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Proceedings of the 19th annual conference of the Mathematics Education Research Group of Australasia (MERGA) June 30-July 3, 1996 at The University of Melbourne

Powerful use of technology has the potential to radically alter the nature of classroom mathematics. Fundamental issues related to the use of technology can transcend the boundaries of the type of technology used and the level of schooling. Examples from the Calculators in Primary Mathematics project will be linked here with parallel examples and discussion from the research literature in order to explore some of the effects of technology on the nature of mathematical activity, classroom practice and the curriculum.

Over the past decade we have been learning how to research learning - in classroom settings and elsewhere. Research into learning in classroom settings is widely practised. Increasingly, research is addressing teaching and learning events in the workplace. Historically, the research methods employed to investigate learning have been refined and replaced as our image of learning and, consequently, the goal of our research has changed. The availability of new technologies offers researchers the capability to compile rich data sets with a level of complexity not previously possible. The analysis of such complex data sets poses a significant challenge. While new technological tools also offer the possibility of new forms of analysis (particularly of qualitative data), it is the sophistication of our theories and analytic frameworks that will determine whether we are able to utilise the potential of the new techniques. Central to the realization of this sophistication is the operationalization of the constructs with which our theories are expressed. This paper explores contemporary approaches to researching learning and some of the questions associated with this type of research.

We examine the educational possibilities afforded by new connections between physical and simulation-based data to build intimacy with function representations. Historically, technology was first used to facilitate actions within notations, then to link them, eventually bidirectionally. Yet recent data strongly indicate that students' difficulties with interpreting and productively using mathematical notations continue. We suggest that students need phenomena expressive linked to the notations, and secondly, that students themselves should be immersed in generating such phenomena.

This paper reports the results of a preliminary investigation into primary school teachers' beliefs about the role of problem solving in learning mathematics. A survey was used to gather teachers' favourite problems and comments about traditional versus constuctivist perspectives on using problems in classrooms. At least half of those surveyed chose problems which they believed promoted learning but most teachers adopted a position that saw problem solving as an end rather than as a means in the learning process.

The emergence of learning strategies as a critical variable in the active learning process is reflected in the promotion of a range of cognitive and metacognitive learning strategies in recent mathematics curriculum documents. This paper discusses the influence of instructional factors on senior mathematics students' development and use of learning strategies. While acknowledging that there are many causes for students' failure to use appropriate learning strategies reported findings suggest that compensatory instructional practices may limit the development and effective use of appropriate learning strategies.

Research into individual thinking about algebra faces a significant problem in the tacit nature of knowledge in this domain. This paper documents a research design which incorporated Repertory Grid principles within an image-based research model. Finely detailed study of perceptions of algebraic images offered a powerful complement to the more usual verbal approaches, providing insights into both student thinking about key concepts in algebra, and into the network of relations within which such concepts exist for individuals learning algebra .

This paper argues for a future direction for studying the learning and teaching of algebra based on collaborative action research between teachers and university researchers. It considers the results of two different types of research· undertaken from social and cognitive perspective of the algebra classroom. This research has shown that, regardless of learning, algebra tends to act as a stratification agent for the social reproduction of economic differences, necessitating alternative approaches to existing research.

The Victorian "Ripple Effect" study documented the impact of changes in mandated assessment in the final years of schooling on the teaching and assessment of mathematics throughout the secondary school, and showed that assessment can be a powerful catalyst for curriculum reform. This paper reports the final stage of a study which sought to examine the different set of influences on teaching and assessment found in NSW. Interviews with teachers from a broad range of schools explored the constraints which they experience as a consequence of the assessment system, and the extent to which they value, and implement, teaching and assessment practices such as problem solving, investigations, journal writing and self-assessment.

In this study the authors investigated the ability of children with moderate intellectual disabilities to compare relative numbers and the strategies they used. Ten children, aged 12 to 18 years, with moderate intellectual disabilities were given a sequence of tasks to assess next number determination; understanding of the n+ l>n rule; and number comparison skills. The results showed that nine of the children were able to correctly determine the next number in a sequence and had a good understanding of the n+ l>n rule tasks, while six of them were successful on the comparison tasks.

