Faculty of Education - Theses
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ItemAssessment of mathematical problem solving: addition and subtraction in arithmetic word problem solving( 2020)Researchers consider mathematical problem solving (MPS) to be a vehicle for teaching and reinforcing mathematical knowledge, solving real life problems, and helping to meet everyday challenges. Like many other countries in the world, Bangladesh has given great importance to MPS in its primary curriculum. The country has achieved well in quantitative parameters of primary education (e.g., net enrolment rate, completion rate, etc.) but there are concerns about the quality of learning. This is particularly the case for primary mathematics, as exhibited in different examinations. One of the issues with mathematics teaching and learning is the lack of formative assessment materials that teachers can utilise to diagnose students’ learning deficiencies and use as a basis for designing appropriate remedial teaching. Furthermore, there is a paucity of research on mathematics assessment at the primary level in Bangladesh that has addressed the issues related to quality of learning. It is the aim of this research to develop and validate assessment materials using the latest developments in psychometrics that can (a) be used for this purpose, and (b) serve as a model for further relevant work. Arithmetic word problem solving (AWPS) is an area within MPS that is often used to help children learn how to apply the formal mathematical knowledge and skills they acquire at school in real-life situations. Although AWPS plays an important role in understanding mathematics, it is an area where students experience great difficulty. It is a complex activity that requires a series of cognitive processes, but current assessment of AWPS in Bangladesh relies exclusively on giving a single score based on the solution being correct or incorrect. There is limited opportunity for understanding students’ difficulties in the component cognitive processes. In the global context, empirical studies exploring the cognitive processes involved in solving arithmetic word problems are scarce. In the Bangladeshi education context, no studies were found that focused on AWPS. This research endeavoured to develop assessment materials that will enable teachers to identify specific difficulties students face while solving AWPS. The research objectives were to (a) develop a theoretical framework for arithmetic word problem solving using an iterative process (as discussed by Wilson, 2005) by (b) constructing tasks aligned with the framework, (c) trying them out in both small- and large-scale studies, (d) investigating the results using qualitative and quantitative methods, and (e) evaluating the success of the process by investigating the validity and reliability of the results. Only addition and subtraction operations for AWPS were considered to limit the scope of the study. Based on review of existing literature, a framework for assessing AWPS was developed. This framework theorises five cognitive elements/strands of AWPS: comprehension of the problem, mapping conceptual relations to a numerical representation, selecting an appropriate strategy for solution, performing the calculations, and verifying the solution and communicating the results. The validity and reliability of the framework were investigated by developing and administering a test (with items representing the different processes). The test was administered in two phases in Bangladesh: a pilot study with 62 students (from one school) and a final study with 808 students (from nine schools). Grade 4 and grade 5 students were selected through convenience sampling for both iterations of data collection. Four different types of validity evidence (based on test content, response processes, internal structure, and relations with other variables) were collected and analysed. In addition to using test design and specification, an expert opinionnaire was used to collect validity evidence based on test content. Analysis of this data showed that overall, the experts’ association of strands with items agreed with what was hypothesised, that is, they nominated items for strands they were hypothesised to be representing in this research. Cognitive interviews were conducted with 20 respondents from three schools to collect validity evidence based on the response processes. Data from these cognitive interviews were analysed using a qualitative method called ‘reduction and synthesis’ (Miller et al., 2014) and showed that the items were mostly working as intended. An Item Response Theory (IRT) approach was adopted to evaluate the internal structure of the test. A dichotomous Rasch analysis was used. The analysis showed that the item difficulties were appropriately distributed, and that the item fit statistics suggested that almost all the items were within a reasonable range (weighted MNSQ 0.88 to 1.14) and did not show any extreme over-fit or under-fit. The fit t statistics for the items also indicated a good fit. Overall, the data showed good adherence to the Rasch model and can be considered to be demonstrating reasonably good evidence of validity in terms of internal structure of the test. Two types of data were collected with the aim of establishing validity arguments based on the AWPS test’s relation with other variables: (a) scores from six textbook items included in the test that was administered and (b) students’ test scores for Bangla and Mathematics from their schools. These scores were positively correlated with students’ performance on the test conducted in this study, and hence showed good evidence of validity in relation to other variables. Test reliability was calculated using both IRT and CTT measures. The WLE person separation reliability was 0.81 and the EAP/PV reliability was 0.82. The Cronbach’s Alpha value was 0.82. The findings of this study can inform the inclusion of appropriate assessment techniques in the primary mathematics curriculum in Bangladesh, and the development of professional development materials for teachers using the assessments. Teachers of mathematics could utilise the framework to design their own tests that would help them to diagnose students’ learning deficiencies and design appropriate remedial teaching. In addition, future research can replicate this study in different settings to refine the AWPS framework and use it in different contexts. Furthermore, the methodology used in this study could be used in future research to develop assessment materials for other areas of mathematics including other areas of problem solving (e.g., arithmetic word problem solving where other operations like multiplication or division are used).
