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The status of mathematics education in schools in Samoa is a major concern to post-secondary mathematics educators. Incoming students from secondary schools are not adequately prepared for advanced studies in mathematics for lack of a solid understanding of basic mathematics principles which is essential for further studies in calculus and linear algebra. This paper provides a general overview of national examinations results and selected comments from examiners' reports to illustrate the state of mathematics education in Samoan schools. The quality of mathematics learning in schools directly impacts on first year programs in post-secondary institutions as time is often taken out to bridge students up to the appropriate mathematics entry level. The paper briefly describes some of the research and professional development activities organized by the University in an effort to impact on the teaching of mathematics within its teacher training programs.

The Numeracy Development Project (New Zealand), now into its third year of implementation, spans years 1 to 10 of schools. By the conclusion of 200 1 the project had involved approximately 3300 teachers and 64000 students. The Number Framework forms the core of the project programme by providing teachers with a knowledge of how students acquire number concepts, an increased understanding of how they can assist students' progress and an effective means to assess students' levels of thinking in number. This paper focuses on the impact that the project has had on the students in the first three years of school and their teachers.

In adult numeracy instruction, the gap between current and potential instructional realities is of great concern. Narrowing that gap might best be accomplished with creative, sly, and surprising combinations of practice, research, theory, and/or policy. In this paper, I describe current policies, practices, and research connected with adult numeracy instruction in the United States, propose a different model, and depict three multi-dimensional interventions which create powerful opportunities to fmd the way from what currently exists to what might be.

Technology allows new kinds of tasks revealing the meaning of theoretical objects. But creating these new tasks is not easy and requires time. We will present the learning tasks of a new type allowed by dynamic geometry software and try to explain why it takes time for teachers to design and use these tasks. Examples will be given in Cabrigeometry, existing both as a software program or application of the calculators TI 92, TI 89 and TI 83.

In the past decade, many countries called for changes that result in students knowing a 'different' kind of school mathematics. This is due to a changed view of learning that requires different forms of teaching which are only beginning to be understood. In this paper, findings from research in primary maths classes with 'reform-oriented' culture are presented; examples drawn from these classrooms illustrate differences in teaching that influence the mathematics children learn. Two dynamic dimensions derived from this research are presented.

A large study tracked four primary teachers during their first year of teaching, with a focus on how a 'fellow worker' professional development support model influenced their pedagogical. practices and beliefs concerning mathematics. The case studies used an interpretive research approach for analysis of data from interviews, observations, reflective journals, and group meetings. The findings highlight how the realities and challenges of beginning teachers' classroom experiences prompted adoption of 'traditional', teacher-centred mathematics pedagogy. Support in the form of a 'fellow worker' helped the beginning teachers survive and then begin to teach in ways consistent with current trends in mathematics education.

This is the second report to MERGA of a project working with senior secondary mathematics in low socio-economic schools in Auckland, New Zealand. It is aimed at enhancing the achievement and participation of Year 12 & 13 mathematics students 'and promoting their transition into tertiary mathematical programmes. This paper outlines the theoretical basis for the project, and describes the 2001 and 2002 Pilot Studies.

This paper engages current thinking about the links between students' early experience with number properties and their future understandings of algebra. It reports on a study of students' understandings and explanations of commutativity. Central to the analysis are both the forms of representation students draw upon and the language they employ to make their understandings explicit. The investigation reveals that despite the rhetoric of educators and the intention of curriculum developers, student learning continues to remain at the procedural level, constraining rather than enabling algebraic reasoning.

This paper discusses data from interviews with mathematics educators from Australasia and Colombia on their views of, experiences in and expectations from the internationalisation and globalisation of their discipline. International collaboration is essential for moving the discipline forward in a globalised world and avoiding the colonialism of the past - and allowing the discipline to play its role in bridging the increasing gap between developing and developed countries. The aim of this analysis is to examine some problematical aspects of this collaboration between countries of unequal "power" and resources.

A sample of 141 Year 8-10 students in a suburban secondary school was given a computation test to complete by using a four-function calculator. Students were required to record all their key presses for each item in order that they could be assessed on their efficiency of calculator use. There was a small increase in effectiveness of calculator use over the three year levels, but efficiency actually fell away in Year 10. Overall, only 62 percent of students used their calculators highly efficiently.

The espoused beliefs of 465 secondary mathematics teachers in Greek State secondary schools regarding the teaching and learning of mathematics were the focus of this study. The data for this investigation were collected using a 34 items questionnaire. There is evidence from this study that there are teachers who may be assumed to espouse a 'constructivist or contemporary orientation', teachers who may be assumed to espouse a 'dynamic problem-driven orientation', teachers who may be assumed to espouse 'traditional-static' and 'traditional-mechanistic' beliefs, and teachers who espouse a 'cooperative orientation' to mathematics learning and mathematics teaching. It was also found that teacher characteristics, such as, gender, teaching experience, position held and postgraduate qualifications possessed, significantly influence teachers' beliefs about mathematics and the teaching-learning process.

