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In this address 1 consider the research undertaken by me and by others which is concerned with how primary school children make sense of mathematics. The focus is on the strategies children use, the effect of processes and materials used by teachers, and the demand that these make on capacity to process information. The issues examined include the use of concrete representations, aspects of number, length and time measurement, and the transition from arithmetic to algebra. The results suggest that the strategies and materials that teachers choose often do more harm than good unless they are used thoughtfully and carefully. On the basis of the results 1 argue for mathematics to be taught in the most meaningful, straightforward, interesting, way possible.

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This paper traces the evolution of some research projects that relate to the use of a particular type of classroom task as the focus of the learners' mathematical activity. The tasks used are, on one hand, open-ended to allow opportunities for thinking and creativity, and, on the other hand, content specific to ensure that the focus of pupils ' activity is not just mathematics generally but the specific content that the classroom program seeks to address. The research grew out of a particular view of learning and teaching, consideration of the motivation of the learners, a perspective on mathematics as the content focus of the learning, and a recognition of constraints operating in classrooms. These perspectives are summarised first, after which there is a discussion of characteristics of the classroom tasks, and some reports of research into their use. Some implications for teacher education are suggested.

A review of the literature to gUide the development of a coherent teaching program on percent for Year 8 students revealed varied alternatives and little consensus. Working from the premise that fundamentally percent is a proportion, the proportional number line method was created to assist students experience success in percent problem solving, to also promote conceptual understanding of percent as a proportion, and to embody the multiplicative structure of percent situations. Classroom research indicated that students readily adopted the method.

In this paper an investigation is reported on whether the trend of sex differences in mathematics achievement of students at the lower secondary school level changed over time by bringing the mathematics achievement of the students to a common scale, which is independent of both the samples of students tested and the samples of items employed. The scale is used to examine the differences in mathematics achievement between the sexes in Australia over a 30 year period. Conclusions are drawn about such differences in mathematics achievement over time in Australia.

This paper describes an approach to the analysis of talk within small groups of students working on mathematical tasks. Two typical ways of talking and thinking are identified: cumulative talk and exploratory talk. An example of each type is given, using data from a study of a Year 10 class. Changes to the nature of the talk when the teacher is present are also reported. The focus is on both the content of the talk and the interactions in the group.

Eight classes (Years 4 to 7) from a Queensland primary school trialed an integrated learning system (ILS) as a means of re mediating students' mathematics learning problems. At the end of the trial, the teachers were asked whether they would recommend the system to other schools. Endorsement appeared to be related to the computer knowledge of the teachers and concomitant experiences of the students, the extent of integration of the ILS sessions in classroom teaching, and the pedagogical beliefs of the teachers.

The development of learning theories has continued since the acceptance of constructivism with enactivism being one emerging theory. Some influences on this development include criticisms of constructivism, ideas about Cartesian dichotomies, consideration of noncognitive knowing, notions from phenomenology, and the neural biological work which emphasises evolutionary or Darwinian notions and systems theory. This paper, based on the literature,puts some of these interrelated influences together to introduce enactivism.

This paper reports on aspects of a study which investigated the writing of explanations and justifications in mathematics with Year 11 students. Six teachers and 36 students from the same school responded to problem solving tasks; 14 of these students were interviewed. Student and teacher views about the writing process are reported. The study suggests that students and teachers need to jointly negotiate an understanding of what is meant by an explanation, a justification, and what makes a quality response.

This paper reports findings from a study designed to investigate the impact of an early numeracy program on the mathematical achievement of young children. Two groups of Year 1 students were assessed using the Schedule for Early Number Assessment – once in May, prior to the experimental group participating in the numeracy program, and once in November. Results indicate that the experimental group performed significantly better than the control group at the post-test phase.

This paper reports some initial findings from a substantial project funded by a UK government agency. The project aimed to investigate effective pedagogy in numeracy and literacy using ICT in primary schools. Some preliminary findings relating to pupils' attainment and a series of lesson observations are reported. Some of the broader findings from the development work in mathematics and ICT are also discussed in the context of two particular case studies which focused on developing effective pedagogy in mathematics.

The Victorian Certificate of Education (VCE) has been subject to ongoing modifications since its full introduction in 1992. These changes largely focused on reducing curriculum choice and the value of the school-based assessments. The recent government initiated Review has maintained this direction. We draw on the experiences of close to 500 senior mathematics teachers -interview and survey data - to consider the impact on them of the evolution of the VCE, particularly on their teaching practices at the junior secondary levels.

