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This paper describes some of the broad lessons on curriculwn which have emergedfrom 'The . Cognitive and Unguistic Demands of Learning To Use Algebra", a research projectfocussing on students' understandings of algebraic notation. Much of the current curriculum in number and algebra is designed principally to support algorithm development, whether skills or concept based. As technology increasingly frees us from an algorithm-driven curriculum, this should be replaced by experiences which expand children's conceptions of numbers and the operations of them. To learn algebra, students need experiences which assist them to become alert to multiple meanings and to explicitly recognise processes. Teachers need to address students' expectations of notation systems.

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Observations of Gareth's learning behaviours reveal a student who is keen to succeed, is on task, and completes homework. However, despite being actively engaged in doing mathematics, Gareth makes little progress in learning mathematics. As part of research examining senior students' use of learning strategies, data relating to Gareth's learning behaviour was collected from questionnaires, interviews, observations and stimulated recall interviews. This paper proposes that some students, and in particular Gareth's, learning difficulties are compounded by inadequate use and control of appropriate learning strategies. Gareth's cognitive learning strategies are directed towards to the goal of collecting information about the 'way' to do a problem, recording the method and hopefully recalling this method on a similar example in a test. Gareth' s metacognitive behaviour lacks appropriate monitoring and checking strategies. Social-support strategies, such as help seeking and modification of a task, are used inappropriately, often inhibiting learning rather than enhancing learning.

This paper describes a preliminary investigation into students' understanding of pictographs and bar graphs. An interview protocol was developed and administered to 30 students (20 primary and 10 post-primary) selected at three ability levels. Students' responses were documented and categorised. They show that in relation to the set tasks, students had fairly well-developed skills in reading, interpreting and predicting from pictographs and bar graphs with increasing facility as the ability level and year level increase. Nevertheless, some interesting results related to four key dimensions of prior knowledge, missing data, scale and pattern were found, and these are reported together with some suggestions for future work in this area.

A class of grade 9 students underwent a period of instruction on the use of concrete materials to represent algebraic expressions and equations and to solve linear equations. Interviews conducted at the end of the instruction aimed at studying the students' knowledge of variables and their ability to solve equations using a variety of methods. In particular cognitive load theories were used to explain the problems in teaching and learning of algebra using concrete materials. The hypothesis that the students benefit from concrete material in algebra is not supported by their use in this study.

Recently the Mathematics Education Unit at The University of Auckland has been involved in three studies concerning calculators in secondary mathematics. Each study, for a variety of reasons, can only suggest hypotheses for future confirmation. However there are general conclusions which can be made by viewing all three studies together. Furthermore, they provide a basis for a theoretical framework for the analysis of the role of technology in mathematics education. Taken in the context of a new curriculum, the studies give evidence for the potential of calculators as a tool for developing more investigative teaching styles amongst teachers, but leave unanswered questions about the effect of these styles on children. More can be said, however, about the process of implementing technological change in the mathematics classroom A first attempt at a theoretical model for the use of technology in mathematics education is presented for criticism

As part of the literature review in my research into professional development of high school mathematics teachers I was looking at models from industry. Deming's work on quality management appealed and it has been useful to consider his guidelines with respect to the planning of staff development in schools. This paper introduces Deming's ideas and considers the implications, with particular emphasis on those implications that are different from normally accepted school practice. These include the need for a vision, considering the school more holistically, planning for a customer focus, blaming the system not the people, doing away with staff appraisal (and summative student assessments) and setting up quality circles.

There have been few recent crossnational studies comparing Korean students with those of other nations, and none reported with Australian students:i which is surprising in view of the large concentrations of expatriate Koreans in various parts of Australia including Campsie in Sydney. This report sets out some brief comparisons between year 6 students in Korea and Australia, and draws some superficial conclusions about observed differences.

Sixteen items were chosen from tests used in a primary school mathematics competition taken, in 1992 and 1993, by over 25000 students in Years 5 and 6. For 5 of the items, males did better than females; for 6, the proportions of males and females who gave correct answers were virtually identical; and for 5, females did better than males. The 16 items were randomly sequenced and shown to primary teachers, trainee primary teachers, trainee secondary teachers, and mathematics education researchers, who were asked to select items on which (a) girls did noticeably better than boys; (b) boys did noticeably better than girls; or (c) girls and boys performed equally well. Analyses showed that (a) female respondents were more likely to give correct responses than male respondents; and (b) the mathematics education researchers and the practising primary teachers were more likely to give correct responses than trainee teachers.

This paper presents the results of a survey investigating the provisions different secondary schools make for teaching mathematics to Non-English Speaking Background (NESB) students. The particular focus was on the ways ESL and Mathematics staff in secondary schools cater for NESB students' needs. The survey was carried out among Metropolitan Melbourne government schools, and among Catholic schools in the Archdiocese of Melbourne. The results reveal not only a wide range of school practices and provisions, but also some important teacher needs which are not being met.

his paper reports on the first and second phases of a longitudinal investigation of the attitudes of preservice primary teachers towards mathematics. An Attitudes to Mathematics questionnaire was used to construct profiles for each participant in three attitudinal domains - mathematics self-concept, attitude toward teaching mathematics and attitude toward teaching with the aid of technology. The profiles allowed monitoring of change in each of these domains as students progressed through their professional teacher education programs. Initial findings indicate that teacher education programs can provide a "watershed" for beginning teachers' to scrutinize and, where necessary, modify existing attitudes toward mathematics and the teaching of mathematics in primary schools.

This paper will report on an on-going project, funded by CAUT (Committee for the Advancement of University Teaching), involving the use of TI-85 graphics calculators in the teaching of a first year mathematics class at Sydney University. The calculator was introduced to the 1993 class as a demonstration tool during lectures. New tutorial material has been written for the current class, and students are using the calculators in tutorials. This paper will discuss findings from the initial stages of the project.

The arithmetic mean is probably the most commonly taught statistical measure. This study of 136 pre- and in-service teachers concentrates on responses to questions about the arithmetic mean in different contexts. It was devised to allow for increasingly sophisticated understanding of the concept to be demonstrated in the concrete-symbolic mode, and also allowed for ikonic mode processing. The results are considered from the perspective of cognitive development. Cycles of response in both ikonic and concrete-symbolic mode are identified and described. A theoretical model to explain the interaction between the modes is proposed. The implications of these findings is briefly discussed.

