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In this paper I argue that we need to analyse critically our currently popular theories of mathematical learning and instruction. I initially raise a number of issues warranting attention and begin by reviewing briefly a selection of theories, confining my discussion to socioconstructivist perspectives, situated cognition, and cognitive psychology science. I then undertake a critical analysis of these theories, with a focus on the curriculum examples that have been used to support these theories. Finally, I propose a working model that might assist us in advancing mathematics education and research into new times.

In recent years mathematics educators have begun to draw on academic disciplines beyond mathematics and traditional, cognitive psychology as resources for theorising. Sociology, anthropology, and even psychoanalysis have formed the theoretical framework for some researchers. Mathematics education has become more conscious of culture, in all sorts of ways. But in cultural studies, writers have been addressing what it is to be and to communicate, and therefore to learn and to teach, in ways which of most of us in mathematics education have not been aware. In this paper I attempt to examine and use postmodem ideas to look, in particular, at mathematics teaching.

This study explores the sources of influence on and the level of coherence of one teacher's beliefs and practices in relation to mathematical problem solving. Rose regularly uses problem solving approaches in her Year 2 class as she believes that this makes mathematics learning meaningful. Her beliefs and practices have been strongly influenced by her own experiences as a learner of mathematics as well as by advice received in postgraduate education and in curriculum documentation. Her beliefs are coherent when explored using a variety of data collection methods.

This proposed study is designed to investigate a rapidly expanding area of activity in mathematics education that has not been researched and theorised. Arguably, mathematics education is one of the most internationalised areas of the curriculum. TIlls paper outlines a longitudinal study to investigate issues in the internationalisation and globalisation of mathematics education in 3 regions with ties to Australia. It is a part of a workshop that aims to obtain feedback from peers and trial the instruments to be used.

This paper discusses the learning's from an action research project conducted in collaboration between beginning teachers and a group of university researchers in mathematics and science education. Participants formed action research cells based on their common interests. This paper presents the overall structure of the project and discusses the resulting benefits to the participants and the benefits and . limitations of the use of action research to assist transition into teaching.

Four groups of Year-8 students completed a set of 2-stage Pythagorean problems under differing conditions. One group completed the set with specific goals (find x) without instruction; the second and third groups were encouraged to either work forwards or backwards from the goal (x); whereas, the fourth group completed the set with unspecified goals (no x's). On a subsequent set of test problems, the no-goal group demonstrated a superior performance than the other groups in applying Pythagoras' Theorem.

The detailed analysis of four probability experiments conducted with Grade 5 and 6 students revealed trends and patterns in both the group and individual data. These results suggested that certain variables in the experiments, such as particular sequences of outcomes and the confirmation/refutation of student predictions, influenced the students' decision making strategies. The use of video recordings of deliberately controlled probability experiments offers the potential to systematically explore these influential factors with large samples of students.

Based on an approach to power derived from the work of Michel Foucault, this paper describes the development of a framework for analysing power relationships in small groups of students working on collaborative activities. Student-student interactions were observed, videotaped and transcribed. Key features that emerged were techniques used to control the flow of the discourse in the group and behaviours which influenced the mathematical knowledge constructed. Other factors include body language and the control of resources.

This paper reports on a study of the decimal-number numeration knowledge held by 112 Year 5 students who had completed formal instruction in tenths and hundredths. It explores the interaction between available and accessible place-value and regrouping knowledge. Available knowledge was elicited through diagnostic test items; accessible knowledge was elicited through an error analysis of the students' numeration procedures used in addition and subtraction algorithms. The results showed that performance varied markedly between classes (indicating an instructional effect), that regrouping was more difficult than place value, and that there was generally a direct relationship between available and accessible knowledge (although there were many instances of high-available and low-accessible knowledge).

This paper explores curriculum development issues in nine regions in five federal countries. The focus is on the curriculum contents, the development process, and the role of teachers in development. The study explores alternatives that others may consider when reviewing their curriculum.

This paper reports the findings of an investigation that evaluated the impact of the Early Number Project conducted by the NSW Department of School Education in 1996. It includes information obtained from a questionnaire distributed to all participating classroom teachers. Generally teachers benefited from their involvement in the project both professionally and personally. The classroom-based model of professional development was determined to be a major factor in the success of the project.

