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Teacher education throughout the 20th century has consistently been structured across a persistent divide between subject matter and pedagogy. The prevalent conceptualization and organization of teachers' learning tends to fragment practice and leave to individual teachers the challenge of integrating subject matter knowledge and pedagogy in the contexts of their work. We assume that the integration required to teach is simple and happens in the course of experience. In fact, however, this does not happen easily, and often does not happen at all. This paper takes up three problems that we must solve if we are to meet this challenge to prepare teachers who not only know content, but can make use of it to help all students learn. The first problem concerns identifying the content knowledge that matters for teaching; the second regards understanding how such knowledge needs to beheld; and the third centres-on what it takes to learn to use such knowledge in practice.

This personal and reflective story presents some incidents from my professional life that have. (or have not) influenced me as a mathematics educator. Looking backwards I see some mistakes I made, opportunities I missed, and lessons I learnt. Looking forwards. I see areas within our discipline that seem worthy of exploration, boundaries that may need to be crossed, and initiatives that need to be taken. From these perspectives a sense of direction is emerging from and conferring with colleagues will help me see this more clearly.

This paper reports on research into patterns of social interaction associated with metacognitive activity in senior secondary school mathematics classrooms. Unsuccessful collaborative problem solving sessions were characterised by students' poor metacognitive decisions exacerbated by lack of critical engagement with each other's thinking, while successful outcomes were favoured if students challenged and discarded unhelpful ideas and actively endorsed useful strategies. The findings have practical significance for implementation of new mathematics curriculum policies that emphasise problem solving, mathematical reasoning, and communication.

Concept mapping exercises were used to determine whether changes to teacher knowledge had occurred as a result of their involvement in an early numeracy professional development program. Findings indicate that the most significant change to teacher knowledge occurred in relation to how children learn mathematics. While some change was evident in teachers' pedagogical knowledge, there was little change to teachers' knowledge of mathematics content.

This paper is an inquiry into the development of mathematical self-concept or how students perceive themselves in relationship to mathematics. The inquiry is centred on two classroom episodes which involve changes in the nature and direction of conversation between students and the teacher. We suggest that if the episodes were typical of patterns of interaction in a class, they might significantly impact on the students' perception of their own competence in mathematics.

In keeping with movements in society generally there has been, over the past several decades, an increasing interest in the social aspects of learning mathematics in school and using mathematics beyond school to meet the demands of life, work and leisure. In this paper I attempt to draw connections between the two; how what is learned in school, and how it is learned, might affect its competent and critical use after school. I argue that classroom practices of socialisation, as in social constructivist, Vygotskian and socio-cultural psychological approaches to teaching and learning mathematics do not tell the whole story; stories of subjectification must also be· told to help us understand how it happens that some school leavers are able to use mathematics to libratory and powerful ends and others in limited ways that deprive them of choice.

Three interlocking papers have been written for the symposium 'Researching a Systematic Early Numeracy Initiative'. The initiative referred to is the mathematics project 'Count Me In Too' (CMIT), developed for the early years of New South Wales government schools. The three papers 100k at the effect this project has had on teachers and students, and how the project is situated within the body of research on the early years of schooling. This p~per is written in response to an invitation to discuss these three papers with a view to forming some questions that may be useful to ask at the symposium.

The 'research-practice gap' has its roots in the concept of Technical Rationality - the 'Positivist epistemology of practice' (Schon, 1987). This is interpreted by some in mathematics education as the hierarchical division of labour, wherein it is the task of the tertiary-based mathematics educator to create the fundamental knowledge and theories,and the task of the mathematics teacher to apply that knowledge to practice. This paper considers the issues that arise from this demarcation, highlights the problems it creates, and suggests ways to alleviate these problems.

Previous research has shown that when students expand brackets, errors tend to cluster around key positions due to increased working memory load. This study found that by neutralising the effects of position, the occurrence of error clusters was reduced. Furthermore, a self-rating mental effort instrument was employed which found a positive correlation between errors and mental effort. This instrument also detected subtle variations in mental effort between groups of varying mathematical ability.

A case study methodology was utilised to document the processes involved in the development and marking of extended response tasks within a system-wide numeracy· assessment. One of the tasks included in the final test is used to exemplify how numeracy and measurement requirements can be satisfied within this process. This paper is a preliminary report of a research project being undertaken as part of the development of the pilot of the NSW Department of Education and Training Secondary Numeracy Assessment Program.

This paper analyses the major concepts and processes within decimal-number numeration in terms of the cognitions inherent in their structure to determine how they are related to each other, how they are different and, if possible, to determine which are more difficult conceptually. It synthesises this analysis into a cognitive model that provides a framework for decimal-number knowledge.

