Display Conference Proceedings

I am delighted to have the opportunity to pay tribute to the work of a highly valued colleague who is sadly no longer with us—Glendon Lean—but whose work and spirit lives on through the bequest to Deakin University.

Perception of the mathematics classroom as an arena in which there are various opportunities to learn mathematics leads to a fine-grained focus on the structure of mathematical tasks. Each mathematical task affords engagement with mathematics in certain ways. Variation within a task is a major factor influencing learning.

Results from the mathematics portion of the Third International Mathematics and Science Study (TIMSS) 1999 Video Study, comparing videotaped Year 8 mathematics lessons from seven countries, were released in March 2003. This paper presents selected findings from that study, with a focus on those results that might be of particular interest to Australian educators. In addition, the paper considers ways in which the results and products from this study can make a lasting contribution to the field of mathematics education. Three areas are described: the innovation associated with the study’s “video survey” research methodology; the networking possibilities for mathematics educators and researchers internationally; and, the opportunity provided to educators and researchers around the world to “visit” classrooms from each of the Video Study countries.1

This paper focuses on the social justice imperative to bring about improved mathematical learning outcomes for Aboriginal students. It provides comment whereby mathematics educators can appreciate more fully the context in which many Aboriginal students learn mathematics. Further, the paper reports on five mathematics education case study projects initiated by educational systems working collaboratively with Aboriginal communities. It examines each program using seven constructs: social justice; empowerment; engagement; reconciliation; self-determination; connectedness; and relevance. As an outcome, possible roles and responsibilities of mathematics educators for working collaboratively with Aboriginal communities to provide appropriate mathematics pedagogy for Aboriginal students are identified.

This inquiry synthesises the stories of one prospective teacher and a teacher educator into a narrative that describes how case investigations and the writing of case stories can enhance learning within communities of practice. The narrative highlights the importance of critical reflection and shared conversation in recognising the silent messages that we hold and send. The writing of case stories is presented as a tool for critically reflecting on beliefs and practices and learning from each other. The implication for teacher educators is that we need to listen more and tell less if we want to shift the ownership of learning to prospective teachers so that their sense of identity is enhanced.

In this symposium, four aspects of one major research project are presented. The project, Overcoming Barriers to Mathematics Learning, is funded by The Australian Research Council and the Victorian Department of Education and Training. Focus groups were used to identify and described aspects of “implicit pedagogy” that may present barriers for some children when open questions are used as a basis for the teaching of mathematics. These focus groups are the subject of the first paper, Perceptions of barriers to numeracy. Mousley describes the focus-group methodology and some findings, and raises issues concerning the use of focus groups. A product of this initial research stage was a manual that presents advice to teachers about the barriers identified. In The potential of open-ended mathematics tasks for overcoming barriers to learning, Sullivan describes some of the considerations in choosing tasks to be used as the prompts for mathematics learning experiences. He discusses the advantages and possible limitations of using open-ended tasks in mathematics teaching, and the pedagogies that teachers can use to avoid potential disadvantages to some students. In the third paper, Teachers’ perceptions of how open-ended mathematics tasks assist in overcoming barriers to learning, Turner Harrison reports data from interviews with teachers who sought to implement the pedagogies recommended as part of the project. The data indicate that it is possible to use open-ended tasks effectively and inclusively. Finally, in Disrupting the pedagogic relay using open-ended tasks, Zevenbergen gives an account of the implementation of the advice in a Queensland school. The research context here includes Indigenous students learning mathematics in a language that is not their home language. The paper reports ways in which a teacher adjusted patterns of language and work to accommodate the students.

This paper reports on one of the initial stage in a project that aims to identify, describe, evaluate, and provide advice on aspects of classroom pedagogy that may act as barriers to the numeracy development of some primary students. The paper describes how focus groups were used and some of the outcomes of this process. Some concerns are discussed, and suggestions for improving the process are made. However, it is concluded that the focus group approach served the research purposes well.

The data reported are part of a project called Overcoming Barriers to Mathematics Learning, which has explored pedagogies associated with maximising student participation when implementing open ended tasks in mathematics lessons. This paper presents data that support a number of the claims made about the advantages of using open-ended questions, and especially about creating opportunities for achievement for all students.

Now that open-ended mathematics tasks have been used in many countries for some years, there is some debate about the implications of the use of such approaches for the learning of all students. This paper makes a contribution to that debate. While acknowledging both the advantages of open-ended approaches, and the potential of these to exacerbate the disadvantage of some students, it is argued that if open-ended approaches are accompanied by appropriate pedagogical supports, then

This study investigated performance of South African foundation year students’ ability to compute rates of change from functions represented (i) algebraically, which are largely procedural in nature and (ii) those represented graphically, which are mainly conceptual in nature. Students’ responses in a test on calculus were analysed. The results show that (i) students’ knowledge on calculus is mainly procedural; (ii) their understanding of rates of change at a point is inadequate, and (iii) procedural knowledge is more preferred than its conceptual referents.

I report the results of a three-year study into my own teaching of preservice secondary mathematics teachers in a combined 15-week methods and practicum course (60 class contact hours and 180 hours field placement). I have two main goals for my students. First is to develop their understanding of the art of “teaching for understanding”. Second is to develop their ability to think reflectively. One activity completed each week is the reporting and writing of a critical incident reflective journal. At the start of each weekly methods class, groups of three or four students report their critical incidents to each other, then choose one incident to report to the whole class. They submit a written report of ten critical incidents for grading. I have collected these journals for three years, and have 34 journals, with over 320 incidents reported. I have analysed the journals and found that the issues raised by students focus on four main areas: teaching and classroom management; student factors such as prerequisite knowledge, understanding, resistance and motivation; issues concerning relationships with colleagues, students and parents; and school organizational issues such policies and access to resources. Through this Round Table presentation, I would like to solicit collaborators to undertake similar activities with their preservice teachers, so that we could combine our data and broaden the applicability of our findings across a wider range of contexts. The ultimate goal of my investigation is to improve the course to better meet the needs of my students.

Developing prospective primary teachers’ knowledge of mathematics is a crucial step towards improving the teaching of mathematics. This round table will explore approaches to raising the personal content knowledge, and the confidence of student teachers. In one College of Education, incoming students were screened using a test that required student teachers to do costing and time calculations in the context of planning a class trip. Students that struggled with this test were required to attend a 30 hour course targeted at developing their personal content knowledge, and willingness to engage with mathematics. The course had a strong emphasis on problem solving and sharing of ideas through discussion. The course was assessed through daily journal entries. The research literature specifies many examples of prospective primary teachers having weak content knowledge of mathematics, suffering from mathematics anxiety, or using algorithmic approaches in mathematics. The students required to attend the courses had similar dispositions and needs. In order to empower the students to begin to address their own mathematical needs and start growth towards being teachers of mathematics, the course was designed to reinforce to the students that they could their extend own knowledge by being active participants in the process. Experiences from this course will be shared as part of a more general discussion on developing prospective primary teacher’s personal content knowledge of mathematics.