Learning to operate algebraically requires assimilation of new mathematical concepts and procedures. Current literature identified a gap between arithmetic and algebra and proposed a pre-algebra level. This paper reports on a longitudinal study that investigated students' readiness for algebra, from a cognitive perspective, to determine what constitutes a pre-algebraic level of understanding. Thirty-three students in grades 7,8, and 9 participated. A two-path model depicts the transition from arithmetic to pre-algebra to algebra; students' understanding of relevant knowledge is discussed.

The invisible wall project analyses problem solving processes of children in grades 3-4 and 8- 9. By now, we have transcribed more than 160 interviews; as a consequence, the qualitative interpretive methods were complemented by quantitative methods. Besides analysis of difficulty of tasks and learning while being tested, we describe the process of problem solving by action profiles. Using the method of divergent coding, we introduce interpretive methods to better understand action profiles.

Students from elementary, intermediate and high schools were asked why they considered they were assessed in mathematics. The small sample (n=559) came from the US, the Netherlands, and New Zealand. Some responses were consistent across countries. Most students thought they were assessed in order that teachers might find out what they know/understand. Some cultural differences are suggested by the data illustrative comments are included.

An extensive study of the literature on teachers' views of mathematics and mathematics teaching revealed inadequacies in the available frameworks for structuring and reporting on research. In this paper a new framework is developed for analysing and describing primary school teachers' knowledge and feelings about mathematics teaching and learning. The framework; 'Teacher Type Table', was developed after surveying 107 primary school teachers in suburban Melbourne. The teacher type table is presented here as a model for use in the professional development of teachers.

This study investigated teachers' understandings of the role of learning activities in secondary school mathematics. A qualitative study involving interviews and questionnaires was conducted in government and non-government schools. Most teachers in the study were supportive of the use of learning activities in the classroom for both teaching and learning purposes but were aware of constraints of time, adequate resources and curriculum requirements.

Students, in triads in a near-classroom environment, were video-taped as they worked on interpreting and representing supplied data. Their responses were categorised using the SOLO taxonomy. Representation skills varied from copying out some of the data to relating two variables graphically; interpretation skills varied similarly. Moreover, there appeared to be connections between the two skills. The study also considered the nature and effectiveness of the collaboration which took place within groups.

An important area of mathematics teachers' expertise is their content knowledge. One way to characterise the quality if this knowledge is to examine the organisation of the various components that form the content knowledge base. In this paper we describe the procedures we have used to generate estimates of the state of connectedness of mathematical knowledge of teachers. We illustrate the procedures by focusing on a single teacher's knowledge of the concept of square.

A, Ministerial Report has questioned again whether beginning teachers are adequately prepared at universities. The Report proposes measures to ensure this does happen such as a testing regime in mathematics and literacy at the completion of courses. This paper provides data that indicates some beginning teachers do not show competence in the mathematics they will soon be teaching. Hence there is a basis for concerns raised in the Ministerial Report. However it is argued that the solutions in the Report are somewhat naive and an inadequate response to this problem.

This study is the second in a series designed to explore how class inclusion concepts evolve. Class inclusion concepts in this context is the ability to have an overview of possible relationships that exist among figures, and is an important characteristic of Level 3 thinking in the van Hiele Theory. In depth interactive interviews were undertaken with 24 capable secondary students concerning how they view linkages among six different quadrilaterals. A developmental pattern is identified.

This paper argues for the need to address children's structural understanding in dealing with mathematical problems. In support of this argument, a study that investigated children's structural understanding of 2- D and 3-D combinatorial problems, via a range of thought-revealing tasks, is reported. The results raise a number of issues for further attention, including the discrepancy between children's accuracy and their structural understanding, and the lack of significant correlation between children's graphic and symbolic representations on most of the problems. Individual profiles of response highlight the importance of rethinking our interpretations of understanding.

This paper explores some of the effects on mathematics education in the VET sector where economic rationalist ideologies prevail, and hegemonic industrial values subsume educational values. In a sector which carries considerable investment of public and private funding; it may be expected that mathematics, which has a high public profile generally and with sections of business and industry in particular, should be an important focus. Somewhat paradoxically, there is an apparent lack of status.

Pre-service teachers participated in peer and self assessment activities during a seminar series focused on current issues in mathematics education. Their feedback comments to each other and the various assessment forms they used were analysed to determine what changes took place in what they perceived to be important issues, how the quality of their comments changed over time, what they felt they learned from peer and self assessment, and how they developed in their capacity to reflect critically.

