Theses abstracts (5)
The purpose of this study was to investigate how primary teachers in Queensland, Australia can make use of the Internet for professional development and to enhance the teaching and learning of mathematics. As a result of this study, implications for using the Internet for the professional development of Indonesian mathematics teachers in primary schools were drawn. The genesis of the study had emerged from reflecting upon my personal experiences in using the Internet for my own professional learning, by exploring education phenomena related to the Internet in Indonesia and Australia, and identifying gaps in research as a result of my literature review. I argue that the Internet has potential as a medium for professional development and for teaching and learning mathematics. However, little is known about the personal and professional characteristics of teachers who use the Internet to promote and renew their professional knowledge and to support their on-going learning process as well as to be good facilitators for ‘new learners’. The literature review establishes the need for investigating how teachers can use the Internet for professional development and for teaching and learning mathematics. The literature review also examines the characteristics of effective professional development, identifies inadequacies in existing professional development programs, and examines the potential advantages and limitations of using the Internet for professional development. The review suggests that there is a need to build a new model of professional development to shed light on how the Internet might be used to support primary mathematics teacher professional development. In this study, two case studies have been conducted. The first case study was of a ‘high use Internet (HUI) teacher’ (a teacher who intensively uses the Internet to sustain his/her professional growth as a mathematics teacher) and the second case study was of a ‘low use Internet (LUI) teacher’ (a teacher who has not made use of the Internet for those main goals but has a willingness to do so). The researcher learned from the HUI teacher and formulated ways to help the LUI teacher. An ethnographic approach was chosen for this study, as the researcher went into the field for an extended period of time. This study employed multiple data gathering methods, namely: participant-observation, interviews, questionnaires, and written and non-written sources. The research reported in this thesis investigated factors (personal and contextual) that support or inhibit mathematics teachers in making use of the Internet for teacher professional development and for teaching mathematics. The findings support the notion that teachers’ knowledge and beliefs are key determinants in embracing technology as a tool for teaching and learning. The findings are also significant in underscoring the non-linear, interactive and contingent nature of authentic professional development. The significance of this research is that it deepens our understanding about what is necessary for primary mathematics teachers to optimise the potential of the Internet for mathematics teaching and learning both for teachers and students. This study established the extent of the positive and negative potential effects of the Internet for professional development and the difficulties of using only this for professional development. Yet another significant outcome from this research is the construction of a theoretical framework for identifying the implications of using the Internet for professional development of Indonesian Primary teachers and for mathematics teaching and learning in Indonesian primary schools.
Computer Algebra Systems (CAS), a powerful mathematical software currently available on hand held calculators, is becoming increasingly available to assist secondary students learn school mathematics. This study investigates how two teachers taught introductory differential calculus to their Year 11 classes using multiple representations in a CAS-supported curriculum. This thesis aims to explore the impact of the teaching on students’ understanding of the concept of derivative. Understanding of the concept of derivative was gauged using an innovative Differentiation Competency Framework that was developed to describe understanding of the concept of derivative. It consists of eighteen competencies for formulation and interpretation of derivatives with, and without, translation between different representations. It clarified the objectives of the curriculum, purpose for using particular CAS activities, and also guided the construction of individual test items on the Differentiation Competency Test that enabled individual and class learning about the concept of derivative to be identified. The Framework also helped identify each teacher’s privileging characteristics and facilitated analysis of the learning in relation to the teaching. This study found that using multiple representations was important in developing understanding of the concept of derivative but that the graphical and the symbolic representations were the most useful and important to emphasize and link. Analysis of the teaching actions showed that the teachers used CAS in ways that were consistent with their teaching approach and preferred use of representations and that a conceptual teaching method and student-centred style supported understanding of the concept of derivative. Teaching is directly linked to learning and each class developed a different understanding of the concept of derivative that related to the combined effect of their teacher’s privileging characteristics: calculus content, teaching approach, and use of CAS. This study also shows that if a CAS-supported curriculum is to be successfully implemented, it needs to acquire institutional status including a corresponding change in assessment to legitimize new teaching practices.
The research study investigates students’ understanding of hypotheses testing by exploring their conceptual and procedural knowledge of the topic.
Eighteen volunteer students from a large introductory service subject in statistics were interviewed three times during their semester of study – at the beginning of the semester, several weeks after their introduction to the topic of hypothesis testing, and after the final exam in the subject. This thesis reports mainly on the results from the third interview, in which students were required to complete a Concept Mapping Task and two Hypothesis Testing tasks while talking aloud. On completion of each task, students were interviewed about their responses on these tasks. The focus is on four main concept (hypothesis, significance level, p-level, significance) and the process of hypothesis testing. Students’ protocols were analysed, and three case students provided a deeper analysis of overall responses. Qualitative methods were used in the collection and analysis of data.
The study revealed that while some students had a good understanding of the hypothesis testing concepts and procedures, there were many deficiencies. In particular, students were often unable to define or explain a concept, give an example, or describe its relationships with other concepts. Less than half of the students could complete an Hypothesis Testing task by an approved method, and several solutions were incorrect at the decision step. There was evidence of some problems with statistical notation, and it was found that each procedural step in the hypothesis testing process had the potential to present difficulties. An overriding problem was associated with expressing ideas with statistical accuracy. The case studies showed that the relationship between conceptual and procedural knowledge was an interactive one. The relationship between conceptual and procedural knowledge was found to be so close that one type of knowledge depended on the other. Metacognition was needed to access the conceptual knowledge, which in turn improved the procedures.
This study provides empirical support for the move away from hand-worked hypothesis tests to an emphasis on the development of concepts. From the research, suggestions are offered for teaching with the latter emphasis, and improving students’ overall understanding.