Student teachers' negative attitudes towards mathematics and the inadequacy of their mathematical backgrounds have been a concern of mathematics educators for many years. In an attempt to understand the interaction of cognitive and affective factors in mathematics learning, this paper presents a case study of one preservice early childhood/primary teacher education student's experiences of learning mathematics. The study identifies issues which are of concern to teachers and teacher educators.

The application of mathematical conceptual knowledge in the solving of problems has been of continuing interest to researchers and classroom teachers. Towards this end, research evidence is providing some insights as to the role of mathematics schemas in the solution process. Little research effort, however, has been invested in the examination instructional methods that will enhance the activation and use of such schemas by students. In this study we report the results of two studies which attempt to generate data relevant to the effect of instructional strategies on the activation and use of previously learned schemas in junior high school geometry. Two forms of training derived from our earlier studies of geometry problem solving were developed. The first, Generation training emphasised searching memory to access information that is related to the problem. Management training directed students' attention to planning and controlling accessing of information. No effects on training or transfer items were observed following Generation training. Management training improved the near and far transfer performance of both high- and low-achieving students.

There is a need to reconceptualise "Change" in I the context of teacher professional development. This paper examines alternative conceptions 0/ change. In the context of teacher professional development, "Change" could be something that: • is done to teachers (teachers are "changed"; that is, change as training); • is experienced passively by teachers (teachers "change" in response to something; that is, change as adaptation); • teachers do purposefully to themselves (teachers "seek to change"; that is, change as personal development); • teachers do purposefully to their environment (teachers "change something"for reasons of personal growth; for example, the mathematics curriculum; that is, change as local reform); • teachers do as agents for others (teachers enact the "change policies" of the system; that is, change as systemic restructuring); • is organic and intrinsic to professional activity (teachers change inevitably through professional activity; that is, change as growth or learning). These alternative perspectives on change need not be mutually exclusive. This paper will contrast the implications of adopting each of these perspectives. The identification of the various teacher change perspectives is of value for the insight it offers researchers and in the criteria it provides for the development of effective inservice programs for teachers.

This paper reports the results of a three-year evaluation of a major curriculum and teacher professional development project, the Junior Secondary Mathematics Resource Schools Project (JSMRSP), carried out in South Australia. The Junior Secondary Mathematics Resource Schools Project was established with a central purpose: "To enhance and improve the mathematical capability of all students." Through investment in expertise at the level of the six Project schools, the Education Department of South Australia hoped to facilitate the development of curricula and resources of value beyond local application.

This paper describes a study of the development of bilingual students in solving problems in mathematics, as they progress from year 4 to year 8. The total sample of 700 children will include Italian, Arabic, Vietnamese and Cambodian speakers with a control group of 200 English monolingals. We are particularly interested in the phenomenon of code switching during the problem solving process. What may prompt a bilingual student to switch languages? How often does it occur? Does it depend on the mathematical context? What changes might occur as the student progresses through year 4 to year 8? The project is being funded initially by a large ARC grant for two years, 1994-95. We would like to discuss the methodology and the practical problems involved in gathering the data.

Thefunction concept permeates many aspects of the mathematics curriculum at both the secondary and tertiQlY levels. Students' use and conceptualization of the function concept have been the subject of a number of research studies (see for example, Arnold (1992), Barnes (1988)). These studies have found that the overwhelming view students hold of this concept is that a function is a rule of correspondence, or more precisely, an algebraic formula, into which values are substituted. However, this rather narrow interpretation begins to breakdown when questions that require the use of more advanced reasoning skills are encountered. The competent use of function notation, given this stance, relies heavily on students' understanding of the symbolism involved. Thus the notion of a variable and how different variables relate are critical factors in students' ability to use the junction concept purely within an algebraic context. The purpose of this study was to explore this feature by examining tertiary students' responses to a series of questions, in which relationships between variables were set within the framework offimctionnotatiol1. To assess such responses, the SOLO Taxonomy was used. The study has highlighted the difficulties students experience with second order relationships in non-routine function questions and the value of the SOLO Taxonomy in interpreting students' responses.

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Diagrams are a form of communication that are particularly useful for conveying geometric ideas. Children experience difficulty with diagram interpretation in geometry and also with the interpretation of graphics in geography. The similarity of children's difficulties in geometry and geography led to an examination of a model of levels of mastery of representation (Liben & Downs, 1991) based on geography, to determine its applicability for mathematics. Diagram interpretation was explored through observation of eight babies' behaviour with a three-dimensional shape and a corresponding diagram, and interviews with eight Grade 2 children and eight Grade 5 children. This study provides evidence of differing levels of children's understanding of geometric diagrams. Although the results support only two of the four levels of Liben and Downs' (1991) model, a refined five level model is proposed. The results indicate that the interpretation of diagrams may be a constraint to effective communication in geometry.

Teachers' ideas, attitudes and beliefs about a subject are thought to influence the way the subject is taught. This paper reports on a pilot study conducted to explore mathematics educator's views about technology education and its relationship to mathematics. Related mathematics education literature focuses on technology's utilitarian value to mathematics education, with particular emphasis being placed on the use of computers and calculators. Little attention is given to mathematics education's contribution to technology education. Results from the study reflected the approach taken in the literature and suggest that the mathematics educators had difficulty in adopting a balanced view of the relationship between mathematics education and technology education. The paper calls for a broader approach on the part of mathematics education and argues that more attention be given to the interrelatedness of knowledge and the boundaries between the school subject areas.

Analogy appears to be one of the most important mechanisms underlying human thought, at least from the age of about one year. A powerful way of understanding something new is by analogy with something which is known. The research community has given considerable attention to analogical reasoning in the learning of science and in general problem solving, particularly as it enhances transfer of knowledge structures. Little work, however, has been directed towards its role in children's learning of basic mathematical ideas. This paper examines analogy as a general model of reasoning and proposes a number of principles for learning by analogy. Examples of analogical reasoning in children's mathematical learning are presented, including children's ability to recognize similarity in problem structure which was investigated in.a recent two-year study. The proposed principles are applied to a critical review of some commonly used concrete analogs and to a brief analysis of more abstract analogs, namely, established mental models which serve as the source for the construction of new mathematical ideas.

