachers, his peers, himself, and the researcher. The student?s negative affective responses to mathematics are explored through these stories. The student was found to have very positive beliefs, values, attitudes, feelings, and emotions about mathematics. He ?loves? mathematics because of his enjoyment of mathematics as a discipline, and because he is good at it compared with his classmates. He is perceived to be in the top group of mathematicians in his school, a role endorsed by himself, the school, his teachers, and his peers. During the year however, he becomes less positive about some aspects of mathematics as he struggles to continue to fulfil this role.

We report on the mathematics competencies of 206 Engineering and Science students commencing an algebra and calculus course at an Australian university in the first semester of 2006. To inform course design in the face of growing student diversity, skills were assessed via a pre-test covering six fundamental areas. These data were also compared with the 1997 to 2001 data. The findings revealed reasonable skills with arithmetic, fractions, and index laws but ongoing weaknesses in areas of algebra, functions, and trigonometry. These findings have important implications for planning in Australian universities. Implications for school curricula are also considered.

Jane?s decision to write her maths-autobiography came as she witnessed the benefits achieved by other preservice teachers at UNDA undertaking the same task. However, unlike fellow students, Jane did not suffer from Mathematics Anxiety. Jane?s autobiographical writing demonstrates the potential uses and benefits for a non-anxious preservice teacher. Her autobiography provides insights for teachers and teacher educators into the everyday experiences of the classroom and students. For teacher educators, it further demonstrates the value of various writing styles as tools for self-growth. Jane?s writing contains a number of examples that demonstrate that her childhood experiences and subsequent writing about those times, directly impact on her emerging teaching philosophy and future professional work. Jane?s writing also demonstrates the transformative potential of writing a mathematics autobiography for preservice teachers.

One of the challenges in research is in understanding processes and systems that enable teachers to build their expertise and commitment to reform-based pedagogies. A qualitative study documented the influence that a set of support mechanisms, or connection levers, had in assisting upper primary teachers over the course of a year in developing confidence in teaching mathematics through inquiry.

Acquiring the mathematics register is often assumed to occur when learning mathematics. However, when students learn in a second language and are taught by teachers who are also not teaching in their native language, this may not be a straight-forward process. This paper describes the strategies that teachers in a M?ori immersion school (kura kaupapa M?ori) used to scaffold and model the mathematics register. Although most strategies could be seen in many classrooms, there were some strategies that seemed to be related to the students and teachers using te reo M?ori as the language of instruction.

A group of practising teachers implemented the Teaching for Abstraction method for the Year 8 topic ?Ratio and Rates?. The authors first constructed materials for a unit in which students explored familiar ratio and rates contexts, searched for similarities in their mathematical structure, defined the two concepts, and learned to apply these concepts to other contexts. After an introductory workshop, teachers taught the topic in six 1-hour lessons. They experienced considerable difficulties adapting the approach to the abilities and interests of their particular classes, but all students showed evidence of learning. It was concluded that, although Teaching for Abstraction shows promise, there are many factors that need to be taken into account if it is to be implemented in practice.

This paper reports on teachers? choice of examples and the role they play in students? construction of knowledge. Selecting an appropriate example is a challenging task for teachers, with both the teacher?s content and pedagogical content knowledge being a determining factor in the selection process. A case study approach was used to document the nature of three different teachers? choice of examples. Qualitative descriptions illustrate the types of examples selected and the understandings the students constructed from these examples. The findings indicate that teachers need to consider carefully their choice of examples to avoid the likelihood of students forming misconceptions about important mathematical concepts.

What a study of the historical development of mathematical concepts can offer teaching is still being debated. This study examines use of a combination of the historical development of number systems and modelling, with concrete materials as a way of deepening students? understanding of positional notation. It looks at place value in different number bases as a way of enhancing students? understanding of the decimal number system. The results suggest that the combination of a historical and a concrete approach helped the students to understand the place value system to the extent that they could generalise it to other bases.

This paper describes the process of developing a survey instrument aimed at measuring aspects of mathematical confidence, value, and the interconnectedness of mathematics as part of a larger study investigating the thinking processes and attitudes towards mathematics of Singaporean secondary school students (aged 12-14) during interdisciplinary learning. Results from exploratory and confirmatory factor analyses on scale items tested revealed six scales with sound validity and reliability properties. The scales are intended for measuring attitudes towards mathematics particularly during interdisciplinary education.

Analyses and commentary for 2002-2005 Mathematical Methods (CAS) pilot examinations in Victoria, on student performance with respect to common items with the standard course have been reported at previous MERGA conferences. In 2006, both Mathematical Methods and Mathematical Methods (CAS) were available to all Victorian schools as equivalent subjects with a new examination structure that comprised a 1-hour common technology-free examination and a 2-hour approved technology active examination. This paper provides some analysis of student performance on the technology free examination, and also with respect to common items in both the multiple choice and extended response components of the technology-active examination.