Mathematics curricula commonly promote the development of problem solving, reasoning and communication skills. Students need to be able to communicate their mathematical ideas; written communication is therefore an important part· of the assessment· process. in mathematics. When writing explanations students. need to consider and choose an appropriate mode of representation for their response. This paper examines students' modes of representation used in solutions for two problem solving tasks arid considers the issue of judging mathematical communication.

This paper presents the findings of a study that examined the results of 120 students selected according to their performance levels in mathematics in Year 1 to Year 5. The results show that while most high-ability students had a conceptual understanding of length, the majority of the lower-ability students did not acquire important concepts relating to the linear nature of units and took longer to acquire basic measuring skills. This knowledge is essential for its application in perimeter, area and volume, and in topics that rely on the understanding of scales, such as directed number gauges, and graphs.

The introduction of portfolios into the mathematics assessment programme of a New Zealand secondary school. was closely monitored. The responses of teachers, students and parents are recounted in this paper. Inconsistencies in teachers' views and behaviour highlight the professionalism necessary for education reform.

The probability concepts of a group of preservice primary teachers were investigated and classified in relation to known misconceptions as well as with a framework that identifies levels of understanding. These results were compared with the misconceptions exhibited by a group of 11 and 12 year old students. The major question investigated was whether these prospective teachers showed more sophisticated probability concepts than 11 and 12 year old students.

One approach to assessing mnneracy is through performance assessment that requires students to create a response. Four specially designed performance assessment tasks, addressing different aspects of mathematics, were used to assess students' numeracy achievement. These tasks were considered for their educational measurement characteristics and the underlying variable interpreted from a numeracy perspective. An overall developmental sequence of numeracy was found, with associated 'productive elements'. From these, a framework for numeracy assessment is proposed and implications for assessment and classrooms are examined.

A quantitative investigation of 18 Western Australian secondary school mathematics classrooms was conducted prior to implementation of the Curriculum Framework. The Classroom Cultural Elements Questionnaire was administered to profile existing learning environments. Results revealed learning environments with varying levels of elements conducive to improved learning outcomes. This finding suggests Curriculum Framework implementation in mathematics requires attention to these elements of the classroom environment and development of a classroom culture focused on the. educative needs of students rather than traditional subject content.

We report on how students deal with some technical aspects of the operation of a graphics calculator. Clinical interviews were conducted with 25 Year 10-11 students as they used graphics calculators to study graphs of straight lines and parabolas. Three common student difficulties were identified: a tendency to be unduly influenced by the jagged appearance of graphs; a poor understanding of the zoom operation of the graphics calculator; and a limited grasp of the processes used by the calculator to display graphs. Implications for teaching are discussed.

Young adults were presented with raw data involving a number of variables and asked to produce a report that highlighted any "interesting aspects" that they observed. The approaches taken varied from simple frequency information through to scatter plots and the use of ratios to consider the relationships between three variables. Although the use of sophisticated statistical techniques was not expected, the limited use of even basic techniques was surprising, given the increased emphasis on Chance and Data in school curricula.

An important feature of students' understanding of algebra and trigonometry is the relations and representations that are constructed between these two areas of school mathematics. In this study we report one students' understanding of the concepts before and after two lessons. The student developed knowledge about the. topics that appears to be consistent with what the teacher was aiming at. However, in a problem-solving situation, the student activated knowledge that was different and incorrect. The results provide tentative support for the argument that students sometimes develop 'garbled knowledge'.

In recent years calculators and computers have been used more often in mathematics teaching. While their potential advantages in terms of dynamic multiple representations has been recognised for some time, it has not. always been clear how to integrate the technology into conceptually· based teaching. While it is generally agreed that technology could play an influential role in enhancing mathematical understanding, there is little data about how students' cognition and knowledge building are affected by the instructional use of such tools. In particular, a theoretical perspective on how to teach in such a way as to encourage students to build inter-representational conceptual structures has,been lacking. In this study we take up this issue by examining how two experienced teachers plan and incorporate calculator and computers in their teaching. The results suggest that these teachers aim to build knowledge schemata that are rich in connections and representations.

This paper describes the development of a framework of key aspects of early numeracy learning. The framework was developed as part of a project seeking to identify processes for supporting and enhancing numeracy learning in the early years of school. The framework built on work of others, through considering key research in numeracy learning and others' attempts in developing such frameworks. Data collected from interviews with over 5000 children led to modifications of the framework.

Values are taught in every lesson. However in mathematics classes this seems to be implicit rather than explicit. This paper outlines methodological difficulties encountered in researching the values teachers teach. One crucial area that has emerged is finding a common language with which meaningful dialogue can occur. We also reflect on differences, due to the different cultures, that have emerged between this Australian project, and a parallel project in Taiwan, including the influence of personal value systems.