This study gathered teachers' descriptions of dilemmas they faced when teaching young children mathematics. It was considered that asking teachers to describe teaching dilemmas would enable them to identify a problematic aspect of their teaching practice, and lead to reflection, evaluation and refinement of their practice so that their students' mathematical learning would be more powerful. In analysing the dilemmas described by teachers, key elements were identified. These elements have important implications for providing effective professional development.

This paper reports on a study which investigated patterns of collaborative metacognitive activity in senior secondary school classrooms. Although peers working together on mathematical tasks may enjoy the metacognitive benefits of being able to monitor and regulate each other's thinking, collaboration does not guarantee that they will achieve a mathematically productive outcome. Analysis of a videotaped lesson transcript illustrates how metacognitive uncertainty, itself a trigger for collaboration, remained unresolved when students did not have the means of validating their solution.

Four mathematics teachers with different preferences for the use of visual strategies in solving problems are compared in their use and support of visual methods in their classrooms. Presmeg (1985) found that no teacher who displayed a strong preference for non-visual strategies in problem solving made substantial use of visual methods in their classroom. In this study, the teacher who least preferred to use visual strategies to solve problems displayed a strong use of visual methods in the classroom.

Research has shown that mental computation is a valid computational method which contributes to mathematical thinking as a whole (e.g., Sowder, 1990). This paper reports on a pilot study of young children's understanding of mental computation, and compares the mental architecture of two mental computers, one flexible and one inflexible. Further questions which have been raised as a result of this pilot study will also be discussed.

This paper reports the analysis of videotape and interview data from four year 8 mathematics lessons from the perspective of student cognitive engagement. The study attempted to contribute to our understanding of cognitive engagement by locating empirical evidence for its occurrence within the classroom. On the basis of the data we have examined, cognitive engagement can be consistently recognised by specific linguistic and behavioural indicators and appears to be promoted by particular aspects of the classroom situation, the task, and the individual.

With the rapid growth of Maori language schooling in New Zealand in recent years, and the development of a Maori mathematics curriculum, there has been considerable demand for Maori primary teachers who are both fluent in Maori and competent in teaching mathematics. Trying to meet this demand has proved a real challenge. This paper outlines and discusses a number of factors faced by Maori mathematics teacher educators in preparing preservice Maori teachers for the primary classroom.

The Mathematics curriculum has undergone extensive change over the past few years and much attention has been directed towards coping with the demands of this change. However curriculum has social implications and it is important to examine the intentions of curriculum innovation, which, in turn, leads us to consider the issue of control of the curriculum. This paper draws on the work of Max Weber as it attempts to fashion an explanatory model that will provide a conceptual framework for the consideration of curriculum change within the context of curriculum control.

Sixth grade students' responses to a task which involved matching common fractions and decimal fractions were examined. Of interest was their understanding of relationships between these symbolic forms, evidenced in explanations offered in an interview setting, and the role of the follow-up question in helping them express their thinking. A most favored explanation involved relating decimal and common fractions to a unit of 100.

Northern Territory secondary teachers' awareness of curriculum change in mathematics and their thoughts on its impact upon teaching and learning are reported. A survey followed by semi-structured, in-depth interviews were used for data collection and analysis was done inductively. Teachers were aware of a number of changes and expressed many concerns about their impact, especially in relation to appropriate in-service opportunities. Discussion of findings in relation to geographical, educational and cultural features of the NT is included.

Current debates about numeracy in the UK focus on the availability of calculators but although standards of numeracy are poor, they have not changed significantly since pre-calculator days. It is the way numeracy is taught which is the problem, with or without calculators. Effective use of calculators should be associated with the concurrent development of students' number sense and mental methods. Calculators can enhance the teaching and learning of mathematics if used as a focus for discussion and reflection on learning.

The research presented in this paper confronts head on the difficult question of teacher change. It was catried out with preservice teachers in a mathematics methods course in the second year of their teacher education program. Collected data reveal how students' prior experiences of institutionalised mathematics reveal patterns of subjectification which actively undermine the future implementation of more investigatory methods of teaching. I use the poststructuralist concepts of knowledge, positioning and subjectivity: initially as analytic tools to expose the coercive operation of the mathematics discourse; and collectively, as a conceptual base from which to think about possibilities for change.