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ItemFrom the general to the particular : connecting international classroom research to four classrooms in Brunei Darussalam( 2009)This doctoral research project set out to investigate whether large-scale international classroom studies have the capacity to connect with and offer insights into the classroom practices of individual teachers in Brunei Darussalam (hereafter, referred to as Brunei). In this study, the categorising scheme and results of the 1998-2000 Third International Mathematics and Science Video Study (TIMSS-99 Video Study) were used to examine the practices of four Grade 8 mathematics classrooms in Brunei. The practices documented in the four Brunei classrooms were then compared with the practices identified in the seven countries that participated in the TIMSS-99 Video Study. The comparative analyses were made possible by the application of the analytical codes of the TIMSS-99 Video Study to the Brunei video data. Adapting the Learner's Perspective Study (LPS) data collection methods (lesson sequences, interviews and an additional questionnaire) in combination with the analytical framework of the TIMSS-99 Video Study, generated a substantial body of detailed data about each of those four classrooms, sufficient to characterise the practices of those classrooms using the TIMSS-99 Video Study coding scheme and to support comparison with the TIMSS-99 Video Study findings. Connecting the generality of the TIMSS-99 Video Study findings to the specificity of the four classrooms studied in Brunei revealed both similarities and differences between the patterns of practice evident in the international and local data sets. In addition, the study addresses the question of how these similarities and differences might be used to inform classroom practice among the four Brunei teachers. Certain characteristics were common to the 20 Brunei lessons analysed: (i) The consistent shortness of the Brunei lessons (about 22 minutes), (ii) The consistent use by the Brunei teachers and their students of short spoken public utterances (typically less than five words); and, (iii) The relatively high "connectedness" of the Brunei mathematics lessons in comparison with those lessons analysed in the TIMSS-99 Video Study. One reading of my findings is that between-teacher variations problematise the usefulness of national typifications of practice. On the other hand, studies such as the TIMSS-99 Video Study can offer us salient dimensions of practice that alert us to characteristics of familiar classrooms that might otherwise go unnoticed.
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ItemPredictors of performance in arithmetic in the middle years of primary school( 2002)While the amount of research into difficulties in mathematics has increased markedly over recent years there continues to be a need for more research into mathematics in the middle years of Primary School. The present study examined the extent to which performance on various maths related processing tasks (e.g. reading numbers, reading number statements, mental arithmetic) and measures of maths understanding (e.g. numeration and counting) predicted maths computation ability as determined by performance on typical Year 3-5 un-timed pen and paper arithmetic tasks. Analysis consisted of a stepwise regression for each of the three year levels. Some of these tasks were found to be highly predictive of achievement in arithmetic. The multiple regression was not only significant at each of the three year levels but accounted for a substantial proportion of achievement criterion variance: Year 3: 61%, Year 4: 59.8% and Year 5: 61.5%. Achievement in arithmetic was best predicted by a combination of factors at each year level with some similarities occurring across levels. The most striking of these is Mental Arithmetic: multiplication which was found to be a predictive factor at all three levels. Other significant predictive factors included Mental Arithmetic: subtraction (Year 3), Numeration: tens of thousands (Years 3 & 4), Processing of 4-digit numerals (Years 4 & 5), and Mental Arithmetic: addition (Year 5).