This paper will relate the experience of four groups of Bruneian technical students working on a project titled “Trigonometric Waveforms Around Us”. This project tries to connect whatever was learnt in class to the real world. The main aim was to show students that mathematics phenomena are around them and that mathematics is useful in everyday life. A second aim was to enhance to make them more interest and motivated in learning mathematics. The paper examines their chosen topic and looks at the problems that students face in carrying out the project. Data was mainly collected through observation, interviews and journal writing. Although the students were unmotivated in the beginning, their interest and motivation increased as they went deeper into the project.

A study has been undertaken to investigate the implementation of an ethnomathematical unit in a mathematics classroom in the Maldives. The research was conducted at two primary schools and involved teaching grade 5 students an ethnomathematical unit of work on measurement. The unit was designed in conjunction with the teachers. This paper discusses ethnomathematical curriculum models and the approach used in the study. Data are presented indicating teachers’ and students’ reactions to using such a curriculum unit. The data show that despite the very traditional education of the Maldives, the ethnomathematical approach was appreciated and understood by teachers and students.

This paper discusses the challenges posed by ethnomathematics as a field of study, and suggests ways of meeting these challenges. Using an on-going ethnomathematical study on the practice of stone walling among the Kankana-ey people of northern Philippines, it shows how an investigation of QRS conceptual systems may be useful in identifying unfamiliar mathematical ideas. It presents four indicators that may be used in uncovering mathematical systems in a cultural practice, and examines stone walling using these indicators.

This paper presents a case study of a primary teacher in her first year of teaching. As a preservice teacher, Ann was a participant in a longitudinal study that investigated the impact on primary teachers' mathematical beliefs, knowledge and practices of a mathematics education course that utilised a constructivist approach.

This paper reports part of a larger study into primary school teachers’ problem-solving beliefs and practices in NSW. In particular, teachers’ selection of problem-solving tasks and the reasons for their choices were investigated. Teachers reported choosing exercises and application problems more frequently than open-ended or unfamiliar problems. Teachers preferred to use exercises for practice of basic skills and procedures, and application problems as indicators of the relevance of mathematics. Open-ended and unfamiliar problems were considered to be appropriate for more able students or for students in higher grades. Reasons given for task choices revealed particular beliefs about how students learn mathematics.

This paper engages current thinking about the links between context and learning. It reports on a study of students’ understandings of sharing and part/whole distributions within the family pizza dinner context, represented by a circular region model. Central to the analysis is the part that informal knowledge plays in the development of rational number understanding. The investigation reveals that for some students whole number operations play a more significant role in the pizza context than do conceptualisations of part/whole distributions. The suggestion is that rational number development requires a gradual pedagogical exposure to a range of structural representations which embody the concept of part/whole.

This paper presents findings from a study in which the bicultural perspectives in a third year mathematics education course within the Bachelor of Education (Teaching) programme at Wellington College of Education were explored. The authors experimented with inclusion of a wide range of bicultural perspectives within lectures and tutorials and students’ perceptions of these were collected. This paper canvasses College requirements and relevant research and compares with these the evidence and perceptions of bicultural perspectives.

A teaching practice today requires an understanding of pedagogic thinking which prioritises the constitution of learning over the execution of teaching. Integral to that understanding is the idea that knowledge is constituted through the learner’s active engagement with mathematics. This paper moves beyond an exploration of the way teachers’ speak about their practice to explore teachers’ actual creation of productive learning communities within their classrooms, as reported by preservice teachers in both Australia and New Zealand; thus providing valuable material for their mathematics education courses.

The espoused beliefs of 465 secondary mathematics teachers regarding mathematics assessment were the focus of this study. The data for this investigation were collected using a 24 items questionnaire. There is evidence from this study that teachers with experience at Junior High schools placed more emphasis on a problem solving orientation to mathematics assessment than did teachers with experience at other levels. The problem solving view of assessment in mathematics was more prevalent among experienced teachers and veteran teachers than among the inexperienced teachers and the socio-constructivist view of mathematics assessment was more prevalent among teachers than among consultants and principals.

One of the first major changes in mathematical technology used in upper secondary school mathematics in Victoria was the introduction of logarithmic tables in the early twentieth century. Information about this change and the reasons for it is difficult to come by, yet much can be gleaned from such examination papers and official publications as survive. Understanding of this process is of interest as an early example of the introduction of technology into the mathematics classroom, a phenomenon of particular interest in our own times.

This paper reports an investigation of collaborative learning in mathematics classes that sought to identify patterns of participation in small-group activities. Videotapes of lessons in three senior mathematics classes were studied, and, using positioning theory, a range of subject positions available to students during small-group collaborative activities was determined. A student’s pattern of participation was defined as the combination of positions regularly taken up by that student, and groups of students with common patterns of participation were identified. Implications for collaborative learning are discussed.

This paper is concerned with the implications that being placed in a particular group has for the range of student identities available to individuals. I begin by discussing some factors that lead schools to group students according to their ‘ability’ and consider how these policies play out in practice. I then consider the ways in which students in different groups experience mathematics in very different ways. Finally, I consider the ways in which individual students are positioned in mathematics lessons.

This is the third report to MERGA of a project working with senior secondary mathematics in low socio-economic schools in Auckland, New Zealand. It is aimed at enhancing the achievement and participation of Year 12 & 13 mathematics students and promoting their transition into tertiary mathematical programmes. This paper focuses on three theoretical approaches used in the project, and describes the way that data is organised and analysed using them.

I am exploring general system theory as a framework for work-in-progress on curriculum and development. This paper outlines the influences that led me to general system theory, discusses some examples of systems, and concludes with related ideas about curriculum.

Teacher beliefs research has promised much but delivered relatively little in terms of improvement in the teaching and learning of mathematics. This is due in part to insufficient account being taken of context. This paper examines literature that stresses the contextual nature of teachers’ beliefs and in light of this examines studies reporting both consistency and inconsistency between teachers’ beliefs and practice. In each case it is argued that appropriate consideration of context renders their findings highly predictable.

This paper reports on data collected from a larger study investigating the effect of game playing on students’ mathematical learning and motivation. The study was conducted in three schools with 240 students participating in four different treatment groups. This paper reports some of the results of interviews with students to gather data on their attitudes towards mathematics and game playing. The interview data collected from a sample of students in the three game playing groups pertaining to attitudes are presented and discussed below.