Problem solving is an important activity in algebra, as all mathematics, and research has shown that affective factors can be crucial to mathematical progress. This research study considered 105 15 year old students, and investigated the relationship between their affective domain and algebraic problem solving ability. Analysing students performing at different ability levels, our test and interview data have enabled us to describe a preliminary model of the interaction between self-concept, self-perception, enjoyment, interest, motivation, anxiety and problem solving ability, describing how they may be influencing students' learning.

As part of a study focusing on mature age students in tertiary mathematics courses, a survey was administered to a large number of fIrst year mathematics students. Data on personal background, self-perceptions as learners of mathematics and perceptions of the tertiary learning environment were gathered. Responses from school leavers and mature age students were compared. Differences were found on several background factors and some students! attitudes and beliefs. Generally, the mature age students were more satisfIed with the tertiary learning environment.

This paper examines the way in which two young children engage in a computer-based problem solving activity that required a degree of visual-spatial reasoning. The case study traces the children's spatial-visual understandings and looks at the way in which these understandings were activated and developed through the particular activity.

Mathematics curricular statements refer to problem solving, in particular word problems, as a means of relating the curriculum to "reality". Research indicates that pre-service primary teachers have little knowledge of woro problems despite the strong emphasis on their importance within the curriculum. This study reports on the understandings of word problems types of pre-service primary teachers. A number of activities were given as part of a mathematics education method course. Educational implications of these findings are discussed and recommendations proffered as to the amelioration this situation.

The tenn protasis is used to describe an epigrammatic expression which, in an appropriate context, stimulates questioning of assumptions and behaviour. By extension, the word protasis is applied to a disciplined technique for fostering professional development through the use of protases, and a theory is put forward of when and why protasic techniques work.

While considerable attention has been focused on forms of teacher knowledge (eg. Shulman, Elbaz, Petersen, Carpenter et al) as essential components of expert and effective teaching, but knowing-that, knowing-how, and knowingwhy are insufficient to enable a teacher to act upon that knowledge. What is critical is that teachers know-to act in the moment-by-moment unfolding of a mathematics lesson. We distinguish different types of knowing, and following Gattegno, relate these to levels of awareness. Our methods use descriptions and illustrations such as a fine-grained study of one teacher involved with innovative materials for teaching algebra to 13 year-olds to help the reader experience distinctions which we have found fruitful. Furthermore, the situation is self-referent, for what we want children to learn is to to know-to act themselves when formulating or solving problems. Drawing on the study, on our experience, and on the experience of others, we move towards a psychology of how knowing-to arises and can be supported.

Since the introductio'n of the Victorian Certificate of Education in 1989, assessment in VCE mathematics subjects has included school-based tasks undertaken over an extended period of time. This paper reports on an evaluation of the latest of many changes that have been made to these tasks since their introduction and examines whether they have retained their validity for assessing problem solving and investigative project work. Student use of powerful technology is identified as an emerging issue in their future conduct.

One perennial unanswered question for debate among mathematics educators is that of 'What is mathematics?'. Polarised views range from those who point to traditional classical Euro-centric knowledge as the definitive answer to those who wish to widen the definition to include mathematics from every instance of mathematical activity. This paper explores the dilemma thus raised and suggests that the inclusion of a third element may help to clarify the issue.

This paper looks at the different ways teachers interpreted and represented algebra for their students. In this study, teachers are viewed as mediating between the student and the mathematics content. The teachers in this study differed widely in their epistemological and pedagogical beliefs about algebra but tended to teach similar content. The results of this study suggest that teachers' choices are a consequence of particular beliefs about algebra and that these beliefs are attributable to cultural influences rather than academic considerations.

Student understanding of odds is explored by analysis of response data collected in 1993, 1995 and 1997. Students were asked to interpret a newspaper headline, "North at 7-2". Responses included interpreting the numbers as the score, and in three contexts of expression involving chance, frequency of wins, and betting. Levels of responses were assigned according to the SOLO developmental model. Longitudinal development for individuals was observed, and females tended to interpret the numbers as the score, while males were more likely to respond in the context of chance or betting. Levels of response from Grade 6 and 9 students in 1995 and 1997 were lower than in 1993.