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ItemRemediating the elementary mathematics of prospective primary school teachers( 1980)Three areas are reviewed: the knowledge of prospective primary teachers of basic mathematics, remediation programs in mathematics and the use of calculators as a learning aid in mathematics. A remediation program is described involving a diagnostic pre-test and consequent assignment to one of three treatments: self-study, group tutorials or individual tutorials. The purpose of the 10 week program is to help prospective primary teachers meet a competency requirement in mathematics which is mandatory for progress beyond the third semester of a six-semester course of primary teacher training at a Victorian university. The program covers understanding and knowledge of place value, whole number decimals and fraction operations, percentages, averages and measurement including metrics. The group tutorials were given one of two treatments: the use of calculators during the remedial program, or the same program without calculators. Seven hypotheses related to the effects of the program are described. Post-testing after the 10 weeks of the program showed very large gains by each of the three groups. However the calculator group showed no difference from the non-calculator group, a finding consistent with the literature. Discussion of the results and pointers for future study conclude the study, with one recommendation being that mathematics specialists be appointed to primary schools.
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ItemAn investigation of Australian OECD Pisa trend results( 2009)This thesis investigates a range of equating-related issues for the Australian data collected under the Organisation for Economic Co-operation and Development (OECD) Programme for International Student Assessment (PISA). The implications for Australia's reported trend results are considered in detail. Following the exploration of differential item functioning (DIE) and dimensionality of the Australian PISA scales, a single scale, over all three PISA cycles (namely 2000, 2003 and 2006) for each major PISA domain (namely Reading, Mathematics and Science) was constructed. Previous published PISA results have employed a common reporting scale across all cycles for Reading, however scales common to all cycles have not been utilised for Mathematics or Science. Two further classes of equating issues are considered in this paper. First four different approaches to equating were used - two different treatments of missing data as well as two different item sets (all items and link items only) were estimated for each scale - and for each approach the implications for trends were discussed. Second, the equating approaches studied here used item parameters which are set at the country level rather than at the international level, thus allowing an examination of the impact of country DIF on the Australian trend results. Australian PISA trends were first explored in terms of means and standard deviations, and then in terms of the overall shape of the estimated performance distribution. This was achieved through the use of Q-Q (Quantile-Quantile) plots. Where applicable, comparisons were made with published trends. While results showed many similarities between models and published results, some differences were found. Australian PISA Reading means were statistically significantly lower when treating all omitted (or missing) responses as not administered at the item calibration stage compared to treating embedded omitted responses as incorrect and trailing omitted responses as not administered in PISA cycles 2003 and 2006. Between 2003 and 2006, published Australian Mathematics means were significantly lower than those found in this study. The published results showed a decline in means between 2003 and 2006, whereas the results reported here showed no change in the Australian means between these two cycles. Published Australian Reading distributions reported a decline from 2000 to 2003 and 2003 to 2006 in the number of Australian students located at the top end of the performance distribution. Between cycles 2000 and 2003 there was a decline from around the 70th percentile onwards and between cycles 2003 and 2006, the decline was even more severe; the higher the ability group the higher the decline from around the 20th percentile onwards. These estimated changes in the distribution shape were not replicated here, where the Australian data is analysed independently of the international data. The reanalysis undertaken here found a decline between the first two PISA cycles, but remarkably in the bottom 15 per cent of the distribution only. Between cycles 2003 and 2006 an almost constant decline across the whole proficiency distribution was found and not a decline that was limited to the top end of the distribution. The reported results highlight some of the potentially important differences that can occur when different analysis methods are used.
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ItemAn evaluation of a short, intensive mathematics program( 1999)This thesis evaluates a short intensive mathematics program for primary and secondary students that was conducted at the University of Melbourne in April 1997. The methodology of the evaluation can be described as action-research: a collaborative investigation where researchers, teachers and students cooperate in order to gain insights into a specific learning environment. This process is conducive to various improvement and modifications that the participants can apply to their respective practices. The primary aim of the thesis was to explore factors affecting the success of such programs and the learning which results. The findings suggest that while mathematical self-esteem is closely associated with mathematical achievement actual changes in mathematical self-esteem might not be easily detectable in programs of such short duration. The strength of the Program seemed to be the 'learning atmosphere' fostered by the instructors as well as in the choice of non-routine tasks. A possible weakness was the lack of clear focus regarding the pedagogical aims of the Program.