This paper presents the results of a qualitative study involving the internet-based software LOGAL Tangible Math. The study was carried out in a secondary school classroom over a period of ten weeks and focused on identifying factors in the software design involved in promoting higher level thinking. Specifically, the roles of cognitive tools, question types, format of student answers, manipulable objects and navigational issues were examined. The authors conclude that higher-level thinking is encouraged by the provision of a rich set of tools, and the presence of manipulable objects that allow students to generate exemplars of a new concept.

The results of a survey of 120 first year mathematics students confirm earlier findings that student approaches to learning mathematics fire related to their conceptions of the nature of mathematics. In addition, a positive initial experience with the computer algebra system Mathematica is associated with a deep approach to study, a cohesive view of mathematics, and a more extensive background in computing. Features of students' initial experiences with Mathematica are described, based on qualitative and quantitative data from the survey.

This paper describes an investigation into the factors which influence students in their choice of tertiary fields of study with major focus on mathematics and computing. The primary research questions were: what are the factors behind their choice, how are they ranked, and are there gender and regional differences underpinning these choices? 541 first year students were surveyed at four Queensland universities in 1998. Notable findings are reported, and the implications for promoting these disciplines are discussed.

Coping with and planning for developments in school mathematics requires informed debate based on factual information. A knowledge and understanding of school mathematics enrolment trends can facilitate planning and identify issues that need to be addressed. The trends in upper school mathematics up to 1990 have been well documented by the authors (Dekkers, de Laeter, & Malone, 1991). This paper presents highlights from that earlier research and reports on year 12 student numbers and mathematics enrolment trends for the period 1980 to 1998, based on the most recent statistics from State authorities. Implications of the trends are also discussed.

Students experience a range of difficulties in generating effective diagrams. Hence, it is important to explore these difficulties so they can be addressed during instruction on diagram generation. A cross-study comparison of the results oftwo network tasks revealed that students experience similar difficulties on feature-similar but non-isomorphic tasks. Students' difficulties on these tasks appeared to be due to a lack of sense-making. in mathematics rather than a difficulty with the problem structure or the generation of a particular type of diagram.

This research paper describes a longitudinal study which tracked a cohOlt of seven student teachers through the secondary mathematics method component of a one year, full-time, preservice university-based teacher education course and then as beginning teachers into their first year of teaching in the Western Cape region of South Africa. Qualitative data in the form of interviews, students' reflective journals and other students' texts as well as lecture and classroom observation were collected over a two-year period .. The study found that in constituting their teaching repertoires, beginning secondary mathematics teachers recontextualized certain discrete tasks introduced on the mathematics method course, as well as a professional argot, a way of discussing the teaching and learning of mathematics. There appeared to be a disjuncture between what student teachers were taught on the preservice course, and the way in which they practiced in mathematics classrooms. There also appeared to be a disjuncture between what teachers said about their classroom practice, and the way in which they actually taught. An explanation for this seeming hiatus emerges from a sociological model developed from the work of Basil Bernstein and Paul Dowling and turns centrally on the extent to which student teachers are given access to the r~cognition and realization rules of "best" practice.

This paper reports a cross-sectional pilot study of 30 students from Grades 1, 3 and 5, that investigated students' visualisations and representations of three-dimensional shapes. The study describes how students focused on critical and non-critical aspects of three-dimensional shapes and whether any differences exist between students' visual images, verbal descriptions and drawn representations. An interview-based assessment showed that although the quality of some students' visualisations may improve with grade level, non-mathematical qualities feature strongly in most students' visualisations.

Tertiary mathematics students reported on the encouragement they perceived that parents and teachers gave them to continue in their mathematics studies. Women had perceived stronger encouragement than men, and correlations between early encouragement and university achievement were stronger for women. Interviews confmned the importance of encouragement for the women, and indicated that their confidence could be shaken by stereotyped views about gender. Men's interviews indicated that, for them, the greatest social danger to their studies was the anti-intellectual peer group.

A previous paper has shown that in the Third International Mathematics and Science Study (TIMSS) teacher effects contribute little to student achievement in mathematics, but that classroom differences are important. The present study used data from TIMSS to examine classroom and school differences in maths achievement at secondary school level to find out which classroom and school variables affect achievement. It found that the pooling of student resources associated with grouping has a large effect on mathematics achievement at primary level, while at secondary level policies regarding pupil management such as grouping according to ability have the greatest effect on mathematics achievement.

We report on research into outcomes that emerge when MAPLE software forms a central component of both the lecture presentation and laboratory activity in a first-year undergraduate mathematics course. We have noted an increase in both the number and type of questions asked by students compared with conventionally taught classes, and have identified factors associated with a symbolic manipulator environment that appear to link with task demand and student success. While enabling capable students to move faster and further, our evidence suggests that the use of such software does not compensate for, replace, nor render irrelevant, flaws in mathematical background.