Learning mathematics involves both its product (body of knowledge) and processes (ways of knowing). The mathematical reasoning process,es enable the products to be developed, applied, and communicated.· The role of- reasoning skill per se in the learning of mathematics has received little attention, apart from studies addressing spatial ability (e.g. Tarte, 1990). Yet the importance of such processes in mathematical learning has often been acknowledged (Australian Education· Council, 1990; Cockcroft Report, 1982). A vast number of research projects have investigated novel problem solving (Lester & Kroll, 1990; Schoenfeld, 1992) and the algebraic domain (Booth,1989; Kieran, 1992; Kuchemann,1981; McGregor, 1991; Quinlin, 1992). Yet there seems to be a paucity of research examining the link between reasoning processes and mathematical performance in these domains. It is generally acknowledged that these processes play a role in mathematics learning (e.g. Resnik, 1987) but the exact nature of this role remains unclear. This paper will report on research which begins to explore relationships and interactions between students' general reasoning processes and their competence in solving algebraic and novel problems.

According to M cLeod (1992) affect plays a significant role in mathematics learning and instruction. Studies by Wood & Smith (1993) and Crawford, Gordon, Nicholas & Prosser (1993) have examined attitudes towards mathematics held by first year university students enrolled in mathematics and engineering courses. Students entering TAPE Associate Diploma courses in Applied Science are likely to differ from those entering university for two reasons: 1. Their intention may be to enrol in a vocational course in order to obtain employment in the areas of laboratory technology or fire technology. 2. They may have obtained lower tertiary entrance scores than those entering university, but may nevertheless aspire eventually to enter university through the articulation process. This study will examine the beliefs, attitudes, and emotions towards mathematics of TAFE students enrolled in the Associate Diploma of Applied Science at Swinburne University of Technology (TAPE Division). A broad questionnaire will be administered and the results analysed.

This paper is concerned with a teaching model for training mathematical thinking through a scheme based on the Information Processing Taxonomy (IPT) Model, an extension of the Newell and Simon's Production System Theory. The Model features two characteristics in teaching thinking: the primary production and secondary production of information; and two types of information available for solving problems. The paper shows the analysis of a set of ratio and proportion problems at the secondary level using the Model's features. Instruments in the form of worksheets for training mathematical thinking based on the Model were constructed for use in the teaching experiment. Three types of worksheets were constructed for the thinking programme: basic-skill worksheet, training-thinking worksheet and practising problem solving worksheet. The stages of development of the programme and its rationale are discussed.

This paper reports a part of an ongoing Pupil Profiling Project in mathematics, which aims to produce a set of six test instruments on number, measurement and geometry for Primary 3 and 5. Item response theory was used for analysis of the test results to provide profiles of pupil performance on their levels of attainment of basic skills and processes across the mathematics curriculum. Such profiles served as semi-diagnostic information for teachers to assess in greater details the weaknesses and strength of individual pupils. The development of one of the tests on Number for Primary 3 will be presented in this paper. For each test a sequence of attainment levels was identified. The lowest levels consisted of elementary knowledge and skills while the higher levels reflect increasing complexity of understanding and the use of more advanced problem solving processes. The two objectives of this study are: 1. to develop and validate tests on Number for use in Primary 3 schoolbased assessment, 11. apply item response theory to analyse data and to present results in a form of "Kidmaps" that are meaningful to teachers and pupils.

Affective variables are included in several models explaining gender differences in mathematics learning. Not much is known about the relationship between students' beliefs and the context in which they learn mathematics. This study aimed to infer attitudes from behaviours in the classroom, to compare these with attitudes determined from more conventional pen-and-paper measures, and to examine learning context factors which might explain inconsistencies and contribute to our understanding of gender differences in mathematics learning. Two male and two female grade 7 students, who sat together, were observed for founeen consecutive mathematics lessons which were videotaped. During the monitored period, the students were engaged in individual and paired tasks and one collaborative project. Analyses revealed that students' behaviours were fairly consistent with their beliefs. Relevant contextual factors which might account for the consistencies and the differences found among and between the students included group composition, the content and demands of the mathematical tasks, and the teacher.

The purpose of this study was to investigate the relationship between students' classroom experiences and their construction of mathematical meanings. Data were collected from classroom observations and video taping sessions and from subsequent video-stimulated interviews with 6 students in each of two year 5 classes. Results discussed here are those from analysis of the interviews. This analysis revealed the existence of four primary sources by which students determined the meaning or correctness of mathematical activity: the teacher, intuition, familiarity, and procedural knowledge. Second, in relation to the social level, the teacher emerged as playing the most valued role in the sense making and ratifying of procedures or answers.

Students' difficulties in learning about limits hdve been documented by many authors. Cornu, Tall, and Dubinsky all emphasise the process of encapsulation, whereby a dynamic process becomes transformed into a static concept image. Tall and Gray also place importance on the "proceptual" nature of mathematical thought. I report on the early stages of a study that is designed to access student schemas in limit problems. 25 students in a second year university numerical analysis class were given a questionnaire that probed aspects of their understanding of limits. The schemas of 5 of these students will be examined using clinical interviewing techniques over a 4 week period. The numerical analysis course uses spreadsheets instead of a programming language, and one of the longer-term aims of this research is to determine the effect of regular use of a spreadsheet on students' concept image of limits.

Mathematics teachers' influence on student learning of mathematics could interfere and limit the learning of higher order mathematics. This paper stages this influence as a concern for teacher educators. To assess the affects of this influence, 1st year University mathematics students were selected as a study sample. These students mathematical understanding was explored using 'mathematical items' designed specifically for this study. The students' responses were assessed and evaluated using the SOLO taxonomy. This paper also reports on preliminary findings focusing on the transfer of abstract thinking in functions. The findings tentatively suggest that prior learning affects the depth and clarity of University students' understanding of mathematics.