The purpose of the study was to develop an explanation why some children are better at addition and subtraction mental computation than others. For the purposes of this thesis, mental computation was defined as “the process of carrying out arithmetic calculations without the aid of external devices” (Sowder, 1988, p. 182). To reflect current views of mental computation as calculating with the head, rather than merely, in the head, the definition was extended to calculating using strategies with understanding (Anghileri, 1999). Thus, proficiency was not confined to accuracy, but also included flexibility of strategy choice. The study investigated the part played by number sense knowledge (e.g., numeration, number facts, estimation and effects of operations on number), metacognition, affects (e.g., beliefs, attitudes), and memory. The study showed that students proficient in mental computation (accurate and flexible) possessed integrated understandings of number facts (speed, accuracy, and efficient number facts), numeration, and number and operation. These proficient students also exhibited some metacognitive strategies and possessed reasonable short term memory and executive functioning. Where there was less knowledge and fewer connections between knowledge, students compensated in different ways, depending on their beliefs and what knowledge they possessed. Accurate and inflexible students used the teacher taught strategy of mental image of pen and paper algorithm in which strong beliefs were held. Combined with fast and accurate number facts and some numeration understanding, their familiarity with this strategy enabled the students to complete the mental computation tasks with accuracy. Working memory was sufficient to use an inefficient mental strategy accurately. The visuospatial scratchpad was used as a visual memory aid. The inaccurate and flexible students compensated for their poor number facts and minimal and disconnected knowledge base by using a variety of mental strategies in an endeavour to find one that would enable them to complete the calculation. Although their limited numeration understanding and memory (including central executive) were sufficient to support the development of some alternative strategies, these were not high level strategies. Finally, the inaccurate and inflexible students who exhibited deficient and disconnected understanding tried to compensate by using teacher-taught procedures (similar to the strategy employed by accurate and inflexible students), but they were unsuccessful, as they possessed no procedural understanding and also had poor working memory. Detailed analysis of students’ knowledge was used to develop frameworks, which explained children’s proficiency in addition and subtraction mental computation. The theoretical frameworks explained the influence of contributing factors and the relationships (if any) between them. The frameworks formed the basis of flowcharts, which explained the process in mental computation for each group of students. The importance of connected knowledge for proficient mental computation demonstrates the need for teaching practices to focus on the development of an extensive and integrated knowledge base. Students can and do formulate their own strategies, but do not always use them accurately. Therefore, students should be encouraged to formulate their own strategies but in a supportive environment that assists them to use strategies appropriately. Because of memory load, students should be permitted to use external memory aids (e.g. pen and paper) to assist mental computation. This has a second payoff in that efficient mental strategies are, at times, also efficient written strategies. By having students formulate mental strategies, they have to call upon number sense knowledge, thus acquiring connected knowledge while they develop computational procedures. This is in contrast to students using teacher-taught procedures, which require little connected knowledg
This thesis is concerned with percent knowledge and instruction. It explores the relationship between instruction, learning and unlearning in actual classrooms for the purpose of developing instruction to facilitate Year 8 students’ access to percent knowledge for solving common percent problems. Research conducted in this study occurred in response to suggestions in the literature that percent is a difficult topic to teach and learn; that no best method for percent instruction has been developed; and, of a more disturbing nature, that many students in senior high school, can not perform two-step percent problems. In this study, a series of teaching experiments was conducted. A teaching program was implemented, consisting of a proportional method for percent problem solving, and metacognitive training. Implementation of the teaching program was guided by a model of diagnostic-prescriptive instruction which states that prior knowledge must be taken into account in any teaching sequence, and errors, misconceptions and naive conceptions must be dealt with to promote forward development of knowledge. The influence of instruction was monitored through analysis of pre-, post- and delayed posttest results; researcher-generated field notes; observations; students’ diaries and artefacts; ad hoc interviews with students and observers.Results of the study indicated that the teaching program developed was extremely effective in promoting students’ percent problem solving proficiency; that the metacognitive training component of the program appeared to enhance the development of students’ principled-conceptual percent knowledge; and that application of “unteaching” strategies were more effective than good “reteaching strategies” in overcoming inappropriate prior knowledge. This study gave rise to the development of a model of percent instruction, a model of percent knowledge, and a model of diagnostic-prescriptive mathematics teaching. The teaching experiments in this study were conducted in actual classrooms and therefore in authentic school environments. The students who participated in the study were from intact classes, and the teaching program was implemented during students’ normally timetabled mathematics classes. The teaching program spanned the typical allocated time for instruction (2 weeks approximately) in the topic of percent with Year 8 students. Within these constraints, the teaching program presented to students in this study resulted in students operating proficiently on all three types of percent problems, including those involving increase and decrease. Trialling of the teaching program in this naturalistic manner underscored the viability, transplantability, and relevance of the teaching program to the mathematics classroom.
This thesis is an institutional study, attempting to account for the current situation of mathematics within the Australian vocational education and training (VET) sector. Contextualisation is provided in the first place through pertinent issues concerning professionalism, teaching, learning, and research ¾ arising from my own and other vocational mathematics teachers’ reflections on practice. The construct of institution is employed to set a more theoretical foundation with regard to, respectively, the discipline of mathematics, the field of mathematics education, and the field of vocational education and training. Technology emerges as a unifying construct for the complex relationships between mathematics and industry, in both production and management discourses, and between mathematics and vocational education. From the meta-analytic stance of this thesis, vocational education itself has become an industry and its political and social structures are explored to elucidate the apparently ambiguous position of the discipline of mathematics within the sector. A recurring theme is public image ¾ firstly of mathematics, secondly of vocational education and training in an increasingly deregulated sector which relies on segments of public opinion for its continuing survival, and thirdly in relation to the discourses of lifelong learning.
Literature reviews are undertaken with respect to mathematics in and for the workplace, adult learners of mathematics, and issues associated with curriculum and teaching ¾ not least the trend towards flexible delivery in the face of a chronic lack of discipline-based professional development. In this sector, where technology is, inter alia, a tool, an object (rarely a subject), and a vector for transmission, there is a dearth of research related to mathematics education. The question arises as to whose interests are being served by the apparent under-theorisation of mathematics education in the Australian VET sector.
The latter part of the thesis attempts to theorise these developments, drawing upon a range of intellectual work, but finding that Basil Bernstein’s concepts of symbolic control, pedagogy, and identity provide the most coherent framing for the terrain covered. The thesis concludes with discussion of unresolved policy, research, and practical issues, briefly considering a selection of relevant vocational and mathematics education research being undertaken in comparable European countries. Possibilities are explored for developments on a structural level and, drawing on the literature of pedagogies intended to combat racism, on an individual level for teachers and students to address issues of identity formation. It is hoped that these might contribute to an enhanced public image of both mathematics and vocational education and training. However, it is argued that even public image has become technologised.
This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools. The analyses were undertaken from past, contemporary, and futures perspectives. Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability.
The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented. Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987. The first of these was characterised by syllabuses that included specific references to calculating mentally. To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken. The statements in the postal, self-completion opinionnaire were based on data from the literature review. This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy.
The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Therefore, the traditional written-mental sequence needs to be reevaluated. As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations. However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation. Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers.
Of significance to the development of efficient mental techniques is the way in which mental computation is taught. R. E. Reys, B. J. Reys, Nohda, and Emori (1995, p. 305) have suggested that there are two broad approaches to teaching mental computation─a behaviourist approach and a constructivist approach. The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching. In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development. Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies. To support syllabus development, a sequence of mental strategies appropriate for focussed teaching for each of the four operations has been delineated.
The implications for teachers with respect to these recommendations are discussed. Their implementation has the potential to severely threaten many teachers’ sense of efficacy. To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined.
Reference: Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental computation performance and strategy use of Japanese students in Grades 2, 4, 6, and 8. Journal of Research in Mathematics Education, 26(4), 304-326.
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In this project university students' orientations to learning statistics is investigated. The students who participated in my research were studying statistics as a compulsory component of their psychology course. The central thesis is that learning develops in the relationship between the thinking, feeling and acting person and the social, institutional and cultural contexts surrounding him or her. How students orient themselves or position themselves to learn statistics is reflected in their engagement with the learning task - their activities. These activities determine the quality of their learning and emerging knowledge. To understand student learning the author draws on the powerful theories of Vygotsky and Leont'ev. The author's investigation consists of two studies. Study One is a qualitative exploration of the orientations to learning statistics of five older students. Study Two is grounded in Study One. The main source of data for this broader study is a survey which was completed by 279 psychology students studying statistics. In keeping with the theoretical framework, my methodology involves a holistic analysis of students and the milieu in which they act. The authors findings suggest relationships among students' affective appraisals; their conceptions of statistics; their approaches to learning it; their evaluations and the outcomes of their actions. In Study One the relationships emerged from the students' descriptions. In Study Two the author quantified the ways in which variables related to each other. Structure for the data was provided by means of correlations, factor analysis and cluster analysis. For this study the author also interviewed students and teachers of statistics. The data supports the systemic view of teaching and learning in context afforded by my theoretical perspective. Learning statistics involves the whole person and is inseparable from the arena of his or her actions. The goal of statistics education is surely to enable students to develop useful, meaningful knowledge. The findings suggest that for many of the participants in the project this goal was not being met. Most of these students reported their reluctance to learn statistics and described adopting primarily surface approaches to learning it. The project indicates the diversity of students' experiences. It raises issues as to why we teach statistics today and how the teaching and learning of statistics is being supported at university.