For many years the research-development-dissemination (RDD) model has been used in education. From my experience and scholarship this model seems unsatisfactory and I have been theorizing about alternative models that recognize the complexity of change. In offering a model with eight activities occurring simultaneously at three levels, and with these activities being interrelated, I have replaced the simple RDD model with a complex model. This model emphasizes involvement by teachers and offers an inclusive and empowering approach to change. The challenge is now to trial and improve this model.

This paper describes a set of cognitive research tasks used to assess the understanding of place value of 21 year 3/4 students. Seven tasks provided enough infonnation about the conceptual development of the students so that a determination of their progress towards understanding could be made. Re-administration of the tasks occurred after six months and then twelve months. The results supported the ability of defined levels of response to the tasks to describe a developmental pathway. Mediation of the tasks during administration identified some validity and interpretation issues. The study confirmed that students of this age are grappling with the development of an understanding of place value and therefore are in need of explicit teaching in this domain.

This paper describes a part of a study of the beliefs about mathematics, its teaching and its learning, held by secondary mathematics teachers. The case of one particular teacher is used to illustrate the difficulties inherent in the study of beliefs, and to highlight both the importance of data triangulation and the caution required in making inferences about a teacher's beliefs.

Over the years, national mathematics curricula have tended to be imposed on New Zealand teachers without regard to their realities. As a means of (potentially) informing future curriculum development the study reported here sought to do two things: (1) identify the actual concerns held, and mathematics teaching difficulties encountered, by classroom teachers, and (2) find ways to support the teachers in addressing these concerns and difficulties. The six month collaborative project with staff in a small urban primary school revealed some insights into difficulties experienced by teachers, the process of teacher professional development in mathematics, and possible directions for mathematics curriculum development.

Mental computation is an important aspect of numeracy. There is little information available, however, about appropriate sequences of learning, or what can be expected of students in different years. The responses of 3035 students in years 3 to 10 to linked tests of mental computation were placed on a single scale using Rasch modelling techniques. An eight level scale of mental computation development was identified, and student achievement across the years was mapped onto these levels. The findings and their implications for the classroom are discussed.

Following earlier interest in mental computation and its relationship to number sense in the early and primary years, this exploratory study investigated the mental computation competence of students in the middle school. The study involved clustering the responses of four students in grades 6 and 8 based on individual interviews. In each grade a student was identified as more or less competent in mental computation performance on a pencil-and-paper class test. Several contrasting content and performance features were identified and examined with respect to implications for teaching mental computation in the classroom.

This exploratory qualitative research study uses observations and semi-structured interviews to examine secondary mathematics teachers' attitudes toward probability and their use of probabilistic reasoning in teaching two activities designed to extend students' models of exponential relationships. Student learning difficulties present problems for the teachers. Teachers' preparation in probability is typically weak and they are uncomfortable teaching probability. Thus, they emphasised the exponential aspects of the activities, concentrating on developing generalised deterministic equations that model growth and decay.

This paper reports on an attempt to measure aspects of pre-service teachers' understanding of middle school mathematics together with their perceptions of its difficulties by using a simple questionnaire. Pre-service primary teachers did not, in general, perform as well as their secondary counterparts, and they rated the items as more difficult. While there were differences between perceptions of conceptual difficulty and computational difficulty, it is not certain that these distinctions are clear in the minds of pre-service teachers. Suggestions for improving the questionnaire are discussed.

The development of students' algebraic understanding is generally accepted to be one of the major goals of K-12 mathematics teaching. In this paper I attempt to examine this understanding by characterising a group of high school students' algebraic knowledge and patterns of use of that knowledge during the solution of selected problems. Results show that these students tended to show acceptable levels of proficiency with problems that involve substitution of values for variables, and simplification of equations. However, students experienced difficulties with the solution of equations and the interpretation of variables both in symbolic and graphical modes. These results are interpreted as suggesting that the participating students' understanding was buttressed mainly by schemas that were dominated by procedural knowledge of algebra.

The aim of the study reported here was to examine the quality of understandings developed by young children in the area of fractions and decimals. Analysis of data showed that the existence of great disparity in Year 3 children's knowledge base of fractions. We discuss these results in light of levels of competence that are expected in K-6 curriculum documents and with reference to past research on students' knowledge of fractions. The results of this small study raise doubts about the progress being made in the teaching of fractions.

This paper presents an overview of the findings of three masters-level research investigations that the writer supervised during the period 1998-2001. Each investigation was based on a "multiple perspectives" design, and analyses revealed that most of the Bruneian secondary school children who were taught mathematics in the traditional "skill and drill" way did not learn much mathematics. Data from the studies suggest that emphasis on preparation for forthcoming tests and examinations does not result in students achieving a genuine understanding of mathematical topics. Lim's (2000) summary of a didactical contract between Bruneian secondary mathematics teachers and students is presented. It is argued that the teachers and the students, and indeed whole school communities, are continuing to exercise uninformed faith in a method of teaching and learning which is not "producing the goods".