This case study reveals difficulties teachers face recognising effective group discussion. The overall study, of which it is a part, examines the discussion occurring during a group activity designed to reduce misconceptions related to division. Groups of children were videotaped, card placement identified and transcripts coded for mathematical aspects of the discourse. Characteristics of effective group discussion were compared with the discussion occurring during the group activity and related to learning outcomes. Once again learning outcomes vary for the group members despite the children's active engagement with, and discussion of, the task. Teachers are alerted to the complex nature of student participation in and outcomes of group learning.

This paper presents a theoretical framework for guiding future research whose purpose is to understand how knowledge is constructed and transacted in collaborative and individual activity involving teachers and students. A synthesis of sociocultural and social constructivist theories has contributed to the formulation of our central metaphor of the classroom as a "community of practice". In creating such classrooms, three inter-related contexts need to be considered: teacher-student interaction, student-student interaction and individual reflection. These contexts are examined in terms of three phenomena of interest: collaboration, mathematical dialogue and metacognitive activity.

This presentation discusses the construction and implementation of a research project concerned with adult learners' formal and social ways of knowing and applying mathematics. It details findings from the literature as well as the researcher's own attempts to begin to investigate individuals' tendencies to disassociate formal and social mathematics.

Teachers use printed material,for example textbooks, workcards, practice examples extensively in mathematics classrooms. They are given little advice on different forms of usage and additionally they are made to feel that they should not be employing these aids. Mathematics educators need to research styles and effective use of textbooks.

The research project discussed in this paper re-opens the issue of teaching negative number concepts and operations. The experimental work in schools has only recently commenced. In conjunction with a detailed evaluation of current teaching strategies, an experimental card/counter teaching strategy will be carefully 'classroom tested'. Short and long term student learning outcomes resulting from the experimental and the more common current teaching methods will be compared and contrasted. It is anticipated that the study will provide enhanced insights relating to the teaching and learning of negative numbers resulting in substantially improved teaching practice and student learning.

This paper explores the possibilities for cooperative group work in NZ elementary mathematics programmes. It discusses effective tasks and appropriate teacher behaviour for promoting cooperative group work. The paper is based on observations made in eight classrooms of children in the first two years at school in the Wellington city area over terms two and three 1993. In particular it highlights what appear to be. key factors of effective small group instruction in mathematics.

Computers have provided many opportunities for new approaches to mathematics in the classroom. This project focusses on the use of a spreadsheet as an algebraic environment in which algebra can be introduced to students at year 7 level. Materials for this have been written and used with a group of mathematically weak year 7 students who had no previous experience of spreadsheets. An assessment task given the students following the use of the spreadsheets required students to explain some of their solutions. The results of this give some ideas of students learning. The project is in early stages and further suggestions for monitoring students algebraic development in a classroom situation would be appreciated.

The results of a two-year professional development progrdl11 in teaching mathematics for intermediate and secondary school teachers showed that the intermediate school teachers involved changed less than did the secondary school teachers. The limited mathematical knowledge of the intermediate school teachers was seen as a major constraint preventing them from making changes consistent with a constructivist approach to learning as recommended in a new national mathematics curriculum. Their limited mathematical knowledge tended to result in a narrow perception of mathematics and mathematics learning. They tended to reflect upon their classroom teaching more in terms of attitudes toward mathematics than in terms of values drawn from a broad mathematical knowledge. This reflection in terms of attitudes was unlikely to lead to improvement in their students' learning.

In recent years much attention has been given to demands for curriculum reform in mathematics education. To date most attention has been placed on developing adequate frameworks for mathematics syllabuses and the need to obtain quality structures for assessing and reporting the mathematical attainments of students. Largely left out of these concerns, however, has been a focus on the understandings of mathematics teachers with respect to (1) content knowledge in mathematics and (2) content specific pedagogical knowledge in mathematics. The primary aim of this study is to obtain a view from the field of the current range and depth of mathematics teacher knowledge with respect to these domains. This pilot study has utilised data obtained from a purpose built survey instrument; 44 teachers in 10 schools (primary and secondary) comprised the sample. Analysis of results has indicated that over 50 percent of the teachers in the study may not be sufficiently prepared in mathematics content, and that almost two third of the teachers in the sample are concerned about their level of knowledge in contemporary teaching methodologies. Key differences with respect to these variables emerge between primary and secondary sectors and within the secondary sector. A study on a considerably larger scale is currently under way; this will provide results of greater validity and details concerning finer structures of relevance to this investigation. It is intended that this study provide much needed evidence relating to debates concerning the appropriate structures, content material and policy for pre-service and in-service mathematics education.

Recent policy documents such as the· Discipline Review of Teacher Education in Mathematics and Science (1989) advocate "constructivist" pedagogy in the tertiary sector so that preservice teachers are enabled to learn mathematics "by constructing their own knowledge through discovery, exploration and problem solving in relevant and supportive environments (p.17}". Having completed a three-year action research project implementing just these recommendations, I am led to critique "constructivist" practice for the dangers inherent in its framing in the mathematical perspectives of cognitive psychology. The persistence of a critical view of the social and discourse relations within "constructivist" practice cannot but prejudice the dual expressed goals of the Discipline Review for "... a literate society, a problem solving society" that "... recognises social justice and equity as major goals" (p.7).

In the early 80s Mayberry (1981) developed a diagnostic instrument to be used to assess the van Hiele levels of pre-service primary teachers. The test which was carried out in an interview situation, was designed to examine seven geometric concepts. The Mayberry study has been replicated as a written test under Australian conditions. Analysis of the Australian results led to the identification of some problems with the Mayberry test items which had the potential to lead to incorrect assignment of a student's level of geometry. The analysis of these results was reported earlier (Lawrie, 1993). The analysis also led to the identification of several Level 4 Mayberry items which were seen as capable of assessing deductive skills. This paper analyses the responses to three of these items and discusses how these responses can be seen as indicators of a student's level of geometric reasoning.

The research reported in this paper is part of a large studyl investigating the cognitive and linguistic aspects of learning algebra in secondary school. Five of the 22 schools participating in the study tested the same cohort of students on two or three occasions. Analysis of the data from these five schools has shown that certain initial difficulties are easily corrected for the majority of students, whereas others are more persistent. Most students by Years 9 and 10 had reached an understanding of algebraic letters as unknown numbers, but their understanding of equations was less secure. Major persistent difficulties were the significance of brackets and the notation for products and powers. A small proportion of students made very little or no progress.

