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Provisional results are reported here from developmental research on negative numbers in the upper grades of primary school. An attempt has been made to deal with learning strands both on concrete magnitudes or quantities and on formal constructs the observation of individual learning process is the core of the research. The results show that - in the learners - a gradual shift takes place either from the realistic context to the mathematical one or from the known mathematical context to the new one.

No abstract available.

No abstract available.

This presentation discusses 9bservationsandfindings of classroom-based research focused on ways of developing children's mathematical understanding through authentic, situated activities. Instructional and heuristic~lly structured activities were used to foster discussion, argumentation and resolution among alternative constructions, allowing children to engage in the social construction of notions compatible with ideas that· had· evolved historically, through social and cultural forces in the wider community.

The aim of this study was to investigate children's perserverance when solving difficult or unfamiliar number problems. It was concerned with those students who are ,referred to as 'perseverers' because they reached a stage in their problem solution where they recognised that they had not reached a satisfactory answer and decided.to take some action - start again, modify their strategies or change to different strategies rather than giving up immediately. The sample consisted of ten boys and ten girls in grade 6 and ten boys and ten girls in grade 10. The tasks consisted of number problems of varying difficulty. Data gathering took the form of clinical unstructured interviews with individual students in which they were asked to verbalise concurrently With solving a set of number problems. Task analysis maps were used to provide overviews of the interview protocols. From observation of the maps of students who were ultimately successful, it became apparent that these children were more inclined than others to be flexible in their use of strategies. A model was developed which described the sequence of strategies used most consistently by successful students. ", This model formed the basis for a small-scale training programme to investigate whether the strategies could be taught. A descriptive analysis suggested that most of these children were able to be trained to use the model independently.

This paper discusses one ~f three case studies to investigate students' conceptualizations of quadratics by solving quadratic contextual problems. The theoretical framework which guided the study was constructivism. The methodology was teaching interviews between a researcher and student solving problems. A multi-representational software, FUNCTION PROBE, was available as a tool to aid students in their problem solving. The results showed that using realistic situations as problem contexts invited multiplicity of interpretations and methods for defining quadratic functional relationships. By reasoning empirically from the problem context, students conceptualized quadratics relationships iteratively and in terms of summation in contrast to the most common view of quadratics as a product of two linear variations. Further, requiring students to verify and justify their strategies by cross-referencing between multiple representations of functional relationships and problem context led them to construct viable schemes to characterize quadratics in terms of rate of change, dimensionality, and symmetry.

This paper describes the trial of a unit of work on linear and quadratic graphing with six year 10 classes. Two treatments were developed. The computer treatment made use of the ANUGraph software package, while the calculator treatment paralleled the computer treatment but used a combination of previously prepared graphs and graphs constructed by the student with the aid of a calculator. The emphasis in both treatments was on the interpretation of graphs related to real situations. Comparisons between pre-test and post-test results and interviews with twelve students showed that students learnt to handle the software proficiently, and that both groups improved on most of the topics taught. However, the calculator group seemed to be advantaged by practising plotting of points by hand. Implications/or future work are discussed.

This paper illustrates the use of the Functional Theory of Language as presented by Halliday to compare the interaction between students and teachers in a mathematics classrooms as a function of gender and social class. The constructs of field and tenor presented by Halliday are used to compare the observation from two classrooms from a high social class boys' school and a low social class girls' school. Conclusions from the data as well as methodological implications are discussed.

Children with Mild Intellectual (IM) Disabilities are' often taught addition and· subtraction algorithms through the use of rote learning models which fail to enhance their problem solving skills. Recent developments in other curriculum areas suggest that there are alternative process oriented models which may be more successful in ameliorating successful problem solving strategies among such students. This research attempts,to contrast one such model, Quality-Sameness Analysis (QSA), with more conventional rote learning systems such as the Computational Skills Program (CSP) in addition it is proposed that the use of calculators to solve algorithms will free cognitive functioning of IM students and enhance the acquisition of problem solving skills. Preliminary results confirm the QSA as a powerful process oriented model with considerable potential in the IM classroom and the major contribution that the use of calculators can make in the acquisition of problem solving skills among IM children.

Mathematical problem solving is a complex activity that necessitates very thoughtful consideration about which are the best ways to teach problem solving. The purpose of this investigation is to seek information about problem solving in Year 12 (VCE) Calculus. A qualitative methodology (thinking aloud) was employed in the study. The verbal protocols of 6 students are currently being analysed for regularities in overt behaviour and for potential heuristics.

This study investigates how math case methods support teachers' professional development by shifting their perception of authority from external to internal and collective sources. The primary data include transcripts of case discussions and interviews, as well as math assessments of teachers. The findings demonstrate that case discussions provide opportunities for (1) realizing that capability and wisdom exist within the group, (2) developing a critical stance, and (3) developing stronger, more refined content and pedagogical content knowledge. Teachers that capitalize on these opportunities have a richer sense of their own autonomy. Lisa freely admits that she was a "drill and kill" teacher, never straying far from, the basics in the sixth grade text book. She blamed her lack of confidence ,on her weak mathematical background saying, "I never/earned it when! was in school". Lisa joined case discussions at the insistence of her fellow' teachers, three of whom, had participated in case discussions the year before. She rarely contributed to the early discussions. When she finally ventured into the conversations, she asked questions that others might have been reluctant to ask, such as, "I'm like the kid [in the case], I don't understand why 100% isn't 100", or "Why doesn't the teacher just tell them how to do it?" Lisa's confidence and trust grew with each discussion. She began to tap her fellow teachers as resources for new ideas and materials. Instead of skipping percent this year, as she had always done previously, she did a six-week unit using the text as a guide supplemented by "real~world" problems and manipulatives. She prepared her unit collaboratively with Elena a sixth grade bilingual teacher, She even admitted to her students that she did not learn percent until she was an adult.

Metacognition is here characterised by three themes: i) 'thinking as the object of thought', ii) 'thinking regulating thought' and iii)'the feeling of knowing'. Traditiona1 formulations of metacognition are critisiced on their theoretical, operational and applied bases via these three themes, mainly because they denigrate the crucial role of affect in metacognition. Metacognition is more usefully defined as double context cueing', firstly to an appropriate ego-state of awareness, add secondly to a target !content emotional state with appropriate knowledge and ability response tendenCies. Examples are discussed of metacognitive context cuing by internal/external co-constructed contexts, internal contexts, finely differentiated contexts and restrictive contexts that inhibit mathematics education. The necessary role of metacognitive context cuing in mathematical thought is discussed. The results of a research exploration into metacognitive context cuing are presented, including a simple credibility index for Likert responses. Directions are suggested for further research applications in mathematics education.

This paper argues that we do not teach students to understand mathematics, We only teach them mathematics, and leave theunderstahding- or lack of it -up to them, The reason is that the affective aspects of students' m(lthematical experiences - feelings that are essential for understanding increasingly ,abstract mathematical concepts - are, perversely, continually reduced as the mathematics becomes increasingly more abstract. The two major causes of this anomaly are i) the mismatch between the structure of, understanding mathematics and its'logical' structure mirrored in mathematics curricula together with ii) the influence this has, in conjunction with Piagetian theory, on mathematics teaching. The role of inner affective contexts. in organising students' mathematical knowledge, developing their mathematical intuition, in their learning and applications of mathematics is discussed. It 'is also considered how these organisational contexts are conceived and developed, and how to explicitly encouraged this during teaching children how to understand mathematical abstraction.