The motivation for this study was the desire to make the path to learning elementary algebra as 'generalised arithmetic' more clearly defined for both students and teachers.In the initial learning of algebra, algebraic expressions are transformed to equivalent other forms and techniques are developed for solving simple equations. Both facets require students to have a thorough understanding of arithmetic equality ' properties' if the developed procedures and techniques are to be adequately understood. The same can be claimed also with respect to arithmetic inequality and the solving of inequations.The specifics of the research described in this thesis entailed: (i) the identification of the properties of the equality and inequality relations considered to be the arithmetic roots from which algebraic procedures emanate; and (ii) consideration of what could constitute 'understanding' of the properties identified in (i).The research activity involved the design and development of an instrument referred to as the Mathematical Equality and Inequality Understanding Survey (the MEIUS). Specifically, the MEIUS has the following design features: (a) for the Equality Relation, the properties are exemplified using 'small numbers', 'larger numbers', and 'algebraic numbers; (b) for the Inequality Relation the properties are exemplified in 'small number' and algebraic numbers' only. The resulting Survey consists of three Stages for the Equality Relation and two Stages for the Inequality Relation.Through consideration of MEWS responses, levels were devised in order to determine 'understanding' of the relation properties. The levels were associated with the developed MEWS Thought Process Model. The MEWS has a tight protocol for administration designed to ascertain, in a valid and reliable manner, the 'thought processing' which a student employs when responding to an Item.The field work of the research involved the administration of the MEIUS to two hundred and fifty seven (257) Grades 7 to 10 students in ten (10) Tasmanian High Schools. Overall the sample consisted of 137 females and 120 males.The experience revealed that the MEWS components can be conveniently administered within the school context. Subsequent analyses of responses, using an elaborate but readily comprehended response 'scoring' procedure, indicate that there is a great deal of potentially useful information concerning student understanding of the relation properties which could be obtained in a specific school setting. Such knowledge could be used to indicate the need for remediation, on the one hand, or to identify 'readiness' to proceed or apply, on the other.Comprehensive analyses of the data gathered have been made with 'implications for teaching' firmly in mind. Links between the various relation properties and procedures for 'simplifying' expressions and solving simple equations are pointed out, in juxtaposition to the information of the proportion of a teaching year group that has demonstrated the various MEWS Levels of Understanding of the properties. Thus, the analyses can be of assistance to teachers and curriculum designers in anticipating the degree of need for remediation, as well as deciding on expressions' and solving simple equations or inequations.In considering aspects of 'remediation' the Study proposes cognitively sound approaches to teaching a number of 'selected' properties of equality. The properties have been 'selected' for their significance to the algebra topics identified.In summary, this Study has two tangible products:1. The Mathematical Equality and Inequality Understanding Survey (the MEWS) with its sound cognitive and content bases, tight protocol for administration and elaborate response 'scoring', leading to the MEWS Thought Process Model articulated in Levels;2. The identification and articulation of links between the analyses of responses in terms of the MEWS Thought Process Model and the application of the relation properties to aspects of elementary algebra, where algebra is considered as 'generalised arithmetic'.It is claimed that both these concrete products have the potential to make a valuable contribution to the teaching and learning of algebra.
Provides insights into the processes involved in teacher professional growth and factors associated with the way in which professional growth programs influence growth through a longitudinal study of primary teachers involved in a Victorian mathematics professional development program. A case study approach was adopted.
This study examined whether professional development programs can act as appropriate vehicles for the professional growth of teachers of primary mathematics. A longitudinal study was conducted of primary teachers involved in a Victorian mathematics professional development program, Exploring Mathematics in Classrooms (EMIC). The professional growth of six teacher participants in one EMIC course was examined over a period of 18 months. A case study research approach was adopted and data gathered through observations, interviews, questionnaire, and the collection of teacher work documents. A theoretical model of teacher professional growth was used to represent the teachers' growth. The data provided evidence of a strong link between the content and outcomes of professional development programs. The study provided insight into the processes involved in teacher professional growth and factors associated with the way professional development programs influence growth.
The first purpose of this study was to validate a modified and personalised form of the College and University Classroom Environment Inventory (CUCEI) and then to use this instrument to examine the actual and preferred classroom environment perceptions of students and instructors at the senior secondary and post secondary levels. A third purpose was to examine students' attitude to their courses on three specific scales, namely, Satisfaction, Difficulty and Speed. A sample of 504 students and24 instructors from Canada and the Australian Capital Territory completed the CUCEI. The students also responded to an attitudinal questionnaire. Statistical analysis confirmed the reliability of the CUCEI. The Cronbach alpha reliability figures, using the individual student as the unit of analysis, ranged from 0.73 to 0.93 and from 0.76 to 0.94 for the actual and preferred versions respectively. Good alpha reliability figures were also apparent for instructor versions, ranging from 0.72 to 0.90 for the actual version and from 0.72 to 0.93 for the preferred version. When the two levels were compared, students at the higher level had a less favourable perception of their learning environment. Hardly any difference in perceptions was seen in the learning environment between male and female students. There were significant differences in the perceptions of the classroom environment by mature students. Mature students perceived task orientation and equity more favourably than did younger students. Senior secondary students were generally more satisfied with their science courses than post secondary students. There was no difference in their attitude to the speed of delivery of science courses. The sample of 24 instructors generally perceived their environment more favourably than did their students, however, senior secondary instructors viewed the learning environment more favourably than the instructors at the post secondary level. The study also suggests that instructors at the post secondary level are aware of the changes in students classroom environment and seem to take into account these changes. Qualitative data collected from class observations and student and instructor interviews complemented the quantitative findings of the study.
At head of title: Science and Mathematics Education Centre. This thesis reports a study of the differences in educational outcomes between selected groups of students. The focus was on sex and Maori (indigenous New Zealanders)/non-Maori differences in mathematics participation and achievement amongst New Zealand secondary school students in the years 1992-1996 and, in the main, used data from the three national school qualifications available at years 10, 11 and 12. Most previous studies of education achievement and participation differences have used cross-sectional data (from students at one point in time) with separate measures of participation and achievement. The purpose of this study, was to investigate how participation and achievement data, collected at a number of points over time (both trends in replicated cross-sectional analyses and longitudinal data), could be used to enhance the measurement and understanding of the fairness of education. As an integral part of this research, a single statistic (the Cumulative Outcomes Index, COI) was developed which combined and accumulated achievement and participation results over time. Cross-sectional trends for the four years, 1992-1995, showed small but consistent sex differences in mean achievement in favour of males on externally examined qualifications, and in favour of females in the qualification with school-based assessment. In all three qualifications, non-Maori students consistently performed markedly better than Maori students (by at least 10 percentage points). At year 12, the number of mathematics papers taken accounted for the sex differences, but not all the ethnic differences, in mean achievement. Non-Maori males were the most likely group to continue in mathematics at all levels of previous achievement, but the end of year 11 seemed to be a decision point for females about further mathematics participation. Application of the COI to the longitudinal data showed that although the participation of females decreased more than that of males across each qualification its impact was reduced at year 11 by the better performance of females in the school-based qualification. The cumulative impact of both differential achievement and participation over the three years was a marked increase each year in the overall disparity between Maori and non-Maori students. A basic premise of the study was that participation and achievement differences that remain consistent (in size and direction) over time indicate some systematic effect or bias, and that such differences could not be demonstrated using the one-off, crosssectional analyses of most previous research. The research demonstrated that longitudinal analyses, especially those that combine both participation and achievement through an index such as the COI, provide critical information about the cumulative and interactive nature of achievement and participation and complement the information gained from cross-sectional analyses.