Student difficulties with the study of algebra have been well documented. The inability of many students to understand variables and formal symbolic manipulation act as a barrier to success in mathematics study. This report documents an intervention that uses a concrete approach to teaching algebra in a Year 9 class. Results indicate that much of the student struggle was associated with a lack of understanding of arithmetic concepts including those associated with equivalence, operations with negative integers, and the distributive law and fraction concepts. Once these difficulties were addressed through the explicit teaching of the links between materials and symbols, materials and language, language and symbols, students made considerable progress in writing, simplifying expressions, and solving equations with variables on both sides.

This paper reports on one teacher?s attempts to teach critical algebra understandings to a Year 9 class in ways that engage the students and help them to develop positive perceptions of their ability to learn algebra in a ?rigorous and symbolic way?. This paper describes a 6-week algebra intervention based upon connecting concrete representations with symbolic expressions and equations through the careful use of formal algebra language. The teacher had expressed her aspirations that interventions such as this would encourage more students to undertake intermediate and advanced mathematics courses in senior secondary years. The study collected data on student perceptions about their learning experiences including perceptions about mathematics as a subject domain, engagement with the activities, development of algebraic understanding, and the quality of discourse within the classroom. This study found that the students valued the classroom discourse much more than they did the normal mathematics learning experiences. These findings have implications for inservice and pre-service teacher education.

Papua New Guinea has hundreds of languages and cultures and each group measures in different ways. This report discusses the informal measurement and contexts for measuring by a range of cultural groups as obtained from a survey. Intuitive approaches traditionally used in villages indicate an interesting use of length for deciding areas. People seem to visualise the areas and rely on lengths for comparing or counting to compare these areas. The use of informal measurement has implications for schooling in that it is a valuable place to begin measurement education rather than smaller formal units. Concepts, such as area, and the structure of measurement units, such as placing length units end to end, can be ascertained and established from these informal measures as a transition to more formal school measurement.

This paper presents results from a survey of secondary mathematics teachers in rural, regional and metropolitan schools across Australia. The purpose of the survey was to compare the major needs of teachers in relation to the attraction and retention of qualified staff, professional development, availability of material resources and support personnel, and the accessibility of a range of student learning opportunities across the three geographical areas. Although differences emerged for some of these factors, the most significant findings were identified in schools with Indigenous populations of greater than 20%.

A case study monitored the development of 53 preschoolers? mathematical patterning skills in two similar preschools, one of which implemented a 6-month Intervention promoting patterning concepts. Pre- and post-Intervention assessment data and follow-up data evaluated the impact of the Intervention on the growth of Repeating and Spatial Patterns. Intervention children outperformed Non-Intervention children across a range of patterning tasks and this trend was maintained 12 months after formal schooling. Intervention children readily identified the unit of repeat and the structure of spatial patterns. Without exposure to Growing Patterns, Intervention children identified, extended, represented and justified triangular and squared number patterns.

Many researchers have noted that students? whole number knowledge can interfere with their efforts to learn fractions. In this paper we discuss a teaching experiment conducted with students in Years 5 and 6 from an eastern suburban school in Melbourne. The focus of the teaching experiment was to use number lines to highlight students? understanding of whole numbers then fractions. This research showed that successful students had easily accessible whole number knowledge and recognised the relationship between the whole and the parts whereas the weakest students had poor number knowledge and could not see the connections.

This paper reports on a project that identified and explored the factors leading to outstanding mathematics outcomes in junior secondary public education in NSW for students across the ability spectrum. Once a sample of mathematics faculties was identified by drawing upon the extensive quantitative and qualitative data-bases within the NSW Department of Education and Training (DET), seven intensive case studies were conducted to identify faculty-level factors. Seven common themes are reported and these are the strong sense of team, staff qualifications and experience, teaching style, time on task, assessment practices, expectations of students, and teachers caring for students.

In this study eight primary school teachers formed partnerships with researchers to investigate the use of questioning during two numeracy lessons. The teacher researchers were encouraged to act as reflective practitioners (Sch?n, 1995) and contribute to formulating their own ?interpretive frames? (Cochran-Smith & Lytle, 1990). Methods of data-gathering, analysis and interpretation were developed to allow the teacher researchers to have control over the research and contribute to the direction of the project as it evolved. This paper describes some of the challenges faced by both the teacher researchers and the researchers in developing partnerships. It also discusses how the methodologies allowed teachers? views about practice to be made explicit. Teachers gained insights into the complexity of their teaching practices and described ways in which the experience impacted on their views of research as a bridge between theory and practice.

Debate about changes in teachers? beliefs and attitudes about mathematics teaching leads us to understand that these changes result from a teacher?s personal experience. Professional learning in its various forms is an attempt to change teachers? practices in the classroom, and hence influence student learning outcomes. The paper uses the responses of one mathematics teacher involved in a professional learning project to examine the relationship among the professional learning, classroom practices, and teacher beliefs and attitudes.

This paper reports the use of three questions to guide students? discussions and reflective writing in a year 5/6 mathematics class. Journal entries and work samples were examined for evidence of students making sense of their thoughts and processes used during the completion of Space-based tasks. Reflective writings were inspected for evidence of the three functions of metacognition and Bloom?s Taxonomy was used to note changes in students? levels of understanding of the content. Preliminary findings suggest that the approach and questions used in this study warrant further investigation.