Twenty year-lO students in a boys' private school in Mel/Journe were interviewed individually while they worked on algebraic taskS. The tasks included solving very simple linear equations and simplifying expressions. The taped interviews provide examples of informal, and often' misleading, ways in which many' students describe algebraic manipulation procedures. Informal terminology (e.g., "You move the 2 over there. and put it on top") was used successfully by some students, but for many others it was associated with making mistakes. Students often appeared to be guided by memory of actions they might carry out rather than by general principles. They showed difficulties understanding the ways in which numbers call be used for checking algebraic work and the purposes of basic algebraic tasks such as simplifying and solving.

Question-asking is a key process in mathematical thinking, and is recognised as such in the 1992 New Zealand National Mathematics Curriculum which emphasizes the need to encourage children to ask questions in various mathematical contexts. However, observations suggest that this is something which few New Zealand primary school teachers do and, further, there seems to be almost no research into children's question-asking in mathematics which might provide some guidance. The present paper reports the results of a small investigation into middle primary school children's ability to ask questions about geometry. It also provides a first glimpse into preservice primary school teachers' perceptions about both the value of children's questions for teaching and learning in mathematics, and difficulties involved in eliciting them.

The purpose of this paper is to report findings of a study intended to investigate the effect of instructional activities designed to facilitate the development of visualisation strategies in young children. Utilising this intuitive capacity can provide a basis for developing number relations and devising strategies to learn basic facts. Given practice children can develop a reasonably large collection of set dot patterns that they recognise without counting. These patterns begin to be related mentally to one another as they enhance part-part-whole relations. Organising the dots into recognisable subgroups and patterns facilitates the ability to recognise and give number names to groupings, especially when the groupings are larger than five. , Instructional activities and materials that were used to facilitate the development of visualisation strategies in two kindergarten classes over a one year period are described. Students' verbal and pictorial responses are analysed and the implications for fostering the development of basic facts in this manner are discussed. The results indicated that, with practice, children can become capable of mentally combining and separating patterns, thus able to instantly recognise the whole and its related parts - a very . efficient strategy for aiding the recall o/basic addition and subtraction facts.

At Swinburne University of Technology we have found that, after the introduction of a graphics calculator in introductory calculus subjects, female students for the first time significantly outperformed males on. calculus tests. Students solution strategies were examined as a possible. reason for the superior performance of females, but seemed to point to the opposite result. Comparison of scores of exam questions with graphical and purely algebraic content revealed that females received their better marks due to their better performance on purely algebraic questions.

This is a report of a study of the representations and strategies for addition, used by a sample of 55 children in years 1, 2, and 3 in three schools in Brisbane. Children were presented with operations represented in symbols and asked to explain their procedures as they worked The teachers were, also interviewed to determine the representations that they were introducing. The general developmental sequence was from use of objects, to use of counting to mental calculations using knowledge of number facts and place value. The results are discussed from the perspective of the demand that the procedures make on children's information processing capacity. We suggest that some of the difficulties occur because teachers introduce procedures that are, recommended. in curriculum documents without being aware of the cognitive load that they impose.

In this paper, we describe the case of a teacher, Christine Brown, as she incorporates new teaching approaches and organisational strategies within her classroom practice. Drawing on a set of interviews and observations, we analyse the effects of the experimentation with classroom organisation, management and teaching approaches carried out by Christine. We find that Christine made significant shifts in her practice that are consistent with constructivism. Furthermore, her classroom experimentation did result in a change of beliefs about how students might best learn mathematics.

Consistent with a growing awareness of constructivism and child centred learning arid teaching approaches, a research project is presented where instructional games were implemented into a classroom setting. The project allowed the children's mathematical concepts to be developed and their understanding consolidated and reinforced by creating an environment which used language as a bridge between the children's informal mathematical knowledge and abstract mathematical concepts.

This study investigated the interface between ikonicand concrete symbolic functioning by studying two different aspects of visual processing associated with mathematical problem solving. The first of these involved the ·use of mathematically based visual. images' and diagrams, while the second was concerned with incidental non-mathematical visual imagery associateq withthe problem story; Four groups ,of grade, J 0 students were studied, representing all combinations of high and low concrete symbolic ability and high and low ikonic ability. The results suggested that Success at problem solving was related to concrete symbolic, but not in ikonic ability, and that the use of mathematically based visual images and diagrams was similarly related to concrete symbolic rather than ikonic ability. Incidental visual imagery associated with the problem story was, however, linked to individual differences in ikonic processing and was not related to concrete symbolic functioning .

This case study research investigated changing teacher roles associated with two teachers' use of innovative mathematics materials at grade six level. Using daily participant observation and interviews with the teachers and the project staff member responsible for providing in-school support, a picture emerged of changing teacher roles and of those actors influencing the process of change. One teacher demonstrated little change in either espoused beliefs or observed practice over the course of the study. The second teacher demonstrated increasing comfort with posing non-routine problems to students and allowing them to struggle together, without suggesting procedures by which the problems could he solved. He also increasingly provided structured opportunities for student reflection upon activities and learning. Major influences on this teacher's professional growth appeared to be the provision of the innovative materials and the daily opportunity to reflect on classroom events in conversations and interviews with the researcher.

Recent research in education, and mathematics education in particular, has led to the identification of independent categorizing systems intended to mirror the structures found in such diverse fields as teacher professional development (Barnett, 1992); student writing in mathematics (Clarke, Stephens, & Waywood, in press); and student acquisition of calculus knowledge (Frid, 1992). There are particular characteristics of these categorizing systems which display a tantalising similarity: • Contextual similarity - the common location of all three studies within educational environments; • Structural similarity - the "three-valued"(triadic) structure of all three categorizing systems; • Conceptual similarity - categories in each system resemble each other in the nature of their conceptual distinctions. This degree of similarity suggests that each categorizing system is an independent manifestation of a more fundamental triadic system (TRIADS). This paper examines the characteristics of these triadiC systems and makes comparison with other systems (or analytical frameworks) found in the research or theoretical literature. in an attempt to establish the significance of the degree of conceptual similarity found in the categorizing systems employed in mathematics education. It is proposed that cognitive sophistication be identified with personally contextualized knowledge rather than with formally abstracted knowledge. TRIADS is proposed as a robust structure having relevance in a variety of educational contexts. It is also proposed that conceptual similarities between the first two levels of TRIADS and Skemp's (1976) diadic structure for mathematical understanding support the addition of a third level to be called Contextual Understanding.

The Junior Secondary Mathematics Resource Schools Project was established with a central purpose: "To enhance and improve the mathematical capability of all students". Its realisation was dependent on the collective expertise and enthusiasm of the project coordinators on the six Project schools, their associated "key~teachers" and other staff, and the coordination and support available to the Project as a whole. The research design or the evaluation of the 1.S.M.R.S.P. acknowledged the existence of distinct communities of interest with respect to the Project. This paper reports those aspects of the study most likely to be of interest to the research community: that is, the study design and the student and teacher outcomes that can be associated with the implementation of this innovative curriculum and the associated teacher professional development. At this stage it appears that Project teachers are reporting a growing satisfaction with their participation in the Project and a growing awareness, understanding, and endorsement of the Project's goals. Student outcomes suggest that the emerging inclusive curriculum is succeeding in both cognitive and affective areas at least as well as other more conventional curricular practices.