This paper explores gender differences within data collected in a study of the factors influencing Year 12 students to study or not to study mathematics. In particular, differences in subject choice, beliefs about mathematics, mathematicians and users of mathematics, and reasons for studying or not studying mathematics are examined. While the results of analysis generally confirm the trend towards a reduction or elimination of gender differences at this level, some differences persist. Possible reasons for this are conjectured.

In this paper we employ three interpretations of the ZPD to frame an analysis of episodes of teaching and learning in a year 7 mathematics classroom. The analysis is concerned with the way authority is constituted locally by the teacher and students within the norms of Collective Argumentation. In traditional classrooms, authority is assumed to reside in the teacher and texts whereas in Collective Argumentation, authority is attained through discourse practices that privilege socio-mathematical norms such as meaningfulness, communicability, and testability. We show how students in the classroom speak with unusual authority as they communicate and negotiate their claims.

This paper reports on a research project which investigated the value of using games to assist Year 7 and 8 students' learning of probability concepts. Games, although generally useful in mathematics for helping children learn, may not automatically be as useful in helping students develop normative probability concepts. The study found that the type of game impacted on the value of using games for learning probability, and that there are implications for the teacher's role when using games for learning.

This paper explores teachers' responses when implementing a teaching approach that encouraged children to develop and use invented algorithms for computation. The children demonstrated number sense and used methods that predominantly reflected left-to-right computational processes. While acknowledging the children's success and their ability to explain and justify the methods used, the teachers faced a number of challenges. They confronted previously held beliefs,perceived curriculum requirements and social expectations regarding the place of standard algorithms for computation.

Development of assessment tasks that both provide quality insights into students' mathematical understanding and produce rigorous data for measurement purposes is a challenge for mathematics educators. This paper presents a case study of the development of three performance assessment tasks. The tasks addressed several strands of the mathematics curriculum and were designed to match the teaching strategies being used. The tasks were refined following the application of a qualitative developmental model, the SOLO Taxonomy. They were also tested against criteria for good assessment.

In this paper, we report two subsets of partitioning strategies utilised by a sample of twelve Grade 3 children when they were asked to solve sequences of partitioning problems. These strategies emerged when the children sought to make the process of partitioning more efficient in terms of generating equal shares, reducing the number of steps involved in partitioning and reducing the load on working memory. Implications for instruction are drawn from the findings.

The increased emphasis on the Chance and Data strand of the mathematics curriculum means students should be approaching adulthood better prepared to interpret data and recognize relationships between variables and between a sample and its corresponding population. This study examines the statistical appropriateness of conclusions drawn by young adults from data in a small sample. While many of the analyses took into account the data's limitations, there were still some erroneous assumptions and omissions.

This paper describes the use of a method for analysing concept map data to generate information on quantitative and qualitative features of a mathematics teacher's knowledge base in the area of geometry. The system of analysis used in this study provides a useful way to represent the state and connectedness of teachers' knowledge in the area of geometry and trigonometry. Use of this system indicates that this experienced teacher has a rich store of knowledge which is complete and accurate. His knowledge base shows a high degree of branching with depth of knowledge within each branch.

An important element in teaching is the quality of content knowledge that teachers use in the design and delivery of their lessons. In this study, we present a framework for investigating how this knowledge is structured. The framework is then used in the analysis of an experienced teacher's knowledge of functions and the teaching of functions. The data show that our teacher has built up knowledge that is dominated by conceptual rather than procedural aspects of functions.

A teaching experiment was designed to facilitate Year 8 students' understanding of algebraic expressions and equations through the use of unknowns, patterns, relationships, and concrete materials. This paper discusses the study's theoretical framework, the teaching episodes relating to expressions, equations, and equals, the students' reactions to this instruction, and their initial and final understandings of algebraic expressions, equations, and equals. Exploration of the successful and unsuccessful teaching episodes emphasises the relationship between instruction, prior knowledge and learning.

The effectiveness of a diagram in problem solving is dependent on its utility as a cognitive tool. In order to develop students' ability to use diagrams as cognitive tools, teachers need to assess the quality of students' diagrams and provide them with the necessary support. However assessing the quality of diagrams is problematic. This paper discusses how theoretical prototypes and exemplars of level ofpelformance provide a practical and effective avenue for assessing the quality of students' diagrams.