This study extends research into the van Hiele Theory by narrowing the microscopic lens and providing a focused analysis on the understanding and development of class inclusion concepts in Geometry. This paper integrates two qualitative frameworks, identified through the utilisation of the SOLO model, that indicate developmental growth in understanding of relationships among figures, and relationships among properties. This is considered via a quantitative approach, using a Rasch analysis model, which provides a comparison of the complexity of seven different interview tasks within the context of triangles and quadrilaterals.

This paper reports the case of two teachers with respect to the positioning of numeracy in a reform curriculum and subsequent student learning across five dimensions of numeracy. By analysing the conversations of these two teachers, their underlying beliefs about numeracy and its value and role in the curriculum were able to be explored. These beliefs were further reflected in the learning outcomes of the six students in this study. The paper describes examples of how the five dimensions of numeracy were evident in the thinking and practice of both the teachers and their students.

During 2006, two teams of preservice teachers spent a week in three rural schools and completed diagnostic assessment tasks in mathematics using the Nelson Numeracy Assessment Kit. The classes that were assessed were all being taught by newly graduated teachers. The results were collated into detailed profiles, which enabled these teachers to identify whole class, small group, and individual strengths and weaknesses. It was anticipated that the new graduates would find these profiles of great benefit in planning for mathematics. However, the teacher-educators who continued to work with the new graduates discovered that this assumption was flawed, and that the new graduates experienced difficulty in planning curriculum based on identified needs. This paper discusses the typical approaches to curriculum planning adopted by the teachers, which were largely teacher-centred.

There is a strong case for arguing that the application of relational thinking to solve number sentences embodies features of mathematical thinking that are centrally important to algebra. This study investigates how well students in Years 5, 6, and 7 in three countries were able to use relational thinking to solve different types of number sentences. There were other students who appeared to rely solely on computational method to solve the same number sentences. The study then examined whether those who had shown clear evidence of relational strategies to solve the number sentences were better placed to solve symbolic sentences than those who had used only computational methods on these number sentences.

A framework to support successful implementation of mathematical modelling in the secondary classroom was developed from transitions between stages in the modelling process and the cognitive activities associated with these. This framework is used to analyse implementation of a task with a Year 9 class. Cognitive activities engaged in during the task and competencies and technological knowledge required to complete the task successfully are identified. This framework can be used by teachers, researchers, and curriculum designers to design tasks and predict where in a given task blockages occur, hence allowing advance consideration of scaffolding for in-the-moment classroom decisions.

Responding to an instrument we developed to give insights into students? orientation to, and motivation for, learning mathematics, Year 8 students showed more confidence in their ability to learn mathematics and in their persistence than observations of their classes would indicate is warranted. They identified a negative influence of peers for some classmates but less for themselves, and had modest career aspirations. We believe teachers can assist students by becoming aware of their orientations to learning, their perceptions of the value of schooling, and their further vocational aspirations, and by finding ways to overcome factors inhibiting their engagement in school.

The Interactive Teaching and ICT project explored the process of interactive teaching and learning with and without ICT. A key technique in our methodology was the use of videostimulated reflective dialogue to assist teachers to reflect on key episodes in their teaching. In this paper we discuss how this technique was extended to encourage pupils between the ages of 5 and 14 to reflect on their learning of mathematics. Analysis of the reflective dialogues indicates that even quite young children were able to articulate opinions about the ways in which they learned and the ways in which ICT supported this.

This paper examines how the activities, discourse, and artefacts in a mathematics classroom may serve to position students as dependents or to objectify them, rather than encouraging the development of subjectivity by apprenticing them into the valued discourse of the mathematics classroom. The paper uses three sociolinguistic approaches to interpret the interactions between Simon, the teacher, and Dean, a student, in a Year 7 mathematics classroom. Although they have very different goals and methodologies, each approach has the potential to reveal the social function of language in a mathematics classroom.

In this paper the pedagogical practices of practising teachers and pre-service teachers when using digital technologies are described and compared. Data were collected by observation of presentations about using digital technology in mathematics by teachers and pre-service teachers and practising teachers were interviewed. Teachers generally used pedagogical approaches involving student-centred activity whereas pre-service teachers were more likely to use technology to teach concepts by demonstration and were not inclined to use the more student-centred approaches, though many used guided tasks. The study enabled some analysis and reflection upon the promoted action in the learning environments of pre-service teachers.

This study examines the procedural complexity and mathematical solving processes required by problems on two topics in seven Year 8 textbooks from four Australian states. The study used definitions from the 1999 TIMSS Video Study. Although variation existed between textbooks, the majority of problems were of low procedural complexity, requiring only the practising of procedures. The general picture was consistent with that painted by the Video Study, with a somewhat stronger emphasis on procedural work.