The project children playing arithmetic games with strategic components concerns with. the existence and with types of strategies used during playing. In a qualitative study with an interpretive background we have observed primary school children of different grades playing arithmetic games with strategic components. In this paper we shall present in detail the procedure of children of grade 3 playing an addition-game with strategic character.

In response to an interview question on features of classroom organisation that support the development of mathematical understanding, all four teachers interviewed referred to the benefits of "ability grouping". This was seen as a means of structuring learning conditions so that teachers could attend to children with similar levels of understanding. Subsequently, achievement grouped classes were videotaped. Analysis of the data showed that interactions within this mode of organisation demonstrated the same problematic features as whole-class teaching. It is concluded that if achievement grouping is to be used, other strategies need to be used to make mathematics pedagogy inclusive.

Some ethnomethodological research techniques that can be used to analyse social activity in mathematics classrooms are examined. A method where sections of video-audio tapes were transcribed to facilitate detailed inquiry, and individual "photographs" were used to reproduce some of the data not available in transcriptions, is described. Snippets of the original data were also available in movie format. This paper outlines some of the advantages and limitations of each of these three forms of visual data, and raises issues of anonymity and subjectivity. Ethnography

This paper describes the integration of multiplication and division strategies within a research-based Learning Framework in Number (NSW Department of Education and Training, 1998). The framework, consisting of six levels of multiplication and division knowledge is described in order of increasing sophistication of modelling, counting, grouping and sharing processes. Assessment tasks show a progression from initial grouping and counting to abstract composite units, to repeated addition and subtraction and to multiplication and division as operations. The framework, which is integral to the Count Me In Too Project is currently being implemented in years K-2 in NSW schools.

This research project investigated the use of computer software (ANUgraph) in the development of calculus concepts. One group of students completed the calculus unit following a traditional teacher-exposition method. The other group covered the same content using a graphical approach, which relied heavily upon the use of the computer software to introduce and reinforce key concepts. Results indicate that the lessons proved successful in increasing students' understanding of the fundamental concepts of calculus.

Children (1 0-11 years) were asked to match short melodies and line graphs. Evidence was found for significant visual/graphical influences in the matching process (as well as previously established melodic factors). Intramodal tasks were performed better than cross-modal task, and within those categories, visual first items were performed better than melody-first items. Mathematics ability was a significant main effect, but evidence pointed towards an effect of mathematics experience more than aptitude. Musical ability had only a limited effect.

This paper describes the development of a questionnaire to identify Separate and Connected Knowers (Belenky, Goldberger, Clinchy and Tarule (1986) in mathematics. Justification for the selection of items is provided from the work of Belenky et aI (1986), Buerk (1985), Becker (1995, 1996), Koch (1996), and Erchick (1996). The questionnaire was administered to 67 university students as part of a larger study. Interviews within the main study showed that the questionnaire had served its purpose; it had accurately identified a group of Separate Knowers in mathematics.

Children in Year 2 and 4 were developing the concept of angle through their responsiveness, manipulating of materials, and interactions with others. Their selective attention to aspects of the materials, interactions, or their own imagery and thinking assisted the development of the concept.

The standard of numeracy skills of young Australians continues to be of concern to mathematics educators. teachers. parents and politicians. This paper discusses whether a mathematics intervention program developed for Grade 1 warrants extension into Grades 3 and 4. The focus is on 1997 data that show that although Grade 3 and 4 students had made significant progress since Grade 1. there is still need for additional assistance in Grades 3 and 4.

Sixteen Year 3 students were interviewed individually about two- and threedigit numeration concepts, before and after ten groupwork sessions. One group of questions asked students about values represented by each digit in a two-digit number. Responses revealed that some students had not developed the idea that the "tens" digit represents a collection of ones, but believed that it represented only its face value. Comparison of results before and after the group work sessions showed that a number of participants had improved understandings of values represented by individual digits at the conclusion of the study. These results have implications for how place-value concepts are developed with place-value blocks.

The paper examines the changing conception of giftedness over the last four decades, and characteristics of mathematically gifted students. Ways of identifying mathematically gifted students in a study conducted with year 6 students are described. The Renzulli Enrichment Triad model for catering for gifted students is outlined. Case studies of a number of school and individual teacher's programs which cater for mathematically gifted students are described. A professional development module [CD ROM] designed for teacher inservice will be demonstrated.