The cognitive, mathematical, and technological processes undertaken by senior secondary students as they searched for a complete graph of a relatively difficult cubic function were investigated. Particular circumstances: that became defining moments in the solution were identified. Those related to students’ responses to particular views of the function are presented. A variety of processes were used, however, a mental image of possible forms of a complete graph, understanding the use of graphing calculator features such as Zoom Fit, and the effect of each value in the WINDOW settings facilitated the students’ solutions.

Teacher educators recognise that subject knowledge is critical for effective teaching. In recent years some pre-service primary teacher education programmes have reduced the class time for students studying subject knowledge. This discussion paper critically examines the history of ‘subject studies’ within one institution and reports on students’ responses to a survey that included questions about their mathematical subject knowledge for teaching mathematics in the primary classroom. These results advocate for pre-service programmes to have increased emphasis in subject knowledge.

The same group of Year 10 students undertook three different mathematics assessments. Two of these were based on developmental continua and addressed higher order thinking, but in different ways. The third was a multiple-choice test of mathematical skills and knowledge, appropriate to the Year 10 cohort. Results suggested that although all the assessments were reliable, and no test bias was detected, there were some differences between the assessment outcomes across subgroups and by mathematics course being followed. The implications of these findings for teachers and test developers are discussed.

With a crowded curriculum, many primary schools attempt to integrate their key learning areas. One primary school in a large regional city has taken this a step further. Using the “environment” as the overarching theme, the key learning areas are interwoven into the teaching of the environment. This has presented some issues when attempting to teach English and Mathematics. These issues and the way the school and teachers solve them are documented in this case-study.

First year university students were provided with two sets of data and told about messages or conclusions that could be drawn from each set. The students were then asked to represent the data so that the representation effectively conveyed the message. These tasks required students to “transnumerate” the data, or, in other words, reorganise data to better understand them. Four phases of transnumeration were identified: recognising the message, choosing the representation, transforming the data, and representing the transformed data. Although many students were able to produce effective representations, other students had difficulties with aspects of the transnumerative process, many at the basic numeracy level.

The digital age requires students to use a range of literacies in order to engage in and interpret an increasing array of multimodal communications. This case investigation describes the ways in which two students use and interpret maps when engaged in gameplay with popular culture multimodal texts. The study encouraged the participants to share their experiences as they played their game. The Pokemon phenomena was used as a catalyst for the investigation as the students (aged 10 and 11) described how they used the interrelated nature of the texts to support and scaffold their mathematical understandings. The students demonstrated their capacity to use a range of literacies across a variety of texts to interpret and navigate their way through different mapping contexts. These processes required strong mathematical understanding and the ability to internalise large amounts of mapping-related information across various 2D and 3D representations.

This paper reports the key findings of an extensive review and critique of international comparative research in mathematics education. A feature of the reports of such research is the interweaving of similarities and differences, and it is proposed that research should address more explicitly the interconnectedness of these similarities and differences. Surveystyle and case study approaches are examined in this regard. Issues raised include representation, appropriation, and exploitation, and the cultural authorship of international comparative research.

This paper reports a multi-stage analysis of 55 lessons taught as five ten-lesson sequences and one five-lesson sequence by three Australian and three American teachers. Despite the assumption of cultural similarity, there were significant structural differences between the Australian lessons analysed and the U.S. lessons. It is our contention that the comparison of lesson components (‘lesson events’) is more helpful than a (national) lesson pattern or script as a guide to the differences between the practices of teachers or nations.

We have argued elsewhere that global collaboration is essential for moving the discipline forward in this globalised world - at the same time avoiding the colonialism of the past - and allowing the discipline to play its role in bridging the ever-increasing gap between countries (Atweh & Clarkson, 2001b). This issue of global collaboration sets the tone for this paper. In this study we conjecture that for those who have had and continue to have an active set of international contacts and/or experiences, their appreciation of the impact of the processes of globalisation would be heightened. To gain some insight into this issue a survey was developed and distributed to Australian and New Zealand mathematics education researchers. Although we found a divergence of views that our colleagues hold across a range of issues, there does seem to be a trend for colleagues who have experienced at some depth non western cultures to appreciate more deeply some of the issues thrown into relief by the notions of globalisation. We believe that more discussion on the notions of globalisation and how it is impacting on mathematics education will challenge us all and give more depth to our thinking and practice. This in turn will position each of us to be better able to deal critically with fundamental issues in the globalised world in which our students and we now live.

Although perceptual inference facilitates reasoning, there has been scant attention to the use of icons in visual displays of qualitative data. This paper uses an iconic data map to provide a wholistic representation of two teachers’ mathematics teaching and to establish similarities and differences in their practice. The map had five advantages for data display and analysis related to visual memory, strategic professional development support, the identification of anomalies, differences between espoused and enacted teaching beliefs, and the identification of further research questions. Thus, iconic data maps are a powerful tool in the display and analysis of qualitative data.

Changes in teaching practice have not kept pace with the changes envisioned in numerous reform documents. In this paper, we present the results of a study of an experienced secondary teacher who taught a sequence of modelling tasks designed to investigate exponential growth and decay. Our results suggest that changes occurred in her practice along three dimensions: her role in listening and questioning, a new emphasis on student explanation and justification, and a more fluid view of the representations of the concepts.

Progress in improving the practices of teaching and learning requires a shared knowledge base for teaching that is grounded in professional practice. In this paper, we argue for an approach to the design of research on teachers' knowledge development that can contribute to a knowledge base by being open enough to accommodate variability in perspectives and structured enough to allow for the sharing of knowledge. We illustrate this methodology with examples of investigations on the development of secondary teachers' knowledge.

While curriculum frameworks are major influences on learning, teachers know that children progress at different rates. Sometimes this is evident within a particular topic, and at other times more obvious across different topics. In this paper, we present the hops, steps, and jumps of numeracy learning of some 3000 Australian children. All were assessed using I Can Do Maths, and their achievements mapped to provide a detailed picture of how children hop, step and jump on their numeracy journey. This mapping provides teachers with information about key hurdles to numeracy learning for Australian children.

Snapshots of classroom practice in middle years numeracy classes for students experiencing difficulty with the study of mathematics often show students undertaking individualised programs of study. Such practices are questioned in light of definitions of numeracy and the degree to which they promote critical numeracy skills for adolescents and their diverse life-pathways. In this paper, models of support programs for numeracy are analysed in terms of pedagogical practices. A call is made for a reconceptualisation of what numeracy means for students in the middle years of schooling.

In NSW, Australia, a new course, General Mathematics, has been introduced for the Higher School Certificate replacing the two lowest level courses. Twenty-five thousand students study this course each year. This article reports on a study of 95 General Mathematics students in six schools where they were asked to comment on their learning preferences. Many of the benefits of collaborative learning with spreadsheets, were perceived by only a small number of students in this study, as many students do not envisage benefits of small group learning or learning with spreadsheets. These results have implications for mathematics learning.