The role of the teacher in the learning of mathematics is one which continues to be refined. Evidence continues to mount for the relationship between teachers' content knowledge and pedagogical content knowledge, and the way they structure their classroom lessons. In this paper we present data from interviews with four teachers about their teaching of differentiation and derivative. The analysis shows that there are both similarities and differences in the pedagogical approaches emphasised, and that these are related to the teachers' conceptual and representational perspectives.

Recent government reports identify the importance of producing individuals with high-level mathematical capabilities who can contribute creatively to Australia's intellectual capital. To address the shortfall in such individuals, mathematically gifted students must be identified and nurtured. Hence, teachers' programs need to be well-grounded in research on educating the mathematically gifted. In this paper, we report on the status of research on these students, draw upon our research to inform practice, and highlight myths associated with the education of these students.

This paper describes part of a continuing program of research and development into improved reporting of student performance in mathematics. The goal of the work reported in this paper is to derive summative and formative information from a single assessment. Developers of multiple-choice test items typically have a good knowledge of the research evidence about children's errors and misconceptions. These common errors are often employed as item distractors. Descriptions of item distractors were made for a mu1tip1echoice test (scored correct/incorrect) to provide descriptions of errors likely to be made by students of different abilities. Through the linking of formative and summative analyses a 'map' was constructed that provides a way of alerting teachers to possible flaws in students' mathematical understanding.

Many students come to pre-service mathematics teacher education programs with considerable anxiety and fear of mathematics. The challenge is to provide students with the necessary skills, knowledge and understanding for teaching, and to positively impact upon their belief systems. In this study 429 pre-service students rated change of beliefs about mathematics through a self-assessment questionnaire. Results indicated a strong positive change, and together with interview data, highlighted key contributing aspects of the program and thus provided a measure of program quality assurance.

This study compares computer supported collaborative learning (CSCL) based instruction and traditional pen-and-paper methodologies and is aimed at investigating the effectiveness of using spreadsheets and collaborative learning methods in mathematics instruction for enhancing student learning. Ten Year 11 mathematics classes comprising of 172 students from six high schools across Sydney participated in the study. This study showed that majority of students found the collaborative learning activities very interesting and enjoyable although positive effects of using spreadsheets were not apparent. Statistical analysis of the pre-tests and post-tests showed significant differences in the mean improvement scores between the schools and also between the teaching interventions when only schools 4, 5 and 6 were taken as part of the analysis. Implications of the findings as well as recommendations for future research are discussed.

Examples of "show that" examination questions highlight the various ways CAS can impact upon question design, possible solution approaches, documented responses and marking schemes. A classification scheme used on some recent non-CAS NSW and Victorian examinations found that CAS and the role of automatic simplification can influence the solution process in a variety of ways. Assessment issues such as control over CAS and the nature of solution accounts are also discussed using student responses from a pilot CAS subject in Victoria.

Mathematics and computing have been stereotyped as 'male domains'. Contemporary mathematics teachers are encouraged to use computers in class and it is believed that students' learning will thus be enhanced. In this paper, findings from a large survey tapping students' attitudes towards computers for mathematics learning are presented. The results were analysed by several equity factors - gender, student socio-economic background, ethnicity, school type and location (rural/metropolitan, and socio-economic location) - to explore the gender stereotyped beliefs of these groups of students.

This paper reports an inquiry into approaches to grapbing required in Physics and Applicable Mathematics tertiary entrance examinations (TEE), in Western Australia. The focus in the paper is on the different graphing practices in Physics and Applicable Mathematics in relation to gradient and transformed data, and in regard to use of graphics calculators. The differences mediate against transfer between the subjects and are one explanation for students' poor performance in graphing in the Physics TEE.

This paper explores student empowerment in the context of graphics calculator usage. The setting is revision and a student has her calculator attached to an overhead projection panel, hence the Sherpa-student role. She is working according to the teacher's instruction. Problematically, she lags the teacher's instruction and another student preempts her answers. In addition, the teacher uses tripartite questioning which can mediate against student empowerment, but a mitigating factor is students disrupt the questioning repeatedly. Use of the overhead panel and the conditions that worked towards and against student empowerment are discussed in the paper.

We continue to report on research into outcomes that emerge when symbolic manipulator software forms a central component of undergraduate teaching. Preliminary data led to the identification of 7 categories that encompassed the types of student questions and blockages arising during interactive laboratory sessions. Examples illustrative of the categories are provided and implications discussed. Additionally we report and discuss the results of successive applications of test items designed to assess the impact of complexity (defined in terms of syntax, and function choice and specification), on task demand.