This paper reports on research being conducted in the area of the identification of characteristics of mathematical ability. Research is being carried out through case studies of ten students with ability in mathematics, with input from their parents, siblings, teachers and peers, as well as from the students themselves. The age of the students in the sample, has allowed observations about the them to cover the period spanning from infancy to tertiary levels of education, and the gap between the events and their reporting has been relatively short. Differences in approach between the students have been identified. These are compared with differences noted in the research of Krutetskii and Osborn, and comment is made on some of the implications that such differences in approach and aptitude have for mathematics education.

This paper reports on a study which sought to determine whether changes could be induced in primary student-teachers' beliefs about and attitudes towards mathematics and mathematics teaching through their participation in a mathematics education course which adopted constructivism as its theoretical framework. The course was designed to facilitate the development of positive attitudes towards mathematics and mathematics teaching and to facilitate the development of beliefs about mathematics and mathematics teaching which are consistent with a constructivist perspective. Two beliefs scales and three attitude scales were used to measure changes in beliefs and attitudes. These were completed by subjects prior to the commencement of the course and again following its completion. There was a significant shift towards a constructivist perspective in students' beliefs about mathematics and in their beliefs about teaching mathematics. There was also a significant reduction in mathematics anxiety and a corresponding increase in mathematics self-concepts for those students who had completed four or less years of secondary school mathematics. Interview data identified several constructivist teaching practices as having been significant in contributing to changes in students' beliefs. The results are discussed in terms of their implications with respect to the instructional practices which students might adopt in their mathematics teaching and in terms of their implications for pre-service teacher education.

While mathematics curriculum developers have attended carefully to issues of content coverage and even to models of cognition, the same attention has not been accorded to student affect. The planning of instructional and assessment practice seldom includes consideration of such factors as student motivation, confidence, satisfaction, or self esteem. It should be stated at the outset that we do not endorse the partitioning of human behaviour into affective and cognitive as mutually exclusive categories. For the purpose of the present discussion, however, we find the term "affect" useful to describe those behaviours conventionally associated with emotive response. This paper adopts the position that it is essential that curricula attend to student affect in an explicit and structured way. Strategies, developed for research purposes, offer the possibility of accessing student affect in classroom situations. The use of these strategies raises all of the familiar issues of construct and content validity, and, most importantly, the contemporary concern with consequential validity. The use of the term "access" in the title to this paper is intended to foreshadow the need to distinguish between "measurement" and ''portrayal''. We find the term "measurement" to have unfortunate uni-dimensional connotations, and for the purposes of this paper would like to advocate the use by researchers and teachers of the term ''portrayal'' in relation to the process of data collection regarding student affect. Approaches to the portrayal of student affect are discussed and illustrated by reference to the results of studies employing instruments with the capacity to access student affect.

Thirty six Year 2 children were each presented with two out of a set of six realistic models of various physical angle contexts and asked (1) to indicate which of a set of 10 abstract angle models could also represent those contexts,' (2) to represent the contexts in drawings; and (3) to indicate· whether they recognised any similarities between the two contexts. Previous papers have reported results for the turns, slopes, rebounds and corners contexts. The present paper reports on the crossings and bends contexts and summarises the specific features of each of the 6 contexts which appear to hinder the recognition of similarities. The basic hypothesis of abstraction theory - that an abstract angle can only be recognised in a particular context when angle-related similarities between that context and other, superficially different contexts are also recognised - is then examined.

A group of teacher educators watched a videotaped mathematics lesson. Their written critiques demonstrated that six specified components of quality teaching were present in the lesson. However the written comments of some of the observers were surprisingly contradictory. This paper outlines some issues raised by these contradictions. It argues that we need to find ways of critiquing lessons which stimulate debate about different teaching styles and which acknowledge a variety of intentions and perspectives of classroom teachers.

Action Research programmes need to have a political dimension involving an examination of dominant ideologies in education. One such ideology, the Technocratic/Bureaucratic ideology, is having a marked affect on teaching practice in mathematics classrooms. It is producing an increase in technique-oriented, behaviourist, approaches to pedagogy, and teacher beliefs and values are adapting to those more consistent with this ideology. Teachers need to be empowered to analyse their practice, beliefs and values against a range of alternatives. One of these· needs to be the dominant ideology and the practices which spring from it. Another needs to be an approach to teaching which sharply contrasts with the dominant ideology. The point of entry for these analyses is the language, and 'currency' teachers are most familiar with: classroom approaches and teaching activities. The goals of the analysis are planned change in teacher practice, a strengthening of the belief and value system which surrounds this practice, and the deliberate adaptation and subversion of the dominant ideology.

Some mathematics educators have pointed out that students seeing or using so-called concrete representations of concepts may not construct the expected conceptualisations. This paper presents research that specifically considered how problem solving with concrete materials could lead to the development of concepts. Students in Years 2 and 4 in primary school were engaged in spatial problem solving. A model developed by the author emphasises the interaction of concepts, imagery, heuristic thinking, and affective processes in problem solving as well as the importance of manipulation of materials and interaction with others. Students' attention to certain aspects of the problem, to other problem solvers, or to the materials was seen as particularly significant in the problem-solving process.

During term I, 1993 Bulleen Primary School in Melbourne established a Mathematics Intervention Progmm for children in Year 2. based on the work of Or Bob Wright. This program was designed to assist those children at risk of not coping with the current Mathematics curriculum as presented in the Mathematics Course Advice -- Primary (1992) and the National Statement (1991) documents. Children were clinically interviewed using tasks based on the stages of construction of the number sequence developed by Steffe, Cobb and co-workers at the University of Georgia (1983~ 1988), and documented by Wright (1991; 1994). The children identified as in need of the Mathematics Intervention program had· mathematics difficulties of a significantly different order of magnitude from the other children. The six year 2 children allocated to the Mathematics Recovery program were found to be in. or have been in, a reading recovery-type program. Further. all year 2 students in a reading recovery program were independently assigned to the Mathematics Intervention program. This procedure was repeated in 1994 at Boorondara Park Primary with six Year 1 and three Year 2 children as in need of Mathematics Recovery, and independently as in need of a reading recovery type program. This implies that there is a need for the hand-in-hand development of language and mathematics and the evidence suggests that reading recovery programs are tackling only one part of linguistic comprehension and mental development.