The process by which classroom teachers change their practices and their knowledge and beliefs about the teacher's role and about their subject is fundamental to the learning process. The purpose of this paper is to outline the characteristics of a new model of teacher change. It must be emphasized that this model deals with a process of change; it is not a model of the instructional process. In this model, the learning aspects of teacher change lead us to characterize the process as "teacher professional growth". In modelling teacher professional growth, our concern is solely with change in each of the four domains which encompass the teacher's world and the mechanisms by which change in one domain leads to change in another. Central to this new conception of teacher professional growth is the significance accorded to the mediating processes of reflection and enactment, by which change in one domain is translated into change in another.

If the use of assessment as a catalyst for systemic reform in mathematics education is to be justified, then research is required which links changed assessment practices with instructional consequences. Based on the Victorian experience,this study provides the first systematic investigation of this hypothesised "ripple effect". This paper reports the results of the document analysis which comprised the first stage of the study. The analysis' of documents relating to the mathematics curriculum, teaching practice, as well as to assessment and reporting in mathematics from eleven Victorian high schools revealed extensive adoption of· . the distinctive features of the Victorian Certificate of Education (VCE) Study Design (VCAB, 1990 ),and its multi-component assessment scheme. The document analysis charts the impact of these changes in nomenclature on the structure and practice of the mathematics curriculum in Years 7 to 10. The first stage of the study strongly confirms the impact of changed assessment practices in Years 11 and 12 on curriculum policy and practice, and on how mathematics is taught and assessed throughout secondary school.

Understanding symbolic notation is usually considered crucial to the study of mathematics. One significant aspect that has emerged from research into this understanding involves the meanings students give to pronumerals (see for example, Collis (1975), Pegg and Redden (1990). Kuchemann (1981), in particular, has identified six different interpretations for the meaning attributed to letters by junior secondary students. These interpretations have been grouped into four levels representing a hierarchy of understanding. The purpose of this study was to explore students' responses to three of the more demanding Kuchemann test items which were identified after an initial sample of 278 students were given the entire set of test items. Twenty one students were then interviewed and the reasons for their responses to these questions were analysed. Students' responses and associated reasons could be divided into two distinct categories, depending upon the mental processes involved in answering these questions. The distinguishing factor was found to be inability to take into account the range of possibilities and limitations associated with relationships involving letters.

The changing nature of modern technological society and the role of mathematics in its functioning is resulting in calls for increased emphasis on mental computation and computational estimation and reduced. emphasis on pen-and-paper algorithms. There is strong evidence that children be9in primary school with many original and creative strategies for the operations,strategies that support mental algorithms and computational estimation. However, the procedures in traditional pen-and-paper algorithms appear to inhibit these spontaneous strategies and the early treatment of pen-and-paper algorithms may be wrongly placed in modern mathematics syllabi. This paper identifies and describes the different spontaneously derived cognitive strategies for one-; two- and three-digit mental computation in addition and subtraction used by 130 children within a longitudinal study to' examine changes in knowledge and use of strategies over Years 2 and 3 of traditional pen-and-paper algorithm instruction.

The equivalence relation exhibited as an equals sign is misunderstood by students who consider it to mean 'do something', to indicate the location of the answer and to act as a separator symbol. This presentation reports on a cross-sectional study of students in years P, 2; 4, 6, 8 and 10 to explore understanding of equals and equivalence, conceptions of same and different and knowledge of the properties of equivalence and approaches to the equals sign. Students appeared to predominantly have an operator ('do something') understanding of equals, and to exhibit differing understandings of equals in situations such as 2+3=?, 2+3=5 and x+3=5 of the three properties (reflexivity, symmetry and transitivity) and of the two approaches (static and dynamic).

This conference paper discusses possible relationships that may exist between social-interaction roles and children's metacognitive functioning. The paper reports on results of a study which compared the effectiveness of two approaches to group problem solving, specifically: (a) non-directed group activity involving limited teacher involvement only in the form of posing problems and engaging in follow-up discussions; and (b) role-directed group activity involving direct teacher intervention through the training of social-interaction roles (recorder-reporter; checker, and leader-judge) and the rotating of these roles, as well as posing problems and follow-up discussion. The role directed groups appeared to: (a) perform better in maintaining task commitment and producing quality solutions; and (b) develop superior metacognitive ability. The paper concludes by proposing relationships between social-interaction roles and metacognitive processes and discussing possible ways to study these relationships.

This paper reports preliminary results from an ongoing investigation to identify the conceptions of mathematics held by beginning first year university students and their orientations to their previous study of mathematics. A questionnaire was administered to approximately three hundred students during their first week at university. The questionnaire contained five open-ended questions designed.to elicit students' own conceptions of mathematics and their orientations to studying it. Two of these are discussed in this paper. Phenomenographic techniques were used to analyse responses and identify qualitatively different categories of description. In-depth interviews of a subsample of twelve students revealed details of the range of conceptions of mathematics and the related qualitative differences in approaches to learning mathematics after several weeks of university mathematics. Analysis of responses revealed that, although a wide range of beliefs was elicited, the majority of students view mathematics as a necessary set of rules and procedures to be learned by rote that are unrelated to other aspects of their lives. The survey results also indicate a relationship between conceptions of mathematics and approaches to studying mathematics at university level. There was no evidence of gender differences in either conception of mathematics or approach to learning. This paper will report on the evidence of relationships between conceptions of mathematics, approaches to learning and course results. These preliminary results raise questions about the impact of prior experience on approaches to learning mathematics at university level and the quality of learning outcomes.

Since 1946 there have been a number of studies reported concerning students' understanding of the concept of speed (e.g., Piaget 1946, Trowbridge 1979). This paper describes one part of a study concerning Year 9· students' strategies in solving problems which involved comparing the speed of two trolleys given the starting and finishing position and the time taken for each one. It was found that students used a variety of strategies ranging from guessing to intuitive use of variables to rather sophisticated uses of ratios and a calculation. Students also revealed some misconceptions concerning the variables that determine speed. Of particular interest was the use of time or distance as a determining variable without mention of the other variable.

This paper describes a research project which brings together current interests of the co-investigators. Anderson has been investigating the metacognitive strategies used by junior secondary students, to solve nonroutine problems in mathematics. Dawe has been studying the use of visualisation by teacher trainees, in the comprehension and solution of higher order non-routine problems in mathematics. The project examines the introspections of teachers and students as they solve non-routine problems, and the implications/or cooperative classroom learning. The increasing use of computer graphics, and graphics calculators in schools and tertiary institutions, has re-awakened interest in the specific role of visualisation in the problem solving process. We are 'particularly interested in the interaction of visual imagery with metacognitive strategies in . non-routine problem solving~ The data and its analysis will provide an indepth qualitative case study, of the role of verbal and non-verbal mental activity in comprehending and solving non-routine problems.