This paper reports on students' choices of and preferences for different hand held technologies when provide with options within a classroom that integrates technology into the teaching and learning of senior secondary school mathematics. Observations were drawn from video and audio data as well as a whole class interview of the focus group in which the video of a previous lesson was used for stimulated recall. This report notes a number of emergent categories of student preference and choice based on this data and comments on the potential impact of such choices and preferences on student learning.

It is an accepted practice throughout Australia and New Zealand for schools to implement Reading Recovery Programs for Grade 1 children. In contrast, intervention programs in mathematics are less well established. As part of the Early Numeracy Research Program (1999-2001), an intervention program was introduced in 21 schools in both Grade 1 and Grade 2. Children's progress was compared to determine whether the program was more effective in either grade. Further perspectives were gained from 21 specialist teachers and the classroom tea<;::hers and parents of two children participating in the program. Although Grade 2 children made more progress in two of the four domains considered, the parents and classroom teachers believed that the program is best implemented in Grade 1.

This paper reports on a study that investigated the pedagogical practices and beliefs of preservice and beginning teachers in integrating technology into the teaching of secondary school mathematics. A case study documents how one teacher's modes of working with technology changed over time and across different school contexts, and identifies relationships between a range of personal and contextual factors that influenced the development of his identity as a teacher. This analysis views teachers' learning as increasing participation in sociocultural practices.

Pre-service primary school teachers come to their initial teacher education programs with beliefs, attitudes, values and feelings about mathematics and mathematics education. These have been formed through their experiences of mathematics prior to tertiary study and they can powerfully impact on their development as a teacher of mathematics. Then, during their teacher education they have another set of experiences of mathematics, which further impact on their views and emotions about the discipline. This study focuses on two pre-service primary teachers in the first year of their initial teacher education and through a series of interviews and tasks it explores their affective views towards mathematics. It seems from the study that tertiary mathematics education courses need to explicitly and continually revisit the pre-service teachers' affective positions if positive development is to occur.

This study aimed to investigate the ways in which one student developed his understanding of concept of function and the use of his concept image of function when solving problems. This student's responses from a function concept questionnaire, a function test and interviews were classified into one of three categories: action, process or object concept of function.

This paper reports on a study of seven Year 3 students' diminished performance in addition and subtraction mental computation. Although all students were identified as being inaccurate, three students used some variety of mental strategies, while the other students used only one strategy that reflected the written procedure for each of the addition and subtraction algorithms taught in the classroom. Interviews were used to identify students' knowledge and ability with respect to number sense (including numeration, number and operations, basic facts, estimation), metacognition, affects, and memory. Two conceptual frameworks were developed, one representing the flexible mental computers, and the other representing the inflexible mental computers. These frameworks identified factors and relationships between factors that influence flexibility in these inaccurate mental computers. The frameworks were compared with a framework of an ideal mental computer. These frameworks showed that inaccuracy resulted from disconnected and deficient cognitive, metacognitive, and affective factors; and in some cases might have been affected by deficient short-term memory. It appeared that students' choices of mental strategies resulted from different forms of compensation for varying levels of deficiencies.

As part of the Early Numeracy Research Project, children's mathematical growth in 295 single grade and 275 combination classes in grades P - 2 was compared. Significant differences were found with negative effects for students in the upper grades of combination classes, though overall differences were not significant. Data also indicates that deliberate assignment of weaker students to the upper grade of a combination class as an exercise in homogeneity can have a negative effect on mathematical growth.

The New Zealand Numeracy Project is based on two kinds of number models. One kind using the Forward Number Word Sequence, the Backward Number Word Sequence, Numeral Identification, and number facts that need to automatically recalled by children are defined as knowledge models. The other using Steffe's counting types and extensions is called a strategy model. Material written to support teachers assume two different teaching models are appropriate corresponding to whether the teaching involves knowledge or strategy. This paper maps the construction of the strategy teaching model. In particular it considers the influence of Pirie-Kieran Theory and Mathematics Recovery on its development.

Key features of developing multiplicative structure were analysed from case studies of twenty-four students representing extremes in mathematical ability. Data drawn from a longitudinal study from Years 2 through 5 of schooling indicated that low ability students represented multiplicative situations without structure and development progressed from the use of pictorial to ikonic representations. Absence of any underlying structures persisted through to the end of Year 5 for half of these students. From the outset in Year 2, high ability students used notational representations with well-developed structures, and dynamic imagery featured strongly in their responses.

Forty-three children, aged 8 and 9, were interviewed about their understanding of informal and formal measurement. Results showed that they visualised units accurately if the number of units was small. For many this was their preferred method of measuring. When asked to iterate an informal unit or use rulers in unconventional ways, they made errors that indicated ignorance of many of the unwritten rules of measurement. Successful children used visualization and the width of a finger to measure.

Responses of 73 students to an interview protocol based on selecting 10 lollies from a container with 50 red, 20 yellow, and 30 green are categorised with respect to centre and spread of numerical answers and to reasoning expressed in justification of the answers. Results are compared to earlier survey research and small-scale interview studies.