The purpose of this paper is to report the analysis of senior secondary students' proof using deduction on an exercise based on a parallelogram. The question was designed so that it could be solved by recourse to a relatively little used definition of a parallelogram. The results showed that the better students did make use of the definition. However, the majority of students who chose to use a deductive approach, relied initially on congruency concepts. For these responses a clear difference in quality was identified. Because of this work a general framework of growth, based on the SOLO Taxonomy, is suggested. Following from this a possible structure of the nature of early Level 4 (deductive) thinking identified by the van Hiele theory is hypothesised.

The New South Wales Minister for Education has decreed that all teachers wishing to gain initial employment from 1996 in New South Wales government primary schools need to have successfully studied the equivalent of 2 units of mathematics at the Higher School Certificate level. In response, the authors are investigating an alternative approach for teaching such material to primary teacher education students who have not reached this standard of mathematics. The approach was trialled in I993.and, in 1994. the approach is being used with further groups. In this paper, the theoretical foundations for the approach are discussed. along with preliminary results concerning critical changes in the students' development of their mathematical ideas and in their beliefs, attitudes, and values pertaining to mathematics.

The Secretary of Education (or other appropriate authority) has not determined that using manipulatives is . either a sufficient or a' necessary condition for meaningful learning. (Baroody, 1989, p. 4) Statements extolling the virtues of manipulatives (concrete materials) for the learning of mathematics abound in curriculum documents, research literature and, even, textbook series. Concrete materials are often seen by teachers as the basis for mathematical ... learning. But is this a reasonable view? In this paper, through a review of the literature concerning the use of concrete materials, we build up an historical view of the place of these materials in the learning and teaching of mathematics. Examples of situations in which the use of concrete materials has constrained children's mathematics learning are discussed along with the notion that materials introduce 'reality' to children's mathematical learning. Current research on children's thinking in the social and cultural contexts of their mathematics learning is used to help explain ways in which concrete materials can help and hinder this learning and how the very notion of manipulative might be expanded beyond concrete materials.

A study in Queensland was undenaken to replicate a British study that examined four levels of algebraic complexity Year 8 students were capable of reaching. The levels had been identified through the British study by Kuchemann. The test items and procedures used were based on the British study. The results were similar to the British findings and indicated that a large proportion of the students had not progressed beyond the first of the four levels of cognitive functioning. The report outlines the findings and their implications within the Australian context.

Since 1990 nine cohorts of preservice teachers in Australia and the USA have been given the task of ordering five decimals and asked to explain their reasoning. Data were collected via written explanations or interviews. The percentages of students in all cohorts with incorrect ordering of the decimals were disturbing but were similar to those from other studies with preservice and practising primary teachers. Written explanations for incorrect response were sorted into 11 categories. Patterns of responses and similarities with the results of the previous studies among upper primary/lower secondary students were found. In particular, evidence was found for usage of three rules found in previous research. The strategies used by students who were successful at the task were also examined for insights into students' constructions of decimals.

This paper reports ongoing research on mental processes in early algebra. Two Year 7 classes were each taught with the aim of developing an understanding of algebraic generalizations which included the distributive law. One class used arithmetic examples leading to generalizations. The other used an objects-and-containers model to assist. Significantly better gains were recorded by the latter class on attitudes and content-specific achievement. No significant differences were detected before the teaching intervention for these mixed ability coeducational classes. The evidence points to the likelihood that the use of a concrete analogue assisted cognitive development.

Papers presented at MERGA conferences in recent years have provided little stimulus for debate on methodological issues relating to research in mathematics education. It seems appropriate that not only should results of research be presented but opportunity for discussion of the processes of research and related issues should be promoted. At the 1993 Brisbane conference of MERGA two papers were presented (Mousley, Sullivan, & Waywood, 1993; Forgasz, Landvogt, & Leder, 1993) that had research methodology as their focus. This focus lay within the qualitative paradigm. This paper continues this theme by considering a strategy for building a conceptual framework for a set of qualitative data and then outlining a procedure for "looking inside" the associations between variables.

This paper reports the preliminary results of work undertaken for a thesis project on control-related aspects of mathematical problem solving. The study uses a non-routine mathematical problem as the focus for the development of problem solving profiles which summarise subjects' knowledge, beliefs and attitudes towards mathematics and problem solving, and uses a protocol parsing technique developed by Schoenfeld to illustrate control behaviour observed during a thirty minute think aloud problem solving session. Discussion focuses on the protocols of three sets of subjects and their achievement on the non-routine problem. The reliability of the parsing technique, and the appropriateness of a think aloud methodology are also considered.

This paper reports on an investigation of students' difficulties in first-year statistics examinations at university. Our hypothesis was that difficulty with language was an important factor in student performance in statistics examinations. Our data consisted of examination questions ranked in order of difficulty based on student performance, lecturers' perceptions of difficulty and a measure of the linguistic complexity of the questions. The examination paper that was analysed was a typical short answer paper for students studying statistics in their first year at university. The paper was not designed specially for analysis but rather was the normal end of year paper for a business statistics subject with an enrolment of 600 students. Three statistics lecturers not involved in teaching the subject ranked questions according to their perception of the level of difficulty and these were compared with the performance of 186 students. The examination paper was also analysed for linguistic complexity as measured by lexical density and this was compared with students' responses. The results were surprising. There was no correlation between student performance and the linguistic complexity of the questions as measured by formal measures of lexical density. The lecturers' rankings were consistent and correlated highly with student responses in most cases. Certain topic areas appeared to cause more difficulties than others and further research will concentrate on these topics.