Drawing on a review conducted of the resources that the mathematics education research community has developed while learning to support teacher learning, I direct attention to researchers? understanding of teachers? current practices. In particular, I argue that designers, facilitators, and researchers of professional development alike would benefit from understanding teachers? practices (a) as reasonable from teachers? perspectives, (b) in a way that can directly feed into the efforts of supporting teacher learning, and (c) as shaped by the institutional context of teachers? work.

From their everyday experiences of life in classrooms, children develop understandings of what is meant by ?doing maths?. This paper draws on the findings of an ongoing longitudinal study following the mathematical learning careers of ten children from the beginning of their third year at primary school as seven-year-olds to the end of their eleventh year as sixteen-year-olds. Over this time, ?doing maths? has changed remarkably little for these students. Using the children?s accounts of doing maths, the paper probes the connections among mathematical content, teaching, and learning, and considers the implications of their stories for teaching practice.

Current curriculum initiatives in mathematics call for the development of classroom communities in which communication about mathematics is a central focus. In these proposals, mathematical discourse involving explanation, argumentation, and defense of mathematical ideas, becomes a defining feature of a quality classroom experience. In this paper we provide a comprehensive and critical review of how mathematics teachers deal with classroom discourse. Synthesising the literature around a number of key themes, we critically assess the kinds of human and material infrastructure that promote mathematical discourse in the classroom and that allow students to achieve desirable outcomes.

This paper reports on a small pilot study conducted in an Indigenous P-13 school in North Queensland. This pilot study occurred over a two day period with the specific aim of exploring the role of oral language and representations in negotiating mathematical understanding. Implications are drawn for the implementation of a large study, commencing in 2007 with 4-year-old Indigenous students as they transition from home to school. All students in this context either speak Aboriginal English or Creole as their first language. The pilot study occurred in two classrooms, one with 15 Year 6/7 students and the other with fourteen Years 4/5/6 students. The preliminary results indicate that explicit consideration needs to be given to the development of precise mathematical language, strategies for linking school mathematics to home environments, the use of questioning in establishing classroom discourse, and the recognition that many of these classrooms are bilingual.

Teachers and students in nine rural Tasmanian schools have been associated with a research project providing professional learning for teachers in mathematics in a reform-based learning environment. Students completed surveys to measure attitudes and mathematics skills and understanding late in 2005 and late in 2006. Teachers completed profiles late in 2005 and participated in professional learning activities from then throughout 2006. The professional learning program is described and change in student attitudes and performance reported.

This study examines prospective ?STEM? [Science, Technology, Engineering, and Mathematics] teachers? motivations for undertaking a teaching career and their perceptions of the teaching profession, for undergraduate and graduate teacher education entrants from three major established urban teacher provider universities in the Australian States of New South Wales and Victoria (N=245). Motivations and perceptions were assessed using the recently developed and validated ?FIT-Choice? [Factors Influencing Teaching Choice] Scale (Watt & Richardson, 2007). Differences are highlighted between males and females, and undergraduates and graduates, including switchers from previous careers. Demographic profiles for STEM teacher candidates are also provided. Findings provide important implications for enhancing the effectiveness of efforts to recruit mathematics, science, and ICT teachers.

Five practising teachers in regional NSW implemented Teaching for Abstraction for the Year 6 topic ?Percentages?. The authors constructed materials for a unit in which students explored familiar percentage contexts, searched for similarities in their mathematical structures and then applied their learnings to more abstract situations. Particular emphasis was given to additive versus multiplicative approaches in different percentage situations. After an introductory workshop, teachers taught the topic in eight 40 minute lessons. The results show that even though this approach is radically different from that to which students and teachers are accustomed, it has the potential to benefit student engagement, learning, and attitudes for both students and teachers. The overall conclusions have implications for how professional development for Teaching for Abstraction is addressed.

Bibliotherapy provides a new approach to eliciting and understanding the affective responses of pre-service primary teachers. This paper further explores bibliotherapy as a reflective tool in teacher education by analysing affective responses of pre-service primary teachers studying an elective number theory unit. Pre-service teachers voluntarily wrote responses to readings about school students? learning, discussed their understanding of their own experiences in the light of the readings, and identified readings that impacted most on them. The paper describes the responses using the five stages of the bibliotherapy and identifies some factors which affect levels of engagement with the process.

This paper investigates students? conceptual understanding of equivalent fractions by examining their responses to questions using symbolic and pictorial representations. Two hundred and thirteen students in Years 3 to 5 from three Sydney primary schools were administered a general mathematics achievement test and a fraction assessment. Five questions from this fraction assessment instrument were analysed. The different types of knowledge used to answer each question were examined and common misconceptions identified. The responses of students with limited general mathematics achievement were compared to those of their more competent peers. The differences that emerged between the two groups in their conceptual understanding of equivalent fractions, were highlighted.

When Year 10 students are introduced to reasoning from box plots the type of classroom discourse that will lead them to understand statistical inferential argumentation is unknown. In this paper the discourse of one teacher and her class is analysed. Although the teacher required evidence for claims and introduced statistical vocabulary, she argued with the medians, lacked uncertainty, did not answer the original question or make sense of the conclusion. The implications for teaching are discussed.