The visual attractiveness of two Queensland mathematics textbooks is examined. Diagrams were categorized according to their type, and each type was examined in terms of perceptual precedence. Series A textbook was found to have one third of diagrams of a cosmetic type, commanding 40% of the perceptual precedence. Such diagrams have no relevance to the subject matter, and therefore the potential to distract a student. Series B textbook was found to have combination diagrams in abundance that demanded perceptual precedence. Each diagram of this type has two or more functions and the potential to be effective in developing mathematical understanding. It is recommended that further research be conducted into student uses and perception of diagrams so as to inform authors and illustrators.

This study investigated how pedagogical content knowledge developed while five preservice secondary mathematics teachers were enrolled in a mathematics methods course. In three phases, data collected were participants' concept maps of pedagogy, teaching and learning, and assigned content and related interviews. Themes of adjusting and reflecting often developed during interactions with students' misconceptions or when timing and motivation were pedagogical issues~ Implications include the metacognitive benefits of focused reflection on pedagogical content knowledge during preservice education.

In this paper I argue that teachers of the twenty-fIrst century need radically reformed professional development experiences if they are to teach in inquiry-based ways consistent with views of learning mathematics as an active, social and constructive process. By carrying out a comparative poststructuralist analysis of two teaching/learning interactions [from a paper authored by Manouchehri and Goodman (2000)], I go beyond the extant argument about the importance of teachers' mathematical and pedagogical knowledge and new images of teaching to suggest that additional factors may need to be considered as part of the professional development process. From a poststructuralist perspective I argue that the teacher's unspoken but influential knowing about learners and learning must also be considered and addressed as this impacts heavily, and often conservatively, on practice. Though contentious, the implications for professional development are contemplated.

What would it mean for mathematics education hi the twenty-ftrst century were teachers and researchers to take even more seriously, to interrogate more carefully, the productive power of the learning process? While the importance of the cognitive aspects of process in the construction of robust mathematical understandings and relationships is well appreciated and articulated in teaching and research, its constitutive force has largely been ignored. In this paper I use the poststructuralist notion of the productive, constitutive force of the process of coming to know in mathematics to (a) extend current understandings of how the teaching/learning processes of the classroom influence participation, knowledge growth and mathematical identity, and (b) contemplate the practical implications of this potentially generative force with regard to instructional practice and further research. The paper is framed by the poststructuralist notion that the ability and inclination to engage in mathematical reasoning and inquiry is not a personal attribute or skill but discursively produced.

Student misperceptions (visual unawareness) constitute a powerful hindrance to successful mathematical problem solving. Using the study's computer software, a number of lower secondary, upper secondary mathematics students, their teachers and other pre-service teachers were required to perform a number of transformations. These exercises demonstrated the extent to which misperceptions confound the process of conceptualisation in problem solving activity and mathematical learning. This presentation highlights the misperception phenomenon and describes remediation processes that are being developed in an effort to counteract the negative impact of the phenomenon.

This paper reports on a year-long technology-rich teaching experiment conducted with four low-achieving undergraduate students. Using questionnaires, classroom assessment tasks, self-report journals, interviews and field notes, the study identified three stages of cognitive development - computational, technician, and multirepresentational - which reflect successively higher levels of tool-based engagement and performance. Four tool-based pathways or "journeys" were also identified, illustrating how these mathematically weak students navigated their way through the stages of the model. They are: a restricted learning pathway, a linear learning pathway, a non-linear learning pathway, and multiple learning pathways.

The construction of concept maps is increasingly seen as a useful tool for mathematics educators. The maps have usually been built either directly by the individual themselves or from their comments in a discussion or an interview. In this paper we describe how a novel kind of concept map was constructed based on questions teachers use in their classroom lessons. An analysis of what we call pedagogical concept maps may assist teachers to construct and determine the goals of a lesson. We present here the results of analysing four lessons from a conceptual and procedural perspective which show their value in discriminating between these kinds of lessons.

This paper examines the problem-solving approaches employed by eight Year 6 students (aged 11 or 12) as they completed a task that involved the construction of a model using computer software. The participants were required to interpret two pictorial elevations of a pre-designed model and then construct the model using the simulation software. The students who were able to effectively use the perspective and orientation functions from the software package were more likely to accurately complete the task.

This paper explores the way in which six young children (6 year olds) interpreted and made sense of screen-based images on a computer. The participants were asked to interpret a screen image as they were introduced to a construction-based software program. Some of the children could not make links between the screen images and the intended three dimensional (3D) representations. The participants who were able to make these connections tended to relate the screen objects to life-like images or used analogies to describe concepts associated with depth perception, perspective and orientation.