It is important for teachers to identify children who are at risk of not learning mathematics successfully as early as possible in their schooling. This enables school communities to put in place strategies to assist these children before they lose confidence. This paper explores assessment interview data for 1667 Victorian Year 1 primary school students with a view to developing profiles that may be used to identify children who are at risk. These profiles are based on growth points within the Early Numeracy Research Project assessment framework.

With the introduction of graphics calculators to tertiary entrance examinations in mathematics in Western Australia, a question that arose is whether rural students were placed at a disadvantage when compared with those in the city. A preliminary analysis of examination data suggests that rather than disadvantage, there was some evidence of an improvement in the achievement of rural students when compared with those in the city.

Maldivian teacher education students' understanding of functions and rates of change was investigated. The written responses of 22 students to 10 examination questions were first analysed, and then 5 students were interviewed in depth as they solved 4 of the most difficult questions. It was found that students had a good grasp of basic concepts,but tended to rely on . remembered formulae or procedures. They also had difficulty visualising tangents and linking them to rates of change and derivatives. Implications for teaching are explored.

This study investigated whether minimal intervention with new teaching resources could make a difference to children's understanding of decimal numeration. Four teachers were supplied with resources designed specifically to address the common misconceptions children have about decimal numbers, and asked to keep a record of their teaching. Use of the resources was. unexpectedly low, but teachers who used them achieved an encouraging improvement in decimal understanding, measured against previous performance of the school over some years, indicating that a small amount of deliberate attention to decimal concepts can make a difference.

This paper describes the development of a web-based resource designed to support preservice mathematics teachers on school practice. The development of the site resulted in part from a needs analysis conducted with a focus group of students and reported on. below. As a result of focus group discussions with students from the target group, the website was developed and refined to provide a comprehensive resource to suit their needs while on professional practice. Elements of the site, together with potential further developments and research, are described.

This study set out to investigate whether assessment strategies for mathematics classrooms transfer to real world practice when learnt within an authentic context. A multimedia program was designed to incorporate characteristics of , situated' learning environments and cut to CDROM. Preservice teachers used the program in their mathematics method classes, and they and their supervising teachers were interviewed regarding their use of strategies while on professional practice. Results show that all students used a variety of assessment strategies, and according to the beliefs of the students themselves; were influenced in their use of strategies by the multimedia learning environment.

Research carried out amongst adolescent students with learning difficulties is consequential to the delay/deficit interpretation of performance: Is performance immature and development simply delayed or does a processing deficit underlie performance difficulties so that development is impeded? This study investigated what strategies three students in Year 10 applied to simple addition problems. Performance was analysed on a trial-by-trial basis and strategy variability within problems was examined. After participating in ten years of formal schooling, these students still did not know the basic addition facts (including tie facts, and facts to problems with an addend of two and ten). Their performance based on strategy use was not consistent with a processing deficit interpretation but neither could it be considered simply delayed. Understanding the extent of the difficulty experienced by some students in acquiring basic number facts is imperative to developing appropriate interventions.

Teachers' beliefs about a subject and its learning and teaching are known to affect students' learning. A great deal is known about the beliefs of primary mathematics teachers in Australia and other countries but little is known about how these beliefs compare across countries or whether it is even useful to make such comparisons. This paper reports on a international pilot study in which the beliefs of Australian and Singaporean primary student teachers towards mathematics, mathematics learning and mathematics teaching are measured and compared using a validated instrument. The findings are linked to the TIMSS results in these countries and some tentative conclusions drawn which will inform future research.

This study arose from a request from teachers for help in evaluating their current methods of teaching decimals. Results reported here are from 14 classes of students, aged 9 through 12, and demonstrate that it is possible to find marked differences in progress of different classes. Students were given a common pre-test and post-test. These results were related to the models and reported teaching procedures used. The main factors leading to students' improvement appeared to be careful planning to meet their needs based on teachers~~ knowledge of the underlying concepts, the use of a clear model that students could use to visualise decimal division, and careful bridging from visualisation to numerical forms.

To be considered numerate beyond 2 000 preservice teachers need to know more than som.ebasic mathematical concepts and skills; they need to know and value the mathematics and know themselves tbbe competent andagentic in its articulation. Teacher education programs attempt to realise these ideals :but are hampered by what I consider to be untenable. assumptions regarding agency for. preservice teachers; that rational reflection leads to progressive change, and that knowledge is power. These assumptions are based on humanist versions of a person where agency is taken to be a feature of any adult, human being. In this paper I argue rather for a poststructuralist understanding of preservice teachers as discursively constituted through power relations, both in schools and in tertiary education, where agency, or the lack of it, lies in the constitutive power of discourse. I explore what this might mean for mathematics teacher education beyond 2000.