This paper reports on a project aimed at developing pedagogical tools for intervention in the number learning of low-attaining 3rd- and 4th-graders. Approaches to instructional design and intervention are described, and the use of the design research methodology is outlined. A major outcome of the project, an experimental framework for instruction, is described. The framework consists of five aspects: number words and numerals, structuring numbers to 20, conceptual place value, addition and subtraction to 100, and early multiplication and division. The descriptions of aspects include a discussion of low-attaining students? knowledge and difficulties, and details of instructional approaches developed in the project.

In studying the use of Interactive Whiteboards (IWBs) we have observed that there are concerns in relation to measures of pedagogy. Using a productive pedagogies framework to analyse the use of IWBs in middle school classrooms, we found very low rating on aspects of pedagogy related to intellectual quality. Using an activity theory framework, and drawing on observations and interview data, we theorise the tensions in the uptake and use of IWBs to support mathematics learning.

The maths anxiety of 29 pre-service primary teachers, which was measured using a questionnaire, was exhibited in writing a response to two different stimulus statements. These students used the word fear on many occasions and other metaphors to describe their anxiety about the sort of teacher they might become and whether they might break the cycle of maths anxious teachers producing maths anxious children. Their words are used to illustrate this desire to break the cycle.

In this short communication, I provide an example of a student?s work to illustrate the power of assessment for learning. This approach has been adopted to support both teachers and students to come to aspects of mathematical learning. Drawing on the literature on assessment and how it is integral to learning rather than as the end product to show what learning has occurred, I provide examples of one case to illustrate the power of assessment for learning. By drawing on aspects of second generation activity, I provide examples of aspects of the learning milieu to frame the analysis of the student?s work. Through the use of activity theory, a coherent approach to understanding the complex milieu of classroom learning environment can be developed.

Students? beliefs about mathematics were the focus of a pilot study based on fieldwork carried out in Norway in early 2005. A web-based Likert-scale questionnaire about beliefs in mathematics was administered to students in 6 schools from one urban area. Two hundred and seventy-six students from grades 7 (12-13 years), 9 (14-15 years), and 11 (16-17 years) completed the questionnaire. Despite lacking interest in mathematics, students acknowledged the usefulness, importance, and need to work hard in mathematics.

Research methods such as narrative inquiry and autobiographical research are increasingly accepted in education. In this paper I discuss why and how I used autobiography in my research, the value of this to my work in mathematics education, and my emerging view of school mathematics, curriculum, and the related development processes.

This paper is a report of a sustained professional development project in South Australia in which a small group of preschool teachers worked with the authors to develop their own skills in facilitating young children?s mathematical learning through investigative approaches and their own assessment of this learning through the use of learning stories. After providing some background information about the project, this paper considers the impact of the project on the early childhood educators themselves and their growth in knowledge, skills, and confidence in early childhood mathematics.

Computer Algebra Systems (CAS) are used in middle secondary classrooms as a tool to support learning and sometimes in preparation for senior secondary mathematics. This paper presents an analysis of strengths, weaknesses, opportunities, and threats identified in the literature and perceived by twelve secondary teachers working with year 9 and 10 students. CAS is valued for calculation and manipulation capabilities, the option of alternative representations, the opportunity for systematic exploration, and for prompting rich discussion. However the technical overhead, initial workload for the teacher, and unresolved questions about the contribution of machine and by-hand work to learning must also be considered.

A study of teacher knowledge, necessary for and used in the teaching of statistics investigations, was conducted in four New Zealand primary classrooms in 2006. This presentation reports on the framework that was developed to describe the components of teacher knowledge with regard to statistics. The framework integrates six dimensions of statistical thinking with four types of teacher knowledge: knowledge of content, both common and specialised; and pedagogical content knowledge, related to both students and teaching. Video and stimulated-recall interview data were analysed in relation to the framework to develop descriptions of knowledge used in the real-time tasks of teaching.

Grade 2 students, experienced in the use of technological tools in learning, were engaged in measurement and data investigations related to an integrated unit ?How I have changed?. Data analysis skills were developed as students were encouraged by the mathematics specialist teacher to pose and interpret questions, experiment with data handling methods, draw inferences, and make connections with prior representations. This pilot study focused on the impact of students? use of Excel software on their understanding of the relationship between the data and different graphical representations. The study also identified both student misconceptions and advanced development of proportional reasoning.

The paper gives a brief overview of the relations between procedural and conceptual knowledge and emphasizes the importance of developing effective mathematical techniques. The paper describes and analyses some teaching strategies used by the authors in a classroom in order to improve students? learning of some mathematical techniques and their understanding of the relevant concepts. We show how the substitution method can be applied to some classes of mathematical problems where traditionally other methods are used. The problems are completing the square, solving quadratic equations, evaluating the limits of indeterminate form 0/0 and integrating rational functions with a quadratic in the denominator. We also analyse how the method of probability trees is used in problems about conditional probability and the ways to improve this teaching strategy.

