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This keynote address was an audio-visual presentation using over 30 overhead projector and colour slide illustrations. It is therefore impossible to reproduce as a written paper in the usual form and I have not attempted to present it as an academic paper. What follows is a lightly edited transcript of the tape recorded talk, with an indication of the visuals shown and their source. Many of the illustrations can be found in just two publications - the speaker's recent book, Mathematics in a Cultural Context: Aboriginal Perspectives on Space, Time and Money (Harris, 1991), and in Dreamings: The Art of Aboriginal Australia (Sutton, 1988). This transcript is intended only for those who attended the conference and should only be quoted as something that was heard, not as though it were a published paper.

This paper is about an eight-year project for bridging the gap between theory and the classroom. As Dewey has written, "Theory is in the end ... the most practical of all things". The process of developing a theory is a lengthy one; but producing a practical embodiment is at least as lengthy; and we still have to get people to use it, and use it sensibly. This paper will give a pinhead introduction to the theory; a brief outline of the process by which a practical embodiment has been developed; and will describe how these classroom activities are currently being used not only for helping children to learn mathematics with understanding, but also for the professional development of teachers.

A preliminary analysis is offered of problems connected with the construal of images from and on screens, whether physical and dumb, like OHP screens, electronic and dumb like TV screens, physical and intelligent like software on computer screens, or intelligent like mental "screens". My interest is in relationships between doing and construing in the space accessed through mental and physical screens. Throwing images onto screens is a popular pastime, but there is a difference between "looking at" a screen and "looking through" a screen, with concomitant educational value.

This paper addresses students' understanding of Geometry by: 1. providing an overview of the van Hiele Theory; 2. considering research, from Australia and overseas, that reflects aspects of the Theory; 3. describing how the SOLO Taxonomy of Biggs and Collis can help clarify and extend ideas in the van Hiele Theory; and finally, 4. identifying future research themes.

This paper outlines the roles of the Australian Research Council (ARC), its funding programs, and the benefits of research supported by the ARC.

The study of functions is central to modern mathematics, and occupies the major part of the time spent on this subject in the senior school. The concept itself is rich and diverse, able to be conceived of using a variety of images, singly and in combination. This study explores ways in which Australian secondary and tertiary mathematics students think about functions in terms of both images and definitions. It further investigates the extent to which these factors influence their ability to discriminate between functions and non-functions, and to solve problems involving functions and their properties. The results have implications for both the teaching of functions, and the use of mathematical computer software which facilitates the representation and manipulation of functions in a variety of forms.

This paper describes a pilot study involving the use of a computer treatment and a calculator treatment in a unit on linear equation solving in year 9. The computer treatment made use of spreadsheets. while the calculator treatment parallelled the computer treatment but used either previously prepared tables or ones constructed by the student with the aid of a calculator. Although both groups were able to generate tables to solve an equation using guess and check. computer treatment students did not discover on their own how to make full use of the spreadsheet capabilities to refine their guess and check approach.

This paper is concerned with the validation of a Structured Classroom Interaction Schedule (SCIS) developed to record classroom interactions between teachers and students. The instrument was constructed as part of a long term investigation into the social context of mathematics education (Atweh and Cooper, 1989, 1991, 1992a and 1992b). The investigation employed a variety of methodologies, one of which was classroom observations. This paper describes the instrument and a trial of its validity and reliability.

This paper attempts a philosophical basis for ethnomathematics by explaining how it develops, how it is, legitimised, and how it is integrated within other mathematical cultures. Ethnomathematics requires a relativistic philosophy. Questioning universal notions in mathematics can be traced back to the French philosopher Gaston Bachelard. He describes an historically relative notion of objectivity which gives rise to changing conceptions of mathematical objects and of rationality. This analysis provides a parallel which is used to explain cultural relativity, not just in mathematical practices, but also for rational thought. The history of navigation is used to illustrate such relativity. The possibility that ethnomathematics exists at the level of rational thought raises important questions in education. Questions about the practice of mathematics education, and about its socio-political function, are discussed.

Starting with apparently unrelated ideas and making connections between them by looking for similarities and differences is one way of gaining insights. In this paper some ideas that are compared are: mathematicalproblem solving, professiohal development, research, and assessment. The cycles that can be used to summarise these processes are compared, some of the steps of the cycles that are neglected are discussed, and examples of insights that affect assessment are considered as examples of those that might be gained by comparing the processes.