The MATH taxonomy is designed to help teachers plan assessment to test a range of skills and concepts. The communication aspects of mathematics and multiple representations are included. This paper describes the analysis of a pre-test and post-test given to 172 senior secondary students studying a new subject, General Mathematics for the NSW Higher School Certificate. The study highlights the types of questions students find difficult and reasons why.

Preparing students for proof type geometry problem solving has become a key issue for mathematics educators. Prevailing instructional strategies have been shown to be inappropriate to address the complexities of deductive geometry. In this report, we propose a design of a learning environment to address the above issue. We argue that this model has a potential to help students to make progress up to van Hiele Level 3 and to acquire skills required to solving a range of proof type geometry problems.

Victorian Curriculum and Assessment Authority Mathematical Methods (CAS) Units 1–4 were accredited in February 2001 for a pilot study from 2001 to 2005. As a consequence of this decision, the VCAA has produced sample examinations and supplementary materials and set examinations for Units 3 and 4, with the first cohort of 78 students taking these examinations in November 2002. This paper provides some initial commentary on student performance, including common items from the Mathematical Methods Units 3 and 4 examinations.

Based on cases of classroom events this paper discusses two perspectives on the issue of teacher informal classroom observation as a means of assessing their students’ understandings. One perspective emphasizes difficulties and obstacles that might be overcome. The other centres on hearing through as an intrinsic characteristic of what it means to understand what someone else is saying or doing. The paper concludes with the suggestion to merge the two perspectives in practice.

This paper describes a conversation in a small group where the teaching is based on a model for teaching the language arts. Students in the group are engaged in a non-routine task and talk about their ideas, revealing their understandings. Some of the students’ responses are discussed, highlighting the ways in which linguistic variations can be used reveal students’ conceptual understandings. It is suggested that such a model may be applicable to other mathematics retrieval classes.

The subject of inquiry in this paper is copying on a graphics calculator by students in a Year 12 Calculus class. They copied interim numerical results and algebraic expressions as part of calculation. The practice called for knowledge of order of operations, was a means for linking representations and generalisation, and was a facet of discernment of function relationships. Implications for teaching are the act of copying could assist the development of understanding in each of these domains.

In this paper I review theory on communicative competence and identify characteristics of competent spoken performance in a Year 12 mathematics class. Key actions of the teacher were that he asked open-ended questions, accepted responses that he did not expect and attended to student interpretation. Key actions by students were that they joined discussion and offered ideas without waiting to be nominated by the teacher. The nature of the teaching can inform teaching practice in all classrooms.

This research examines the learning of 4th-year pre-service primary teachers (N=62) engaged in development of a mathematics teaching portfolio. An interpretive research approach was used to analyse data from classroom observations, draft and final portfolio work samples, student reflection sheets, and external assessor interview reports. The findings indicated the pre-service teachers developed substantially in their confidence and reflective capacities as mathematics teachers as well as their capacities to clearly articulate and justify ideas for innovative mathematics teaching and learning practices.

Scales designed to assess approaches to studying tertiary mathematics were used to compare approaches of females and males. Scales assessed active study, intrinsic motivation, expected future use of mathematics, confidence and anxiety. Comparisons controlled for entrance qualifications showed more active study among females, less anticipated use of mathematics, lower confidence, and higher anxiety. Multiple correlations between scale and achievement scores were highly significant. The dominant connections with achievement involved intrinsic motivation, high confidence, and absence of debilitating anxiety.

We apply an adaptation of Valsiner’s Zone Theory to the structuring of activities as they occur within teaching-learning contexts. A challenge exists to develop frameworks within which to describe, theorise, and interpret the range of learning activities engaged by teachers and students in technology enriched settings. We illustrate the approach through applications to the analysis of literature comment, to classroom episodes, and to professional development experiences.

The Victorian Curriculum and Assessment Authority (VCAA) Mathematical Methods Computer Algebra System (CAS) Units 1–4 pilot study 2001 – 2005, integrates the use of CAS in the curriculum, assessment and pedagogy of a senior secondary mathematics subject, with end of final year examinations. This paper reports on some preliminary reflections from a pilot teacher – researcher perspective and outlines directions for qualitative research into teacher and student responses to the use of CAS as part of a new doctoral study.

This paper continues a long term investigation into students’ views about and attitudes toward the learning of mathematics within a technologically rich classroom environment. The data was sourced via a series of questionnaires that focused on the role of technology in student/student and teacher/student interactions within this environment and, in particular, the collaborative aspects of these interactions. The advantages/disadvantages of technology in investigating novel mathematical problems were also explored. Links are drawn between the results of this survey and earlier work which developed categories of students’ use of technology.

This paper explores the counting errors produced by 40 Grade 1 children from 16 Victorian schools prior to their commencement of a mathematics intervention program. Analyses of the errors highlighted several common difficulties and issues related to learning to count. It is anticipated that if teachers are on the lookout for these difficulties, then they will be able to identify children who are in danger of being “left behind’ their peers, and may provide the type of experiences that will assist these children to construct a more powerful understanding of counting and the number word sequence.

It has been some time since it was identified that student perspectives on equations and their use of the equals sign have not mirrored those of mathematicians. This paper describes some of the understandings of the equals sign displayed by secondary school students and seeks to analyse them in terms of properties of the constituent parts of equations. We find that students display a number of incomplete or pseudo-conceptions, and are sometimes influenced by representational aspects of the properties. A start is made on constructing a framework for understanding of the mathematical equation object.

This three year study is investigating the pedagogical practices and beliefs of pre-service and beginning teachers in integrating technology into the teaching of secondary school mathematics. A case study of one student teacher explores the influence of beliefs, attitudes towards technology, and constraints and affordances in the practicum school environment in shaping her identity as a teacher. The opportunity to experiment with technology-based lessons may assist beginning teachers to persist with the innovative approaches promoted by pre-service methods courses.

There has been a concern for some time that preservice primary school teachers hold negative views of mathematics and their views influence their mathematical teaching practice. The participants in this study were involved in an initial course in mathematics education that deliberately set out to challenge their affective views of mathematics. The findings of the study indicated that some positive affective change occurred in the participants’ beliefs about mathematics and their feelings and attitudes towards the subject. The aspects of the course that seemed to facilitate these changes are presented, followed by a discussion of some of the issues surrounding affective reform in mathematics for preservice primary teachers.

This paper reports on a study of Year 3 and 4 students’ addition and subtraction mental computation. The purpose of the study was to investigate the cognitive aspects of the mental computation conceptual frameworks that the author formulated from a study of Year 3 students who were accurate in their mental computation. These frameworks had accounted for differing levels of flexibility in mental computation.