This paper considers the potential of Computer Algebra Systems (CAS) to enhance the processes associated with mathematical modelling and application tasks. In doing so, the role of technology in the cyclical development of mathematical models will be theorised. Finally, a theoretical framework will be outlined for a classroom-based investigation into the implementation of CAS technologies into classroom contexts where mathematical modelling and applications are a focus.

This paper reports on the school transfer of 15 mathematically gifted Year 6 and Year 8 students. The data are extracted from a longitudinal qualitative study that examines student and parent perspectives, and programme provision for mathematically gifted and talented students before and after a change of schools. Two groups of primary school students made the transfer to intermediate school (Years 7 and 8) or to a Years 7-13 high school. Another group of students from intermediate school made the transfer to high school. The students? and their teachers? and parents? perceptions of the transfer are described.

This study investigates the effectiveness and efficiency of mathematics instruction in two languages (English vs. Cebuano/English Code-Switching) on the performances of Filipino algebra students in 11 classes of a Philippine private high school. Conducted in 2005-2006, a quasi-experiment addressed the question: Between the two languages of instruction, which promotes better algebra learning among Filipino bilingual students? The analyses are limited to evaluating the performances of high-ability students (comprising 3 classes) by using Multi-level Nested Modelling to compare the impact of languages-in-use on student achievement, and Data Envelopment Analysis (DEA) ? a novel economic modelling approach ? to measure the relative efficiencies of student learning outcomes.

This paper explores misconceptions revealed when pre-service teachers locate negative decimals on a number line. Written responses from 96 pre-service teachers to tests and group worksheets and video-recorded observation of their classroom discussions and interviews provide extensive data. Three misconceptions are identified. Two relate to incorrect analogies between the positive and negative parts of the number line. The other is a ?repair? to overcome an inconsistency, made more likely by intuition that negative decimals are very small. Implications for teaching are drawn.

School mathematics values abstract reasoning over practical knowledge, propagating the myth of reference. School mathematics also makes claims that mathematics is essential for effective functioning in society, thus propagating the myth of participation. This paper uses hermeneutics to examine a worksheet used in a year 7 mathematics classroom to illustrate the myths of reference and participation. The continuation of these myths, together with discourse that is localising and limiting, devalues students? informal knowledge and positions them as subservient to mathematics rather than as subjects having mathematical agency.

There is ongoing concern about the negative attitude large numbers of pre-service primary teachers have towards mathematics. The participants in this study were students in year-long primary mathematics curriculum studies courses that focussed on beliefs and attitudes, alongside content and pedagogy. Throughout the year, pre-service teachers personally experienced supportive and effective mathematics communities. As a result, many pre-service teachers developed a deeper conceptual understanding of mathematics. The majority of the pre-service teachers expressed increased confidence in their ability to teach mathematics and a willingness to continue gaining skills and knowledge in mathematics pedagogy.

The study of 29 year 8 students investigated whether the development of proportional reasoning was localised or generalised. It found support for localised early development and a generalised later development. Multiplicative thinking with whole numbers was found to be a necessary, but not sufficient, condition for proportional reasoning.

Public education in Victoria is currently undertaking a major reform process involving a number of initiatives including reporting and assessment. This paper will describe the identification of a gap between assumptions made by Department of Education and Training about teachers? assessment practice and about what teachers actually do when they assess student progress. A description and findings of a pilot study is included justifying the need for a wider investigation.

A teaching model based on Pirie-Kieren Theory was devised for the teaching of strategic thinking for the New Zealand Numeracy Project. This was used in the design of teacher material that was progressively available to most New Zealand primary teachers in the period 2001 to 2006. The ways in which teachers mis-implemented the model in their classrooms led to some significant alterations. In particular, the meaning of imaging is better defined and support for real-time formative assessment of students to inform the teaching is revised and extended.

Maori medium education in New Zealand has been in existence for more than 20 years. Te Poutama Tau (TPT), similar to the mainstream Numeracy Project, is responsive to the Ministry of Education?s strategy for improving levels of literacy and numeracy in New Zealand schools. The major difference between the Numeracy Project and TPT is that TPT is situated within the context of M?ori development, including the maintenance and revitalisation of the M?ori language. Since 2002, there have been four evaluation reports tracking the progress in Te Poutama Tau. The reports have been used to show trends in student achievement, and the teaching of mathematics via Te Reo M?ori.

Computer Algebraic System (CAS) calculators are currently being trialled in a number of junior secondary New Zealand mathematics classrooms. In this case study, two geometry lessons are described where the teacher has successfully incorporated the technology into his teaching so that a problem that might otherwise have proved to be a stumbling block is surmounted, and insights into the nature of mathematical proof are explored. The observed lessons reflect a problem solving approach advocated by the teacher himself and they appear to be engaging yet challenging for the students. The lessons illustrate, in a positive manner, what can be accomplished once a teacher has attained a solid grasp of the relatively new technology.