The purpose of this ongoing study was to examine just a small element of the complex area involving students' errors and misunderstandings in elementary algebra, namely the relationship between students' thinking, based on their visual perception of the physical movement of algebraic symbols and their awareness of the conceptual link thus denoted. Other matters of interest to the researchers included: students' approaches to reading and transforming relationships expressed by formulae; students' ability to see and "express generalisations; the behaviour of capable students when faced with problems beyond their manipulations; the performance of students on a number of paradigmatic problems to identify points of breakdown, and the capability of average students to acquire and retain effective strategies for checking their work on algebra problems.

This year a large group offirst-year student teachers at the University ofWaikato began their introductory mathematics education course by each interviewing a five-year-old child in a school. During the interview they explored the child's understanding of number and the intellectual strategies used by the child to generate such understanding. Data from an investigation into the effects of this approach show that it took most student teachers by surprise and changed the views of many about how children learn in mathematics and about the teaching . role. The more important of these changes are outlined in the paper .

The research reported here concerned itself with determining, whether the introduction of technologies in the algebra curriculum improved conceptual understanding of the concept of variable. To define what it means to understand variables after a first-year algebra course, . algebraic activities and the role variables play in those activities were examined. This process resulted in a division of conceptual knowledge of variables into four aspects: 1. knowing the ways that symbols can be used to represent elements of a numerical domain; 2. interpreting variables in different contexts; 3. using variables in modelling problem situations and in translating one representation of a problem to another; 4. using variables in justification and proof.

In 1991 the TI-81 graphics calculator was introduced at Swinburne University of Technology to all first-year Applied Science students in their calculus courses. At the same time as the calculator was introduced, research was initiated to study the impact of the graphics calculator on the teaching and learning process. Several sources of information were tapped. These were: student surveys, a student 'brainstorm' session about the . calculator, and the study of final examination scripts. In this article we will discuss some r of the results of this research. Before presenting the research methodologies, results and discussion of the results, the calculator itself and the calculus subject in which it was used will be described.

The assumption that mathematics is learnt by the individual construction of ideas, processes and understanding rather than through the transmission of pre -formed knowledge from teacher to student is now a commonly held belief among mathematics educators. An essential feature of this view is that existing conceptions, whether gained from everyday experiences or previous learning, guide the understanding and interpretation of any new information or situation that is met. As a result, there is often a resistance to adopt new forms of knowledge or to give up or adapt previously successful thinking, and the intuitive conceptions of children may appear very different to accepted mathematical practice. While much of the early support for constructivism has come from observations of situations where new knowledge has arisen from concrete situations, constructivism also , needs to account for the more complex mathematics which has been formed by the processes of abstraction and generalisation of earlier ideas. The conventions that have emerged cannot simply be replaced by the idiosyncratic building of a host of <" individual learners; they need to be acquired in the same social context from which the mathematical concepts are to be drawn. This paper reports research which set out to establish a constructivist approach to the learning of initial fraction ideas, focussing on the social setting and activities which could lead to the negotiation and reconciliation of mathematics formed from historically and culturally determined generalisations.

What are these students' approaches to mathematics education and, in particular, what are the differences in approach, between passing and failing students? The term "approach" is used to suggest a broad underpinning of a student's thinking about knowledge, a relatively persistent characteristic, changing only gradually, and, as proposed here, plays a powerful part in a student's learning behaviour.

Research has shown that young children possess a considerable store of mathematical ideas (particularly about number) when they start formal schooling. This paper describes an attempt that was made to use these ideas as the starting point for a course in mathematics education for first year students at university. Students were given an interview schedule and were asked to interview one or two five year olds over two sessions. Between interviews the students met back at university to discuss and reflect upon the children's responses, and to prepare for the second interview. In a questionnaire administered one week after the second interview, the university students were asked to comment on what surprised them about the five year olds' responses. These were, in order of frequency, that the children knew so much "maths", that they were so open and co-operative, and that their ideas and strategies were so different from those of the interviewer. The implications of these are discussed in the paper.

Although. over the last few years there has been much effort devoted to introducing problem solving and modelling into our teaching of mathematics, up to' now there has been little progress on this at the tertiary level. At this level there appears to be just too many problems to overcome and little research has been undertaken. In this paper reasonably efficient problem solving guided approach that addresses some of the issues is proposed and some accompanying resource material presented.

This paper reports a case study of the development, implementation and outcomes of the "Active and Reflective Teaching In Secondary Mathematics" (ARTISM) professional development program. This development arose through the collaboration of teachers on three school sites, the school system (Catholic Education Office), and a tertiary institution (Australian Catholic University). An evaluation has been undertaken to document the development process, the presentation of the program, and participants' responses, including the long-term take-up of the program's key features in the participants' classroom practices. Details are presented of development and delivery and the preliminary findings of the evaluation.