At times in the past ten years, politicians, educators and business people have deplored the decline in the mathematical ability of adolescents and older young people. Generally it has been the arithmetical skills which have been criticised. This· paper attempts to compare the understandings of primary teacher education students in their first weeks at University in 1994 with those of a similar group of students at two NSW Teachers' Colleges in 1964. The 1964 sample completed an achievement test and a test of mathematical understanding. As well, a small group were interviewed to determine any obvious attitudinal factors related to achievement and understanding. The 1994 sample consisted of 162 students in their second week of a pre-service teacher education course. In this case, as the students were to be given an achievement test later in the semester, only the survey of mathematical understandings was administered. The responses were analysed using data related to the upper, middle and lower thirds of the scores. This provided information about the topics which students found difficult and enabled a comparison with the previous results.

Students often feel constrained in their attempts to do mathematics because of their fear of failure or their lack of confidence. As well, some contextual factors inhibit understanding and enjoyment in mathematics. The way in which they perceive mathematics and learning mathematics has an impact on their success in the subject Following a smaller scale survey of beliefs concerning mathematics held by primary and secondary students and teachers, this study looks in more detail at a larger sample of over 2000 secondary school students, makes further comparisons and suggests trends in beliefs of students at secondary school level. The analysis of data is presented under several headings, including: Students' beliefs concerning (i) their mathematical success and/or failure; (ii) the nature of the mathematics learned; (iii) the learning of mathematics in relation to other subjects; (iv) learning geometry; and (v) parental expectations. This current study confirms most of the findings from the previous study, particularly in relation to students' beliefs that mathematics is mostly facts and procedures which have to be memorised and that everything important about mathematics is already known by mathematicians.

The Calculators in Primary Mathematics project was a long-term investigation into the effects of the introduction of calculators on the learning and teaching of primary mathematics. A wide variety of qualitative and quantitative data was collected. This paper analyses differences on a test of arithmetic using calculators, between Grade 3 and 4 children who had been in the project for at least 3 years and a control group. All children handled whole number calculations equally well, the difficulty being determined only by how many transfers from paper to calculator and vice versa were required. Project children were better able to handle calculations involving decimals or negative numbers and to identify the appropriate operation to be used in a word problem. This very important observation is supported by data from another source.

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. The focus of this research is the impact on mathematics instruction at Years 7 to 10 arising from the introduction of mandated changes in assessment at Years 11 and 12 as part of the Victorian Certificate of Education (VCE). Based on the Victorian experience, this study provides a systematic investigation of this hypothesised "ripple effect". The overall study consists of three phases: a document analysis; a survey by questionnaire; and a series of selective interviews. Reporting the second phase of this study is the principal focus of this paper. In the second phase, a questionnaire was given to fifty teachers drawn from participating schools. A key issue addressed in the questionnaire was "How is the Ripple Effect characterized in terms of teachers reported classroom practice in Years 7 to 1O?" Data from the questionnaire substantiated the general conjectures and conclusions arising from the document analysis with respect to changed instructional practice replicating, at junior levels, assessment practices and work requirements associated with VCE mathematics. In addition, the questionnaire revealed different patterns of implementation between such groups as experienced teachers and those relatively new to the profession.

This study investigated the relationship between senior secondary students' written responses and the mathematical and cognitive processing used when solving a complex problem. Response maps were used to categorise scripts by global strategy. These categories were stratified into clusters displaying common elements of mathematical structure and. cognitive demand. The maps revealed difficulties faced by students in applying known facts and formulae to complex word proble'ms. The overwhelming reason for failure on the task was inability to construct a satisfactory model of the problem. The majority of students were able to identify and record the essential elements of the problem and to recall necessary formulae and procedural skills for a satisfactory solution but they had difficulty establishing crucial links between the data in their representation. This did not necessarily reduce the cognitive demand of the task the student attempted to solve. Cognitive demand appeared to be more related to global strategy chosen.

Taplin (1992) addresssed the notion of effective perseverance by exploring managerial strategies used by children in their attempts to solve mathematics problems. The focus was on those students who were tenned "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 give up immediately. A model was developed which described the sequence of strategies used most consistently by successful students. This paper will describe the next stage of the project, investigating the application of this theoretical model to a teaching experiment Evaluation of the instructional sequence is based primarily on observation, with comparisons being made between pre-treatment and post-treatment performances. The paper will focus particularly on evaluating teachers' perceptions of the model.

This paper considers the difficulties which many students, across a wide spectrum of the ability range, experience in correctly translating word problems into mathematical symbols. There is particular reference to problems which give rise to linear algebraic equations, and it identifies the process-oriented ordering preference which procedural students have been found to display when attempting such translations. It also shows that for students who have not fully encapsulated the concept of equation there is an apparent relationship between this preference and the well documented reversal error for translations of word problems such as the 'students and professors' problem. It suggests that for some of these students, as the structural difficulty of translation problems increases, then the students' tendency to revert to the process-oriented ordering preference also significantly increases, and that this has a primary effect on the occurrence of syntactic translation, resulting in the reversal error.

An exploratory study of 92 high ability 3rd through 6th Graders was conducted to investigate links between their understanding of the structure of numeration and their representations of the counting sequence 1-100. From children's explanations and drawings of the numbers from 1-100, we seek to infer their internal imagistic representations. Our observations indicate that children showing evidence of dynamic internal representations and access to a variety of internal images have more developed relational understanding.

This paper reports the development of a typology for describing mathematics assessment items, with particular reference to aspects of the items which impinge on their gender-fairness. The typology is in two parts - one part to describe the context of the item and the other part to describe the format of the item.

Some writers have argued that mathematics curriculum change in Australia has been a mimicking of overseas trends until recent times when a more mature approach been developed. This claim is examined for the teaching of probability in South Australian schools from 1960 to the present. This recency and probability's distinctive requirements mean that the forces underlying its introduction are likely to be more accessible to the researcher than those for less radical changes. It is argued that neither a "colonial echo" nor a "coming to maturity" model are adequate to describe what happened. The forces for change have often been idiosyncratic, poorly integrated and poorly justified. They have been concerned more with content than with research findings or pedagogical difficulties. "Muddling through" may well be the best way to describe the process of change.

Research into children's understanding of probability has revealed a number of significant misconceptions. But much research has been eclectic, with piecemeal influence on classroom practice in both teaching and assessment A good test of probabilistic understanding is sorely lacking. This paper uses the results of previous research to propose a multi-dimensional classification of the very wide range of situations and appropriate responses which together make up what we refer to as "an understanding of the idea of probability". Ways in which the classification may be applied to classifying assessment questions and complete tests are discussed briefly.