How to convince teachers (other than those who have been involved in calculator projects) of the merits of integrating calculators into primary mathematics programmes is an issue that needs to be addressed. This paper reports effects on the beliefs of a group of pre-service teachers who participated in a short but challenging, action-focussed calculator course. The results suggest that pre-service teacher beliefs can be changed in a positive direction.

This paper explores the nature of Collective Argumentation talk in the primary classroom. Interpersonal, intrapersonal and discursive data collected from three Collective Argumentation classrooms is analysed within a framework which recognises the linguistic, psychological and cultural nature of classroom talk. Findings suggest that Collective Argumentation talk functions to assist students to view the development of mathematical knowledge as occurring within their own community of discourse. Suggestions for employing talk as the basis of a classroom community of practice are provided.

The relationship between beliefs and practice is often discussed in research. On the one hand it is perceived that one's beliefs will influence the instructional practice in the classroom. On the other hand professional development focuses on practice, encouraging the reflective practitioner who is prepared to reorganise beliefs as a result of practice. Is this a case of "the . chicken and the egg" or does one of the ingredients in this complex recipe carry greater weighting than the other? This paper explores the relationship between beliefs and practice· as perceived by primary classroom teachers implementing a new curriculum document in mathematics. The teachers shared the viewpoint that practice was the ingredient which carried greater weighting. They perceived their beliefs as shifting or being reorganised as a result of practice. They acknowledged, however, that for shifts to take place they did hold one over-riding belief: the belief that educational practice was not fixed and that children's performance could suggest new approaches and ways forward.

Assumptions are made about what primary school teachers need to make them better mathematics teachers. ProfessIonal development courses and conferences are held based on these assumptions? Who decides what teachers need? What do teachers want for professional development in teaching primary mathematics? Do they know what's good for them? This paper presents preliminary results of a study of primary school teachers' views of what's important in teaching maths and outlines the inherent difficulties. . Teachers views of the influences on their teaching and their ideas of the perfect· professional development for maths teaching are also included.

This paper considers the effectiveness of the traditional "first principles" approach to differential calculus and investigates some of the conceptual difficulties associated with it. Materials developed by Mary Barnes were adapted to produce a sequence of lessons characterised by investigative exercises and realistic calculus problems for a group of year 11 students. The lessons proved successful in improving students' conceptual understanding without impairing their ability to perform standard calculus techniques. They also reported positive feelings towards their study of calculus.

In this paper, we report on our initial application of a theoretical model (Nason, Chinnappan & Lawson, 1996) which explicates the relationship between the organization and accessability of teachers' subject-matter knowledge, the nature of their teaching and the nature and quality of student learning within the domain of Analytical Geometry and Trigonometry in a Pilot Study. The major aim of this study was to begin the process of validating the hypothesized relationships and causal connections between the source knowledge elements of the model.· This was done by examining the nature of a pre-service student teacher's (a) substantive mathematical knowledge, (b) pedagogical content knowledge, (c) knowledge about the learner, and (d) the relationships and causal connections between these three types of knowledge in the domain of geometry and trigonometry.

Two teachers, good friends and colleagues, and yet with quite different approaches to the teaching of mathematics, exploring what· was for them ' "new territory" (pedagogically and mathematically) as they taught a problem based unit of work for the first time. Using classroom observations and post-observational interviews, this study described the challenges of the teaching of the unit through teacher identification of "critical incidents" that were encountered. This paper tells the story of professional growth through .. highlights and challenges, and explores issues of support. provided by colleagues and the researcher.

Using email, academics, consultants and other teachers of mathematics were surveyed as to the most difficult question they had ever been asked by a mathematics teacher. The 110 questions that were received in response to this request were then coded using "Hyperqual". A range of themes emerged, with interesting similarity across the four contributing countries; assessment issues being the most common theme. Implications are discussed.

This paper provides historical and comparative perspectives on current pressures to standardise mathematics curricula. Two main questions are considered: (a) What are the forces behind the move towards core mathematics curricula? and (b) Are core curricula likely to generate more equitable school mathematics programs? The first question is discussed in terms of accountability, colonialism, and politics. For the second question, evidence on equity considerations is provided from developments in the United Kingdom and in the United States.