Recent work on classroom and school effects in Australia has suggested that teacher effects account for up to 50 per cent of variation in mathematics achievement. The present study used data from the Third International Mathematics and Science Study (TIMSS) to look at classroom and school differences in mathematics achievement at both primary and secondary school levels. It found that while, classroom differences account for about one-quarter of the variation in student achievement, little of this was due to teachers. Most of the classroom variation was due to compositional and organisational factors. This has important implications for policy regarding the improvement of mathematics achievement in schools.

The authors formulated three research questions in association with the development of a test in 3-dimensional geometry in response to Mariotti's study on the complexity in mental imaging using nets of solids. This paper reports on the coding of responses to a question on nets, using two frameworks, the SOLO Taxonomy and the van Hiele levels of understanding. The findings show that students' descriptions of nets can he coded in both frameworks, thus supporting the three research questions.

In this paper, we present findings from a new instrument, "Who and Mathematics", devised to measure the extent to which mathematics is now stereotyped as agendered domain. A large sample of Victorian grade 7-10 students participated in the study. Gender trends elicited from· the responses to each of the 30 items comprising the instrument are examined and gender differences among the· respondents are reported. Many of the results were inconsistent with previous findings in the field. Students' gendered beliefs about mathematics and the relevance of mathematics as an area of study appear to be changing.

Considerable emphasis is placed on teachers' informal assessment of students, yet little has been written about the assessment that teachers actually use in their classrooms and its effect. This paper presents a framework for analysing teachers' use of informal assessment questions. A sample of the teaching of two secondary school teachers is presented to demonstrate the usefulness of this framework. The' analysis of these two lessons demonstrates the possibility of discriminating lessons based on teaching procedures from those emphasising more complex conceptual relationships. The potential value of the framework is discussed.

With support from undergraduate university students, Grade 1 children (aged 6-7 years) were challenged to pose and then solve their own open-ended tasks. The way in which the children engaged in problem solving prior to formulating or posing a problem was explored. An overview of the types of problems posed by the class is presented, along with two detailed case studies. The case studies show that the children were able to identify and discuss the type of mathematics content and strategies they would need to employ in order to successfully complete the task.

This paper examines the type of specific understandings and general strategies students of different ages (Grade 3 and Grades 516) employ to solve an open-ended mathematics problem. The task required the students to construct a map that would allow a friend to get from a ,specific destination to their house. Although the Grade 6 students had a more sophisticated understanding of scale and proportion than the Grade 3 students there was not a great deal of difference between their work samples.

Number sense is generally regarded as an important aspect of general mathematiCs knowledge, while mental computation performance is frequently thought to link to number sense ability. With current national numeracy testing, placing an emphasis on number sense, this study explores the relationship between number sense, mental computation and general mathematics performance of primary students. Results indicate that performance in these three areas was often disparate, and raise some interesting questions about number sense and mental computation and assessment of mathematics curriculum.

This paper reports trials of two computer games designed to enhance learning about decimals. In two exploratory studies, students used especially designed computer games that focussed on aspects of decimal understanding. These games were effective in challenging children's misconceptions about decimals. Students developed strategies for dealing with decimals, assisted by teachers. This article considers the nature of teaching assistance that could be programmed into the games to further strengthen their usefulness as a tool for increasing understanding about decimal numbers.

This paper examines Br'uner's notion of scaffolding in one-to-one teaching of Maths Recovery. The interactions of two teachers working with students were videotaped and then transcribed. The sessions are described and discussed. The results of this study showed that scaffolding is an important characteristic deemed to be suitable in one-to-one teaching of Maths Recovery. However, more study of one-to-one teaching. is needed to further understand the types of characteristics that would enhance the teaching and learning process in one-to-one teaching.

This paper .reports on a study to measure the. effectiveness of an integrated learning system in improving mathematics achievement for low achieving Ye~rs 5 to 9 students. The study found that statistically significant gains on the ILS were not supported by scores on standardised mathematics-achievement tests. It also found that although student attitudes to computers decreased (significantly for some items), the students stilI liked the ILS and felt that it had helped them to learn.

Mathematics is abstract in the sense that mathematical concepts are formed by a process of abstraction-the recognition and reification of a similarity between a set of previously unrelated objects, events, or ideas. It follows that effective mathematics teaching should promote the abstraction process. Traditional.m:ethods seldom do so. Constructivist methods theoretically encourage abstraction, but this is rarely emphasised and the result may be an inefficient teaching procedure. Three examples are given of how explicit attention to the abstraction process could let'd to radically different approaches to· teaching; Some principles of what we call Teaching for Abstraction are abstracted from these examples.