Significant technological change has impacted on the representational modalities employed in mathematics learning. Yet, studies evaluating their impact and efficacy are not entirely unequivocal. This study investigates the unique contribution of interactive, digital technologies in the learning of early mathematical concepts in the first years of schooling. The study describes the pedagogical and technological affordances upon differences in observed learning outcomes. The impact of digital, pedagogical interventions on children?s internalised representations of concepts is explored. Case studies of the development of children?s mathematical representations are presented through the use of digital agents, such as learning objects, interactive whiteboards, and wireless mobile learning devices.

Using a design approach, this study monitors the influence of patterning tasks on the mathematics learning of 10 Kindergarten children. The children were engaged in a Pattern and Structure Mathematics Awareness Program over 15 weekly teaching episodes. Children were pre- and post-tested using an interview assessment of pattern and structure (PASA) and a standardised mathematics test. Nine of the 10 children showed impressive growth in their ability to represent and symbolise simple and complex patterns, arrays, grids, partitions, borders, and growing patterns. They also showed substantial improvement in mathematical skills such as counting in multiples, sequencing, similarity and congruence, and co-linear structure.

This paper considers how the didactical contract between students and their teacher is influenced by students? foregrounds and backgrounds and the impact that this has on their learning of mathematics. It uses examples of two students in two primary classrooms, one in Denmark and the other in New Zealand. It is possible to see in both situations that students develop non-conventional mathematical learning. We speculate on how students? perceptions of the didactical contract are affected by considerations such as those to do with social interactions and the impact of these perceptions and consideration on possibilities for learning.

A teaching experiment was conducted to explore the impact of Cabri Geometry on 6 Year 7 students? understanding of parallelograms. Students worked in pairs, and were guided through activities designed to introduce the software and to encourage discovery and exploration of specific geometric properties. Students were also required to use this knowledge to solve geometric problems. Over several lessons, data were collected via taperecorded observations of student interactions and discussions, work samples, and test results. The dynamic investigation promoted deeper understanding of the underlying geometric properties of parallelograms and students? abilities to solve geometric problems.

The paper presents and analyses students? attitudes towards using counter-examples and paradoxes as a pedagogical strategy in teaching/learning of a first-year university course in probability theory and applications. Our intentions of using this strategy were: to achieve deeper conceptual understanding; to reduce or eliminate common misconceptions; to advance one?s statistical thinking, which is neither algorithmic nor procedural; to enhance generic critical thinking skills ? analysing, justifying, verifying, checking, proving; to increase motivation and interest in the subject; and make learning more active and creative. The majority of the students reported that the strategy was effective and made learning more challenging, interesting, and creative.

Communicating mathematics to distance students is often difficult. This presentation reports on preliminary research in three areas. (1) The trial of an electronic handwriting tool in MSN Messenger in a large mathematics service course in Semester 1 2006 for online synchronous group work and individual consultations. (2) The use of a tablet PC and a computer software program ?Camtasia? to record live lectures involving electronic handwriting for transmittion to distance students. (3) The use of pre-recorded online ?Breeze? presentations to highlight particular ?tricky questions?. The presentation will demonstrate the use of these technologies, discuss some of their advantages and disadvantages, and report on initial results of the research.

The New Zealand Teachers Council identifies factors important for teacher registration. These factors must also be considered the foundation criteria for effective teachers of mathematics. What is it that makes an effective mathematics teacher? This article presents a picture of the ideal mathematics teacher as described by a group of Year 7 and 8 Pasifika students (11-12 year olds). The criteria identified by these students fell into two categories. One category addressing factors that were associated with personal qualities of the teacher and the second category that related to the learning environment the students saw as most supportive to their mathematical learning.

Australia?s future prosperity and our ability to compete in the global arena demand that our education systems develop human capital that is highly skilled and knowledgeable. Participation in mathematics, in particular higher mathematics, is an important prerequisite for young Australians if we are to develop the range of skills that underpin this capital. This is a report on a study in progress focusing on the concern that participation in Mathematics at Year 12 and at universities in Australia is declining. In the context of this larger issue, we identify a number of questions we will be investigating with students, teachers, universities, and other stakeholders.

The purpose of this roundtable is to describe a proposed research project and to seek advice, ideas, and suggestions from participants. The project will address two main questions. Firstly, how do mathematical strategies used in traditional school contexts compare with strategies used in real-world contexts? Research continues to highlight the lack of connections that students make between mathematics used in school and in everyday life. The National Council for Teachers of Mathematics emphasise the importance of students being able to use mathematics in varying situations (2000). In this study students will be presented with basic mathematics tasks in different contexts. For example, they will be asked how much change they would get from $5 for something that cost 65 cents. Next, they will be asked to answer a similar question presented traditionally: 500 ? 65. The second research question associated with this project is: What perceptions do teachers have of students? mathematical strategies? Fennema and Franke (1992) indicate that it is important for teachers to understand students? mathematical thinking and to have good knowledge of various instructional approaches. Teachers will be interviewed concerning their perceptions of students? mathematics learning. During the interviews they will watch DVD excerpts of students solving problems and be asked to make comments on the strategies used.