This paper investigates the effectiveness of technology-based instruction in secondary mathematics, by comparing students' achievements resulting from technology-rich assignments with those achievements resulting from equivalent assignments presented in traditional format. In addition, the development of the technology-rich assignments, from traditional paper-based instruction and within existing curricula, provides an example of the relative ease of integrating technology into the curriculum. Within the context of mathematics, issues of attitude towards computers, motivation and gender differences are examined.

A new instrument aimed at measuring the extent to which mathematics is stereotyped as a gendered domain was recently trialed. Included among the participating schools were two with distinct student populations from strong, but different, ethnic and religious backgrounds. In this paper we report the findings from comparisons made between the responses of students from the two schools (N = 75 and N = 67) and those of students from other schools participating in the trial (N = 394). The results indicate that ethnic/cultural backgrounds influence stereotyping of some dimensions of mathematics education.

Students' participation in classroom activities is considered to constitute cognition, and is evidenced in student-teacher interaction. In this paper, elements of the complex phenomenon 'student participation' are described with reference to an episode where a student was discussing her solution of a three-dimensional vector problem with her teacher. The multipleparadigmatic research method used is considered in relationship to how the episode has been portrayed, and is proposed as being suited for both the teaching and research purposes of a teacher-researcher.

"It is afar,far better thing that we do now than we have ever done" Well, probably not quite - but we report on the extension of a project that addresses the potential of attitudinal factors to impact on situations involving the use of computers to learn mathematics. The construction of six attitude scales is discussed, together with response data from students in London and Brisbane. The data point to separate attitudinal dimensions related respectively to mathematics and to computers. This has implications for learning situations in which they interact.

This paper describes our sociocultural perspective on learning in secondary mathematics classrooms where technology is integrated as a central resource. We propose four roles for technology in relation to student learning: Master, Servant, Partner and Extension of Self. One classroom episode is analysed to reveal the different 'voices' that emerge through the interaction of mathematics, people (students and teacher) and technology. We are using this approach to develop a framework for describing and analysing the characteristics of a classroom community of practice.

This paper discusses the effectiveness of the professional development model used by the Catholic Education Commission of Victoria Numeracy Strategy Project. The project used an action research methodology within a team structure to explore numeracy learning and teaching. Three case studies of early years teachers indicate that the teachers developed some important insights about numeracy learning and teaching in the early years of schooling and provide evidence of reported change in beliefs and practice.

This paper reports on research that investigated processes through which students come to understand mathematical text. Transcripts from a Year l11esson are analysed to illustrate three contexts in which students interrogated mathematical text: reading assigned by the teacher, spontaneous reading, and reading peer-produced text. Although current moves/or mathematics education reform discourage over-reliance on textbooks as a source 0/ knowledge and authority, this study demonstrates that reading as a social practice can stimulate students' critical engagement with mathematical ideas.

This paper reports research in progress. It brings together some issues and themes emerging from the literature review (to date), preliminary interviews, personal reflections and earlier research. These will be extended and refined in further research and will be included in my thesis which focuses on the search for possible (shifting) paradigms within which statistics education (research and practice) is framed.

This paper tracks two children's mental strategies over a 5 year period (Years 2 to 6). Although the children were students in traditional classrooms, where teacher-taught algorithms may have conflicted with the children's spontaneous strategies, they continued to develop their own efficient strategies throughout the period of the longitudinal study. However, by Year 6, both children were also employing taught pen and paper algorithms which were less effective for mental calculations. Finally, some implications for teaching are discussed.

This paper is proposed to stimulate discussion in the area of algebra. It reviews relevant literature in algebra and proposes a framework of growth points to inform future research and to enable monitoring algebraic development through the early years of secondary school through understanding rather than through outcomes. It is work in progress in preliminary stages with some areas still to complete.

The creation of the Victorian Curriculum and Assessment Board marked a significant departure from prior approaches to curriculum determination. Responsibility for the development of senior secondary mathematics curriculum was entrusted to a select group of mathematics educators. However, attempts to develop a new approach to curriculum were challenged by other groups within the community. This study examines the processes and influences that helped mold a particular curriculum.

There has been little research into the intersection of language and arithmetic performance of deaf students although previous research has shown that deaf and hearing-impaired students are delayed in their language acquisition and arithmetic performance. This paper examines the performance of deaf and hearing-impaired students in South-East Queensland in solving arithmetic word problems.

This paper addresses issues relevant to an analysis of mathematical knowledge in a mathematics teaching episode. Moving from a concern with what it terms the 'teaching paradox', the paper seeks to analyse an episode taken from a year 11 mathematics classroom: In this analysis three main concepts are used: knowledge base, knowledge artefact, and functional knowledge claim. The main finding of the analysis is that mathematical knowledge stands in a peripheral relation to mathematics teaching.