In this paper we review major theoretical perspectives on differences between students in their thinking and classroom talk, and conclude that Bakhtin's concept of voice provides a powerful way of theorising difference that draws attention to the multivoca1ity (inherent diversity) of individual ideas and utterances. It is the orchestration of the diversity of voices that enables new insights to emerge both collectively and individually. We apply analytical tools (derived from Bakhtin) to an episode of collective argumentation focussed on the concept of infinity, and we identify key elements of everyday classroom practice that enable challenging and productive talk to occur.

Changes in attitude have been measured over a semester of mathematics in the fIrst year of Early Childhood and Primary teacher training. The unit of study is not a curriculum unit, is taught by mathematicians and explores mathematical ideas and experiences. The Fennema-Shennan Attitudes Scale was used to measure changes in confidence, effectance motivation, and usefulness. Analysis of the results indicates a challenging outcome - the only signifIcant change was a drop in their perception of the usefulness of mathematics.

In 1997, Victoria became the first state to permit the use of graphics calculators in final external examinations. The action was seen as radical for both social and educational reasons. Concerns were raised about the propriety of using the calculators and whether their use would add to existing educational disparities. With the support of the Board of Studies, a survey of secondary schools was undertaken to gauge the response to this decision and inform further action on graphics calculator use in mathematics courses.

This study explores student understanding of the table of values representation for the function concept. One hundred and seventy eight students in Years 8, 9 and 10 across three schools in Melbourne undertook two pen-and-paper tests which sought to uncover the nature of the links students made between different representational forms for function. For the table of values representation there are many students who are operating in an arithmetic or pre-algebraic frame across the three year levels. There are low levels of response for algebraic patterning and indications that most students are at a rhetorical or syncopated stage of symbolism.

This paper presents research into the effect newsletters had on parental perceptions of junior school mathematics. Information on classroom content allowed parents to recognise and individualise mathematics that occurred spontaneously in the home. Interviews of parents and teachers revealed discrepancies in the way mathematics was viewed. All the teachers and participating parents were positive about the benefits of the newsletters.

Prospective primary school teachers often see mathematics as a rigid, inaccessible subject. Their fixed ideas about the nature of mathematics can often impede their learning, and future teaching. This paper investigates a web-based intervention which encouraged dialogue about mathematics between an international community of mathematics educators and the prospective teachers' local learning community. Data from two surveys and from the prospective teachers' reflective journals show that this intervention encouraged prospective teachers to examine and evaluate their own beliefs.

The set textbook has a strong influence on what takes place in the mathematics classroom, especially at the secondary level. This paper reports on the preliminary stage of a project aimed at developing more effective textbook presentations. In this phase of the project, textbooks are being examined to identify the types of messages about mathematics and its teaching and learning inherent in their presentations and to identify the sources of the underlying messages.

Teacher education students in their flrst year of university were invited to take part in an eight week program involving solving problems and posing problems on the basis of the given problem. Data concerning two of the eight students who agreed to participate, have been reported in terms of their problem solving strategies, their progress in problem solving skills and their expressions of cognitive and affective factors related to the study. It was found that the requirement to pose a similar problem to a given problem helped clarify thinking and that other important factors were the realism of the problem, the capability for visualisation and the nature and speed of feedback.

The paper describes some preliminary findings from a project in which teachers from two primary schools were observed over one school term as they implemented their own plans for encouraging number sense in their classrooms. Ownership of the entire project was invested in the teachers involved. Comment is made on some of the impediments to change and on one teacher as she implements mental computation activities.

The project pupils work on problems with a goal which cannot be reached ("invisible wall project") analyses problem solving processes of children in grades 3-4 and 7-8. So far, we have described basic components of problem solving abilities by using interpretive methods. We now want to apply statistical methods, e.g., for comparing abilities of younger/older children. Consequently, we have complemented the qualitative interpretive methods by quantitative methods. The paper describes the methodological background of both sides of the project.

This paper reports the results of a test of decimal understanding based on choosing the larger number from pairs of decimals. Ten incorrect ways of thinking about decimal notation are described. The testing of 2517 students from Grade 5 to Year 10 enables reporting of the incidence of misconceptions according to both a primary and a refined classification. Some variations from previously reported results are noted.

In 1997. 866 Kindergarten students, including 47 Aboriginals or Torres Strait Islanders. participated in the Count Me In Too Project. Their progress in early arithmetical strategies, forward number word sequences. and numeral identification was examined. Progress was compared with expected syllabus outcomes and it was found that the majority of the students met or exceeded those expectations. Consistent with earlier and smaller-scale studies. many students began the Kindergarten year with relatively high levels of knowledge and there was much diversity in levels.