The focus of this round table is to invite colleagues to inform, contribute to, and explore some of the key themes that arose in this study. These being the transition from primary school to secondary school, the role of students? perspectives in mathematics research, and mathematics education in an indigenous language. This study set out to explore the perspectives of 10 year 8 students on their mathematics learning and in particular the transition from Kura Kaupapa M?ori (M?ori medium primary schools) to secondary school (M?ori medium education in the Waikato region has extended from K?hanga Reo to wharekura). The Kura Kaupapa M?ori in this region are at least 10 years old. The 10 students interviewed in this study were identified as successful learners of mathematics in their year 8 programme, with outcomes at Stage 7 or above on the number framework. The students? teachers had participated in professional development in Te Poutama Tau/Numeracy project. Kaupapa M?ori methodology and interviews were key approaches undertaken in this study. Interviews took place with students and a questionnaire was given to the students? year 9 teachers. The interview process was a powerful vehicle in accessing student voice. Results indicated there were differences between the students? perceptions of their year 8 and year 9 mathematics programmes. The students viewed mathematics positively and had formed opinions regarding mental strategies and where and how to use them. Some students were able to discern what constituted effective teaching and had formed ideas on the value of the mathematics taught they had learned. An interesting area to emerge in the research was the self report by the girls in the study on their ability.

In the following project, we investigated students? use of mathematical thinking skills in an interactive, computer problem-solving task. Within the program Between The Lines (BTL) (Ainley & Hidi, 2002), a range of statistical information was presented. Students were required to integrate these different types of mathematical information in line with constructive assessment. This approach promotes problem solutions that move beyond the simple reproduction of answers and formulae and are representative of higher order thinking (Clarke, 1996). Note-taking is a process known to facilitate learning and the development of ideas (Kiewra, 1989), thus the program also asked students to take notes as they worked through the information. These notes were then used to create a problem solution. Students? motivational reactions (i.e., their interest levels) were also monitored throughout the task. Two-hundred and eighty-six Year 8 students (151 males and 135 females) participated in the project. The SOLO taxonomy (Biggs & Collis, 1982) was used to code students? notes whereas their solutions and notes were marked for understanding of the material presented. Associations between students? level of thinking and understanding were investigated in relation to levels of on-task interest and other indicators of motivational engagement. In this roundtable presentation, examples of students? problem solutions and notes will be presented. Profiles of students? thinking, for example, higher order versus low order thinking, and their relationship with on-task motivation will be examined. Discussion will focus on interpretation and implications of the data.

The Home School Partnership Numeracy facilitators will start the round table discussion by presenting the findings from three small studies: i) New Zealand Council for Education Research (NZCER) National Exploratory Home School Partnership Research ii) A case study on increasing parents? confidence in order to support their children?s learning in numeracy iii) A case study on the clarity of communication between home and school in relation to student achievement in numeracy. The international evidence cited in Alton-Lee (2003) positively supports the enhancement of student learning through home and school partnerships. In New Zealand, two Best Evidence Syntheses research studies also highlight the importance of establishing effective relationships between home and school (Alton-Lee, 2003; Biddulph, Biddulph, & Biddulph, 2003). The Home School Partnership project reflects the acknowledgement of parents as first teachers and the desire to continue to encourage parents to confidently interact and communicate with their children about mathematics. Effective relationships within the school community encourage parents to take an active role in the shared responsibility of their children?s education. Immigrants, refugees and parents who sometimes speak English as a second language are the focus of the community partnerships, especially Pasifika families. Pasifika students are identified through National Numeracy data as achieving well below National benchmarks. Aspects that might be considered in this discussion include: the establishment and sustainability of home and school partnerships; successful learning communities involving facilitators, lead teacher, and lead parents; mathematics as a ?frightening focus? for parents and community sessions in parents? first language.

A study about students? beliefs and attitudes towards mathematics was carried out in Estonia in early 2006. The study aimed, firstly, to investigate students? views towards mathematics and the underlying rationale, and secondly, to gain an understanding of the regular mathematics classroom activities in Estonia. The study used methodological triangulation, that is to use different methods on the same object of the study (Cohen, Manion, & Morrison, 2000). Firstly, a web-based Likert-scale questionnaire with 98 statements about beliefs in mathematics was administered to seven schools in an urban area in Estonia. Five hundred and eighty students from grades 7 (14 yrs.), 9 (16 yrs.), and 11 (18 yrs.) completed the questionnaire. Secondly, 26 semi-structured interviews were conducted with students and teachers for illuminating their thoughts about mathematics, and the learning and teaching of mathematics. Thirdly, during a three month period I participated in 11 teachers? lessons observing 55 mathematics lessons. Field notes gathered reflected a general picture of the classroom activities ? teachers? activities, methods, behaviour, relationship with students, and so on; and also personal impressions from different situations that I found interesting. Participants of the round table will be asked to consider the following questions: What are the appropriate tools for these kinds of investigations? What are the strengths and weaknesses of the methods used in my study? What are the possible strengths and weaknesses in combining quantitative and qualitative data collection methods?