This study looks at two groups of children - those for whom the calculator is a part of their everyday mathematics at school and those for whom it is not. The research methodology adopted was that 'used in the innovative Victorian science study (1990). Children at grade three level were asked to complete statements about calculators, sometimes by writing and sometimes by drawing. The analysis categorized the children's responses into mutually exclusive types, which were then assigned integer 'level' labels. These categories were then tested for cohesion, using an IRT partial credit analysis. The last stage of the analysis was to construct descriptors for each of the categories, thus establishing a developmental continuum for beliefs about calculators. While the number of children was only a few hundred, it is clear that further investigations of children's beliefs in this area would contribute significant information for those implementing a 'calculator aware' mathematics curriculum in their school.

This paper reports on a current study investigating 9 to 12 year-blds' strategies and reasoning processes in solving novel combinatorial and deductive problems. Equal numbers of children were selected from low, average, and high achievers in school mathematics an.d were individually administered five sets of problems, namely, two sets of combinatorial problems(2-dimensional and 3-dimensional) and three sets of deductive reasoning problems. Both problem types were presented in "hands-on" and written formats. The written problems were isomorphic to the hands-on examples. The nature of children's strategies and reasoning processes in solving these problems is addressed. Of particular interest are the differences in the responses' of children classified as low and high achievers in school mathematics.

Our purpose is to exploit linguistic evidence derived from the use of number words in languages in order to formulate conjectures of how the number concept emerged in human thinking. Comparison of structures in naming numbers gives us a hint about what thoughts are at the background in the evolution toward the mathematical concepts. Two distinct procedures seem to activate the process; the first consists in developing an awareness of quantity and leads into the cardinal number concept, the second relies on order and introduces the ordinal numbers. Four successive phases may be distinguished in the development of a number system. Phase I is related to the recognition of a number of objects, involving awareness of the concept without exteriorisation, without using names for it. Phase 2 comprises the introduction of number words with classifiers: the number concept is closely associated with objects of a certain kind. In phase 3 the development of a number system occurs through association with a standard: the· human body. Eventually, in phase IV abstraction yields the creation of an abstract number concept. A word for a concrete object which is part of the vernacular is totally freed from any concrete connotation and becomes an abstract measure.

When women return to study mathematics they frequently have to overcome obstacles, both long-term and short-term, in order just to be present at classes, Voluntary attendance at mathematics classes can provide both mental stimulation and an escape from the reality of everyday existence. It can also provide a support to sagging morale when self-esteem is under attack elsewhere. Further it can. be emancipatory when a woman discovers that she is capable of actively helping her children learn mathematics in a productive manner, making it an enjoyable experience for all. This paper will discuss the positive effects of a constructivist mathematics classroom on one woman's life.

Videotaped records can have distinct benefits over other methods of data collection. The technique also has limitations and frustration can be encountered in its use. Inherent in its apparent simplicity is the temptation to use the medium inappropriately. Judicious consideration of the research question and, related methodological and technical issues will determine whether videotape is the suitable choice. In this paper we describe three quite different mathematics learning situations in which videotaped records were used. Common to the three situations was the recognition that no other data collection technique would provide the information sought as effectively. Both visual and auditory information were essential. Yet in each case the researchers were faced with a unique set of interacting difficulties before. during and after videotaping. Our discussion also outlines some consequences associated with using and retaining videotaped information.

The study reported here, was designed to investigate student learning in calculus with a focus on language use and the ways truth and validity are determined. Results reported here are those related to students' processes of construction of particular mathematics conceptualizations as a result of exposure to three different approaches to calculus instruction: technique-oriented, concepts-first and infinitesimal instruction. When students used infinitesimal language and used it in conjunction with everyday language they generally did so as a foundation by which to construct mathematically valid problem responses. This finding indicates that instruction emphasising connections between everyday and technical language is likely to guide students to build mathematically appropriate inter-connected conceptualizations. Also" the use of infinite magnification in a ,variety of problem situations by students who received infinitesimal instruction demonstrates that instruction emphasising visual interpretations can influence students' conceptualizations.

Shared meanings and constructions are necessary if purposes embedded in documents such as the National Statement on Mathematics are to be realised. One starting point is to establish where the profession and the public stand in relation to values and goals. This paper builds on a previous study and reports on perceptions of the role and purpose of school mathematics as obtained respectively from samples of community members (N=662) and mathematics teachers (N=97). Viewpoints were sought by structured interview or questionnaire across the thematic areas of content. individual values, additudinal aspects, and folklore. The data from the study suggest that public and professional views in relation. to school mathematics have more in common with the competency movement than with the broader ideals of mathematics education.

This study investigates the linguistic problems ESL post-secondary students have when communicating through the content area of mathematics. The majority of these students have the mathematics register in their own tongue but need to transfer this to English so they can communicate effectively in tutorials, lectures, practicals and assignments. To investigate these linguistic problems, students from a pre-tertiary (UN/PREP) course were interviewed and their mathematics pretests and the problems from mathematics vocabulary tests were analysed. Findings from the pretests suggest there are a number of levels of understanding and types of methods used by students in solving the problem. The results of the vocabulary study outline the frequency of occurrence of the types of mistakes students make. Preliminary findings from student interviews and research suggest that while some of the mathematics learnt through their native tongue is easily translatable into English other features are more complex and further research needs to be undertaken in this area. The overall results of this paper clarify the problems faced by ESL students and can be used for ESL educators in their curriculum design and assessment.

This study explores how music might provide an effective aid to higher achievement in mathematical development. An experimental group of 35 preschool children was chosen from children enroled in a community music program. A comparison group consisting of 39 preschool children with limited musical background was employed as a contrast group. By testing both groups on early number concepts, initial results indicated that the experimental musical group performed better than the non-musical comparison group. However post-analysis indicated that musical experiences to be found in the home and other pre-existing differences were more likely to contribute to the experimental group's higher mathematical achievement than the music program treatment.

This paper argues, with examples, that as a result of how mathematics is presented, especially in textbooks, explanation is highly implicit. Students are given mathematical problems and are told how to do them, but the whys - the underlying principles and connections - are understated. Thus students are left to infer these from examples, visual representations and manipulations, with the result that mathematics makes little sense to many learners. A strategy for determining at what points explanation breaks down is suggested.

Mathematics bridging programs and related assistance schemes for mathematically underprepared students have become essential features of many tertiary institutions. Often unstated, but nevertheless intended, the principal goal of such programs is to provide a service which gives the mathematically underprepared student the same opportunity to satisfactorily complete their chosen tertiary qualification as the mainstream student. Evaluating the effectiveness of each bridging program therefore necessitates determining whether this goal is being reached. Long-term or longitudinal studies are strongly recommended. However, there is little evidence in the literature that formal evaluation of such programs is occurring to any great degree. Why is this so? This paper takes up this question and reports on one aspect of a large study which involved the synthesis of research evidence concerning bridging mathematics programs in the USA and Australia. Several important reasons for the limited nature of evaluations are identified. The paper concludes by suggesting that evaluation, in the traditional sense, may be incompatible with the successful conduct of tertiary mathematics assistance programs.

Structured, task:.based individual interviews are· one asiJict of an ongoing mathematics education research study of elementary school children at Rutgers University.· The gorlls of the interviews are· to observe complex, mathematical problemcsolving behavior in detail, and to draw inferences from these observations about the children's thinking and development. This paper discusses the scientific underpinnings of the methodology, the role of cognitive· theory in structuring an interview; constraints and limitations imposed by the social-psychological context of an interview, and the interplay between task variables, observed behaviors, and inferred cognitions. Some principles of interview design and construction are suggested for consideration by the mathematics education research community.