Qualitative data analysis has become increasingly important in mathematics education research. However the conceptualisation of the data and the fulfil1ment of the interactive assumption of qualitative analysis are a concern. An alternative to a linguistic approach to analysis is a visual approach. The visual analysis of data utilises wholistic data displays, referred to as data maps, and employs visual reasoning. The data maps, which are produced using drawing software, can enhance conceptualisation and facilitate the interactive process of analysis.

This paper describes open assessment tasks and their place in interview-based research. Open assessment tasks are designed to elicit children's understandings and are interview-like in their intent. Responses to open assessment tasks are extremely diverse, and our analysis provides an order that allows inferences about children's understanding to be made. Our experiences to date with this approach indicate that open assessment tasks offer a highly reliable, valid approach to situations where it is not possible to implement interviews.

To implement the national profile for reporting student mathematical achievement requires alternative assessment strategies. A team of classroom teachers was brought together to prepare a package of rich assessment tasks (RATs), evaluated for their potential in assessing mathematical performance and aligning the profile. The search process highlighted issues as the need to match assessment to the classroom context, and the changing role of teacher and students through varying assessment procedures, resulting in production of tenets for developing and evaluating RATs.

Teachers were brought together to produce a package of exemplary assessment tasks, rich in their potential to provide data on students' mathematical understanding and knowledge, and thus link with the national profile as a reporting framework. Through involvement in the writing team, it was found that teachers actually guided their own professional development. The writing team situation provided teachers with (i) support, . (ii) feedback, (iii) opportunities for reflection; as well as assisting them gain (iv) confidence in using, and (v) knowledge and understanding of, the national profile.

This paper reports on research in progress that explores primary school teachers' ideas about statistics and the teaching of statistics. Data on teachers' attitudes and beliefs was collected through interviews and the use of belief and attitude scales. Issues related to teachers' lack of statistical training, confidence in teaching statistics and their views about essential knowledge for teaching statistics are explored.

After 206 Year 7 students-145 in 4 schools in Malaysian, and 61 in 2 schools in Australia-had answered 24 mathematics questions. they were interviewed, in accordance with the Newman interview technique. Types of errors made by the students in the two countries were analysed and compared. Data revealed that (a) around 70% of all errors were in one of the Comprehension. or Transformation. or Careless categories; and (b) strikingly different error patterns occurred for different questions.

Fewer females than males opt to study mathematics at the tertiary level, particularly at postgraduate levels. Factors influencing students' decisions to pursue tertiary studies have been identified. Less is known about the factors associated with studying mathematics in particular. Reported here are results from the first phase of a three year study aiming to uncover relevant contributing factors. Enrolment data for 1995 were gathered from La Trobe University (Bundoora campus) and several students taking undergraduate mathematics courses that year were interviewed. While the results are not generalisable, interesting gender-related trends were· apparent. The implications of the findings are discussed.

The Internet is a new and exciting technology. It offers enticing and seductive possibilities as a research tool. Simultaneously, it presents a number of challenges to current accepted research practices. Access to its capabilities and the amount of information available have expanded rapidly. In this paper we discuss several issues in the use of the Internet to conduct research. We also outline some of the difficulties we encountered in our initial explorations of the Internet's potential as a source of gathering survey data.

Transition to secondary school mathematics often poses problems for particular students. This paper reports on the initial stage of a larger longitudinal study that examines students' beliefs about strategies for success in mathematics and whether they are personally capable of succeeding. The aim is to identify students in grade 6 who are at risk of failure in secondary school mathematics. Students will be followed from grade 6 into year 7 to see what effect transition has on their belief systems.

Two circumstances of major significance are presently impacting on undergraduate mathematics courses. These are respectively increased participation resulting in a wider spread of abilities among entering students, and the increasing use of symbolic algebra software in course delivery. This paper reports on preliminary work in projects at two Universities. Responses to sample questions are discussed in the context of their purpose which is motivated by the need to address the dual . circumstances indicated above.