In recent times higher thinking skills have received much attention. In this paper a definition of higher thinking is proposed and their importance in non routine algebra problem solving is discussed. A rationale for the author's doctoral study is presented followed by a brief description of the study itself. Some preliminary results are also included.

There have been many studies investigating sex differences in mathematical performance among Caucasian students but to date, few such studies involving Asian students. A number of studies have been carried out comparing mathematical performance of Asian and non-Asian students and these have showed a general superiority of Asian students. This paper reports the results of a study of Chinese students from the central Chinese city of Wuhan. It compares the performance of boys and girls in the first year of secondary school in three mathematical areas - logic, space and numeracy. The results from this study reveal no significant differences in the mean scores in the logic sub-test but significant differences in favour of the boys in space and numeracy sub-tests. A significant difference in the variance was revealed in the numeracy scores of the girls compared with the boys but there was no significant difference in the logic and space sub-tests.

Drawing diagrams is often recommended as a strategy for problem solving. The mathematics education literature suggests that teachers use diagrams as a problem solving strategy but provides little guidance on the kinds of diagrams teachers should draw and how teachers could teach pupils to generate their own diagrams. The findings from two separate studies, one with primary and the other with secondary school children, on the nature and use of diagrams in solving problems are described. The paper discusses the effectiveness of diagrams in helping children solve word problems, and implications for classroom instruction.

This paper presents some results of a pilot study which devised a 20-item paper-and-pencil, short answer/ multiple-choice questionnaire to assess students' understanding of statistics and probability in Grades 3,6 and 9. The items are presented, with discussion of response differences over the three grades, and the level and type of cognitive functioning associated with responses.

This paper presents a nwdel devisedfor the development of a clinical interview protocol to assess children's understanding of probability. The nwdel is based on the work of both Australian and overseas researchers, and attempts to address the many issues that need to be considered when undertaking research of this nature. As the research project concerned is in its early stages, it is hoped that the paper will stimulate some debate of the theoretical issues and draw suggestions for the development of interview tasks.

This paper investigates indicators of formal thinking in geometry problems. A sample of 20 tertiary students were given a test item which restricted the information that was available to them for use in the proof The test item was administered before and after a course on the development of non-Euclidean geometry. Before the course, the students were not able to deal adequately with restrictions to the knowledge they could assume. After the course, students appeared to cope well with the notion of restrictions to assumed knowledge, but this accommodation did not translate into the writing of acceptable proofs. The conclusions drawn are: (i) writing proofs where there are restrictions to assumed knowledge clearly involves a sophisticated form of deductive reasoning; and, (ii) an indicator of advanced formal thinking is the ability to develop a proof H!here cases, which take into account different relationships (as opposed to properties) between necessary and sufficient conditions, are accommodated.

Hypothesis testing is conceptually complex. It builds on a basic knowledge of probability, sampling distributions and the central limit theorem. It includes new concepts such as nuWaltemative hypothesis, alpha level, Type I and IT errors, power, sample size, statistical tests, critical value, p-value, observed value and decisions about hypotheses. The web of interrelationships between these concepts requires their holistic understanding for rational decision making. Firstly, this paper briefly summarises how these concepts may be linked together; and secondly, presents a methodology for research into what students know in these areas, their understanding of simple design concepts, and the sources of their difficulties with hypothesis testing.

This paper provides a progress report of a four-year project which began in August 1992 and focuses on preparing specialist teachers in the provision of long-term, individualised, mathematics teaching programs for low-attaining first-graders (6- or 7- year-olds). The meaning of 'recovery education' is outlined as are theoretical bases and earlier research studies relevant to the current project. Using afive-stage model of early arithmetical learning, the progress of 24 participants in 1992 and 32 in 1993 is summarised and compared with that of counterparts, who like the participants, were initially prenumerical. The participants underwent eight-week teaching cycles consisting of 30-minutes of individualised teaching, four mornings per week. In both 1992 and 1993, almost all of the participants made major progress and overall, the progress of participants notably exceeded that of their counterparts. A planned implementation of the Mathematics Recovery program in the United States in 1995 is also outlined.

In this paper I will seek to argue that teachers have been constituted through practices and discourses of teacher education programs so as to perceive and construct their world through psychologistic frameworks. Part of becoming a legitimate practitioner within mathematics education! is the embodiment of these discourses as recognised through the credentialling processes of teacher education programs. In such programs there is a focus on pedagogical processes whereby the learning of mathematics is seen as the internalising of knowledge and . concepts. These programs are dominant within the broad field of education, so that many, if not all, of the teachers emerging from teacher education institutions enter the school context holding sets of beliefs and attitudes about the nature of teaching and learning which support liberal views of schooling. As such, there is a strong propensity for teachers to explain their students' successes and failures in mathematics education within psychological terms of references where references are made to concepts such as ability or intelligence; levels of development or cognition; motivation; self esteem; and (positive) reinforcement. The focus is on a individualised subject devoid of the social and political context within which meaning making is occurring. By undertaking a critique of these discourses it is possible to understand the subtle ways in such discourses provide the frameworks through which teachers translate student behaviours as well as in the organisation of learning environments (Walkerdine, 1988). Such a framework does not take into consideration the wider context in which the students are located, and the political consequences of such interpretations, particularly in relation to the construction of social and gendered differences.

Graphics calculators will be allowed in VCE mathematics examinations from 1997, but questions must be set so that candidates who only have access to a normal scientific calculator are not disadvantaged. A sample of mathematics teachers, and two VCE examiners, were asked to assess the impact of the graphics calculator on thirteen multiple-choice questions used in a 1995 VCE mathematics examination. The teachers generally agreed with the assessments of the examiners, but where clear differences existed, tended to attribute less potential advantage to a graphics calculator user than did the examiners.

This study investigates how the linguistic devices of metaphor and metonym produce and regulate mathematical knowledge in the performance of work related tasks. Data are drawn from the front desk operation of three metropolitan district motels in various parts of Australia and analysed making use of 'critical incident' methodology. ·Findings indicate that within the workplace, language use and mathematical knowledge are deeply entwined.