Twenty years after the first pilot projects began to develop Student Active Learning (SAL) in Indonesia, and four years since it was adopted for use in the last provinces, this research investigates the implementation of Student Active Learning in Indonesian primary mathematics classrooms. A study of the relevant literature indicates that teaching based on constructivist principles is unlikely to be implemented well in mathematics classrooms unless there are high quality teachers, readily available manipulative materials, and a supportive learning environment. As Indonesian schools often lack one or more of these aspects, it seemed likely that Student Active Learning principles might not be ‘fully’ implemented in Indonesian primary mathematics classrooms. Thus a smaller scale, parallel study was carried out in Australian schools where there is no policy of Student Active Learning, but where its underlying principles are compatible with the stated views about learning and teaching mathematics. The study employed a qualitative interpretive methodology. Sixteen primary teachers from four urban and four rural Indonesian schools and four teachers from two Victorian schools were observed for four mathematics lessons each. The twenty teachers, as well as fourteen Indonesian headteachers and other education professionals, were interviewed in order to establish links between the background and beliefs of participants, and their implementation of Student Active Learning. Information on perceived constraints on the implementation of SAL was also sought. The results of this study suggest that Student Active learning has been implemented at four levels in Indonesian primary mathematics classrooms, ranging from essentially no implementation to a relatively high level of implementation, with an even higher level of implementation in three of the four Australian classrooms observed. Indonesian teachers, headteachers and supervisors hold a range of views of SAL and also of mathematics learning and teaching. These views largely depended on their in-service training in SAL and, more particularly, on their participation in the PEQIP project Typically, participants’ expressed views of SAL were at the same or higher level as their views of mathematics learning and teaching, with a similar pattern being observed in the relationship between these latter views and their implementation of SAL principles. Three factors were identified as influencing teacher change in terms of implementation of SAL: policy, curricular and organisational, and attitudes. Recommendations arising from this study include the adoption of reflection as an underlying principle in the theory of SAL, the continuation and extension of PEQIP type projects, changes in government policy on curriculum coverage and pre-service teacher training, and more support for teachers at the school and local authority levels.
The study upon which this thesis is based aimed, first of all, to document the history of mathematics curriculum change in Western Australia. Although curriculum development in mathematics in this State has been an ongoing process for at least two decades, the outcome of an extensive literature review conducted as part of the study revealed that only a cursory evaluation of the current upper school mathematics curriculum change process had ever been undertaken. Neither has any formal appraisal of the suitability or otherwise of the variety of new upper school mathematics courses introduced during the last decade ever been carried out.This study was designed to 'fill these gaps' by not only documenting the history of the change process, but also by seeking out teachers' and other educators' views about those curriculum and strategy changes as well as the views of the students who were so intimately involved in the process.Tertiary lecturers' perceptions regarding the mathematical preparedness of first year university students were also considered a relevant source of information in this quest to first, record the events that preceded the establishment of the current State mathematics curriculum, and second, record those events that occurred subsequently. Major reports which have influenced the direction of mathematics education were examined, and underlying didactical principles were identified to determine the origins of previous and current educational policy.To determine upper school mathematics teachers' attitudes to curriculum and strategy changes, and the impact of the present curriculum upon students' choice of mathematics subjects, use was made of a variety of instruments - questionnaires and interview proformas - which were used to interview students prior to questioning them on such matters as their reasons for selecting specific units.Upper school mathematics teachers were also surveyed and interviewed in order to obtain the practitioners' views on new topics which were introduced, such as complex numbers and vectors. Five of Western Australia's most high profile mathematics educators who played significant roles during the period of this study were interviewed to determine their recollections of major points of discussion and concern in mathematics education at that time. Feedback from these interviews was used to compile a questionnaire for upper high school mathematics teachers to determine their opinion on such issues as the introduction and practicality of the new courses, teaching and learning strategies introduced, and the degree of support for the new curriculum.Ten teachers were interviewed one year after the survey to determine any changes in their perceptions about the new upper school mathematics courses. By means of a questionnaire, students' reasons for choosing specific mathematics subjects in either Year 11 or 12, and their perceived success in mathematics in general were sought. In order to determine the effectiveness of the new curriculum in terms of further studies, students' level of mathematical preparedness was investigated by means of a questionnaire for university mathematics staff. The results of this research indicate that the most recent curriculum change in the upper high school has been successfully introduced by the Education Department of Western Australia, although this has not always been the case with curriculum change in this State.Though initially daunted by the number of new topics which were to be taught, teachers were appreciative of the in-service courses available, the resources present and the general support they received from the Education Department. Traditional teaching strategies, such as 'drill-and-practice' and teacher-centred environments have been largely replaced by a problem-solving and investigational approach to mathematics in a student-centred classroom environment. Clearly, the constructivist theory of learning has been a major influence on current teaching and learning strategies used in the upper school mathematics classroom. Teachers' opinions about the practicality of the new courses and approaches to teaching them were positive, though the view was held that previous traditional teaching methods should not be discarded. Specific weaknesses in the various mathematics courses introduced were identified (for example, inadequate attention paid to basic algebra and, in particular, to trigonometry), and many teachers were adamant that certain changes should be made for the benefit of the students (for example, reorganisation of parts of the course content). Improvements in the nature of the information provided to students at the time they make their upper school mathematics subject choice were strongly recommended. Information on influential factors regarding students' subject choices was obtained, and interviews with university mathematics staff showed that many first year students remain underprepared because of incorrect mathematics subject choices made in either Year 10 or 11.
This study investigates the beliefs of Aboriginal children, their parents, Aboriginal educators and non-Aboriginal teachers towards the learning and teaching of mathematics in years 5 and 6 in a rural community in New South Wales. Areas explored include the beliefs expressed by the students, their parents and educators about mathematics education, how these sets of beliefs compare and contrast, and what the pedagogical consequences are for mathematics education based on these beliefs. The study was conducted in a rural school following trials in other sites. Conversational interviews were conducted and from the transcript sixteen core categories of beliefs across all participant groups were identified. The belief statements demonstrate the complex nature of the social, cultural, economic, historical and political contexts in which the learning of mathematics takes place. A number of actions intended to enhance Aboriginal children's learning of mathematics are proposed. Non-Aboriginal teachers need to share their beliefs with the Aboriginal community, and conversations need to occur between Aboriginal and non-Aboriginal people about mathematics education. Teachers require pedagogical strategies that address Aboriginal children's learning of mathematics, and educational systems need to include an Aboriginal perspective in mathematics curricula. Future collaborative research in mathematics education has to be based on the premise of researchers working in close co-operation with Aboriginal people.
Probability is an area of mathematics that remains a mystery to many people and is problematic for others who engage in its study in secondary school or its use in fields such as science and business. Yet, ordinary people frequently encounter the informal application of aspects of probability in daily-life situations that require decision-making under uncertainty, understanding of random behaviour and consideration of likelihood. With the purpose of discovering more about children’s ‘natural’ probability strategies, in a particular part of the world, task-based interviews were conducted with 74 children aged four to twelve years from three schools. These children had not received any formal instruction in probability, as it was not part of their school curriculum. The children’s interaction with games involving random generators prompted a range of intuitive strategies for making probabilistic judgments. These invented strategies are related to the development of proportional reasoning, but are also interlaced with the development of understanding of randomness. Examination of the strategies revealed an age-related hierarchy of sophistication of reasoning and mathematical precision. The study confirmed the presence of three developmental stages, but also revealed two distinct transitional stages not reported in previous research. The characteristics of children’s thinking at each stage of development provide a further research on probabilistic thinking and contribute to planning for teaching.