This paper is a brief description of a larger study conducted to examine mathematical responses from 90 ftrst year university students. The main part of the study was concerned with identifying and describing the quality of existing mathematical knowledge that these students brought with them to university. The SOW taxonomy (Biggs & Collis, 1982) technique was use d initially as the evaluation tool. However, certain limitations of the SOW technique were identified which led to the development of a knowledge element coding technique. The focus of this paper is a brief description of this coding technique, its use and value in evaluating mathematical knowledge elements.

This paper reports on a study that examined student interaction and discussion while working on computer-based tasks in a senior secondary school classroom. Analysis of verbal and observational data suggested that the task itself was an important variable influencing the degree of collaboration between students, and that the teacher's intervention could change students' engagement with the task. The findings have implications for specifying the teacher's role in relation to the use of computer technology in the mathematics classroom.

The potential role of student discussion in developing mathematical knowledge continues to interest researchers and teachers. This paper reports on a study that investigated patterns of student discussion and interaction in . senior secondary school mathematics classrooms. Observations of one group of students are used to illustrate three factors that appear to influence the collaborative quality of mathematical discussion: students' orientation towards the task; their relative task-specific expertise; and the degree of challenge the task presents.

(A report of progress in a Doctor of Education research project supervised by Professor Kaye Stacey.) The paper provides an overview of some of the outcomes of an investigation comparing the teaching of negative number at junior secondary level using tiles, as discrete integer entities, with other teaching approaches more commonly used. The experimental approach seems to have facilitated better performance for average ability level students. For more able mathematics students the topic does not appear to be difficult and such students, in . both experimental and control groups, indicated good levels of general topic mastery.

Two major developments in mathematics curricula have been the 'New Maths' of the 1960s and the recent changes of the 1990s. A framework developed by Howson (1979) is used to compare and contrast these two periods of mathematics curriculum change. What are the influences which shape curriculum development and to what extent are these changes under the influence of anything or anyone? It is suggested that mathematics educators have made a difference in curriculum development through placing constructivism on the curriculum agenda, countering the effects of current reforms in education. From this comparison, further research agendas are proposed.

This paper reports on a longitudinal study across YeMS 2 and 3 of 104 childrenfs petfonnance and strategy use for two-digit separation and missingaddend word problems and vertical and horizontal algorithmic exercises with and without regrouping. The study showed that while Year 2 children predominantly used a subtractive removing strategy for separation problems and an additive building-on strategy for missing-addend problems, Year 4 children were more mixed in their strategy use. It also showed that children used subtractive strategies predominantly for vertical and horizontal algorithmic exercises.

This paper reports on research which attempts to conceptualise, operationalise and docmnent instances of meaningful learning in the secondary mathematics and science classroom, and the conditions under which this learning occurred. A "linkage Criterion" for meaningful learning is proposed and applied to the data from 67 interviews of students reflecting on their classroom learning experiences. Instances of meaningful learning were few and comparatively impoverished where they occurred. The data suggests that the role of out-of-class experiences as classroom focussing mechanisms is a significant one. If teachers plan their instruction with the goal that students should acquire significant new knowledge in every lesson, our data suggests that this goal is achieved sufficiently rarely as to call it into question.

This paper discusses the potential of video as a means of discovering young children's explanations of mathematical tasks. This is illustrated with an observation made as part of a larger study into promoting mathematical processes in New Zealand junior classrooms. It highlights the potential power of video as a means of providing the teacher with an insight into the thought processes of young children doing mathematical activities independently of the teacher.

This paper describes an AAMT project which is endeavouring to develop better understandings of what it means for a person to use mathematical ideas and techniques when completing practical tasks. To help us we have collected some snapshots of young people at work. This was done by using teachers, from around Australia, to go into the work place, shadow a worker for up to a day, followed by an interview and then to write up their findings. This paper looks at the processes used, the issues that have arisen as a consequence of doing it, and at the usefulness of the experience.