A small rural school introduced single-sex classes in an attempt to increase the participation rates of girls in highest level mathematics classes at the senior level. The study revealed differences in the learning styles of the students. This led to the adoption of a new construct to evaluate the single-sex classes. This construct offers suggestions in the way mathematics classes could be conducted in high schools to assist students with non-traditional learning preferences.

This paper investigates Grades 4,6 and 8 students' confusion between area and perimeter. Students were given area judgment tasks involving rectangles and their responses analysed in terms of the operations used and the strategies exhibited. Students were found to use both additive (Perimeter) and multiplicative (Area) judgment rules, and 7 different strategies. Those using an additive judgment rule tended to rely on rulers or fingers for measuring length and to align the rectangles vertically, whilst those using a multiplicative judgment rule tended to use overlay and partitioning strategies.

The key issue canvassed in this paper is whether or not current teaching practices in mathematics are likely to foster or inhibit numerate behaviours in students. I understand numerate behaviours to comprise both intellectual (mathematical) and self (social) know ledges (Willis, 1998) which are interwoven and interdependent. Numerate behaviours are fostered where student initiation and construction of mathematical ideas is genuinely valued such that they are authorised or enabled to speak and write their developing constructions with respect. A poststructuralist analysis of classroom practice in three (3) primary mathematics classrooms shows that although students are actively engaged in the construction of knowledge, what they learn of mathematics and of themselves as numerate individuals may not be conducive to the construction of a sense of agency so necessary for lifelong learners of the new millenium.

This paper focuses on the difficulties encountered in writing good questions for the assessment of students' understanding in geometry. A set of questions considered suitable for use in the classroom as a written test, were trialed on 106 Year 10 students. Following assessment of the students' responses, the questions were analysed with regard to their ability to elicit statements demonstrating students' understanding in geometry.

Since its inception in 1990, the Victorian Certificate of Education (VCE) Mathematics study has incorporated extended school based and assessed common assessment tasks (CATs) in its assessment regime. These tasks have been externally set by panel, graded by teachers and reviewed by various formal mechanisms. Key issues associated with the use of such tasks include validity, reliability, authentication and equity. We discuss several of these issues in this paper.

This paper investigates the way in which two children constructed and designed mathematics problems for friends to solve. The two children, of different ages, designed problems for one another and for other friends over a ten week period. The way in which the children engaged in problem solving prior to, and after,formulating or posing a problem was explored. Insights into the children's mathematical abilities were identified through the problem-posing activities.

This paper investigates the performance of Grade 3 children's generation of word problems from photographs. It looks at the relationship between this performance and performance on a grading instrument; whether children tend to generate characteristic categories of word problems in relation to specific photographs and to what extent student generated word problems reflect their mathematical understandings and thinking.

Four teachers of secondary school mathematics in small rural schools, an adviser and a participant researcher are using Farsite audiographic equipment to develop a learning network. A description of the patterns of interaction within the project leads to a discussion of issues for future planning within this project and more general principles for developing later networks. We ident{fy some skills for networking in the cyber-age, and discuss some advantages of audio-graphic networks over face-to-face contact.

The analysis reported in this paper documents the teacher's role in supporting the emergence of notational schemes from the students' problem-solving activity in one first-grade (age six) classroom. Initially symbolizations were offered by the teacher as a means of clarifying and communicating students' thinking. Later, the teacher worked to achieve her pedagogical agenda by using notational schemes to highlight certain solution processes. As a result, the introduction of notational schemes served to support shifts in the students' mathematical development.

A CAS-active introductory calculus course was trialled with three year 11 classes of students who each had a TI-92 calculator. Access to the CAS enabled the students to perform "year 12" differentiations with the same level of success as year 12 students, but without affecting their conceptual learning or their development of by-hand skills. The use of the CAS was influenced by teachers' cognitive "privileging". Some observations are made on the suitability of the TJ-92 for such a course.

This study suggests that classroom research into students' knowledge construction is incomplete without data about teachers' knowledge constructions and practice. Teachers' knowledge related to algebra and teaching algebra were identified and categorised using three knowledge domains identified by Shulman (1987) as knowledge of epistemology, pedagogy and pedagogical content of a algebra. Diverse and idiosyncratic examples of algebra knowledge were articulated by each of the study teachers in relation to the epistemology and pedagogy of algebra, but examples of pedagogical content knowledge were seldom easily identified.