Ocean, Judith
No abstract supplied (PhD)
Ousby, Joseph
No abstract supplied (MEd)
Information technology development has driven many New Zealand educational initiatives. Projects have been undertaken, and strategies released to integrate computers into classroom teaching and learning processes. Pre-eminent among them was Interactive Education: An Information and Communication Technologies Strategy for Schools setting out a 'National Strategy' for integrating computer technology in schools. Other initiatives included: (a) The New Zealand Curriculum Framework, released in 1993, defining official policy for teaching, learning and assessment in New Zealand schools. (b) The statement of Mathematics in New Zealand Curriculum (MiNZC) (c) Information Technology Professional Development (ITPD) initiative to fund schools to organise and manage their own training and development. (d) Financial Assistance Scheme (PAS) to provide schools with at least half of the cost of approved cabling projects for local area networking, and (e) NetDay to provide practical help for schools wanting to create local area network. Extensive funds were allocated to implement these and other projects and initiatives. However there is a need for further research to reveal to what extent teachers of mathematics and other subjects are actually making use of computers. This is because since the release of the National Strategy no nationwide studies, specifically related to mathematics, have been carried out to investigate the achievement of its goals in the field of secondary mathematics teaching. Now, three years later, this present research attempts to fill that gap, and to provide government, educators, and all concerned people with deeper insight into current practice and application of computer use in the daily teaching of secondary mathematics. This research aims to contribute towards a solid foundation for further research and future planning. This research attempts to answer these questions: - To what extent and for what purposes are computers being used in secondary mathematics teaching? - How do teachers envision the use of computers in the classroom teachinglearning process? This research explores consistency between computer usage and Ministry goals as stated in MiNZC and other official statements on Information and Communication Technologies in teaching and learning. This research reveals that actual use of computers in classroom processes for the teaching and learning of mathematics remains small, comprising less than 1% of actual teaching time. Their use tends to be devoted to extending or practising pre-taught material, and serves mostly the statistics strand within the curriculum. Students seem to have unequal opportunities for use of computers. Junior students' teachers are, in general, more likely to use computers than senior students' teachers. The higher thinking mathematical process skills such as reasoning, exploring and discovering are unlikely to playa vital role in the use of computers. Results also indicate that a large majority of teachers have positive attitudes and perceive a constructive role for computers in the teaching of classroom mathematics. However, they remain cautious and are mindful of barriers to computer usage such as hardware availability, accessibility, software suitability, and professional training. The Ministry of Education hopes to achieve several goals from its computer initiatives. The most important of these is to provide opportunities for students to gain confidence and become competent users of computers in mathematical contexts to prepare them for a technology permeated future. On present evidence this goal is not currently being achieved in many classrooms. In summary, this research indicates that the use of computers in classroom teaching of mathematics is not fully meeting the governments' goals. To clarify and overcome obstacles, and to align classroom practice in using computers to teach mathematics with government goals, still requires further research, debate, cooperation, and determination.
This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools. The analyses were undertaken from past, contemporary, and futures perspectives. Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability. Little has been documented in relation to the nature and role of mental computation in mathematics curricula in Australia (McIntosh, Bana, & Farrell, 1995,p. 2), despite an international resurgence of interest by mathematics educators. This resurgence has arisen from a recognition that computing mentally remains a viable computational alternative in a technological age, and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R. E. Reys, 1992, p. 63). The emphasis needs to be placed upon the mental processes involved, and it is this which distinguishes mental computation from mental arithmetic, as defined in this study. Traditionally, the latter has been concerned with speed and accuracy rather than with the mental strategies used to arrive at the correct answers. In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered. Little attention has been given to teaching, as opposed to testing, mental computation. Additionally, such attention has predominantly been confined to those calculations needed to be performed mentally to enable the efficient use of the conventional written algorithms. Teachers are inclined to associate mental computation with isolated facts, most commonly the basic ones, rather than with the interrelationships between numbers and the methods used to calculate. To enhance the use of mental computation and to achieve an improvement in performance levels, children need to be encouraged to value all methods of computation, and to place a priority on mental procedures. This requires that teachers be encouraged to change the way in which they view mental computation. An outcome of this study is to provide the background and recommendations for this to occur. The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented. Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987. The first of these was characterised by syllabuses which included specific references to calculating mentally. To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken. The statements in the postal, self-completion opinionnaire were based on data from the literature review. This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy. The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Therefore, the traditional written-mental sequence needs to be reevaluated. As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations. However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation. Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers. Of significance to the development of efficient mental techniques is the way in which mental computation is taught. R. E. Reys, B. J. Reys, Nohda, and Emori (1995, p. 305) have suggested that there are two broad approaches to teaching mental computation,,Ya behaviourist approach and a constructivist approach. The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching. In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development. Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies. To support syllabus development, a sequence of mental strategies appropriate for focussed teaching for each of the four operations has been delineated. The implications for teachers with respect to these recommendations are discussed. Their implementation has the potential to severely threaten many teachers�f sense of efficacy. To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined.
Mathematical modelling problems are embedded in written, representational, and graphic text. For students to actively engage in the mathematical-modelling process, they require literacy. Of critical importance is the comprehension of the problems' text information, data, and goals. This design-research study investigated the application of top-level structuring; a literary, organisational, structuring strategy, to mathematical-modelling problems. The research documents how students' mathematical modelling was changed when two classes of Year 4 students were shown, through a series of lessons, how to apply top-level structure to two scientifically-based, mathematical-modelling problems. The methodology used a design-based research approach, which included five phases. During Phase One, consultations took place with the principal and participant teachers. As well, information on student numeracy and literacy skills was gathered from the Queensland Year 3 'Aspects of Numeracy' and 'Aspects of Literacy' tests. Phase Two was the initial implementation of top-level structure with one class of students. In Phase Three, the first mathematical-modelling problem was implemented with the two Year 4 classes. Data was collected through video and audio taping, student work samples, teacher and researcher observations, and student presentations. During Phase Four, the top-level structure strategy was implemented with the second Year 4 class. In Phase Five, the second mathematical-modelling problem was investigated by both classes, and data was again collected through video and audio taping, student work samples, teacher and researcher observations, and student presentations. The key finding was that top-level structure had a positive impact on students' mathematical modelling. Students were more focussed on mathematising, acquired key mathematical knowledge, and used high-level, mathematically-based peer questioning and responses after top-level structure instruction. This research is timely and pertinent to the needs of mathematics education today because of its recognition of the need for mathematical literacy. It reflects international concerns on the need for more research in problem solving. It is applicable to real-world problem solving because mathematical-modelling problems are focussed in real-world situations. Finally, it investigates the role literacy plays in the problem-solving process.
A sample of 817 Grade 4-5 mathematics students in the diverse school district of Miami-Dade County Public Schools (MDCPS), Florida, USA was involved in an evaluation of the use of hands-on activities in terms of students' achievement, students' attitudes and students' perceptions of the mathematics classroom environment. Other aims included validating generally-applicable measures of classroom learning environments and students' attitudes to mathematics, and investigating associations between the classroom learning environment and the student outcomes of performance and attitudes. The study was conducted in two phases. Phase 1 had a sample of 442 participants and classroom environment was assessed with scales selected from the My Class Inventory, Questionnaire on Teacher lnteraction and Science Laboratory Environment Inventory. Factor analysis provided a degree of support for the factorial validity and internal consistency reliability (using Cronbach's alpha coefficient) for each of five classroom environment scales. Because of the small number of items per scale (15 items in five scales for the My Class Inventory, 12 items in four scales for the Question on Teacher Interaction and 15 items in five scales for the Science Laboratory Environment Inventory) in Phase I, it was not possible to replicate the a priori factor structure of each instrument scale. Scale reliabilities generally were acceptable. Phase 2, involving a sample of 375 Grades 4 and 5 students in four elementary schools, was necessary because questionnaires in Phase 1 had too few items to enable the researcher to establish satisfactory levels of reliability and validity. The What Is Happening In this Class? (WIHIC) was modified to four scales and 29 questions for use in Phase 2. Factor analysis supported the structure of the WIHIC and internal consistency reliability was satisfactory for two units of analyses, namely, the individual and the class mean. In Phase 1 of the study, differences between an experimental group (that used manipulatives for 60% of the time) and a control group (that used manipulatives for less than 40% of the time), were described in terms of the effect size (magnitude of the difference in standard deviations) and statistical significance for each learning environment, attitude, and achievement scale. Differences between the pretest and posttest for the set of six dependent variables (Student Cohesiveness, Teacher Support, Task Orientation, Cooperation from the WIHIC and Adoption of Mathematical Attitudes and Enjoyment of Mathematics Lessons for the TOMRA) were analyzed in Phase 2 using a MANOVA for repeated measures. Effect sizes were used to describe the magnitude, as distinct from the statistical significance, of prepost changes. In Phase 2, associations between student attitudes and their perceptions of the learning environment were relatively weak for both pretest and posttest data with either the individual or the class mean as the as the unit of analysis. These results were unexpected and are inconsistent with past research, therefore highlighting the need for further research.