Professional growth is a form of learning. As such, it is not surprising that the response of a teacher to a professional development program is a very individual one. This paper reports the professional growth of two teachers involved in the same professional development activity, and with many common elements in their personal histories and work situations. However, the teachers' responses to the professional development program were significantly different with respect to: the level of classroom experimentation (both reported and observed); the degree to which each reflected on their teaching and the consequences of the program; the changes they reported in their practices and beliefs; and their engagement in collaborative activity with their colleagues. The challenge for the researcher investigating teacher professional growth is to find a suitable mechanism to describe the growth process. This paper employs "change trajectories" as a suitable descriptive device. The metaphor of a trajectory offers an image of movement over time, and draws attention to issues like: the dimension along which change occurred; the rapidity of the change; and the factors facilitating and inhibiting the change. In the case of the two teachers reported here, the change trajectories are sufficiently different to provide a useful insight into the individuality of a teacher's response to a professional development experience.

There has been no reported investigation in mathematics education involving the views of Aboriginal educators about the learning and teaching of mathematics for primary school Aboriginal students. This paper reports on such an investigation. It concentrates on two interviews with Aboriginal Education Assistants teaching in a geographically remote primary school in New South Wales. These two Aboriginal educators identify the importance of personal relationships, relevance and humour in the learning and teaching of mathematics to Aboriginal students. They set a pedagogical agenda for teachers in their interaction with Aboriginal students and communities as they facilitate the learning of mathematics.

Before the 1960s, introductory trigonometry was taught in Victorian schools using the ratio method, where trigonometric functions are defined as ratios of sides of right angled triangles. With the advent of "new maths", the unit circle method was introduced. This study explored differences between the two methods for teaching introductory trigonometry. Eight. classes of students were randomly allocated to either teaching method. The ratio method was found to be much more effective, resulting in better performance and retention in trigonometry and algebra, and more favourable attitudes.

This paper proposes and justifies a program of research to extend knowledge of how children use judgements to determine the area of a rectangle. It looks particularly at the relationship between perimeter and height + width rules used in making judgements. Children's responses to a variety of tasks in a pilot study are described and analysed and inferences drawn for further study.

This paper highlights two m~or research approaches into studying how students judge area, logical-operations and information processing. The logical-operations approach focuses on determining the order in which students acquire area knowledge based on a framework developed by Piaget. The information processing approach focuses on perceptual judgement. The paper assesses the two approaches in terms of appropriateness for future research.

In the early 80s, Mayberry (1981) developed a diagnostic instrument to be used to assess the van Hiele levels of pre-service teachers. The test was designed to be carried out in an interview situation. The Mayberry study has been replicated under Australian conditions in a written format, testing sixty first year primary-teacher trainees. This paper presents the results of the study, comparing them with the results of the Mayberry students, and relating them to their geometric backgrounds. Responses, in general, show that many of the students who had completed a recognised senior secondary geometry course could not display better than Level 2 understanding.

This report combines the results of two small studies of probability based on the concept of fair. The outcomes of the first study led to additional opportunities in the second and finally to a hypothesis of conceptual development. In both studies students from grades 3 to 9 were interviewed using a protocol designed to assess their understanding of fair in relation to dice. The theoretical framework used to analyse student responses was the SOLO Model with Multimodal Functioning developed by Biggs and Collis.

We report an investigation of students' attempts to formulate equations for word problems. Ninety students in Years 9. 10 and 11 were tested twice over 10 months. We trace a progression of stages from naming quantities through describing relationships to writing equations and solving them. Even when all relationships were recognised and correctly symbolised, integrating them into an equation was a common difficulty.

In curriculum statements provided for teachers, teachers are encouraged to present to their pupils a rich variety of word problems. Teachers may draw their word problems from reference texts, "real-life" problem situations or generate their own word problems. An analysis of teacher generated word problems reveals that these word problems not only fail to provide as diverse a range of problems as suggested in the resources made available to teachers, but also reinforce some misconceptions about the four arithmetic operations. The implications of these findings are discussed and proposals advanced to overcome this problem.