For whole-class discussions, teachers need methods for orchestrating them with their students. This may require the design of tools to meet specific needs for a whole-class discussion. As teachers design tools for their practice, they can evaluate the effectiveness of those tools. As the tools are designed, documentation of teacher development is generated. So, tool design in a design experiment can meet teacher needs for useful tools and researcher needs for knowledge about teacher development.

In 2002, a new K-6 mathematics syllabus was introduced in NSW schools. The Aboriginal Curriculum Unit of the NSW Board of Studies commissioned a research project to investigate appropriate ways in which Aboriginal communities might become involved in the planning and implementation of mathematics curricula based on this new syllabus. Through an intensive process of professional development of teachers and community involvement in two pilot schools, the project has devised some strategies that work in the particular contexts of these schools and have promise for other communities of Aboriginal people and their schools. This paper reports on one of the sites.

This paper describes a process in which a practising teacher developed a personal teaching model for numeracy. Influences included Vigotsky’s Zone of Proximal Development, Steffe’s counting types, a mathematics education paper which included study of Wright’s Mathematics Recovery and Burns’ cooperative group teaching, a trial of the Count Me in Too project with subsequent developments in New Zealand, and an explicit strategy teaching model derived from the theory of Pirie and Kieren. The teacher uses her teaching model in a classroom setting. The teacher’s espoused teaching model is compared with its implementation. Some implications for assisting teachers to construct personal teaching models are made

This study involved a six-month teaching experiment aimed to improve children’s understanding of decimal fractions. Data indicated that the learning activities based on the work of Moss and Case (1999), which promoted the use of percentages as a visible introductory representation, significantly influenced the students’ emerging understanding of decimal fractions as quantities. Sense making was maintained as the students increasingly translated across rational number representations—using and extending their informal understandings into more formal decimal knowledge.

This paper describes developmental changes as children move from additive to multiplicative thinking. Five broad phases through which multiplicative thinking develops were synthesised from the research. These were labelled as one-to-one counting, additive composition, many-to-one counting, multiplicative relations, and operating on the operators.

This paper reports on the journey of one teacher involved in a Professional Development program that encouraged her to undertake teacher research in her classroom. Case study methods were used, and data were collected by interviews and some observations in classrooms. By investigating her own teaching and reflecting on her practice Julia was able to renew her teaching practice.

In earlier times, the stereotyping of mathematics as a male domain was identified as a factor contributing to females’ decisions not to persist with higher-level mathematics courses, and a weak positive relationship was typically found between females’ achievement levels and rejection of this stereotype. Using a new instrument designed to measure the extent to which mathematics is stereotyped as a male, female and neutral domain, we explored the relationships between perceived achievement levels and the gender stereotyping of mathematics. We compared the results for males and females. Our findings confirm some previous results and challenge others.

This paper reports preliminary findings on a study that investigates secondary school mathematics teachers’ usage of the internet. A convenience sample of Mathematics teachers in Australia (N=63) was administered a questionnaire: the Use of the Internet for Teaching Secondary School Mathematics Survey. The paper discusses who uses the internet, how, and for what purpose, in teaching and learning mathematics.

This paper examines teachers’ use of the Internet in the teaching and learning of Mathematics. The study draws upon data collected via an online survey and interviews with six teachers. It reports on their beliefs, strategies for use and their perceptions on how it impacts on students and their learning of mathematics. Some comparisons are made between the ways teachers used the Internet

This paper explores the way in which a case study participant (11 years of age) constructed problems for friends to solve in an Information Communication Technology (ICT)-rich context. The individual posed problems using construction-based software, justified the design and discussed the approaches and mathematical ideas the problem solver could employ to complete the task. The participant was encouraged to pose problems in three distinct categories including tasks of 1) varying complexity, and with the view of posing problems that had either a 2) technology or 3) mathematics focus. It is argued that the meaningful and empowering nature of the ICT-rich context was influential in the success achieved by the participant in being able to pose appropriate problems in each of the three categories.

This paper describes the 2002 Western Australia Monitoring Standards in Education system-wide random sample assessment of student performance at Years 3, 7 and 10 in mathematics. It presents the design of the sample, outlines the methodology used to analyse the data collected, and summarises the results. Students found the Working Mathematically tests harder than the Content strand tests, and there was variation in their performance across the Content strands. Strong evidence was found that the Content and Working Mathematically items fitted on a single measurement scale.

This paper investigates tensions faced by indigenous parents in developing a mathematics curriculum. These included an uncertainty about their role in regard to their contribution and what they could gain from being involved. We suggest that a community, which exists because their children attend a school, needs to have opportunities for shared activities first. These can be used as starting points for curriculum discussions so that the tensions can be alleviated and the possibilities taken advantage of more fully.

Mean scores on the New South Wales Basic Skills Test (BST) in the period 1996-2002 among 71 Count Me In Too (CMIT) schools from across the whole of the state were analysed. In conjunction, background data on the schools and their staff perceptions of CMIT were gathered. The results showed that the mean Year 3 BST numeracy score increased steadily and significantly during the period, while all other BST scores remained approximately constant. The increase in the Year 3 numeracy score was not consistent across all the schools sampled, and various background factors were found to be associated with the size of the increase. The implications of these findings are discussed.

Part of a broader research project on the development of mathematical understanding in primary classrooms, the research reported in this paper focused on how opportunities to learn about percentage in two classrooms were shaped by worksheets. The words in the text, the way that the teachers replicated these orally, a grid used to illustrate the concept of percentage, the genre of both mathematical and non-mathematical aspects of the worksheet’s presentation, and the calculation methods presented were all potentially influential aspects. It is argued that the “knowledge” conveyed by the texts of the worksheets was distributed over a much wider field than the classrooms involved.

A cross-sectional descriptive study of 103 Grade 1 students from ten Sydney schools investigated the use of mathematical and spatial structure across 30 numeracy tasks. This report describes students’ levels of structural development across two key tasks on visual memory and area as emergent, partial or identifiable structure. Lower-achieving students who lacked structure in their responses did not appear to be located on the same developmental path as other students. Qualitative analysis supported the findings of Gray, Pitta and Tall (2000) and Thomas, Mulligan and Goldin (2002) – that in the abstraction of mathematical concepts these students may concentrate on idiosyncratic non-mathematical aspects of their experience.

A plausibility argument is offered in support of the assertion that mathematics education is unduly dependent upon the forensic metaphor, and that the jazz metaphor is a useful and contrasting alternative. Five components of jazz playing are briefly outlined: structure, improvisation, playing outside, pursuit of the ideal, and ‘ways of the hand’. The third of these, playing outside, is outlined more fully and applied to mathematics education via a discussion of the role of the mathematics curriculum as public knowledge policy.