A qualitative study was conducted to investigate the impact of prior knowledge of task context on students' approaches to applications tasks am the effect of engagement with context on penormance. Prior knowledge mainly enhanced students' understanding of the task having most impact when students constructed a mental picture of the situation. Moderate to high engagement with a task context was not often associated with poor performance. Poor performance was more likely to be associated with no to low engagement. High engagement was not a necessary condition for success as the degree of engagement necessary for success appeared to be task specific.

Responses of Year 8 students to open-ended and closed mathematics questions, which addressed similar aspects of the curriculum, were compared. The responses of the students were examined and the elements of the tasks were listed. In two out of three pairs examined, the closed question was easier for students. In the other, the open-ended task was easier. It seems that closed and open-ended items may contribute productively both to classroom and specific assessment tasks, although their contribution may be different.

This study was designed to refine an instrument to explore the relationships between students' mental, written and calculator strategies of computation. A theoretical framework for studying the relationships between these three main methods of computation was postulated. This paper describes this model of the computational processes and some preliminary work carried out with middle to upper primary students in Western Australia. In addition, the questions of factors such as access to calculators, emphasis on written algorithms and mental computation influencing choice of computation method are also discussed.

This paper discusses a continuation of the study previously conducted by these authors in which a strategy based on Piaget's notion of cognitive conflict was successfully employed to reduce the incidence of mathematical misconceptions in a group of tertiary students. A question which remained unanswered in the authors' previous study was "Will the improvement persist or have we seen a short term gain which will disappear in time?" The students were tested again one year later to decide this issue.

The mathematical thinking skills project (Tanner & Jones, 1995) reported that classes which followed a course emphasising metacognitive skills were not only more successful than controls in assessments of those skills, but also in assessments of mathematical development. Ethnographic data revealed significant variations in the teaching style from teacher to teacher and was used to classify the teachers into four groups. This paper discusses the teaching styles of the two most successful groups: the "dynamic scaffolders" and the "reflective scaffolders".

Learning algebra is very difficult for many students, and one of the major obstacles identified in the research literature is a 'cognitive gap' presented by equations of the form ax+b=cx+d, which require operation on the variable for solution. In this small-scale, preliminary study we assessed the value of a computer environment which enables students gradually to investigate algebraic expressions and equations. The initial results are very encouraging and suggest that this may be a way to assist students to learn algebra in a versatile way.

During the last four years, a survey instrument was administered to 603 primary and 336 secondary teachers in both government and Catholic schools across an urban and a rural school region in New South Wales. This paper reports on comparisons concerning tile espoused beliefs about mathematics, mathematics learning and mathematics teaching of the teacher respondents and demonstrates significant differences between the teachers across the regions, types of school and gender.

This paper reports on an investigation into the impact of the availability of a hand held computer algebra system (CAS) on student performance and patterns of algebraic thinking. In particular. their performance in a range of symbol manipulation tasks in the early stages of the middle school algebra curriculum is discussed.

Secondary students who participated in a computer enhanced mathematics program expressed positive attitudes about the use of computers. They viewed computers ~ a source of pleasure, success, relevance and/or power in mathematics. Girls were more likely than boys to qualify their support for the use of computers and more likely to view computers as a source of success in mathematics. Boys were more likely to claim that computers brought pleasure or relevance to mathematics learning.

The concepts of middle-schooling and curriculum integration were examined in this study, the aim of which was to provide a research base to inform the development and implementation of models of learning in mathematics, technology and science. A number of the case studies of integration conducted with students aged 11 to 15 years are reported, and issues arising from integrated practices are discussed along with their implications for teaching and learning.

The influence of the dynamic geometry tool, Cabri-geometre, on the learning of geometry was investigated. Twelve Year 7/8 pupils were divided into two groups on the basis of van Hiele pre-tests: Group I, pupils at Level 0 or 1, and Group H, the remaining pupils. Both groups completed six lessons with structured Cabri worksheets then Group II pupils completed several construction tasks, documented by means of Cabri files and tape-recorded conversations. All· pupils increased their van Hiele levels for some or all of the concepts involved.