This paper discusses beliefs of one young learner of mathematics about the meaning of the terms "mathematics·' ruld "maths". The latter term is generally considered to be a common abbreviation of the fonner but with the same meaning. Data collected from one child demonstrate that for her the meruJ.ings of the two terms are different. Use of a range· of interview procedures has facilitated the child's articulation of her beliefs and the accessing of them by the researcher.

Each student in a sample of 144 Year 2 students was presented with models of three physical angle situations chosen from a set of nine. A bent straw was used as an abstract angle model. Students were questioned about how they would use the abstract angle model to represent each situation and what similarities they recognised between each pair of situations. The major factor influencing students' interpretations of the angle situations was the number of visible arms of the abstract angle. Implications are drawn for the teaching of angles.

What are students' views of odds? Students were asked to interpret a newspaper headline, "North at 7-2". Three different perspectives were distinguished: (1) a probability view often using traditional part-whole ratios, (2) a frequency view involving scores and frequency of wins, and (3) a social view, usually involving betting and money exchange in part-part ratios. Each view followed a developmental sequence, with interaction between them.

This paper recounts how preparation of case study material for a multimedia resource raised questions about traditional power relations in the control of communication in mathematics classrooms. It aims to provoke discussion on whether more natural patterns of interaction should be promoted, who should shape communication channels, and the roles of researchers and teacher educators in transforming classroom interaction. Ethical issues related to forms of presentation of the data are raised.

This paper reports the developmental phase of a 2-year longitudinal study of 120 Grade 2 children, which is designed to explore the relationship between their representations and understandings, of key number processes. The study focuses on the structure of children's representations of estimating, counting, grouping, re-grouping, partitioning and multiplicative processes. The research methodology, including videotaped interview tasks used during trialing in March and April 1996, and preliminary findings will be presented for discussion and critical analysis.

In this paper. we describe a Pilot Study in which we investigated how a computer-mediated collaborativeleaming environment called CSILE could be used to mediate the establishment and the maintenance of a knowledgebuilding community of mathematics practice within two elementary school classrooms. We conclude that if CSILE is to successfully mediate knowledge-building during mathematical investigations. then it needs a human-computer interface which: (1) preserves the concreteness and the spontaneity of the students' synchronous, face-to-face investigative work and (2) helps to recreate or maintain the sense of community of practice that is engendered during face-to-face investigative inquiry.

This paper explores the possible effects of gendered ethical systems or moralities on participation in mathematics education. In particular, it suggests that the prevailing morality in the classroom significantly affects the attraction, retention, and success of female students.

There is an emphasis in primary preservice teacher education that Key Lemning Areas should be integrated. In this study primary preservice teachers' abilities to fonnulate mathematical links to teaching activities related to science and technology topics was investigated. Two factors seemed to affect the quality and number of links suggested by preservice teachers~ their age and the amount of science, but not mathematics, they had studied in the senior secondary school. Overall, student teachers had great difficulty in perceiving and describing mathematical concepts that could be integrated with science activities.

This research examines misconceptions in probability held by a sample of pre-service primary teacher education students. Questions were selected and modified from those reported in the research literature in order to examine; the misuse of heuristics, a false assumption of equal likelihood, misunderstanding of independence, awareness of counter intuitive probabilities, and belief in other fallacies. Questions were accompanied with a request to explain the reasoning employed. Explanations were partitioned into disjoint categories according to the type of misconception and cognitive level of response. Results of selected questions and a summary of the response levels are presented here for discussion.

In this paper we draw on data from the second stage of a three year study conducted in eight Victorian, state co-educational secondary schools. In 1995, one Year 7 and one year 9 class, from each of four schools with high numbers of non-Anglo cultural background students, participated in the project. Four sources of data were collected. This paper focuses on the analysis of questionnaire responses and where possible snippets of student interviews have been included in an attempt to clarify issues raised.

This paper has been developed as part of a continuing investigation of the use of manipulatives in mathematics learning and teaching by K - 6 teachers. The focus is the beliefs held by teachers about mathematics, mathematics learning and mathematics teaching. The grouping of particular beliefs which allows the development of a profile of these teachers is considered.