In recent years, the role of metacognition in mathematical problem solving has begun to attract research interest as metacognitive processes are considered to be an important factor in influencing problem solving performance. The study described in this paper investigated the metacognitive strategies used by a pair of upper secondary school students, while working on mechanics problems. The bulk of the data consisted of verbal protocols from uninterrupted think aloud paired problem solving sessions. Metacognitive decision points were identified in order to examine the monitoring contributions of each individual student, and the significance of student-student interactions. The main findings of the study were: I. The subjects assumed differing, but complementary, metatognitive roles during, problem solving. 2. The quality of the subjects' metacognitive decisions had an important influence on problem, solving outcomes, but their decision making was sometimes adversely affected by the social, interaction between them.

In a paper presented at Merga-15 in 1992 at University of Western Sydney - Hawkesbury, N.S.W., the author reported on the beginnings of a study of teacher's attitudes and possible changes in assessment practices in secondary school mathematics classrooms within this state. It was reported how content laden the secondary mathematics curriculum is in N.S. W. and just how prevalent is the utilisation of timed pencil and paper tests in this state. Because the assessment procedures are so conservative, methodology employed within the classroom is similarly traditional and does not really reflect changes occurring in procedures in secondary mathematics classrooms in some overseas' countries and interstate within Australia. Nonetheless, some changes are occurring. This paper will report further on a series of questionnaires that have been conduded with practising secondary mathematics teachers on the subject of alternative modes of assessment; and endeavour to make some recommendations for the future.

This paper examines the. extent to which seven teachers changed their expectations of· 'children'smathematical performance during their first two years of teaching at the. preparatory level in the . Calculators· in Primary Mathematics project - a long-term investigation into the effects of calculator use on the learning and teaching of primary mathematics. Questionnaire results' show a slight increase in expectations for most of the 21 items dealing with counting and "large numbers", with a greater increase for the 5 items· dealing with negative numbers. In general, teachers' increased expectations reflected their observations of children's performance during the previous year. Nevertheless, teachers' predictions remained conservative compared to actual levels of performance. However, data obtained from interviews and teachers' comments suggest that teachers are adopting a more open-ended approach to their mathematics teaching in order to cater for the increased level of understanding revealed by the presence of the calculator.

The role and value of concrete mnterials in teaching and learning mathematics is uncertain, yet mathemntics educators tend to assume their uSe is essential. Is this an act of faith? This paper describes a Procedural . Analogy Theory which attempts to explain the value of concrete materials in the teaching of mathematics. Given the range of teaching possibilities for using concrete materials to help the learning of a particular concept or skill, this theory claims to be able to help teachers develop a teaching approach which will be superior to others. Aspects of both cognitive science and mathematics education are discussed in relation to this theory. The paper reports on findings when the procedural analogy theory was applied in a number of Year 4 classrooms where Multibased Arithmetic Blocks were used to support the teaching and learning of subtraction algorithms.

The research reported here focussed on "the mathematical learning of "at risk" preschoolers in computer supported interactive environments. Specifically, our investigations have explored the effectiveness of metacognitively-rich computer-supported contexts on early mathematical learning. Results indicated that children who participated in mathematical learning activities within these contexts performed significantly better in tests of mathematics competence than did children engaged in more regular instructional approaches, including those situated within computer-based settings. This paper presents some of the theoretical background to our research, some findings and considers a range of implications for teaching mathematics in early childhood contexts.

We begin with a discussion about how we come to know the concept of a square. The problem of how best to teach fractions, and by what means we approach this daunting task follows. The example of the CopyCat; a prototype computer-based learning environment is used to illustrate strengths and weaknesses. Other didactic materials such as Steffe and Olive's Sticks, and Kieren and Pirie's paper folding are discussed, and we conclude that there is no possibility of finding physical material with the properties that will mirror the precise essence of the mathematical concept we have in mind to teach. A view of the role of physical materials in an enlarged pedagogy involving what Cobb called the interactive negotiation of mathematical meaning is suggested as a solution. We are reminded also that students operate on material of the mind including representations of past actions, memories, and visual images. Remaining unanswered questions are posed.

An ARC funded project of the Institute of Mathematics Education is to develop and validate a clinical instrument suitable for an initial assessment of a student. Priorities need to be set about what constitutes a "critical task", since there is simply not enough time to systematically sample every content cell from a traditional scope and sequence analysis of the curriculum. We provide background to the project, including phases in the validation process, issues raised by consultant experts, practitioners, and the development team in making decisions about revision. Inclusion and exclusion of tasks will be discussed.

An analysis of the steps involved in forming the idea of an empirical sampling distribution and the nature of the methods and/or images used in most computer based strategies to teach this idea suggest that this way of using the computer adds little insight to the usual text based explanations that they are designed to complement. This analysis suggests reasons why a more recent approach which uses the computer to model and dynamically display the processes that underlie the idea is more likely to be successful.

III the usual constructivist view of learning mathematics the student is engaged in the· active process of constructing meaning for instructionally given target concepts. Cobb, Yackel, and Wood (1992) and others propound an alternative view, social constructivism, which treats mathematics both as an individual constructive activity. and as a social practice. On this view, learning operates as an individual cognitive action (construction of interpretations) "iade compatible by social interaction with the colleCtive interpretations of mathematically acculturated practitioners. One argument used by Cobbet al against. standard forms of constructivism is that they commonly rely on an essentialist theory of knowledge. In this paper, however. it is argued that the notions of interpretation, meaning, and construction crucial to the position of Cobb et alalsodepend on essentialist theories afknowledge. Approaches to validating this claim . are made in three lines of inquiry drawing upon notions of theory of practice, philosophy of language, and the science of signs or semiotics. Important parallels between these discussions, and between certain· poststructuralist formulations are noted in the process of argument. An outline of elements of a new postconstructivist theory of learning emerges. A touchstone for these ideas is provided by Lave and Wenger's theory of learning as legitimate peripheral participation; allusions to this theory are made throughout the paper, although no direct study of it, or its relation to the theory figured here is attempted,

This paper reports on a comparative analysis of research presented to the 1979 and 1993 MERGA Conferences. Two main issues are considered: the nature of research methods and methodologies utilised by researchers; the content focus of principal questions made the subject of research. In considering the first of these, papers were stratified by research methodology (positivist, interpretivist, critical), data type (quantitative, qualitative), and research method (experiment, survey; clinical interview, discourse analysis, etc). Analysis of the data indicated that whilst the positivist paradigm predominated in both conferences, a significant switch to qualitative methods and a diversification of methodologies was observed in 1993. In addressing the second main issue, dominant research questions for each conference were identified it in terms of subject/domain clusters. Methodology used in this task involved the construction of a subject matter matrix, within which each submitted paper was allocated to a single cell. Papers were thus stratified by two variables accounting for educational agency (student, teacher, classroom, researcher) and content (knowledge, beliefs/attitudes, context). Major findings were an increased emphasis in 1993 on the student vis a vis teacher, and a switch of emphasis from beliefs/attitudes to the study of contextual domains of education. The paper concludes with a discussion of the limitations of the methodology used in this study, and an identification and discussion of principal findings.