This paper reports on the outcomes of one component of a study carried out during 2001 to assess the impact of graphics calculator use on Year 12 school-based assessment in a situation where access to the calculator is assumed in the external tertiary entrance examination. The focus of the paper is on test and examination items that form part of the school-based assessment programs contributing to a student's final grade in the Western Australian Year 12 Tertiary Entrance Examination (TEE) subject Applicable Mathematics. Assessment items developed by participating schools during 2001 for use with Applicable Mathematics were collected and coded. While it was found that each of the participating schools was incorporating use of graphics calculators into questions to approximately the same extent, there was a wide variety of usage apparent within some curriculum components. This suggests there is scope for wider incorporation of the technology.

A qualitative investigation of young children's beliefs included the gaining of insights into beliefs about measurement and its relationship to mathematics, the focus of this paper. Through the use of a range of procedures in one-to-one interviews, conducted with eight children over a period of five months, beliefs were found to be complex and idiosyncratic. Children may see measurement differently from how adults see it, and may hold different meanings for measurement terminology. This has implications for teachers of mathematics in the primary school.

Error patterns in computation have been well researched for written computation; but there is little research into the error patterns common among students computing mentally. The responses made by 3035 students in Grades 3 to 10 were analysed and the most common errors made at each grade level ascertained. Common patterns of errors are described. Possible reasons for these errors and implications for the classroom are discussed.

European mathematics is offered to indigenous people around the world, 'gift-like', as a passport to success and future development. Most Maori believe in this idea; but some ask at what cost, both culturally and intellectually. The promotion of mathematics may be ethnocentric, but it is not, we believe, offered with malice. The bearers are probably unaware of the inherent dangers as the receivers (Barton and Fairhall, 1995, p. 1).

This paper examines two early high school students' beliefs about mathematics, writing in mathematics and their writing audience. Students who do not value writing as being part of mathematics or are unable to conceive of the needs of an examiner as their audience are unlikely to perform in mathematics assessment tasks to their ability level. Teachers may need to increase students' awareness of the value of writing in their learning of mathematics and how different audiences will have different requirements for their writing.

Results from part of an earlier survey of teachers and academics were compared with student teachers' responses to the same open question. The question asked respondents to detail features of quality mathematics lessons. The students identified many of the same components of effective teaching as did the experienced respondents, but there were some notable differences and omissions. The resulting information has implications for planning of future teacher education experiences. Most importantly, it raises questions about the relative emphases in pre-service mathematics education courses that need to be placed on general pedagogy as opposed to mathematical content and pedagogical content knowledge.

In this paper, we report on a study where students engaging in model-eliciting mathematical problems with collective discourse mediated by Computer Supported Collaborative Learning (CSCL) software achieved the kind of authentic, sustained, and progressive knowledge-building activity that until now had not been achieved in CSCL software mediated mathematics communities.

It is argued, using Levinasian ethical philosophy, that dominant approaches in curriculum design and implementation are more than just an ineffectual system forced on teachers, they are also injurious to those who work in schools because they manipulate and erode a primary ethical relation between people. Although these approaches are used in several subject areas besides mathematics, mathematics is the key subject area for a more general critique of these approaches because it is the "paradigm case" subject area.

Ten children in Year 7 of primary school were given two numerical data sets to represent graphically - one with 10 pieces of data, and one with 30. Increasing the size of the data set had little effect on the children's organisation of the data. In general, the children were no more likely to represent the larger data set in an organised form than the smaller data set. The majority of the children represented the smaller data set without any reorganisation. Similarly, the majority of the children required prompting to organise the larger data set. Mathematical ability did have some effect in that some of the more mathematically able children found it easier to organise the data than their less able counterparts. Possible explanations for the results are explored and the implications for teaching and the curriculum are discussed.

An important aspect of concept development in primary school students is the development and use of models, symbols, and representation. This paper deals with conceptual development of notions of multiplication and ways primary school students use representations such as equal groups, arrays, region models and stories to extend primitive or simple conceptions of multiplication to more complex, sophisticated ideas leading to more effective transfer to examples involving decimal fractions.

When starting German 6th grade, students already have broad prior experiences with decimal fractions as numbers for measuring quantities. Diagnostic written tests and individual interviews were used to examine how far this experience promotes the transition to abstract decimal fractions shortly before they are taught systematically. Results showed that this transition is still only just beginning in many students, and that the necessary extension of the place-value system is still hampered by incorrect transfers from natural numbers.

Counting On is a systemic mathematics program that targets low achieving students in the final year of government elementary and the first year of government secondary schools through the professional development of their teachers. During 2000, Counting On was implemented in 40 secondary schools, involving more than 600 students, 120 school teachers and 40 district mathematics consultants. In 2001, the implementation was extended to approximately 80 secondary schools and 80 final year elementary classes which 'feed' these secondary schools, involving some 1400 students, 320 teachers and 40 district mathematics consultants. This paper reports on evaluations of the program in 2000 and 2001, with particular reference to students' movement across the levels of conceptual development in place value contained in the numeracy framework which underpins the program.