The research reported in this paper represents one component of a larger project addressing the consequences and implications of the use of a new class of open-ended mathematical tasks called "Good Questions", devised by Sullivan and Clarke (199Ib). The particular characteristics of this type of task are the content-specific focus. and the opportunity for answers at different levels of sophistication. An extended research program is underway to investigate the use. of this form of open-ended mathematics task for instruction and assessment in mathematics.

This paper outlines the goals, methods and progress to date of the Interactive Mathematics Program (IMP).

Problem-solving in school mathematics has traditionally been considered as belonging only to the mode of thinking concerned with making logical connections between data and the mathematical model and then teasing out the relationship between the variable in the model and the concrete symbolic mode. Little. if any. attention has been given to the place of the intuitive processes in the context of mathematical problem~solving. The paper will present some of the results obtained in the early stages of the study.

This study investigates the certainty and uncertainty that students feel as they work on a mathematical problem and how this relates to the checking that they carry out. It is hypothesised that the over-confidence in decisions that characterises reasoning in many fields of human endeavour is also exhibited in mathematical work and that it may partly explain why students generally are reluctant to check their work. Students who feel certain that their work is correct would see little reason to check it. In the problem used in this study, students became uncertain when they moved from a particular case where they could count to a much larger case where a general rule was required. They also became uncertain when the arithmetic became harder - the size of this effect had not been expected. Students with wrong methods that gave easy arithmetic were, in the end, almost as certain·that their answers were correct as students with the· correct method. Students often did not know how to use extra information to check their answers. About half of the students who were correct became less certain after being given supporting information.

This paper describes an exploration of children's responses to problems in the tests in search of any underlying general cognitive performance in mathematics. Some conclusions about order of difficulty within strands and links between strands are discussed. The data used is the responses of about 160000 students to over 120 problems from the 1989, 1990 and 1991 year 6 tests.

In many countries there is a strongly held public view that the most important goal of elementary school mathematics is for young children to acquire a working knowledge of the . four operations and, in particular, to be able to obtain correct answers to pencil-and-paper number questions that require the application of standard algorithmic procedures (such as vertical addition and subtraction, long multiplication, short and long division). The main purpose of this. paper is to challenge this view by summarising data from Newman error analysis studies carried out in several countries which suggest that society in general, and teachers of mathematics and mathematics teacher educators in particular, urgently need to revise the traditional view of what constitutes "basic skills" in mathematics education.

No abstract available.

This paper reports on the unexpected findings when a sample of year 2 students' perceptions of their abilities in mathematics were compared with their teachers' perceptions. No significant differences by student gender were found for teachers' assessments, but gender differences in favour of males were found for students' own perceptions. Notable was the finding that in all five classes, including some in which the teachers had rated the females' achievements higher than the males', male students rated themselves higher than the females. Another interesting trend noted was that where the number of males and females in a class was approximately the same,females generally rated themselves more highly than in classes where the females were greatly outnumbered.

Research findings from extensive task-based interviews with introductory calculus students are discussed. The research was conducted from a constructivist perspective. Students' sources of conviction were the focus of analysis, with sources of conviction referring to how one determines mathematical truth and validity. The existence and characteristics of three groups of calculus learners were revealed: Collectors, Technicians, and Connectors. The groups differed in the nature and role of their sources of conviction and manner of construction of calculus conceptualizations.

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This paper summarises our earlier research in the field of computers and young children, presents data and analyses from our current research, particularly in the context of mathematics education, and discusses possible future directions for this research.

This report examined the findings of a study into primary teachers' present attitudes toward the student use of calculators in primary (Kindergarten-Year 6) mathematics classes. Data were collected from a questionnaire administered during 1990 to"a sample of 147 teachers undertaking their fourth year of study for a Bachelor of Education (Primary) at three university campuses in New South Wales, Australia.

No abstract available.

This shift in Wittgenstein's view of language from representation to activity, from "purest crystal"(1953/1991, §97) to "blurred edges"(§71) has important lessons, I think, for research in mathematics education. This paper attempts to apply ideas such as these to mathematics education (cl Bloot, 1976, 1983; Confrey, 1981; Hamlyn,1989; Kanes, 1991a, 1991b; Watson, 1988, 1989). In doing so, the discussion introduces a model (set out immediately below) which attempts to trace out the key relationships between critical elements in any theory of mathematics education: in the first instance these are taken to be variables in the domains of practice (mathematics, learning and teaching) and theory (epistemology, learning theory, pedagogy). Following an analysis of these elements and their interrelationships a third domain of interior practices, based on concepts drawn from Wittgenstein's philosophy of language, will be identified and discussed. Note that availability of space has lead to a substantial condensation of the exposition and analysis of the ideas presented here.