A comparison of three consecutive implementations of a Mathematics and Statistics subject suggests that a focus on the self as a learner of statistics in addition to one on learning the course content may result in preferred outcomes. Modification of students' perceptions of themselves as learners can alter the acceptability of the experience of learning Mathematics and Statistics, whilst maintaining high performance levels. The body of the paper explores the exercises and manner through which the focus, on learning how to learn statistics, has been created and suggests ways in which the experiences of one class of students may be used to improve the experiences of successive generations of students.

This paper reports on the some findings of a longitudinal study into children's understanding of number patterns by considering the changing use of natural language and symbolic notation over time. The study had its roots in an earlier study that provided a classification system for the responses. The two findings reported here suggest that responses to pattern stimulus items change over time and that the use of natural language in the highest category seems to be a necessary precursor to the emergence of algebraic notation.

One strand of curriculum implementation research concerns itself with the faithful implementation of an innovative curriculum. From this perspective. it is reasonable to suppose that the successful implementation of a curriculum change will be significantly affected by the congruency between the goals of the curriculum change initiative and the goals of the professional development program designed to achieve that change .. There are at least five stages in the development and implementation of a professional development program designed to support cunicular refonn. These can be identified with the actions of specific individuals: Program initiators; program developers; trainers of workshop presenters; workshop presenters; and. teachers. The goals of the curriculum and consequently of the professional development program must be reinterpreted at each stage. This study documents the ways in which the goals of a professional development program were transformed in the course of its development and implementation.

The aim of this study was two-fold: first, to describe exemplary practice in teaching with computers; and secondly, to isolate criteria appropriate for the identification of expertise in instructional computing that might be useful for future research in this area. Specific criteria for identifying expertise in instructional computing were derived from the literature and from the findings of a case study. It is suggested that documenting expert teaching procedures will provide educators with examples of and ideas for improving practice in mathematics education.

Growing interest in statistics and probability in schools has resulted in a change to developing concepts rather than carrying out calculations. However, statistical reasoning is not easily acquired by all students and common errors (misconceptions) reflecting difficulties have been identified in several studies. This paper presents the results of a study which explored form five (14 to 16- year-old) Fijian students' ideas of statistics and probability and how these related to their previous school and cultural experiences.

A previously reported coding scheme for describing student expository writing in mathematics has been developed into a set of descriptors aimed at reflecting the level of understanding being demonstrated in the writing. The original coding scheme provided a detailed analysis of the content of the writing to enable the application of the descriptors. The descriptors are intended to give researchers and teachers indications of the content of writing tasks which may help advance students' thinking. Examination of writing examples mainly from year 8 students has shown that it demonstrates little conceptual understanding of mathematics.

Despite being involved in a mathematics education subject emphasising student centred methods of teaching, some trainee teachers revert to 'chalk and talk' and rote learning methods. To partially overcome this practice, second year students in a primary teacher education course were asked to write their mathematical autobiographies over a period of eleven weeks. The analysis of these revealed increased awareness of mathematics as a life skill, changes in opinions and greater understanding of the influences shaping mathematical development.

Research into the potential of problem posing as a means for developing of students' understanding of mathematics has been hindered by the absence of a framework which links problem solving, problem posing and mathematics curricula. This paper presents an overview of the frameworks· used by researchers for investigating problem posing, and proposes a framework for research into students' problem posing in mathematics. Examples of problem posing situations used in a classroom with mathematically able students are presented.

This report arises from a project which uses an interactive multimedia resource to support learning about teaching mathematics. As one of a range of data collection instruments, pre-service teacher education students completed a card sorting task (using Q sort technique) before and after using the resource. The technique provided data to -supplement other more general qualitative data. Results indicate that use of the resource helped focus students' attention on some key characteristics of quality mathematics teaching, and that the instrument was useful as one of a range of sources for qualitative data.

In terms of subsequent learning of mathematics, the importance of mastering a limited number of basic concepts cannot be over-estimated. This paper examines the nature and frequency of mathematical misconceptions which have been commonly exhibited in tests at two tertiary institutions. The frequency of mathematical misconceptions seen is of great concern. Hopefully information about which misconceptions occur commonly will lead mathematics educators to place greater emphasis on the teaching of these concepts, thereby causing a decrease in the frequency of the related misconceptions.