This is a paper about girls in mathematics. In it I give an account of a piece of research which I consider to be theoretically adequate for the work that I wanted to conduct about gender. It addresses the complex questions that people who do any research in mathematics education want to understand better: questions about learning and individuality, about thinking, about expertise. It links these with questions of power, control, dependence and change, and in so doing provides a different story about girls in mathematics.

This paper explores the use of written tests and semi-structured interviews in ascertaining students' understanding of the concept of a variable. A written algebra test was administered to 379 students and from the results students were selected for a semi-structured interview. The types of questions asked and the medium in which they were asked appeared to influence the responses given. It is conjectured that these issues must be considered when endeavouring to reach a richer understanding of the students' perception of the concept of a variable.

Since January 1997 there has been much debate within Australia on the type of numeracy benchmarks which should apply to children in Years 3 and 5. As well as debate on the breadth and depth of understanding, there has been difficulty in some areas establishing what children actually do know and can do. Although the data included in this report were not collected to answer questions related to numeracy benchmarking, they may help inform the debate about what children in Years 3 and 5 know and can do in the area of chance and data.

Three case studies of children are used to illustrate the variety of strategies employed by children when asked to make probability judgments in several different game contexts. The children's responses ranged from idiosyncratic and intuitive reactions to the deliberate application of proportional reasoning. It was found that certain combinations of variables in the task designs stimulated different mathematical thinking.

This study surveyed a sample of teachers (N = 115) with a questionnaire using the theory of planned behaviour (TPB). It examined primary school teacher intentions to allow students to use calculators in the classroom and the influences upon these intentions. Key salient behavioural. normative and control beliefs held by the teachers are analysed and evaluated according to their respective contribution to attitudes, perceptions of influence by the social environment, and perceived behavioural control.

The importance of metacognition for student learning has been widely acknowledged (Biggs, 1987, Birenbaum, 1996, Brown and De Loache, 1983, Pintrich, and De Groot, 1990, Schoenfeld, 1987, Wilson and Wing Jan, in press). But the practicalities associated with teaching for metacognition and monitoring metacognition are not clear. This paper is concerned about the assessability of metacognition within mathematics. It is asserted that unless metacognition can be assessed then it will exist as a theoretically sound construct but never be considered a viable part of the mathematics curriculum.

This paper provides an overview of a research-based framework for assessing and teaching in early number which has been used extensively in research and practice. The framework includes the following aspects: arithmetical strategies, forward and backward number word sequences, numeral identification, base-ten strategies, combining and partitioning, subitising and spatial patterns, temporal sequences, finger patterns, and quinary-based strategies. Five of these aspects are presented in tabular form as stages or levels. Also described are guiding principles for studying early number knowledge.

What do students need to know if they are to be constructed as effective learners of mathematics? This question provides the stimulus for this paper where the question is rephrased to: "What do students need to know in order to operate in a manner which is acceptable in the mathematics classroom?" Such a question is not without political implications and so needs to be extended to include questions about the consequences of participation in mathematics classroom. It is widely recognised that success in mathematics is not random, but rather falls into quite distinct patterns whereby students from certain social groups are more likely to be successful in the study of mathematics than others. The focus in this paper is the examination of why students from socially disadvantaged backgrounds are less likely than their middle-class peers to be successful in the study of mathematics. The paper uses two key notions. The first is the classroom interaction patterns noted by Lemke (1990) in which students must be conversant to be able to participate effectively. This knowledge becomes a form of cultural capital (Bourdieu, 1983) which can be transfered later to academic success.

In this paper I propose that the literature on situated numeracy has an important role in the reconceptualisation of mathematics education in general and workplace learning in particular. There exists a substantive literature which documents the idiosyncratic knowledge and procedures used by participants in situ to resolve tasks. This literature has been powerful in challenging the orthodoxies within the field of mathematics education whereby the power of mathematics comes from its capacity to be applied across a wide range of contexts. However, this situatedness of mathematics must be considered in conjunction with the degree of mathematics employed within a context. To this end, I propose and develop the notion of "mathematical saturation" which permits the breaking of the dichotomy of school-mathematics and non-school mathematics, and in its place proposes a continuum in which the degree of "pure" mathematics is considered to be a critical element. Using the data collected from a number of worksites, it will be argued that situated numeracy challenges notions of transferability but must be considered alongside the degree of formal mathematics which is used within that context.