The National Statement on Mathematics for Australian Schools advises that algebra learning begins with the study of sequences and patterns leading to their description as algebraic rules relating dependent and independent variables. This study assessed the success of 512 students in seven schools at year levels 7 to 10 in· recognizing· and describing algebraic rules relating two variables. When given a relationship described by a table of values for two variables, the majority of these students were unable to write an algebraic rule of the form y = ax + b. Comparison of the test results from schools using different approaches to algebra suggests that the pattern-based approach (as implemented at the schools taking part in the study) was no more helpful than traditional approaches.

In the current study a drawing and description instrurrient has been developed for use in the primary school which~ most importantly, can expand the research data base for claims regarding effective learning environments, particularly for the learning of mathematics. It is considered that the instrument cart have the potential to be used on a small scale by the primary level class teacher as a diagnostic tool providing insights into the preferences and needs of the individuals in his/her class, with its larger scale use in the present study providing data for comparison by the individual teacher or other researchers~ The instrument provides insights into pupil preferences for learning environment factors different from those which have been the focus of many previous studies.

This paper describes a pilot study for a research project investigating aspects of autonomous learning behaviours, attitudes towards mathematics, and problem solving performance of students enrolled in a year-long bridging mathematics course. Six attitudinal variables were selected for investigation. A problem solving exercise was used as the context for investigating three autonomous learning behaviours. Further directions for research are discussed.

After successfully learning to identify pairs of parallel lines. samples of Year 1 and Year 4 students looked for parallels in geometrical figures and copied figures containing parallels. Oblique parallels were usually. but not always. more difficult to find than horizontal and vertical parallels. 62% of the children copied at least one parallel with an error of more than 5 degrees. but mo.st of them judged they had drawn parallels. Asked to check. most realised their error and were able to draw an accurate parallel. The results suggest that young children attempt to preserve parallels in their 2D drawings but are hindered by the complexity of the figure from noticing inaccuracies. Failure to check more carefully might also be a general trend resulting from current teaching methods.

This paper explores the notion of conceptual change in mathematics classes being a product of theory construction by both teachers and students. Its main thesis is that while there might be a growing recognition of constructivism as epistomology, the institutionalisation of practices which reflect such philosophies is inhibited by our 'knowledge' of what mathematics schooling is.

Children's solutions to ten different multiplication and division word problem structures were analysed at four interview stages in a 2- year longitudinal study (Mulligan, 1992). The study followed 70 children from Year 2 into Year 3, from the time they had received no formal instruction in multiplication or division to the stage where they were being taught basic multiplication facts. A Teaching Experiment that encouraged children to represent a range of multiplication and division situations through language, modelling, drawing, symbolising and reflective writing, was conducted with 10 children in the later part of the Longitudinal study. This paper reports the findings for the division Partition problems revealing that children very rarely used a sharing one-by-one (dealing) strategy at any stage in the Longitudinal Study or Teaching Experiment. Instead, a variety of counting and grouping strategies such as estimation and grouping, one-to many correspondence and trial-and error' grouping procedures was used. Knowledge of addition facts and skip (multiple) counting assisted children in forming equivalent groups.

The study reported in this paper consisted of two interrelated sections. In the first part of the study, the aim was to consider whether a series of spatial problem-solving activities would have a significarit effect on performance on spatio-mathematical tasks. In addition, consideration was given to the effect on performance of other factors, namely classroom organisation, gender, and Year at school. An experimental design was used to investigate these effects. In the second part of the study, consideration was given to the nature of the thinking processes, together with the effects of student-student and student-teacher interactions associated with different classroom organisations.

This research examines the heuristic of representativeness and availability. The subjects were a sample of 24 students in the Graduate Diploma of Education (Mathematics and Science) at QUT, Semester 1, 1992. These all have some formal education in mathematics and most will presumably teach mathematics in Queensland schools - including the topics in probability.

This paper addresses the question of why so many first-year students at Khon Kean University experience difficulties in coping with their mathematics courses. Factors apparently directly related to difficulties associated with students' transition from senior secondary school to university mathematics are a special focus of the study. Also, the varying extents to which school teachers, university lecturers, and first-year students differ in their perceptions of how school and university mathematics courses should be related are studied.