The purpose of the project is to document the progress, over one school year, . of a mathematics teacher in a secondary school in Hong Kong, as he attempts to enhance mathematics learning through the introduction of a discussionbased approach. The results of the study will give insights into the kinds of difficulties experienced by a teacher trying to implement a non-traditional teaching approach, as well as giving some understanding. of the role of discussion in a Hong Kong mathematics classroom. This paper presents a progress report of the teacher's feelings about the innovation after the first five months of its implementation,and the factors affecting these feelings.

In this paper a quantitative analysis of the results of a nationwide survey into the use of computers in New Zealand primary and secondary classrooms is described. The hardware and software distribution and pattern of classroom use in mathematics is outlined. The reasons teachers perceive for not being able to make even greater use of both calculators and computers in their mathematics teaching raises some important issues which have strong implications in terms of school funding and teacher training.

Students' ability to understand calculus has been of concern for some time. Students are often locked into a process-oriented style of thinking which is an obstacle to their understanding of important concepts. In this paper we describe the results of an integral calculus questionnaire given to senior secondary school students and designed to measure their understanding of concepts associated with the Riemann integral. We describe some misconceptions in their understanding and the instrumental, process-oriented· thinking underlying them.

In this paper we discuss the roles of, and the relationship between, arithmetic and algebraic thinking in algebraic problem solving. We report the results of a study where we recorded students' mode of thinking, problem solving ability, mathematical anxiety and self-concept. We found a positive correlation for girls between anxiety and problem solving performance, but for boys the positive correlation was with self-concept. We also found a reluctance on the part of students to use algebraic thinking.

Although learning must arise out of students' own interpretations of their classroom experiences, their voices are not often the focus of educational research. This paper locates students' experiences of mathematics lessons within the framework of 'curriculum genres', the term Christie (1993) uses to define the distinctive ways teaching-learning activities are staged and socially structured through language. The usefulness of this language education construct for analysing mathematics learning is discussed.

This paper discusses the results of diagram drawing instruction on pupils' ability to understand and solve word problems. The results show that after instruction, diagrams drawn were. more relational and resulted in better performance. There was a conscious attempt on the part of the pupils to illustrate the relationship among variables in their diagrams. The pupils also demonstrated positive attitude towards this approach of solving problems.

Responses of 24 children from pre-Grade 1 to Grade 4 were used to illustrate the range of understanding of the partition of continuous fractions in the early years of schooling. The SOLO developmental model with multimodal functioning was used to classify responses and it is hypothesised in this paper that ikonic and early concrete symbolic functioning are exhibited in these . grades as fairness becomes a mathematical criterion for work with fractional parts. Implications are drawn for further research and for the classroom.

This paper reports some preliminary results of an ongoing studyy of children's intuitive understanding of basic probability concepts. The conceptual framework of the study is outlined. Task-based interviews, conducted with fifty K-6 children in N.S.W., have revealed some developmental trends as well as some interesting insight into the types of strategies applied by children when attempting to quantify their perceptions of random devices and to compare probabilities.

The notion of 'Writing-to-learn' brings into conjunction an understanding of language and of learning. It is suggested that a particular view of language has unwittingly determined a view of the learner. By turning to a dialogical view of language a new view of the learner is presented.

This paper reports upon a study that examined the responses of NSW upper primary teachers of mathematics to a series of questions about typical teaching behaviours. A phenomenographical approach was used to conduct the research. The study uncovered a range of conceptions held by the teachers and some of the findings are presented. The results provide information which can assist teachers in their critical reflection upon current practice whilst increasing our understanding of how teachers understand and experience their world.

A two-part study of the difficulties experienced by senior secondary (Year 12) students in writing inequalities from graphical linear programming problem statements is reported. The first part of the study identifies possible difficulties and the second part evaluates a teaching programme designed to alleviate these difficulties. The teaching programme, based around a set of specific heuristics, is successful in alleviating some of these difficulties. The "pronumeral-as-object" notion and certain misconceptions of "constraint" prove resistant to remediation.