This paper is in three parts. In the first part, I compare and contrast three typical methods of teaching generalisations in mathematics. In the second, I describe the theory of generalisation expounded by the Soviet psychologist Vasily Davidov. In the third part, I re-examine the three methods of teaching mathematical generalisations in the light of Davidov's theory, and make some general conclusions.

Two survey items askingfor estimates of probability or frequency of everyday events (A), (B), and their conjunction,(A and B), were completed by 2719 school students in grades 5 to 11. Cross-sectional and longitudinal analyses revealed chance expression improved with grade, but no change in incidence of conjunction errors. Gender differences favouring males occurred for some grades. Comparisons with responses to other probability items indicated incidence of conjunction errors is independent of development of basic chance measurement.

Australasian mathematics educators responded to a questionnaire that sought views on what mathematical understanding is, how it can be identified, and what teachers can do to develop it. This paper presents the responses to two questions about indicators of children's understanding and differences between doing and understanding. Responses were grouped. as verbal, cognitive and physical indicators but it is recognised that these are inter-related.

Graphics calculators were allowed in all tertiary entrance mathematics examinations in Western Australia for the first time in 1998. In this paper we present an analysis of students' answers to four of the examination questions in TEE Calculus and a discussion of misconceptions attributable to the technology and misuses of it.

International studies have indicated that mathematics teachers have been slow in taking up the use of computers in their teaching even where the resources have been available. In this study, survey and case study procedures were used to determine how frequently, andfor what purposes, secondary mathematics teachers used computers in some Australian schools. It also investigated teachers' beliefs about the potential of computers in mathematics teaching and learning. The findings showed that the teachers in this study rarely used the computer resources available, particularly if they taught junior (Years 8-10) and less able senior students. Only teachers with special expertise used computers regularly and, then, mostly for the teaching of calculus and statistics and usually in a calculational way.

This paper summarises the findings of a study into the extent to which students in introductory statistics courses use diagrams when solving problems. The results of the study showed that many students did not use diagrams but those who used diagrams were more successful. Use of diagrams and university entrance score appeared to be better predictors' of success than the level of mathematics studied at secondary school.

This paper reports details concerning the performance of capable Year 12 Mathematics students in their final public examination in NSW The focus of this analysis is on basic algebraic skills tested in the first question of the 2 unit paper. This topic has been encountered by students for a number of years and provides, potentially, an upper bound on what might be expected from such students in an examination. Success rates and the types of strategies employed by students are documented and discussed.

Many students have difficulty with the language used in both the learning and assessment of mathematics. This paper investigates the use of English in the assessment of senior secondary mathematics in New South Wales. Using input from students, along with a linguistic perspective, it analyses a question from a recent NSW Higher School Certificate examination and suggests how the language of the assessment might be changed in order to make the mathematics in it more accessible to the students.

Refining mathematical understanding through the use of multiple representations and negotiation of meaning has been considered important for students. This paper reports on an observational study of students using a Computer Algebra System in an introductory calculus unit at undergraduate level. There was strong evidence that the use of this technology was a catalyst in students using these positive learning methods. However students felt that the use of this technology aided but did not underpin their learning of mathematics.

The relationships between the beliefs and practices of mathematics teachers have been shown to be complex. In this case study of one mathematics teacher, it was found that while the teacher had clear and quite strongly held beliefs about mathematics and its teaching and learning, his classroom practices did not always reflect these beliefs. In some lessons other . factors meant that the lessons was not conducted in a way which reflected the teacher's beliefs. However, the teacher was aware of this contradiction.

This paper reports a longitudinal study of children's understanding of decimal notation. The understandings of 3211 students were classified into four categories and changes over long periods were tracked. The progress of sixty-four students over about three years is reported. When expertise is attained it is generally retained. Over about a year, many students remain in the same misconception category, but in the longer term they move between misconceptions. Improvements and hypotheses to be investigated in the future are noted.

This study identified the conditions that impeded or facilitated students' access to applications tasks at the senior secondary level. Impeding conditions comprised language related, representational, memory related, organisational and task specific conditions. Facilitating conditions comprised memory related, perceptual and engagement conditions. Some conditions were the result of the complexity inherent within the task whilst others arose from students' resource levels and as such acted as intervening conditions between task complexity and the difficulty the students experienced in accessing the tasks.

This study was designed to explore the computational choices made by 25 students in Years 5-7. The data was collected in individual interviews and the results showed that many students have a very limited repertoire of methods at their disposal, Students often tended to lack confidence in their approach and often it was only then that they reached for a calculator. A number of students used two or more computational methods for some items.