This paper reports an action research project that spans more than two years and chronicles the transition from an unquestioning acceptance of constructivism as the "one best solution" for the problems of student teacher empowerment in mathematics education to a questioning of its intuitively convincing canons. One of the many outcomes of the study centres around my changing notions of "empowerment"of students to construct their own knowledge, to an interrogation of constructivism itself/or its evasion 0/ consideration of the historical and social situatedness of the nature of knowing an4learner subjectivity. Implicated as 1 am as the teacher in the project, 1 offer one interpretation of the data which to me renders constructivism vulnerable to censure for a perceived lack of recognition of the effects of Culture and power inherent in all discursive practices. At the completion of two cycles and upon entering the third it appears that students, after participation in, "constructivist"practices, are getting the same epistemologica! messages concerning mathematics and mathematics teaching as previously established during the reconnaissance phase.

In the early 8Os Mayberry (1981) developed a diagnostic instrument to be used to assess the van Hiele levels of preservice primary teachers. The test which was carried out in an interview situation, was designed to examine seven geometric concepts. There has been no reported attempt to (a) rerplicate this work in Australia; (b )consider the items in some alternative format; or (c) analyse the validity of the test questions. To address these issues, a detailed testing and interview program of 60 first year primary teacher trainees was undertaken at the University of New England. This paper considers one aspect of the findings of this study. It concerns the potential for certain aspects of Maybery's work to lead to an incorrect assessment of a student's level of understanding in geometry. In particular, four main features were found to account for major problems to the test validity.

The fraction program described in this paper is characterised as an environment with resources ' of knowledge and skill that are available and appropriate for classroom use. The main goal of the program is that learners might be enabled to use a fraction environment confidently and in appropriate ways. The program presents activities for classroom groups to develop, select and manage fraction knowledge in purposeful tasks. It aims to encourage a learning partnership among 10 to 13-year-olds and their teachers. The intention is that knowledge and skill in fractions are developed interactively and in authentic contexts. The project is informed by research into cognition, especially as it is manifested in social and workplace activity. Three stages completed so far have involved (l) an investigation into perceived classroom learning needs for the selected age group and their teachers, (2) the charting of a fraction environment in terms of relationships between concepts and procedures, ' and (3) the design and publication of a book presenting tasks involving fractions and suggestions as to how the activities might be managed in classroom groups. This paper indicates how theoretical and empirical information gathered have been integrated to provide a framework and action plans for classroom implementation.

Secondary mathematics teachers have historically taught centrally based, skills oriented courses at Year 11 and 12 levels. Introduction of the Victorian Certificate of Education (VCE) mathematics study design represented a Significant shift in personal philosophies. and practices for many mathematics teachers. This report focuses upon the effects that VCE implementation is having on mathematics teachers in the Ballarat area. Individual teachers, and schools, were coping with implementation in different ways, and at differing rates, supporting the Northfield findings (1992). The major concerns of mathematics teachers centred around the needs of more time, more consistent and specific advice from the Victorian Curriculum and Assessment Board (VCAB) and better in-service for country teachers. Many have readily accepted the underlying philosophies of the VCE but are unhappy with the way in which it is being implemented. There is evidence of a growing contempt and cynicism brought about by a perceived failure to acknowledge the importance of the feelings and attitudes of mathematics teachers. The resulting loss of faith, if unchecked, may well see little effective change in mathematics education at these levels in the long term.

The primary purpose of this study was to investigate students' ability to communicate their understanding of mathematics terms in writing. A second goal was to examine the performance of students in rural environments as compared to urban students. Consequently locality of upbringing, gender and, to a lesser extent, culture were variables which were considered as influential on the set task. Data was collected from 2371 year 8,9,10 and J J students/rom twelve urban and rural schools in Western Australia and the Northern Territory. Results indicate that students in urban environments were better able to define mathematical terms that their rural peel's. Both urban and rural male and female student results were very similar with the exception of the Year 10 group where female students outperformed their male counterparts.

Twenty four Year 2 children were presented with realistic models of either cricket or tiling and asked firstly, to indicate which of a set of 10 abstract angle models could represent the path of the ball or the corners of the tiles, and secondly, to draw the paths or corners. Responses were analysed to indicate obstacles to abstracting the angle concept. In the cricket context, children needed to conceptualise the path of an object as a straight line segment and then to link the segments to the sequence of actions. In the corners context, children needed to abstract the two ides and ignore the shape at the point; ideas of size were apparently abstracted concurrently. Similar investigations in further contexts promise to uncover other difficulties which children face in abstracting the angle concept.

This project was commissioned by D.E.E.T.for the Australian Education Council's Working Party on an Annual National Report on Schooling and conducted during 1991-2; Its aim was to document school assessment and reporting practices across Australian schools. A substantial survey instrument was developed and used to profile assessment and reporting practices ~ influences, decisions and actions. In addition, a national network enabled access to individuals and schools with acknowledged good practices and a number of these were supported to prepare case studies of their work in the areas of assessment and reporting. The survey instrument and results are discussed, along with some suggestions about influences on and connections between aspects of practice. Of particular note is the general reluctance to value assessments made informally and/or involving students' self-assessment in comparison with more traditional means. Areas of difference between primary and secondary teachers' responses are noted and discussed. The strong links between teaching and learning mathematics and, particularly, assessment are evident. Consideration of some of the case study material illustrates how a balance of assessment styles can be achieved and how such an approach is intimately linked to a contemporary teaching and learning program .

Qualitative research produces a wealth of complex, interactive ideas which are difficult for a an educational inquirer to control. NUDIST, a qualitative data analysis tool, speeds both categorisation of data and statistical analysis of data, while maintaining an ability to re-visit the original material. This paper illustrates the use of NUDIST to synthesise ideas resulting from one open-response item from a questionnaire; The major issue raised is the subjective nature of decision-making about categories used to form a conceptual framework for data analysis using NUDIST.

An earlier study analysed responses by 70 young children to a variety of multiplication and division word problems at four interview stages in a 2-year longitudinal study (Mulligan,· 1992). 75% of the children were· able to solve the problems using a wide variety of counting, grouping and modelling strategies· and the semantic structure of the problem strongly influenced solution process. A follow up study Was conducted with the same sample (n=45) in Grade 6, prior to entry to secondary school. Identical problem structures were used including problems involving decimals, A similar pattern of performance to the longitudinal study revealed a 60-85% success rate except for Sub-division, and poorer results for decimal problems. Informal methods used by children in the. early stages of the study dominated their understanding. of multiplicative. structures in Grade 6. Evidence of children's lack of conceptual understanding o/multiplication and division was revealed.

Access to effective learning opportunity for students who have mathematics learning disabilities is an ongoing concern for teachers, students, educational policy makers and the community at large. The types of opportunity offered range from the traditional "clinic"approach (for example, as described by Englehardt (1985), Irvin and Lynch-Brown (1988) and Scheer and Henniger (1982)) to the delivery of support within the student's classroom, often via teacher in-service, The Mathematics Learning Centre (MLC), developed in the former Melbourne College of Advanced Education, lies between these two extremes in its approach to opportunity provision. This paper reports a recent evaluation of the effectiveness of the service delivery in the MLC. It examines the guiding mode/of mathematics learning on which the program is based and the evaluation procedures used to measure change.