The Individualization of knowledge representation requires a knowledge base in which teachers can organize contents according to their personal needs. The objective in teachers education is to combine authoring activities with the embedding of existing contents. Student teachers have different mathematical and pedagogical experiences in mathematics education. These experiences determine their focus of interest and their authoring activities in the knowledge management system. They develop for example lesson plans in geometry embedded in the pedagogical and mathematical knowledge in the system. Individualized knowledge representation is a constructivistic approach to build up knowledge according to the progress in teachers education.

Two classes of Year 8 students were asked to organize and represent sets of numerical data with large variation of scores, to determine the extent to which the students had learned to produce grouped data and represent them in displays such as a histogram (as stated in the current Queensland syllabus for Year 7). Analysis of the students’ responses in terms of a statistical-thinking framework revealed wide variation in students’ ability to re-organise the data and represent them in organised graphs. Only 21% of students were able to complete the task successfully. Interviews with other students indicated that a little prompting in terms of grouping the data assisted many to produce grouped data with convenient interval sizes. The results of the study have implications for the teaching and learning of the organisation and representation of grouped data.

As part of a wider evaluation of the Counting On mathematics program in NSW government primary and secondary schools, the authors conducted a case study at Bankstown Girls High School which specifically considered the structural arrangements needed to maintain the program when its specific funding from the NSW Department of Education and Training was withdrawn. The effects of these arrangements on the staff and students in the school are discussed and the success of the arrangements in terms of enhanced cross-faculty teacher interaction and respect, and increased student achievement in mathematics are celebrated.

Animated graphics can provide a basis for discussion of calculus concepts. This paper reports analysis of the responses from nineteen undergraduate mathematics students who were asked to write a work sheet based on a simple animation linking the graph of a function with its gradient function. Their responses were benchmarked against those of experienced teachers. This open assessment task provided valuable insights into students’ perceptions of important aspects of both the applet and the derivative concept.

This study examines participant students’ points of view on accelerated programmes in mathematics from four state secondary schools in New Zealand. Contrary to fears expressed by educational practitioners, this research does not support the commonly held belief that students who are accelerated will suffer from undue stress that may hinder their social and emotional development. Coupled with these findings is the fact that, almost without exception, participants felt that involvement in an acceleration programme had been beneficial to their learning needs.

This paper describes the findings from three items of a 63-item questionnaire, which is part of an ongoing larger study. The larger study seeks to identify similarities and differences in the effective teaching of literacy and numeracy in the early years of schooling and includes identification of beliefs and intentions of fourth-year pre-service and experienced teachers in their teaching of literacy and numeracy. In this paper, pre-service teachers indicate similar intentions for aspects of their teaching of literacy and numeracy. In particular, they intend presenting an integrated curriculum, encouraging children to talk about contexts and using cooperative group work in both literacy and numeracy lessons

Concerns about students' difficulties in statistical reasoning led to a study that explored 14 to 16 year old Fijian students’ ideas of statistics. Existing models developed for investigating students' thinking in statistics education were not completely satisfactory for describing these results, so Shaugnessy's model was adapted to explain the data . This paper presents how students made sense of information in tables and bar graphs. The results of the study confirm some findings of other researchers. The paper concludes with implications of the findings for mathematics teaching and research.

As part of a larger study aimed at identifying and evaluating a range of numeracy teaching approaches in a structured sample of Victorian primary schools, three groups of teachers participated in an activity referred to as “behind-the-screen”. Teachers took turns to teach a small group of children from their own class in a room with a one-way mirror. Observing teachers were asked to comment on what they noticed and suggest labels or metaphors that captured the essence of the teacher’s communicative acts. Preliminary analysis suggests that this technique is a valuable tool in identifying and describing scaffolding practices in mathematics teaching and enhancing teacher’s understanding of their professional practice.

A questionnaire was developed to examine understanding of chance and probability concepts. Measures were taken of gambling behaviour, mathematics skills and understanding of a range of probability concepts. Correlations were calculated to examine relationships between gambling behaviour and mathematical knowledge. Responses to some questions will be described. Key misunderstandings in pure chance gambling, e.g. electronic gaming machines, include a lack of understanding of independence, randomness, and short and long run expectations.

Children’s whole number schemes can interfere with their efforts to learn fractions. To what extent do these schemes persist for secondary school students? This paper reports on the development and piloting of an interview designed to identify and probe inappropriate whole number strategies for working with fractions among secondary students. The interview showed that these strategies are still prevalent among Year 8 students. Among others who use appropriate multiplicative strategies the interview showed that some of these are still not confident in challenging instances of inappropriate whole number thinking.

This paper reports on a strand of a larger investigation into the effect of metacognitive training on 11 to 12-year-old students’ mathematical word problem solving in a cognitiveapprenticeship- computer-based environment. Empirical results from the quasi-experimental and case study designs suggested that treatment students outperformed control students on the ability to solve word problems on their individual written measures and collaborative “think aloud” interactions in a computer environment. Moreover, treatment students in the case study elicited more well-regulated metacognitive decisions compared with control students.

Children sometimes struggle to learn that counting will tell them “how many”. The literature in relation to children’s early number knowledge is reviewed in order to identify components in children’s understanding of number. A model is then proposed of how these different number components come together for children as they develop an understanding of numbers as a representation of quantity.

The study described in this article compared the attitudes of girls and boys in two junior secondary classrooms regarding the use of computers in mathematics. Data were gathered by questionnaire. The relationship between the attitudes of girls and boys to computerbased mathematics and measures of their self-efficacy in mathematics and computing were explored. Boys held more positive attitudes to the use of computers in mathematics than girls. For both girls and boys the relationship between attitude to the use of computers in mathematics was more strongly associated with their attitudes to computers than mathematics. The implications for teaching practice and gender equity in mathematics requires more research.

Year 8 students were introduced to the concept of mathematical proof before participating in conjecturing-proving tasks involving dynamic environments. This paper focuses on the argumentations of two pairs of students who completed a Cabri-based Quadrilateral Midpoints task. Supported by dynamic feedback and teacher intervention, even those students with limited understanding of geometric properties and relationships were able to engage in productive argumentation, conjecturing, and proving, with the dynamic geometric software serving as a cognitive bridge between empirical justification and deductive reasoning.

This paper describes the methodology used for a three-year ethnographical study of children’s expanding awareness of mathematics and their growing mathematical identities during the middle primary years. It explains how the term ‘sociomathematical worlds’ was adopted to represent the network of social contexts within which children learn about mathematics, and how an understanding of these worlds was constructed by the researcher through a process of broad and detailed data-gathering, rich in triangulation.