Over seven months, one group of Year 7 students showed little, if any, progress with algebra while another group improved markedly. The latter were significantly better in developing substitution skills, in identifying when two algebraic expressions were equal, and in understanding algebraic symbols as generalized numbers and/or variables and as representing numbers rather than objects. The former were significantly more inclined to persist with incorrect pre-algebra .ideas and to interpret conjoining (e.g., in '2n') as addition. They were not coping with the new ideas being presented to them.

No abstract available.

This paper reports on the general impact which professional development has had on mathematics teachers in Victoria, gauged from evaluations into the effectiveness of three different models of staff development. It is argued that professional development programs should aim to foster in teachers a professional development ethos which is ultimately self-sustaining and empowering.

The study undertaken here looked at difficulties associated with the first principles approach to the derivative of a function and concentrated in particular on the first five lessons in calculus as experienced by a typical group of nineteen year 10 students who were preparing to take calculus at year 11. A traditional teaching approach was contrasted with an alternative computer teaching approach and both approaches were analysedfor success in terms of conceptual understanding, skill acquisition and student perceptions of whether the work was easy to understand. The traditional first principles approach was found to be too cognitively demanding for the students who demonstrated a 'rush to rule' for meaning. Students undergoing the computer treatment also demonstrated this rush to rule and therefore very gradual development is recommended for students in their first encounters with calculus.

This paper reports on one aspect of a study designed to explore the role of metacognition in fourth grade children's mathematical problem solving. A response mapping technique was developed to represent individual problem solving episodes in terms of Flavell' s (1981) model of cognitive monitoring. The consistency of the patterns observed in the individual models suggested a relationship between conceptual and procedural knowledge of a cognitive and metacognitive kind. A meta-model was proposed to describe andexplain the nature of this inferred relationship. The paper provides an outline of the response mapping technique, a summary of the meta-model, and an example of the response map data. The analysis supported the view that children's approaches to mathematical problem solving can be described in terms of the four generic approaches identified by the meta-model, that is, a Solver's approach (high conceptual-high procedural), a Diver's approach (high conceptual-low procedural), a Player's approach (low conceptual-high procedural), and a Survivor's approach (low conceptual-low procedural).

No abstract available.

This is the report of an investigation of one classroom teaching strategy which has the potential to facilitate the achievement of the stated curriculum goals within the constraints of the classroom. There are three criteria which are necessary for any classroom strategy to be effective. First, the strategy must acknowledge the way that children learn the discipline. Second, it must accommodate the sociological constraints operating in classrooms. Third, the strategy must address the content goals of the discipline.

In this paper I report on a pilot study which has been carried out to investigate children's understanding of numeration. In . particular I am int((rested in exploring how young children develop an understanding of the structure of ones, tens, hundreds, ... that is embodied in the mental, verbal and written notions of the number system that we use.

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Responses to problems involving rates of change were compared across four data collections throughout an introductory calculus course given to a group of first year university students, all of whom had studied calculus at school. The course focused on derivative as instantaneous rate of change, and employed a method based on examining graphs of physical situations. The number of students who could symbolise rates in non-complex situations increased dramatically, but no improvement was seen in complex items or in items which required algebraic modelling. The results point to the critlcal role of a developed concept of a variable in learning calculus, and are interpreted by showing the inadequacies of abstract-apart concepts as opposed to abstract-general ones.

For the past twenty years Piagetian theory has dominated mathematics and science curricula. In primary schools, teaching - learning sequences and activities in mathematics have tradition,ally been built around the ages and stages promulgated by Piaget as being appropriate to the particular performance level of cognition a child might have attained. Conservation - according to Piaget, is the precondition for all measurement. Consequently, exposure to 'real' measurement has usually been delayed until Years 2 or 3 in primary schools. Recent research indicates that children have quite well developed notions about measurement prior to their ability to conserve. This paper examines procedures that were dev.eloped to successfully test the ability of young children to measure length. Findings from a study carried out be,tween 1985 and 1987 indicate tha/the ability to conserve is linked to the ability to measure, the knowledge of particular dimensional adjectives and experience with concrete material. The new NSW K - 6 Curriculum in Mathematics, released in 1989. reflects the findings of this research and has changed the theoretical basis of the teaching -·learning sequences in the curriculum.

The purpose ofthis paper is to review several principles used to design a Hypercard stack for the learning of introductory calculus and the research questions related to the stack. The review, development and generation of research questions, being important phases of any research project, must be dealt with carefully to ensure that the research will produce significant and valid findings. As this project is still being developed, no empirical data will be reported here; feedback from the audience is most welcome to enhance the quality of this project.

No abstract available.