This paper reports two investigation in which mathematics underachievers were taught to use cognitive and metacognitive strategies to facilitate information processing in two areas of mathematics learning: to read symbolic statements and to categorize their mathematics knowledge. One group of students learnt to use the relevant cognitive strategy while a matched group learnt, as well, associated metacognitive strategies (to evaluate the effectiveness of strategy use and to decide when they might use the strategy in the future). Both groups out-achieved a control group immediately after teaching. As well, the group taught both cognitive and metacognitive strategies were more likely to transfer the cognitive strategy to other areas of mathematics. The results are discussed in terms of models of human cognition and performance.

This paper describes the on-going research and development activities of the PRODIGY project. The major aim of this project is to develop inteUigentcomputer-based simulation systems which facilitate the develOpment of high leveldi~gnostic andremediation skills in the domain of common fractions. The development and the functioning of the present generation of PRODIGY simulation systems (PRODIGY] ) is first described. The methods used in and results from the formative evaluations of the PRODIGY generation of simulation systems then are described. Some of the modifications which had their genesis in the results from the formative evaluations are briefly discussed. Three serious limitations of the PRODIGY simulation are. identified and discussed. The paper concludes with a description of the architecture 0/ next generation of PRODIGY simulations together with a discussion about how It is hypothesized that they will overcome the serious limitations of the PRODIGY simulations.

This paper focuses on the function of contour in the association of melodic and visual patterns, and investigates the role played by ability in mathematics and music in children's performance at matching melodies and music notation, and at matching melodies and graphs. Initially, tests in mathematics and music were administered to 101 children in Year 5. Factor analysis confirmed specific sub-scales in the tests and revealed a positive and significant correlation between results in the two tests. However this correlation was no greater than the correlations between other school subjects and mathematics and music. The melody matching tasks (four modality conditions altogether) revealed that format (conventional/non-conventional) was a significant main effect for· graphs but not for music notation. Modality condition and level of contour complexity were main effects for both types of matching (melodies, music notation and melodies, graphs). Success at matching melodies and both types of visual contours was related to ability in music more than ability in mathematics.

The development of young children;s representations of a rectangular array as [perpendicular] intersecting sets of parallel lines is the focus of this paper. Drawing an array using lines, rather than drawing each rectangular area unit separately should assist children to represent the correct numerical structure and to perceive a row as a single, composite unit. Without a knowledge of the numerical structure of an array, it is unlikely that children will be able to apply repeated addition or multiplication skills to determine the number of elements in an array. To draw a rectangular array, it seems necessary for children to grasp three properties of an array: that the area units: must be congruent, that the units are aligned, and each row has the same number of units. A teaching experiment was undertaken to determine the effect of stressing these features on children's array drawings. Emphasising that the units should be the same size did not help the children to draw the correct array structure, while stressing the alignment of units or that there is an equal number of units in each row did seem to assist children to draw an array as lines.

The benefits and disadvantages of cooperative, competitive, and-individualistic learning in school subjects has been considerably discussed in recent years. A coordinated set of three instruments (the "Learning Preference Scales") is now available for use by teachers and researchers in investigating the preferences of students, teachers, and parents: The Learning Preference Scale· Teachers was administered to large samples of primary and secondary teachers in Sydney (N=619) and Minneapolis (N=342), and secondary teachers in the English Midlands (N=278). Differences among teaching subjects and between sexes are discussed. In all three locations the learning preferences of Mathematics teachers were strongly oriented to competitive learning. Discussion is focussed on the pedagogical epistemology o/mathematics teaching and learning, and on the belief systems of teachers.

This study examines the mathematical concepts of "fairness" and "expectation" in probabilistic situations. The subjects were 40 high school students in Semester I, Year 11. Maths in Society classes in three Queensland high schools: Twenty "gamblers" were identified by questionnaire and subsequent interview. A control group of similarly achieving "non-gamblers" was selected. The research compares the ability of each group to contract a working definition of the concept of mathematical expectation and to use this concept in determining the fairness of a number of games of chance.

The study of teacher professional development frequently has been limited in its subjects, its tools and the duration of the research. This paper describes a study to examine teacher change from the perspectives of the teachers, their students and the researcher through the use of a variety of research tools. Each of these instruments was used in a variety of different forms addressing different specific issues. The guiding aim of the study is the comprehensive portrayal of the change process in teaching practice, knowledge and beliefs as experienced by eight junior secondary mathematics teachers; The longitudinal nature of this study was a deliberate attempt to describe the process of teacher change in the long-term. A professional development program for secondary mathematics teachers (the ARTISM program) provided an appropriate context because its design was informed by recent research into teacher professional development and because the program took a long-term perspective on teacher change. ThiS combination of design factors was conceived to give due recognition to the complexity of the teacher change process.

At the Auckland College;f Education, secondary teachers are prepared for the classroom in a one-year postdegree course. This involves lectures and 13 weeks classroom practice. In 1991 it was realised that the model being used was at odds with the classroom experience of the students. An action research programme involving lecturers, students and associate teachers in the schools is in process. Its aim is to link the classroom experience to the model of teaching demonstrated at College. The programme has gone through four cycles in the action-research model. This paper describes the four cycles and reports on lessons learnt during the research. The exciting part of the emerging model is a collegial process which involves lecturer, students and associate teacher working together in the classroom with more experimentation and reflection on teaching as an art. Other benefits are increased cost effectiveness and closer links between College and schools.

This study resulted in the development of two instruments for teacher and student use in assessing the affective development of year 5, 6 and 7 Primary school students. Eighty-seven teachers rated the students in their class on a set of 22 affective characteristics which they deemed important to develop in their students as a result of their teaching. Factor analysis of these ratings resulted in two factors. Twenty-two teachers wrote classroom descriptors for the 11 characteristics making up the two factors which formed the basis for the instruments The descriptors were validated and the reliability of the two instruments was determined using a 25% sample of the children in these 22 classes. These instruments will help sensitise teachers to achieve a more appropriate balance between cognitive and affective objectives in their classrooms, and where there is dialogue between teacher and student following the student's self evaluation, will result in student empowerment.

The main features of a professional development program for inservice teachers (ITAM) are described and a subset of the data from two case studies is presented to highlight some of the strategies used by teachers while undertaking the program. The case studies demonstrate an isomorphism between the stage of growth experienced by the teachers in the program and the four problem-solving phases popularised by Polya. These phases provide a useful framework to analyse the success or failure of inservice experiences and provide some guidelines for those designing professional development programs. This research also highlights the usefulness of using our knowledge of student growth in acquiring mathematics concepts to assist in our understanding of teacher growth.

The year 1990 was officially declared "The Year of Enchantment of Mathematics" in Brunei Darussalam. The Fifth South East Asian Conference on Mathematics Education (SEACME 5) was successfully hosted in June. School based promotional activities were carried out effectively. Chief among them was a National Exhibition, "Mathematics in School". A National Mathematics Quiz was also held. It has since been established as an annual event. A National Mathematical Society has also been formed. Overall, an attempt has been made-to ensure that the benefits of hosting SEACME 5 would be lasting. That private enterprise too, can stimulate national educational growth has been demonstrated.

A year-long teaching experiment explored the possibility of changing fourth grader's approaches to mathematical problem solving. A metacognitive question-and-answer technique was used to negotiate meaning, explore problem representation, discuss possible solution strategies" and reflect on the, problem solving, enterprise. Analysis of the transcript data, classroom observations and childrens' work samples revealed that while those with most to gain and nothing to lose demonstrated the greatest shift in approach, each child's approach was successfully challenged to some extent.