This paper examines the influence certain number combinations have on children’s ability to abstract arithmetic structure. The study replicates an investigation of Israeli and Canadian students, which found that certain number combinations significantly influenced the order in which operations are performed. Seventy-six Australian children participated in the study. The results indicated that, while the number combinations did not significantly influence the order in which the Australian children performed the operations, many exhibited misconceptions that reflected a very procedural approach to mathematics. Implications are drawn for the introduction of mental computation and early algebraic understanding in the early years of schooling.

Pictographs are often used in the media to draw attention to data that would likely be ignored in a table. In school, however, pictographs disappear from the curriculum by the middle primary years. The outcomes of the research reported here indicate that pictographs can provide a basis for rich tasks displaying not only students’ counting skills but also their appreciation of variation and uncertainty in prediction. The range of responses is discussed in relation to other research and classroom implications.

This study considers students’ predictions and explanations for outcomes when a normal six-sided die is tossed 60 times. Changes are monitored after lessons on chance and data and/or after two years. The study is motivated by two approaches to probability advocated in the mathematics curriculum: one stressing expectation based on theory and the other acknowledging variation during experiments. The outcomes are discussed in light of other research and the dilemmas created for students by these two approaches to probability.

Task-based interviews were conducted with 74 children aged four to twelve years from three schools, who had not received any formal instruction in probability as it was not part of their school curriculum. The study confirmed the presence of three developmental stages, but also revealed two distinct transitional stages not reported in previous research. This paper focuses on the characteristics of children’s strategies for making probabilistic judgments in each stage, and, on the implications for further research in regards to teaching.

Lesson Study is a model of professional development designed to assist teachers produce quality lesson plans and gain a better understanding of student learning in mathematics Years 7 to 12. The process involves a small group of teachers as a Lesson Study team, who meet regularly to plan, design, implement, evaluate and refine lessons for a unit of work that they had selected. In 2002 it was implemented in 36 NSW departmental secondary schools from across the state in Semester 1 and another 77 schools in Semester 2. This paper briefly describes the program and considers certain aspects of the evaluation, particularly in terms of teacher learning.

As part of a broader study of characteristics of situations that promote or inhibit spontaneous student exploration of novel mathematical ideas, student inclination to display these spontaneous behaviours was studied in conjunction with resilience. Resilience was operationalised using the dimensions of optimism (Seligman, 1995). Indicators of optimism were identified in post-lesson video-stimulated reconstructive interviews with year eight students. Students who demonstrated the pursuit of novel mathematical ideas were found to also display indicators of resilience.

Forty-eight year 7 and 8 students, aged 10 to 12 years, completed written tasks that required them to generalise a given strategy for multiplication or division. Their responses were analysed to ascertain which strategies they found most difficult to explain and apply. Eleven of the students were interviewed to find out how closely their written responses reflected their understanding of the strategies. The results indicate that the use of written tasks for assessing generalisation has limited reliability and that advanced division strategies are more difficult to apply than advanced multiplication strategies.

Teachers’ perceptions of technology in mathematics were measured with a questionnaire administered immediately before, and after, a programme of professional development targeted at encouraging the use of technology in mathematics. Thirty teachers commenced the programme in July, 2002. Over a five month period, significant positive changes were found in teachers’ (a) access and use of technology in mathematics teaching and learning; (b) confidence in using technology, and (c) attitudes and beliefs about the role and value of technology in mathematics.

This paper reports on the perspectives of two ten-year-old children selected from a sample of 77 children attending four schools that had participated in the Advanced Numeracy Project in New Zealand. The two girls’ ideas about solving a particular multi-digit addition problem that was given to them on paper, their perceptions of the importance of obtaining a “right” answer, and their views about discussing solution strategies with others are reported here. This analysis has raised some interesting questions and issues for us to explore further with the data from the remainder of the sample.

This paper reports on an ongoing study of errors in the use of fractions exhibited by a sample of 396 Primary 5 pupils in Brunei Darussalam. Among the five types of errors identified, those demonstrating a lack of understanding of basic facts predominated. The quantitative data reported here were obtained from a diagnostic pre-test administered in a longitudinal study. The test was administered during the first phase of the study, and coupled with qualitative data obtained from pupil interviews. After having identified the different errors exhibited by the pupils, the study set out to investigate in which components of the fractions syllabus significant errors occurred. Data from the pre-test was shared with the 15 teachers whose pupils participated in this study in order to assist the subsequent teaching of fractions. The presentation will include a discussion of possible reasons for the identified errors.

Current policies – educational and employment – focus on the importance of literacy and numeracy among young people. Simultaneously, new theorisations of education and society suggest that contemporary times are symbolic of radical changes in knowledge, work and leisure. Within these juxtaposing views, consideration needs to be made of what might be the new numeracy skills required by young people and employers in these new times. This paper reports on a large-scale survey designed to access young people and older people’s views of numeracy demands of contemporary workplaces. Differences and similarities in perceptions suggest support for theorising numeracy in New Times.

This paper employs an ethnographic approach to researching the implementation of a reform within a secondary school mathematics department. The study was conducted in a junior secondary/middle school context. Drawing on teacher interviews data, it is argued that there appears to be three main themes in teacher responses to reforms – the Conservatives who prefer the status quo, the Pragmatists who are concerned about practical issues related to implementation of reforms; and the Contemporaries who see the value and need for reform.

This research report is based on data from a larger study whose overall purpose is to investigate the relationship between mathematics teachers’ conceptions of their teaching practices —especially of beginning algebra— and of improving their practice. Data collected from 13 secondary school mathematics teachers show that although the teachers’ conceptions of school mathematics and their portrayed teaching sequences in Grade-8 algebra were very similar, teachers differed widely in their conceptions of their own teaching.

This paper reports findings from a research study which examined students’ strategies for deriving modelling functions from numerical patterns with rates of changes in contrast to the equation-graph matching approach prevalent in schools. Students involved were final year mathematics undergraduate students some of whom were practicing teachers of mathematics or were intending to teach. Students had already examined the cases of linear, quadratic, cubic and some exponential functions and were requested to extend their projects to quartics, other exponential functions and a trigonometric or logarithmic function. This paper presents and discusses the data from the quartic project of one of the 8 students involved in the study.

This report describes a recent case study research which provides evidence that student learning, and student achievement can be accomplished by teachers working with a greater knowledge of student development. The key elements investigated in the study include both classroom and learning features. In particular an understanding of Kohlberg’s (1963, 1973) stages of moral development is addressed. Giddens’ (1984) concepts of the reflective cycle and its ability to lead to empowered action and to the uncovering of the range of choices (for the students and teacher) to act, or not to act, to make a difference to events is included. The data collected via personal observations, students’ perceptions and voice, emphasized that an understanding of Kohlberg’s and Giddens’ work can add new dimensions to the middle years of schooling debate regarding adolescent teaching and learning. An understanding of young adolescents, especially in Year 9, requires greater knowledge of developmental and learning theories with a holistic approach to teaching and learning.