The authors have been working, separately as well as collaboratively, on the area of teachers' beliefs about mathematics and its learning and teaching for many years. In this paper, they report on the use of one instrument to ascertain measures ofthe beliefs of a total of 1254 primary school teachers in a number of different cultural contexts. The stimulus for this combined, cross-cultural study has been the variations in results achieved by students in international studies such as the TIMSS and TIMSS-R and the strong evidence that teachers' beliefs about mathematics and its learning and teaching play a critical role in determining how teachers facilitate their students' learning. Hence, differences in teacher beliefs may be one reason for the measured differences in student achievement. The results of the study clearly show that teachers' beliefs in all of the samples can be adequately described in terms of the statements in the survey and that there are there are marked differences in response to these statements across the different groups of teachers.

The study reported in this paper is concerned with the investigation of primary children's real-world problem solving where particular research interest relates to the dynamics of group problem solving in an authentic classroom setting. Initial interpretative analyses of the video-taped group interactions revealed a kind of "interweaving" of the individuals' lines of thought. The application of an interaction model from social psychology lead to the identification of four basic types of interaction sequences in the group work episodes.

This study explored transnumeration-type thinking exhibited by eleven and twelve year old students as they worked on a statistical investigation. Some aspects of this thinking are identified and described. The analysis raises issues about the development of such thinking and about how teaching might build on students' thinking strategies.

Computer Algebra Systems (CAS) potentially offer students a powerful tool. However the mere availability of CAS does not enhance teaching and learning; students (and teachers) need to use it effectively. This paper presents a framework for components of Effective Use of CAS and illustrates its use as a guide for monitoring the progress of students. The results presented highlight the importance of considering the interaction between technical and personal aspects of Effective Use of CAS.

A Year 3 angles unit using the Teaching for Abstraction paradigm was developed on the basis of earlier empirical research and trialed by 12 NSW teachers. Analysis of data submitted by the teachers showed that the unit was generally successful but a number of minor areas for improvement were identified. An analysis of independent classroom observations of students working on the angles lessons is reported in the present paper. It was found that the difficulties students experienced could be classified as matching, measuring, drawing, and describing errors. The results clearly support the hypothesised steps in learning by abstraction.

Sixteen Year 3 students, in groups of four, worked collaboratively on numeration tasks, using either conventional blocks, or computer software. Participants were interviewed individually before and after group sessions to ascertain place-value understanding. From a teaching experiment design, using a grounded theory approach, emerged previously reported categories of response, including counting approaches, grouping approaches, and "face-value" thinking. A previously unreported construct emerged, the "independent-place construct". Teachers of young children should familiarise themselves with these common approaches to place-value tasks.

A survey ascertained teachers' knowledge about and use of hand-held calculators in primary mathematics classes. This paper focuses on 341 teachers' experiences of Professional Development(PD) on the use of calculators in teaching mathematics. Only nine had experienced such a workshop in the past two years while 10% had during their career. Approximately 40% regarded themselves as poorly prepared for calculator use in their teaching. A face-to-face hands-on PD session conducted by an 'expert' was the teachers' choice to develop their calculator pedagogy.

Many topics within the middle school mathematics curriculum connect to the concept of proportion. Interpretation of proportion situations and understanding of methods for solving proportion problems provides a structure than can be applied to other related topics. As a major resource for secondary mathematics, the extent to which popular textbooks link proportion-related topics was the focus of this study. Our analysis revealed little connectivity of ideas, confusing definitions and frequently illogical calculations. Questions are raised as to the messages texts send to students.

This paper will provide an overview of the Middle Years Numeracy Research Project conducted in Victoria from 1999 to 2000. Results from the final stage of the project indicates that teachers working in professional teams in a coordinated and purposeful way do make a difference to numeracy outcomes for the majority of students. However, evidence from individual interviews with a sample of 'at-risk' students suggests that schools still. face a significant challenge in recognising and dealing with difference at this level. Implications for further research and current practice are discussed.

This paper presents a narrative inquiry into the personal dimensions of mathematics teacher education. It describes my experiences as a teacher educator exploring the use of personal theories to develop and share pedagogical conversations with prospective teachers. Personal theories (PTs) can be thought of as a set of beliefs and practices that are personally meaningful, and based on experience. This narrative describes the developmental process involved in establishing personal theories, and outlines three levels of authority and ownership that were evident in final versions of prospective teachers' personal theories.

Over 3000 students in Grades 4 to 10 completed about 10000 tests to identify and track their conceptions about decimal notation. This paper reports the incidence of two particular error-patterns on these tests and how this incidence changes over time, from cross-sectional and longitudinal viewpoints. At some time, approximately 3% of students answered almost all test items incorrectly and over 10% were unable to compare decimal numbers with the same tenths and hundredths digits. Explanations for these error-patterns are given.