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ItemMathematics and the preoperational child : why and how to use stories didactically( 1986)Premathematical concepts such as more/less are important for an understanding of mathematics in the primary school. The literature concerning the preoperational child's acquisition of concepts, their level of intellectual development, and the use of stories to introduce concepts is reviewed along with the literature concerning the way young children form an understanding of the concepts "more/less". Thirty preparatory children from a working class area were divided into two groups - one group was taught mainly with stories and the other group with the activities described in the Boehm Resource Guide for Basic Concept Teaching (1976). They were pre- and posttested with the Boehm Basic Concepts Test (1971). It was found that teaching using selected stories is as effective as using Boehm's special activities and that girls learn better with stories than boys. The children's knowledge of "more/less" and "most/least" was assessed in individual testing sessions using toy cars and sweets as stimuli. Approximately fifty per cent did not know "less", while a majority of the children had a stable knowledge of "more" in the sense of "greater in number". Approximately 65% of the. children understood "most" and approximately 19% understood "least". The Boehm Test was critically analysed and lists of books currently available from children's book shops or libraries, suitable for introducing and reinforcing children's knowledge of concepts of space, quantity and time, and miscellaneous concepts, have been compiled.
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ItemUse of computers in mathematics teaching and learning : transition from grade 6 to year 7( 2001)Over the past ten years computer resources within Victorian schools have improved. Adequate level of facilities and resources has enabled teachers and students to use computers in the teaching and learning of mathematics. However it appears that computer skills acquired during primary years are not always the basis for further and continuous development of skills in the early years of secondary schooling. Research shows that during transition from grade 6 to year 7 there is need for stability and a sense of continuity in the adolescents' education and this applies to the area of mathematics as well. To gain further insight, this research investigated the use of computers in mathematics in a group of feeder primary schools and their linked secondary school. The study initially investigates whether the computer skills introduced in primary schools were known or built upon in secondary schools. The research then makes recommendations to the network of schools involved concerning continuity in teaching and using computer skills in the teaching and learning of mathematics during the transition years. This study was qualitative and involved parents, students and teachers. Questionnaires, interviews and classroom observations were used to obtain data. Recommendations include the need for continuous communication between grade 6 and year 7 mathematics teachers to form and maintain links regarding the development of computer skills, the need to develop a common goal for all grade 6 teachers in terms of teaching computer skills to prepare students for secondary school and finally, the need to increase computer literacy of primary and secondary mathematics teachers and make hardware and software available and accessible to all.
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ItemMontessori mathematics education : its contribution to contemporary mathematics theory and curriculum( 1987)Montessori education has a long and continuing tradition. The mathematics education component is founded upon an extensive sensory education programme developed by Dr Maria Montessori and implemented in schools throughout the world. This thesis traces the historical precedents that provided the framework for the development of an educational programme firmly premised on the need for a sensorial education. The translation of the first part of Montessori's 'Psicoaritmetica' provides an illustration of an introductory mathematics programme using structured manipulative materials. This programme develops as a direct extension of the sensory-motor emphasis of Montessori pre-school education. Finally, the Montessori mathematics programme, developed initially over fifty years ago is considered in relation to contemporary mathematics education theories, many of which have reiterated principles and practices first developed by Montessori.
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ItemTeacher beliefs about mathematical problem solving in Free Wesleyan Church secondary schools in Tonga( 2001)It is clear from the literature that attention should be paid to teachers' beliefs about how the mathematics classroom operates and about the nature of mathematical problem solving. The introduction of a mathematical problem-solving curriculum which ignores these aspects is likely to be frustrated. This research seeks to learn more about the use of mathematical problem solving by Tongan teachers and what these teachers consider to be good teaching of mathematics and mathematical problem solving. Five mathematics teachers in the Free Wesleyan Church of Tonga education system were interviewed about their beliefs using the Kelly Repertory Grid technique through Enquire Within software. Surveys were also conducted of the material produced for schools by the Tongan Government's Ministry of Education. It was found that the Ministry of Education has implemented a minimal problem-solving curriculum throughout the Kingdom and that the beliefs of the majority of teachers interviewed were compatible with this minimalist model. Recommendations for the Free Wesleyan Church's Education Department are made on possible approaches to fulfilling the Ministry of Education's stated aim of preparing students to apply the principles of mathematics to unfamiliar situations.