It is widely accepted that teachers’ knowledge of students’ thinking in acquiring concepts and procedures in a specific mathematical domain can be a powerful tool in informing instruction. The framework of growth points in the understanding of function developed in the present study may provide such a contribution. This paper is a progress report of the development of the framework of growth points. The basis of the framework was an initial survey of the literature, which was progressively revised, using data from students in Years 8 to 10 in Victoria and The Philippines.

The aim of this project is to link research to the improvement of mathematics teaching practice by investigating ways in which mathematics educators and teachers can foster Pacific Islands learners’ motivation to learn mathematics. An important area to investigate is the ways in which Pacific Islands learners’ motivation is shaped by their culture. A small study involving two students was conducted with the specific aim of exploring cultural influences that contributed to their motivation to learn mathematics. The factors that appeared to have the most influence on motivation were; the aspiration of students to do well so that they can help their families financially, the need to do mathematics to obtain a job, and the disparities between home and school.

A specific difficulty in memorizing basic arithmetic facts has been well established as a persisting problem for students with learning difficulties in mathematics. Theories for underlying causes range from low working memory capacity to a failure to encode numbers semantically. Understanding the quantity meaning of the teen numbers is a particular difficulty for some students. This paper will present an intervention designed for a Year 2 Queensland student with persisting teen/ty confusions and a self-acknowledged difficulty in memorising the large doubles facts. Making the tens/ones structure of the teen numbers more transparent to the student provided a foundation for him to learn his large doubles to the point of fluency.

Seventeen Year 1 and Year 2 teachers participated in a professional development program focusing on the teaching of area. The teachers were offered three different levels of consultancy support. A comparison of results from the students and teachers indicates that the success of the teacher professional development, as measured by student learning and teachers’ change in practice, was determined by teachers’ ability to work in school-based teams, and an initial desire to improve their teaching of mathematics. The success of the program as a teacher professional development activity was not dependent on the level of consultancy support provided for teachers.

In this paper data gathered from teachers who participated in a professional development program designed to improve the quality of questioning in mathematics classrooms are presented. Teachers from five primary schools participated in the program. It was designed by the teachers and funded as a Quality Teacher Project. At the beginning and end of the school year, data were gathered by questionnaire about teachers’ practice and, in particular, the types of questions that they used in their mathematics teaching. The types of questions that these primary teachers used when teaching mathematics are discussed.

The purpose of this research was to examine the possible associations between the perceived classroom environment of high school students in Southern California and the level of mathematics anxiety that they possess. Data were gathered using a revised version of Plake and Parker’s (1982) Revised Mathematics Anxiety Ratings Scale and the What is Happening In This Classroom learning environment survey created by Fraser, McRobbie, and Fisher (1996). This research involved both quantitative and qualitative data obtained via the research instruments and interviews with those having extremely high or low math anxiety.

This research is concerned with the nature of the growth of mathematical understanding, and more specifically with how a group of learners can develop a collective understanding for a mathematical concept. We seek to characterise collective mathematical understanding as a creative and emergent improvisational process, through drawing on theoretical perspectives from the fields of jazz (Becker, 2000; Berliner, 1994; 1997), theatre (Sawyer 1997; 2000) and conversation (Sawyer, 2001). In considering video data, taken from an initial pilot study, we extend improvisational theory to begin to consider collective mathematical understanding as a process with a similar nature and characteristics.

New Zealand teachers’ use of equipment has increased as a result of their participation in the Numeracy Development Projects. However, the equipment choices of the four teachers interviewed in this study were not strongly consistent with the equipment use recommended in the NDP materials. In the teachers’ reasons for equipment choices, the surface features of equipment seemed equally important as the conceptual development it can support. In contrast, the reasons given for equipment choices by the 34 Year 3 children who were interviewed were almost exclusively concerned with how the equipment might help them to solve the given problem. The children’s success rates at solving the problem declined as the equipment became more structured; this paralleled the teachers’ equipment choices. The ultimate goal for teacher educators must be for all teachers to have a richly connected conceptual map of numeracy, in order for teachers to be able to effectively use equipment in ways that help children to construct their own meaningful connections as they learn about number. Rather than talking about equipment as ‘bright, shiny stuff’, teachers must have a clear focus on the role that equipment can play in the development of children’s part-whole thinking. In this round table presentation the findings from this study, which was conducted during 2002 as part of a Masters thesis, will be discussed.

As mathematics educators, we frequently speak of the professional development needs of mathematics teachers. Many of us run professional development sessions or courses. Others of us conduct research and in our scholarly writings reflect on the implications of our findings on teacher professional development. Less often do we think about our own professional development needs. In my capacity as MERGA Vice-President (Research), I have often thought about how MERGA might assist in promoting the range of skills that mathematics education researchers might need to serve as the providers and nurturers of the next generation of researchers in our discipline, and to function as more effective and fruitful researchers whose findings are widely disseminated, highly acclaimed, and broadly implemented for the betterment of mathematics teaching and learning at all levels. I am proposing this round table session as the means to commence a discussion on what the professional needs of mathematics education researchers might be and what MERGA might do with respect to them. Some of the ideas floating around in my head include: various types of reviewing (conference papers, scholarly articles, book chapters, ARC grants), supervising higher degree students, examining theses, preparing grant applications (large/small/other), developing tenders, writing for different audiences, approaching publishers, learning about new/different research approaches/techniques, using computer software effectively for conducting research and/or analysing data, mentoring others, developing teaching/research portfolios, and promoting interviewing skills (as interviewer and/or interviewee). I’m sure there are other needs. Come and share your concerns and ideas

This round table discussion will focus on a proposed study of middle school children’s beliefs about their participation in mathematics classrooms. In the study the motivation of students when undertaking mathematics tasks, and the influence of motivation on strategies for coping with frustration when experiencing difficulties, will be investigated. It is suspected that some students may not have established perceptions of the benefits of being competent in mathematics, nor be aware that there is potential for them to be empowered by competency. One determinant of participation in education is student perceptions of goals, and the influence that perceptions play on motivation. Students who feel in control of their lives are more likely to have opportunities for success both within schools and without (Lapadat, 1998). Dweck (2000) investigated perceptions of intelligence and contended that students may hold beliefs that inhibit their participation at school; that students can be taught that both intelligence is incremental and a mastery orientation can be taught through explicit instruction. Students of one grade six class will complete an assessment in which each task is incrementally harder to complete. Once each task is completed, they will be asked to evaluate their work. If correct they will continue to the next task. If not, they will be asked how they feel, and what teaching they require in order to continue. Various background data will be gathered to seek to identify contributing factors, and a survey adapted from Dweck’s instrument will seek data on their beliefs concerning mathematical intelligence.