A major aim of the study reported here was an investigation of the relationships between mathematical and cognitive processing and metacognitive activities during problem solving by Year 11 Mathematics I students. During the firs! year, one problem was used for video-taping, sessions whilst another was used in an examination situation followed by free response interviews where students reviewed their examination scripts retrospectively. In the second year, the role of the problems was reversed and a structured interview was used. The interviews probed the students" metacognitive knowledge, strategies, decision making, beliefs and affects. Results' indicated that the students possessed quite a store of metacognitive knowledge which had the potential to influence their problem solving activities. Orientation activities were crucial with many students failing to inhibit impulsive responses to initial reading of the problem. Students were more concerned with the mechanics of solution execution and the tyranny of time than with planning, monitoring and verification strategies. The study supported the notion that students acquire and develop their store of metacognitive knowledge through metacognitive experiences and social interaction.' There were indications that classroom practice and assessment techniques emphasized the use of automatic routinized application of formulae and procedures at the expense of experiences where students needed to reflect on, monitor and evaluate their progress.

Survey responses of 125 teacher educators and experienad teachers on an open-response item on aspects of mathematiC$ teaching are presented. A qualitative analysis of the responses using the NUDIST program resulted in six major categories of responses. These were Communication, Problem solving, Building understanding, Engagement, Task orientation, and Teacher concern. Within each of these categories, the frequency of use of particular phrases and descriptors indicates general beliefs about the important characteristics of quality mathematics teaching. These categories and qualifications are presented as a starting point for further discussions about, the development of a common language for describing teaching.

The researched discussed in this paper reports on the findings of a survey of Queensland mathematics teachers. The survey sought to determine the methods used by these teachers to obtain curriculum information. The results suggest that as a group Queensland mathematics teachers do not undertake professional development activities.

Modern primary mathematics curricula explicitly aim to develop children's understanding of basic mathematical concepts, in contrast with the more traditional rote learning of procedures. However there has been continuing concern that many children still do not achieve an acceptable level of understanding of these mathematical concepts and most seem to have difficulty generalising their understanding to relevant contexts. An investigation of the mathematical understanding of tweive students during their transition from primary to secondary school suggests that there may be little educational value in basing curricula on a presumed common learning hierarchy of concepts within mathematics which can be transferred to the learners' minds through a controlled sequence of learning experiences. The conceptual understanding that students were able to demonstrate (though quite limited) revealed that their mathematical conceptualisations often consisted· of unique interpretations associated with personally perceived contexts, rather than common and clearly recognisable generalisations.

In 1991 all Victorian year I2 students undertook the new Victorian Certificate of Education Mathematics Study designed by the Victorian Curriculum and Assessment Board. This paper presents the results of a study into sex difference in achievement in the new VCE Mathematics study in Victoria. An important goal of the study designers was to encourage more equal participation in senior secondary mathematics by females and males and to include assessment of mathematical skills previously not assessed in a year 11-12 course in Victoria. These new tasks could conceivably change the degree and direction of sex difference in achievement in senior secondary mathematics.

Over the last two decades a vast number of research projects have Identified areas of students' misunderstandings in the algebraic domain. It appears that the main focus of this research has been on developing an understanding of variables, the translation of word problems, and on"doing" algebra - the manipulation of symbols. Although these aspects are important to algebra, Booth (1989) and Kieran (1989) believe that a critical aspect is understanding just what the algebraic statement represents, both visually and symbolically. When linking and using various representations, spatial skills and higher order thinking skills play key roles, and are therefore crucial to the acquisition of algebraic understanding. This paper illustrates the importance of these skills to the, algebraic domain, reviews the literature pertaining to these skills within the algebraic domain, and identifies the research implications drawn from this literature.

Following the questions raised by Watson (1992) concerning research in probability and statistics education in Australia in the 1990s. This paper reports on the initial trialling of items with 64 Grade 9 students. The analysis supports the belie/that misconceptions observed in other countries also are present in Australia. Further. the application of a developmental cognitive model offers promise for classifying responses to items and structuring remediation procedures.

This paper reports on the further development of an instrument designed to measure self-concept in mathematics and the attitude of pre-service teachers toward the teaching of mathematics. It presents the results of the questionnaire's application to teacher education students across various programs at the University of Western Sydney, Nepean. The findings of the study revealed that the students' attitude towards teaching mathematics improved as they progressed through the programs, but that their self-concept in mathematics remained unchanged by the experience. In addition there were clear and significant differences in attitude to teaching mathematics and self-concept in mathematics based on gender and program of study. Males compared to females were found to display consistently higher levels of self-concept and more positive attitudes. Secondary students also displayed significantly higher scores on both constructs than did primary students who in turn scored consistently higher than early childhood students.

Research in mathematics education and cognition suggests that knowledge. its acquisition. and its transfer are complex processes. In this study, end of semester totals are recorded for a large group of students in a traditional introductory subject in statistics at the tertiary level. A group of eight students are selected for observation while solving four statistical problems during the semester. This paper illustrates the capacity of students to pass an exam. yet highlights their limited understanding of statistical concepts, their narrow mechanical focus, their poor study habits and the lack of statistical transfer in atypical problems. Alternatives to the traditional approach are suggested.

This study examined students' attitudes to and ideas about mathematics at the beginning of a mathematics or engineering degree at university. The instrument used was one of Schoenfeld (1989) considerably modified for use with university students. Seventy four students took part in the study. Several interesting results were· noted. Both male and female students. attributed their success in mathematics to intrinsic values such as hard work. There was a significant difference between the ideas of students who had studied 3 unit mathematics and those who had studied 4 unit (the highest level in NSW). Students who had studied 3 unit described school mathematics as "just memorising". However 86% of the students described mathematics as "interesting" and gave that as a reason for doing mathematics. The most interesting part of the study was the answers to 10 open-ended questions. Here, this group of students demonstrated a mature understanding of the links between areas of mathematics, of everyday applications of what they chad studied at school and were at ease with discussing their feelings about mathematics and learning mathematics. The mathematics they had studied so far was widely perceived as useful in the development of personal skills - teaching clear and logical thinking, fostering persistence, patience and discipline.

Drawing on current research the author explicates twelve assertions about current curriculum and teaching practice in lower primary mathematics. Topics discussed include the under-challenging curriculum, differences in children's knowledge, the need for compensatory programs, curriculum constraints on teachers, 'anti-interventionism' and discovery learning, verbal versus written arithmetic, the role of problem solving, and the need to better understand how children learn mathematics.

In this paper I employ classroom transcripts to develop Bourdieu's writings with respect to the construction of gendered life in the mathematics classroom and the disempowering consequences of many practices for a significant number of girls within mathematics education. Using Bourdieu's concepts of habitus, practice and dispositions, the construction of a gendered mathematical habitus will be proposed. Bourdieu (1977) describes habitus as being a "matrix of perceptions, appreciations and actions". Bourdieu does not employ gender as a primary category of capital where capital is central to the construction of social space. Feminist writers, such as McCall (1992) have been able to reapply Bourdieu's notions of cultural capital to incorporate a gendered habitus. Through a gendered habitus, students learn to take on certain dispositions which will be influential in their success in schooling and . their later positions within the social structure. For females, a gendered habitus is paramount to the construction of their marginality in mathematics.