The research problem reported in this thesis is an investigation of the teaching and learning of area measurement in the early years of school. Research indicates that children confuse the measurement of area and perimeter and also the use of linear and two-dimensional units of measure. The first phase of the study investigated the knowledge and skills which underpin an understanding of the L x B formula for calculating rectangular area. Those factors were used to plan a teaching program of four lessons for Year 1 and Year 2 children, focusing on: establishing the attribute of area; making, describing and drawing the spatial structure of arrays of repeated informal units to measure areas; and methods of counting to determine the total number of units. The effectiveness of the program was evaluated by implementing the lessons in four classes. Lessons planned from the then current syllabus were implemented in another four classes. Comparison of student learning outcomes from all classes indicated that the research lessons were more effective in assisting children to develop an understanding of a grid pattern or array of repeated informal units. The second phase of this study described the researcher’s investigation, design and trial of teacher professional learning strategies which would assist teachers to adopt successful methods of teaching young children to measure area. Seventeen volunteer teachers in seven school teams participated in one of three models of professional learning, based on varying levels of consultancy support. The models were based on the provision of lesson notes and teaching materials, facilitation of team meetings to discuss the implementation, and the provision of additional time to interview individual children following each lesson. Participation in the project assisted all of the participating teachers to develop their content knowledge and to modify their teacher-centred teaching practices. The key strategies and factors which contributed to this success included ongoing school based professional dialogue and support, the provision of a teaching program which emphasised students’ conceptual development within a sequence of activities, the role played by teacher leaders within each team, opportunities to develop questioning techniques and the motivation and disposition of the participating teachers.
Research has shown that worked examples are superior to problem solving in many domains, particularly for novices. However, most research has only been conducted in individualised learning environments, despite a large body of literature indicating that learning can be effective in groups. This experiment aimed to engage students in a group work activity using worked examples or problems in geometry learning and compared the effect of the two approaches on numeric and reasoning abilities using both near and far transfer tests. Whether learning in a group work setting is beneficial compared with individual setting was also examined. One hundred and one Year 7 students were randomly allocated into four experimental groups: (I) problem solving in an individual setting; (2) worked examples in an individual setting; (3) problem solving during group work; (4) worked examples during group work. Each group received three consecutive instructional learning phases: worked example study, group work skill induction and an acquisition stage. Numeric and reasoning abilities of all groups using both near and far transfer tests were measured and analysed by 2 x 2 MANOVAs. A questionnaire was distributed to obtain information on students' interaction intensity and their impression of the learning activities. The results indicated a significant superiority of the worked example approach in both the individual and group work setting for numeric and reasoning abilities and most students stated that they preferred this study approach. The questionnaire data on interaction intensity revealed that the worked example condition fostered interaction between participants as much as the problem solving condition, nevertheless, a possible interaction effect was found favouring the group work condition under worked example conditions. This experiment adds evidence on the benefits of a worked example approach but does not suggest group work is advantageous. The group work effects are discussed using a cognitive load theory perspective.
This study focused on the teacher interpersonal behaviour in the teaching of Mathematics, compared to English. It investigated: differences between student perceptions of their Mathematics and English teachers' interaction styles using the actual and ideal QTI; investigate associations between students' attitudes to Mathematics and English and their perceptions of the teachers' interpersonal behaviour; investigate whether any factors exist that contribute to students' perceptions of teachers' interpersonal behaviour, determine what the typical Mathematics and English teacher in Singapore is like; and what makes an effective teacher from students' and teachers' viewpoints. The QTI, together with the Attitude to Mathematics and Attitude to English, was administered to 913 students and 37 mathematics and English teachers from an independent school in Singapore. Student and teacher interviews were conducted to further substantiate the quantitative results. Both QTI and attitudinal scales were found to be valid and reliable instruments with alpha coefficients ranging from 0.69 to 0.92. In terms of leadership, helping/friendly, understanding and student responsibility, teacher behaviour as perceived by students, fell short of the ideal. Positive associations were found between students' attitudes to Mathematics and English and their perceptions of the teachers' interpersonal behaviour. Teacher experience and students' grade level were factors that contributed to students' perceptions of teachers' interpersonal behaviour. The typical Singaporean Mathematics teacher is that of the directive and authoritative type and the English teacher is the tolerant-authoritative type. Finally, an effective teacher is one who, besides having the positive qualities of good leadership, helping/friendly, understanding, has a good sense of humour and a passion to make a difference.
There has been much debate over many years in the Australian Federal Parliament on the implementation of a national curriculum in mathematics. In 2004, the Government, under the direction of the then Minister for Education Brendon Nelson, initiated a national mathematics program for students in lower secondary high schools and primary schools. The Australian International Centre for Excellence was commissioned to implement a pilot program and called for expressions of interest to participate from high schools across the nation. At that time I was working as the Acting Head of the Mathematics Department at a senior high school in a large Western Australian country centre. I was concerned with the content and level of difficulty in many of the textbooks that were available for our students and also the processes used in these textbooks (or by teachers) to assist students to gain mastery of the basic mathematical concepts in the Outcome Number. I decided to apply to participate in the pilot program on behalf of my school, and my application was accepted. In the first stage of the program two classes of both Year 8 and Year 9 students were selected. One of my cooperative colleagues and I found out very early that the Year 8 ICE-EM textbook was too difficult for many of these students as they lacked the skills to do much of the work in the Outcome Number. These students had very different learning experiences in their primary school mathematics, with schools and teachers placing different emphases on each of the Outcomes in mathematics. The opportunity to modify our school's Year 8 program and to implement change in the high schools' feeder primary schools occurred with the second stage of the pilot program's Transition Phases 1 and 2, due for implementation in 2007.
The research explores the practice of mathematics teaching in Malawian primary schools – such as its relevance, teacher’s mathematical knowledge, assessment practices and teaching styles in massive classes – as well as the context in which it takes place – including languages used, attitudes towards gender, ideas of the purpose of education, massive class sizes but high dropout rates. It also draws together the policy documentation related to all these issues, such as government policies, the official curriculum and textbooks, and explores the extents to which policy influences practice, and practice determines policy. It concludes with a simple model suggesting that policy, properly conceived and implemented, might help overcome some of the constraints that presently overwhelm the system.