Students' graphical representations of statistical associations are explored through analyses of responses from 97 students in grades 4, 5, and 6. The task involved representing (a) the association "People grow taller as they get older," (b) this association with an age modifier, and (c) this association with sex differences. Three levels of representing bivariate associations and of representing multivariate associations are described: unsuccessful, partial, and complete. Issues for developing student representations are discussed.

A survey item based on a newspaper article about coin tosses was administered to 1256 students from grades 6 to 11. Few students determined the probability of four successive tails. Most students considered that heads and tails were equally likely for a subsequent fifth toss, often describing the probability as "50-50". Students in higher grades were more likely to respond appropriately. Results are discussed with reference to equiprobability, independence, the gamblers' fallacy, and the outcome approach to probability.

Practicum sessions allow student teachers to learn about the profession through participation in it. This paper uses Lave's notion of situated cognition, and argues that academics need to be aware both of changes that are taking place in school contexts and of how such changes are shaping student teachers' lived experience. The paper reports results from two surveys of students' observations of primary mathematics education-the first implemented in 1990 and the second a decade later. Some changes are noted, and it is proposed that these are worthy not only of discussion in tertiary classrooms but also of further research.

In this paper we present a comparative analysis of the 1996-1999 Western Australian Calculus Tertiary Entrance Examination papers. A classification scheme from the literature for question characteristics proved to be a useful tool of analysis. We show that the introduction of graphics calculators led to an enhanced role for diagrams and changes in the structure of questions on graphing, with implications for teaching practice.

Teachers of Mathematics are encouraged to use a variety of methods and activities in their classroom practice but situational constraints lead to their preference for routines that are conservative and involve little risk. Regardless, some teachers do try new ideas with their students and improving the efficacy of these attempts may encourage further activity exploration. This paper presents a survey and graphing template that was developed by the first author while employed in Brunei Darussalam. They help teachers obtain indications of students' perceptions following any activity. Five characteristics are considered: Understanding Content, Communication, Relevance, Work Output, and Enjoyment. Practicality is central to success. Students require about ten minutes to complete the survey in the first instance and responses from 25 students take about twelve minutes to process.

This study uses qualitative research methods to examine why a senior secondary teacher who was experienced in teaching mathematics and using computers would not use either generic software such as spreadsheets, or exploratory software (e.g. Maths Helper) in his teaching of mathematics. The teachers' beliefs about the nature of secondary mathematics, the nature of teaching, his own teaching practice and assessment issues underpinned his rejection of the use of computers in the classroom.

This paper describes a framework for teaching length, area and volume in the early years of school. The framework is based on recent research findings that indicate the importance of students' knowledge of the unit iteration structure. The results of interviews with a sample of 16 Sydney teachers about their understanding of area measurement indicate that they place little importance on structuring in their teaching of measurement. Lessons from the framework were given in four schools and work samples based on these are discussed.

This paper reports on an investigation, commissioned by the Disadvantaged Schools Program (DSP) of the NSW Department of Education and Training, to identify the best ways to provide support for mathematics teachers in DSP secondary schools, so that they can assist their students to meet the literacy demands of mathematics. Case studies in four markedly different secondary schools were carried out by the authors. Through these, the role of literacy in mathematics learning, teaching and assessment, as perceived by the mathematics teachers and students, has been considered.

This paper presents a model for statistical thinking in empirical enquiry, based on practitioner behaviour. The conjectured types of thinking that are fundamental to the statistics discipline are then used to analyse an assessment item and teaching situation. It is concluded that such an analysis is possible and that more discussion and research is needed on defining, clarifying, and articulating the statistical thinking that is embedded and inherent in statistical practice.

This is a report on part of a larger study about teacher assessment practice in New Zealand. The interviews with teachers of Year 3 and 6 students are analysed. These interviews indicate that teachers are using a variety of assessment strategies for a mastery-based system of assessment. The judgement of students' mathematical performance appears to be dominated by the belief that all students must feel that they are achieving. It is conjectured that a system effect determines teachers' assessment in the school environment.

A framework for teaching and assessing statistical thinking comprising four constructs and four levels for each construct, has been developed and the framework validated using data from 20 US students in Years 1 - 5. The same validation· procedures· were implemented in two different cohorts, totalling 40 subjects, of Australian students in Years 1 - 5. Lower levels of coherence were found. This paper reports the Australian data, seeks to address reasons for the differences and compares the levels of performance between the Australian and US students.

This study describes the effect mathematics newsletters had on the home learning environment of two six-year~old boys with low achievement in mathematics. One boy's mother maintained a high level of contact with the school, the other a low level. The mothers of the low achievers appeared to use the newsletter as a means of assessing progress rather than a stimulus for mathematics experiences in the home. These results are contrasted with an earlier study undertaken with high-contact parents.