Previous studies conducted collaboratively by the author have shown that a strategy based on Piaget's notion of cognitive conflict was very successful in causing a reduction in the frequency of mathematical misconceptions exhibited by a group of very bright tertiary students and that much of this improvement persisted over time. This study investigates whether the same method effectively reduces the frequency of mathematical misconceptions exhibited by average or 'middle band' first year university students.

A cross-sectional study of 132 NSW rural children from grades K-6 assessed counting, number sense, grouping/partitioning, regrouping, place value and structure of the number system. Task-based interview data exhibited lack of understanding of the base ten system, with little progress made during Grades 5 and 6. Few Grade 6 children used holistic strategies or generalised the structure of the number system. Grouping strategies were not well linked to formation of multiunits; addititve rather than multiplicative relations dominated the interpretation of multidigit numbers.

Research indicates that a teacher's beliefs have a powerful impact on the practice of teaching, since a teacher's beliefs constitute the reasons which account/or the differences of individual teachers in teaching mathematics, without ignoring the influence of his or her content knowledge. This paper describes the cross-cultural aspects 0/ a teacher's beliefs and its effect on his or her instructional practices under the constraints and opportunities provided by the social context of teaching.

Since Victoria permitted the use of graphics calculators in final external examinations, their use has become quite widespread. A survey of secondary schools, undertaken to gauge the response of teachers to these tools, provides information on how teachers view graphics calculator use in secondary mathematics courses.

In a recent study (Vincent, 1998), van Hiele levels were used to monitor students' progress in geometric understanding when learning with Cabri geometry. The following report focuses on the experiences of two of the twelve participants in the study: Student D who was initially at van Hiele Level 0/1 and Student L who was at Level 2 for most concepts.

The purpose of this study was to explore teacher's beliefs about the uses of assessment and their actual assessment practices. The sample consisted of 387 primary teachers. The results indicate that these teachers are using a wide variety of assessment methods, but assessment is predominately used to provide feedback for the teacher rather than the learner and parents. The types of assessment methods chosen appear to support these beliefs about the uses of assessment.

Two survey items asking for estimates of probability or frequency of conditional events, (AIR) and (BIA), were completed by 2719 school students in grades 5 to 11. Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and in distinguishing conditional events. Conditional events were better distinguished for the frequency item than the probability item. Comparisons with responses to other probability items indicated understanding of conditional probability was related to development of basic chance measurement.

The Harristown State High School project investigated the effect of more time via voluntary after-school, core mathematics tutorials on Year 8 student learning outcomes. Six factors (additional time, assessment structure, colour booklets, supportive learning environment, rewarded competitions and qualified tutors) were identified from a pilot study as critical and were chosen to be central to the study. Pleasing outcomes included strong parental support, the number of student volunteers (115), and improved confidence and test results.

Although group work is widely used by primary teachers in most Key Learning Areas, this is not always the case in mathematics. Why do teachers behave in this way? Argyris and Schon (1974) claimed that behaviour was driven by individual action theories. This study used the Theory of Planned Behaviour (Ajzen, 1988) to uncover teacher action theories with a sample of NSW upper primary teachers (N = 115). It will discuss these theories in order to assist teachers in their critical reflection of current practice.

Recent developments in the Level theory of van Hiele have provided a framework in which the thinking of young students can be explored. This study involved 12 Year 2 students and explores in detail the diversity in responses of six students as they explained their understandings of simple 2D shapes. The focus of the research was on interpreting and exploring the implications of students' language and it highlighted important issues relating to the relevance of the van Hiele model to primary school education.

Examination of the statistical literature shows that consensus on definition, terminology, and interpretation of some hypothesis testing concepts is elusive. This makes hypothesis testing a difficult topic to teach and learn. This paper reports on the results of a study of novice students' conceptual knowledge of four hypothesis testing concepts through talking aloud and interview methods. While some students seemed to have a reasonable understanding of some concepts, many students seemed to have more limited understanding. The study explores students' faulty conceptual knowledge.

Optimism, pessimism and achievement in mathematics in primary and lower secondary students were measured on two occasions separated by almost three years. The students' Grade level and gender were also considered. The hypothesis that, relative to more pessimistically oriented students, those with a more generally optimistic outlook on life would evidence a higher level of achievement in mathematics was confirmed. Students' Grade level and gender were also significant variables. The implications of these findings for mathematics education are discussed.