Teaching a subject requires a teacher to understand its language, epistemology and traditions, and how these characteristics govern what is appropriate for teaching and learning. This research examines how teachers' experiences of mathematics and science subject cultures, including traditions of practice, beliefs, and basic assumptions, influence their secondary school mathematics and science teaching. Six teachers from two secondary schools were interviewed and their classroom practice observed over a period of eighteen months. The research involved observing and video recording teachers' mathematics and science lessons, then interviewing them about their practice, their views of school mathematics and science, and how they see themselves in relation to these subjects. Four themes emerged which highlight similarities and differences between the subject cultures of mathematics and science: the nature of curriculum organisation across the two subjects; the role of learning experientially through hands-on experiences; the translation of 'relevance' as a school culture imperative into teachers' conceptions of, and practices in, the subject; and the role of aesthetic understanding in how teachers experience, situate themselves within, and negotiate boundaries between the two subject cultures. Significant cultural and individual differences were found in what teachers considered to be at the core of their subject teaching. Cultural differences make the subject identifiably mathematics or science. In mathematics, supporting students to move through sequentially organised curriculum content, and the importance placed on mathematics in the school curriculum, led to a Pedagogy of Support. In science, the more topic-based curriculum, and an imperative to foster student interest in science, led to a Pedagogy of Engagement. A school culture imperative to link the subject matter to students' lives was translated differently in mathematics and science. Individual differences between teachers resulted in a diversity of practices across and within the two schools, particularly with respect to how teachers related practical work to theory. The two schools' different approaches to open-ended problem solving resulted in varying degrees of latitude for teachers to move away from traditional teaching modes. In addition, whether or not teachers had stories to tell that related the subject matter to students' lives influenced their approach to making the subject relevant. Teachers' passions, coherence in their understanding of content and pedagogy, and their identity, were shown to be integral to the way they positioned themselves in relation to the subject, and in shaping their confidence and competence. Teachers experienced different traditions within the subject cultures. Some traditions perpetuated practices that might be considered 'outdated'. Emerging traditions challenged current practices through innovation and new ways of thinking about teaching and learning. Local traditions developed within the school as expectations for practice. Teachers experienced these different traditions in the process of moving forward from basic assumptions that they saw as characterising the subject, while translating school culture imperatives, and as they developed a sense of self in relation to the subject. The significance of this research lies in its contribution to improved understanding of the demands associated with subject teaching. Findings relating to the demands associated with negotiating subject boundaries have implications for the support of teachers who are teaching 'out-of-field'. In addition, teachers' experiences of the demands associated with translating school culture imperatives into their subject teaching raise questions about the usefulness of generic descriptions of pedagogy. These findings indicate that teacher and school change processes can be informed by describing subject and individual pedagogies.
Interdisciplinary Project work (PW) was introduced as an educational initiative in Singapore schools from primary to pre-university levels in 2000.PW as posited to (a) enhance perceptions and use of inter-subject connections in real world problems, (b) promote knowledge application, and (c) provide a platform for the use of thinking skills.
The main goal of this thesis to explore how these objectives are inter-related with factors influencing the quality of group collaborative mathematical thinking processes and mathematical outcomes during a mathematically-based interdisciplinary project. In this study, high quality mathematical thinking processes occur when the flow of group interactions is purposefully directed towards the enhancement of mathematically accurate, logical, reasonable outcomes.
A Sequential Explanatory Mixed Method Design consisting of consecutive quantitative and qualitative data collection and analysis procedures was used to answer the seven research questions in the study. A researcher-designed mathematically-based interdisciplinary project was implemented over 14-15 weeks with 16 classes of students (aged 13-14) belonging to two educational streams (higher and average-ability) in three Singapore government secondary schools. No teaching intervention was administered. Six scales were developed for pre- and post-project measurements of students’ mathematical confidence, perception of the value of mathematics, and perception of the interconnectedness of mathematics (N=398).Ten student-group cases (n=38) were selected for further in-depth qualitative data collection procedures pertaining to the nature of mathematical knowledge application, use of metacognitive monitoring and regulatory strategies, and core thinking skills application during three tasks in the interdisciplinary project.
The findings of this study clearly demonstrate the complexities of using PW to promote holistic and connected use of knowledge. Five substantial contributions to research on interdisciplinary learning arise from the thesis:
- An empirical framework synthesising factors influencing the quality of group collaborative mathematical knowledge application processes and outcomes was developed.
- The social influence of the group member activating applications of core thinking skills and metacognitive monitoring and regulatory strategies is mediating factor influencing the flow of cognitive-metacognitive group interactions, and therefore, the quality of collaborative mathematical knowledge application processes and outcomes.
- Leaders of high stream groups who were socially non-dominant but mathematically active were more likely to apply a higher frequency of core thinking skills than a group members in other roles (i.e., questioner, recorder, and encourager) during a mathematically-based interdisciplinary project.
- The types and complexities of mathematical knowledge and skills applied during a mathematically-based interdisciplinary project did not correspond with stream.
- Whilst student were more able to appreciate the use of mathematics for inter-subject learning after participating in a mathematically-based interdisciplinary project, their beliefs about inter-subject connections and efforts at making these connections only marginally changed.
These outcomes enhance our understanding of the challenges involved in the successful use of interdisciplinary tasks with middle school students and provide focuses for future teacher facilitation of mathematical learning during interdisciplinary education.
Lack of mathematical content knowledge (MCK), pedagogical content knowledge (PCK) and the ability to translate this knowledge into practice are recognised as major issues for pre-service teacher education today. Multimedia has been suggested as a way of facilitating the transfer of MCK and PCK to the classroom. In this context the Foundations for Teaching Arithmetic (FTA) CD-ROM was developed in 2001. The aim of this study is to evaluate how pre-service teachers in the Faculty of Education, University of Melbourne have used FTA, if at all, to improve their MCK and PCK and to support the translation of these into practice. Also under investigation is what conditions facilitate or obstruct student use of FTA as a self-help resource in improving pre-service teachers’ conceptual understanding of, and confidence in, their ability to do and to teach mathematics. Questionaires designed to find out why students did or did not use FTA and their rating of particular features of the resource were completed by 389 students in various education courses. Forty-four students users and non users of FTA were interviewed individually or in focus groups about their experience of FTA and the factors that contributed to their decisions to uses it or not use it. The impact of these factors was determined through the development of a framework which mapped the action profile of each student. A four phase needs-based progression model was proposed to explain the factors which contributed to students being able to make the successful translation of PCK on FTA into practice. The design and content of FTA facilitated students’ use of FTA for the purposes under investigation. Factors hindering student use of FTA did evolve from the content of FTA, but were attributed to circumstantial factors.
This study explores the impact of experience on the affective views of preservice primary school teachers towards mathematics. In particular, it focuses on their mathematical experiences in three particular contexts:
- Prior to commencing their initial teacher education programme, in particular their own schooling.
- During their initial course on the teaching of mathematics.
- During their school practicum placement.
As part of the investigation, the study sought to understand how preservice primary teachers experienced mathematics in the three contexts listed above . Concurrently, the participants’ beliefs, values, attitudes and feelings associated with their mathematical experiences are explored so they could be described and possible patterns identified.
Because the study focuses on experience , a phenomenological framework was employed to underpin the data collection and analysis. Data were gathered using both qualitative and quantitative modes, although primacy was given to the qualitative data as it best captured the essence and nature of the participants’ responses . A questionnaire was used to gather the quantitative data and interviews, participant journals and class activities were used to collect the qualitative data.
The findings show that overall the participants were initially apprehensive and negative about mathematics but after the tertiary course in mathematics education, their views generally improved. However, through their school-based practicum about half of the participants regressed to their initial beliefs and feelings, thus indicating the fragility of their changes. A key factor in the development of the participants’ affective responses to mathematics was the relationship they experienced with the teachers or lecturers concerned.
The findings of the study indicate that it is possible in mathematics education to bring about positive affective change, although long-term sustainability still seems rather dependent upon a complementary practicum experience. It therefore seems likely that if long-term positive change is to occur in peoples’ affective responses to mathematics, then the practicum experience requires close scrutiny and the affective dimension has implications for inservice as well as preservice teachers.
The study addressed the need to inject a values component into the mathematics curriculum taught in religious schools. To this end, a values-based set of student outcome statements was developed and then incorporated into the syllabi of two Perth schools. The outcomes of the study included a set of guidelines for teachers which provided a framework for the choice of appropriate values and also strategies for incorporation.