Research was conducted in the Northern Province Primary Mathematics Projects schools. In view of the high failure rate among matriculation students, the researcher believed that it was necessary that ways to improve mathematics understanding should be devised from the foundation level of schooling. He further believed that children learn better in a cooperative non-threatening classroom environment where they are free to participate. Research was conducted to evaluate the use of constructivist and investigative teaching and learning methods as employed by teachers in a number of primary schools in an area in one province in South Africa, the poorly performed Northern Province. A pilot study was conducted with two experimental schools and two control schools. The experimental schools were operating under the PMP (Primary Mathematics Project) and were supported by expert key teachers who attended courses at Leeds University in England while others were trained locally in South Africa. The methods employed to conduct the research were both qualitative and quantitative, embracing questionnaires done by 174 teachers, interviews with 55 people, written tests by 350 pupils and observations in classes. The results revealed that student performance in the experimental schools where a constructivist and investigative approach was used was better than the control schools. They indicate that children from PMP schools developed a better understanding of mathematics.

Numeracy has become a key priority of government policy in recent years generating a number of numeracy-related projects. This paper will provide an overview of the Middle Years Numeracy Research Project commissioned by the three employer groups in Victoria in 1999 and report-on the design and development of a Student Numeracy Profile incorporating rich assessment tasks and involving teacher judgements in the assessment of student numeracy performance. Preliminary analysis of the Student Numeracy Profile instrument suggests that it is an effective measure of numeracy at this level with the potential to provide useful insights into instructional strategies for middle years' students.

This paper is the result of a school/university partnership that was formed to enhance teachers' professional development. The partnership explored pedagogical practices that promoted students' mathematical thinking and sense making. The links between assessment and instruction are described using 'accounts of practice' that are grounded in classroom contexts and provide guiding principles for mathematics educators seeking to make assessment more student centred and purposeful. The findings contribute to a growing body of empirically based knowledge related to the nexus between assessment and instruction.

This paper focuses on a component of a model to encourage and support teacher reflection in a professional development programme attempting to change the teaching and learning of mathematics. Our goal is to educate teachers to become analytical, self-reflective and adaptive agents in the classroom. The purpose of this paper is to report on the initial attempts by participant teachers to construct mathematical portfolios to promote reflectivity.

This study investigated changes in adult Aboriginal students' attitudes during a year-long mathematics unit taught by the researcher. Students completed an attitudinal survey before and after the unit and also kept regular diaries. Data was also collected from a focus group who were observed, interviewed and visited at home on a number of occasions. Although attitudes fluctuated during the unit, the general trend was positive. Factors identified by the students as contributing to this change included the separation of students into ability groups, negotiation of content and learning style, relevance of content, and community visits by the teacher.

This paper describes a new research project, which aims to investigate how computer algebra systems can be introduced into the Victorian Certificate of Education mathematics subjects. Lessons drawn from the experiences of the introduction of scientific and graphics calculators are reported along with current moves in some other countries. Options for the role to be played by technology in assessment are canvassed. It is proposed that technology should be permitted in all components of assessment but that explicit attention will need to be given to including questions that test algebraic insight. A plan for minimising inequity is proposed.

This paper represents some of the results of a study carried out to explore the computational choices made by 75 students in Years 5-7. Data were collected to determine the initial and subsequent computational choices made by students. A record was made of the success rate for various computational choices and students were interviewed to determine the reasons for making particular computational choices. Reasons given by students were placed in four categories of number magnitude, efficiency, knowledge of multiplication facts, and teacher influence.

Secondary and tertiary mathematics teachers often bemoan the fact that many students cannot solve complex questions due to misconceptions involving simple concepts. The authors have earlier reported that the incidence of mathematical misconceptions in both bright and weaker first-year tertiary students was reduced by using a technique based on Piaget's notion of cognitive conflict, and that most of this gain persisted. This study sought to determine whether the same technique diminished the frequency of misconceptions in longer, more complex questions.

This paper examines the extent to which MERGA's 1999 Conference refereeing procedure was fair to the authors. The short answer to this is "yes". But while conceding that the realities of preparing a conference mean that fairness must be balanced against several constraints, the paper shows how some weaknesses may be ameliorated by detailed planning and a little more editorial authority.

This paper follows from a belief that the activities of an educational organisation should model good educational practice. MERGA's refereeing practice is examined as a way of improving the quality of teaching and learning about mathematics education research for both authors and referees. While many positive features are already present, some weaknesses or omissions are identified which, it is suggested, are more likely to be addressed if MERGA's educational role is seen as an important focus of its work.