This paper reports some initial results from a research projectl aimed at identifying and describing teaching approaches which are effective in improving numeracy learning· in primary classrooms. To ascertain the influences on the teachers' intentions we posed a task that sought some indication of what aspects of a topic the teachers might consider when planning a set of experiences for their students. It seems that the teachers had a limited understanding of the ways that the concepts develop and the types of experiences that they might choose for their students. We argue that teacher education could focus more on key aspects of teaching the important topics of the mathematics curriculum.

This paper reports analysis of discussions between mathematics educators about ways in which the contexts in which classroom tasks are embedded can have the effect of alienating some students, especially those who may have less appreciation of the purposes of schooling. The results suggest that some care could be taken in the choice of and introduction of contexts of mathematical problems, in order to ensure that all students relate appropriately to the contexts as well as the mathematics.

Perceived pressure to cover content in limited time often results in secondary mathematics teachers valuing "quick-ftx" professional development activities that focus on classroom materials over extended programs that promote professional growth through discussion and reflection. The Australian Capital Territory Quality Teacher Program (Mathematics) commenced as a collaborative exercise in developing curriculum materials, but evolved into a process of reflection. This evolution, and the resulting change in teachers' beliefs, is described and discussed from two perspectives: that of the facilitator and a participant.

The principal purpose of this paper is to critique concepts of 'content knowledge' and its 'pedagogical application' as they are currently fashioned in the field of mathematics education. Drawing upon work which is primarily philosophical in character, I propose a view of teaching based on my empirical study with preservice teachers in schools. In the first instance I introduce theoretical ideas and suggest how they might function in the process of data analysis. Second, I use these ideas to analyse instances of teaching knowledge in production. Finally, I argue for a conceptualisation of pedagogical content knowledge which recognises the reciprocal constitution of knowledge and subjective experience.

This paper investigates instruction that assists young children generalise their mathematical thinking in terms of quasi-variables (Fujii & Stephens, 2001). Sixty-five year 2 children participated in a lesson on investigating the patterns in multiples of 5 and 3. From the results it seems that young children are capable of understanding generalising and that specific features of instruction assist this process. The materials, the types of activities and the questions asked by the teacher all play an important role in assisting young children abstract underlying mathematical relationships.

As part of a larger project studying school students' understanding of statistical variation, 92 students in grade 7 and 90 students in grade 9 participated in a unit of work related to the chance and data curriculum, emphasising variation with respect to the topics covered. Students completed pre- and post-tests devised to assess understanding before and after the lessons in the unit. This paper reports on the teaching arrangements for the classes taking part and the change in performance after the unit.

Research into teacher beliefs has a long and troubled history and there is evidence that it is ignoring the lessons of the past. Why do researchers need to continually report the mismatch between espoused beliefs and enacted beliefs? Behaviour is driven by action theories that are different from espoused theories and both are difficult to unravel, but attempts have been made with varying degrees of success. What can history tell us about the definition of belief that focuses upon a global statement or upon a specific behavioural belief statement? Is it still useful to attempt to place teachers upon a continuum based upon their espoused beliefs? Research into the planning, thinking and decision making of teachers should provide insight into classroom events and be worth understanding because of the connection with student learning and performance. This paper discusses these issues and their implications for teacher education.

A tool constructed to examine a task's potential to stimulate and support creative mathematical thinking was derived from a hierarchy of cognitive activities students employ as they solve unfamiliar problems. These cognitive activities informed the formulation of categories of task features that support creative mathematical thinking. Use of this tool is illustrated in a study of the mismatch between potential student responses to a task and actual student responses to the task as implemented. The sufficiency of the tool is discussed.

In this paper geometric eigenproductions of primary school children are analysed in spatial reconstruction experiments: A donor draws an addressee- and purpose-specific picture of a spatial building, and an acceptor tries to reconstruct this object only by means of the partner's drawing. Comparing the original building with its reconstruction, the two partners evaluate the informational substance of the drawing. This paper discusses, in how far the drawings are reconstruction-efficient and how they are improved in the course of the interaction.

Boys' task involvement and ego orientation motivational goals in mathematics were measured prior, during and one year after the introduction of co-education to a single sex boys' school. All primary and secondary boys enrolled in Grades 3 to 10 in the single sex school were followed over the three-year period. Comparisons were made over time across grades and cohorts with analysis of variance and hierarchical linear modelling. While there were significant effects for boys' task involvement in the transition from single sex to coeducation, particularly at the primary school level, there were no significant changes in boys' ego orientation goals.

This paper reports the patterns of performance of almost 10,000 Year 4 to 6 students whose teachers participated in a project (ANP) designed to improve their professional knowledge, skills, and confidence. An important component of the project was the number framework, informed by work with Count Me In Too, and extended to cater for students at Year 4 and beyond. Data collected at the beginning and end of the project provided validation for the upper levels of the framework. A hierarchical sequence of development was evident, with additive part-whole thinking acquired before multiplicative part-whole thinking, which came before proportional reasoning. Support was also found for the idea that knowledge about numbers provides a necessary foundation for strategies.