The evaluation of the study confirmed the compatibility of values with the mathematics education provided by the schools. Staff and student responses to interviews and questionnaires demonstrated that the theistic teachers experienced an enhanced level of satisfaction in teaching mathematics, and students indicated an increased perception of the relevance of mathematics they studied to what they considered to be important to life inside and outside the classroom.
Proponents of middle schooling believe that schools exist to cater for the needs of students through the better understanding of the nature of young adolescents. They claim that a structure should be provided that best suits these students in the age group 10-14+ who are in the process of moving from dependency to independence. The motivation for this study arose from my observations over a considerable number of years that many students when entering Year 9(late middle school years) seem alienated and disengaged from the study of mathematics.
This thesis involves a discussion of constructivist approaches to teaching and learning mathematics at the Year 9 level. The intention was to maximise the students’ learning using a constructivist approach as a theoretical reference. This was achieved by recognizing that students are not passive recipients of knowledge but actively construct knowledge from their own experiences. At the same time, the development needs and characteristics of young adolescents, along with a holistic approach to teaching and learning, provided the theoretical framework for the study. Changes in mathematics teaching practices from a ‘teacher-centred’ to a ‘student-centred’ approach were investigated. In particular, the study attempted to uncover barriers to students’ positive attitudes and to investigate whether their attitudes to mathematical learning were a result of issues related to schooling or more related to the students’ developmental stage.
Specifically, the purpose of the study was to examine how effectively a student-directed program, as opposed to an existing teacher-orientated approach, was in changing students’ attitudes and beliefs about learning mathematics, and whether or not increased achievement and student engagement occurred. The study involved a specific group of Year 9 Independent School students in an Australian rural regional context, and it placed a particular emphasis on mathematical learning. Student perceptions of, and attitudes towards, the learning of mathematics were studied, along with an examination of students’ preferred learning conditions and the influence of the classroom environment on mathematical learning. The data was obtained through a triangulation technique using several methods of data collection including my journal notes, students’ journal notes, questionnaires, surveys, class discussions and interviews - all of which was undertaken within an interpretive paradigm.
The information this study provided regarding student alienation and disengagement informed my analysis of student responses to a classroom environment based on the theoretical framework. What emerged were two major themes, each with three sub-themes . These were : ‘Learning Features’ with sub-themes ‘Metacognition’, ‘Group Work’ and ‘Negotiation’ and ‘Classroom Features’ with sub-themes ‘Feeling Safe’, ‘Clear Objectives and Guidelines’ and ‘Pace of Work’.
The two themes that emerged suggested that student engagement in the context of this study’s conceptual framework was concerned with the commitment to a set of shared values and statements such as ‘learning to learn’, which then became the heart of the classroom culture. There appeared to be connections between both the classroom and learning features of the Year 9 mathematics class that enabled students to experience a sense of empowerment and engagement in the learning process and in the management of the class as a social system. The outcomes of the study indicate that learners were empowered when they were in control of their learning. This control was largely gained through a metacognitive understanding of their own learning processes which occurred within negotiated classroom learning conditions.
Mathematics curriculum and policy documents for upper secondary schooling in many countries including Australia are placing increased focus on mathematical modelling and applications and forms of assessment that are more connected to real-world tasks . However, many fundamental theoretical issues relating to the location of mathematical tasks in meaningful contexts for both teaching and assessment purposes remain unresolved. The resolving of these issues becomes even more crucial in the context of a school based assessment system.
The general aim of the thesis is to develop an analysis system for the complexity encountered in application and modelling tasks in the senior secondary school. Such a system, being based on a sound theoretical framework, can then facilitate the framing of assessment tasks of known cognitive and contextual demand . Where assessment is school based, relying on the expertise of local teachers rather than a central examiner, such a system is particularly important when addressing questions of comparability of tasks set within different schools, districts and regions . Specific aims for this study are to:
- develop a framework for the analysis of applications/modeling tasks and
- a scheme for profiling tasks which
identifies essential mathematical content and processes within the task,
highlights the cognitive and contextual demand of the task, and
investigates how task context, prior knowledge and experience, and mathematical content interact with performance on contextualized tasks.
Finally, a grounded theory of teacher and student construal of task difficulty and complexity in practice in presented.
A cognitive/metacognitive framework for analysing application tasks was developed from a review of cognitive science literature. This was used with research tools based on the SOLO Taxonomy to develop a cognitive demand profile for applications tasks. A preliminary study was undertaken to examine and document current classroom practice through document analysis and an analysis, using the cognitive demand profiles, of scripts from the final two years of one secondary school’s assessment program. This was followed by the main study employing Strauss and Corbin’s (1990) reformulated grounded theory method. Individual task solving sessions were videotaped with 43 senior high school students from two schools. These sessions were immediately followed by clinical interviews using the videotapes as stimulus. In addition, eight teachers from the two schools were interviewed.
Intensive analyses of the task solving sessions revealed that the students used a variety of cognitive strategies to take advantage of memory-related, perceptual and engagement conditions facilitating task access, and to overcome major conditions impeding task accessibility. Metacognitive strategies for monitoring, regulating and coordinating the use of these cognitive strategies were identified. A separation of task difficulty from task complexity was achieved with task difficulty determined by students’ personal attributes and their interaction with the task, and the complexity related to the attributes of the task itself.
Student use of task context to keep their mathematics on track throughout a task did not appear to be related to the particular task but was student specific. Most students showed in the study that they would use the context to keep on track when the situation warranted it despite the claim of many that they only used it as a final check.
Prior knowledge of the task content was classified according to its source as being academic, encyclopaedic or episodic, a classification that proved useful in examining the nature of the prior knowledge reported which was task, rather than student , specific. The study highlights the largely unpredictable and differential nature of the effects of prior knowledge on student solutions. However, episodic prior knowledge was found to be more likely to have a positive influence than academic or encyclopaedic knowledge.
Moderate to high engagement with the task context of an application was not often associated with poor performance, which was more likely to be associated with nil to low engagement. However, high engagement was not a necessary condition for success as the degree of engagement necessary for task success could be task specific. A sense of realism and having an objective to work towards were identified by students as facilitating their engagement.
Some students in the study who attempted tasks with similar mathematical structures in both abstract and contextualised formats had very little success in reproducing standard procedural knowledge in the contextualised setting. Such students believed success on applications resulted from remembering methods and recognising when to apply them. Other students were much more successful in the contextualised settings . These students were able to successfully integrate task-defied information and constraints into clarifying diagrams. They either applied standard mathematical procedures which they recalled easily, or were able to select from their range of mathematical tools and adapt methods to fit novel aspects of an unfamiliar task. The successful students mostly appeared to have a more adaptable, relational understanding of mathematics and how it could be applied in contextualised settings.
Key differences in comparing in-school and out-of-school mathematics practices appear in relation to the flexibility of handling constraints. It is suggested that the salience of cues such as task constraints is mediated by how knowledge is acquired, with experiential knowledge derived through observation proving less effective than that derived through action.
The construction of a grounded theory of how teachers and students construe complexity and a difficulty of application tasks, involved the identification of numerous attributes of a task that contribute to its overall complexity. These encompassed the mathematical, linguistic, intellectual, representational, conceptual and contextual complexities of the task and it was found that both teachers and students based their assessment of task complexity and difficulty on only a small subset of these possible cues. The results of this study clearly demonstrate the complexities of teaching and using applications and modeling as a major teaching focus. They include four major contributions to the research in this area: 1)mthe construction of a cognitive/metacognitive framework for analysing applications tasks; 2) development of a new research tool, the cognitive demand profile; 3) identification of significant insights into the role played by extra-mathematical knowledge in solving applications of different complexities, and 4) construction of a grounded theory of teacher and student construal of task difficulty and task complexity in practice. These outcomes make a substantial additional contribution to our understanding of the challenges involved in the successful teaching and assessment of applications tasks at the senior secondary level .