This paper summarises the final report on a study first described to MERGA conference delegates in 1998. Following an extensive review of research on integration spanning the past 30 years, a number of middle school settings of integrated teaching and learning from across a range of demographic and socio-economic areas were studied over three years in order to provide data upon which a number of assertions related to curriculum integration were formulated. Using a number of approaches to data collection, the facts about curriculum integration were documented and six assertions were complied concerning what we believe we know about the process. Another five assertions about what we believe we still need to understand about this process were also compiled. Arguments for and against integration are discussed from the viewpoints of the students, teachers and administrators.

This paper presents part of the findings of a study into the effect of teacher intervention, designed to improve metacognitive skills, on learning transfer between MicroWorlds programming and mathematical problem solving. The study compared two classes of Year 6 students both extensively involved with programming. Students who were exposed to the teacher intervention activities in addition to programming outperformed the group who did not receive the intervention activities on a test of mathematical problem solving.

This paper is a report on the beginning of a much larger study that examines students' understanding and knowledge about the structures of arithmetic and generalising problems. Ninety-four students in their fmal years of primary school responded to a series of tasks designed to probe their knowledge of associativity and commutativity. The responses indicated that these students had failed to abstract from the experiences in arithmetic some of the mathematical structures believed necessary for the successful transition from arithmetic to algebra.

This paper is a report on the development of an instrument to identify and measure factors associated with primary teachers' purported beliefs about mathematics and teaching and assessing mathematics. Current literature relating to teachers' beliefs provided a theoretical framework for the development of the questionnaire. The sample consisted of 387 primary teachers. The results of a factor analysis identified seven factors that appear to represent teachers' purported beliefs - three relating to beliefs about assessment, two relating to beliefs about mathematics, and two relating to beliefs about teaching mathematics.

This paper considers 33 preservice secondary mathematics teachers' solutions to a famous sampling problem, well known for confounding educated adults. Of particular interest is the Bachelor of Teaching students' use of intuition and/or formal mathematics in reaching a conclusion. The relationships of solution strategy to students' background in formal mathematics and to gender are also considered. Implications for teaching statistics at both the secondary and pre-service teacher education levels are discussed briefly.

Research studies have reported upon the difficulties teachers' display in their articulation of classroom practice and teaching values. Other studies have reported inconsistencies between a teacher's espoused theory and their theory of action. Argyris and Schon (1974) claimed that behaviour was driven by individual action theories. This study used a modification of the Theory of Planned Behaviour (Ajzen, 1988) to uncover the teacher action theories regarding the use of stencils in primary mathematics classrooms from a selected sample of NSW upper primary teachers (N = 115). It will discuss these theories and their use in assisting teachers in their critical reflection and articulation of current practice.

Two collaborative groups of senior secondary mathematics students with similar ability to solve unfamiliar challenging problems demonstrated different· levels of engagement and different levels of conceptual development when they worked with the same task. Study of the nature of task complexity led to the formulation of discovered complexity as a useful tool to analyse student response to tasks. It was found the task undertaken provided the opportunity for discovered complexities that possessed the potential to enhance student learning. Use of such a task was found insufficient to ensure discovered complexity resulted.

Research indicates that appropriate mathematics teaching is connected to an understanding of and art interactive relationship with the cultural environment in which it takes place. The cultural environment of Australia's mathematics classrooms has changed dramatically in the last fifty years due largely to an influx of migrant students of diverse origins. This paper investigates the changing patterns of migration over this period and their links with mathematics classroom environments through documentation of the perceptions of a group of Victorian mathematics teachers. Also investigated were perceptions of any responses through the provision of migrant related resources by mathematics educators and of their own roles in generating responses to the migrant presence in their classrooms.

This study looked at Year 3 children's understanding of the number system at four schools with varying socio-economic status. Just over one third of the children were Maori (the indigenous people of New Zealand). In total, 221 children were interviewed using a task-based interview. Children's understanding of the number system varied as a function of ethnicity and socio-economic status. Performance on several key tasks was used to assign children to a level on a developmental framework for numeracy.

Primary preservice teachers usually do not have strong knowledge nor positive attitudes towards the study of mathematics. In a context where educators need to explore effective and cost-effective modes of delivery, forms of pedagogy that support learning and affect are needed. This project investigated the use of study groups within a primary preservice program in supporting students learn aspects of mathematics. Guidelines and future areas of concern are provided.

A systemic initiative in early years mathematics-Count Me In Too (CMIT), is briefly described. This is followed by a review of links between the research base of EMIT and other research in early number. The review is organised into five areas: assessing students' knowledge and its progression; counting strategies; abstract composite unit; development of tens and ones knowledge; and knowledge of numerals. The review serves to locate the theoretical bases of CM IT more broadly in early number research.