Intersection and partition graphs

This thesis reviews previous work done on both general partition graphs and existential partition graphs, which are a generalised form of general partition graphs, and extends some of the results.

Financial development, shocks, and growth volatility

This paper uses spectral theory to develop the following two testable hypotheses in a unified framework for the predictions of business-cycle and endogenous growth models: (i) financial development affects only business-cycle volatility; and (ii) shocks affect both business-cycle volatility and long-run volatility of GDP growth. In other words, volatility caused by shocks is more persistent than that caused by financial underdevelopment. We decompose the business-cycle and long-run volatility by the spectral method and then test the hypotheses at the cross-country level. Empirical evidence provides support for both hypotheses. Higher private credit, a bank-based measure of financial development, dampens business-cycle volatility but not long-run volatility. Volatility of shocks, as measured by the volatility of changes in the terms of trade, magnifies both business-cycle and long-run volatility. The results are robust to accounting for endogeneity, a market-based measure of financial development, and an alternative method of volatility decomposition.

Innovative science education assessment : linking theory with practice

This paper presents an assessment approach which demonstrates the characteristics of excellent assessment. In a third year Science Education unit in an undergraduate degree in teacher education, assessment is seen as an integral part of the learning process. Learning outcomes are improved for the students and they appreciate the validity of the assessment. Assessment is an important aspect of any unit in higher education but should be seen, not as an end in itself, but rather as a means for educational improvement. Using the assessment task as an illustration, this paper will discuss the theoretical framework of assessment. It will describe how the unit, delivered in a school-based mode, provides opportunity for students to link their theoretical understandings with practical applications, drawing together the academic debates with the first-hand experiences. It will illustrate how the assessment task itself embeds the principles of constructivism and conceptual change models in science education. Finally, the paper will highlight the benefits of this assessment approach with particular reference to the student learning.

Using activity theory in researching young children's use of calculators

Constructivist and socio-cultural perspectives in mathematics education highlight the crucial role that activity plays in mathematical development and learning. Activity theory provides a socio-cultural lens to help analyse human behaviour, including that which occurs in classrooms. It provides a framework for co-ordinating constructivist and socio-cultural perspectives in mathematics learning. In this paper, we adopt Cole and Engeström's (1991) model of activity theory to examine the mediation offered by the calculator as a tool for creating and supporting learning processes of young children in the social environment of their classroom. By adopting this framework, data on young children's learning outcomes in number, when given free access to calculators, can be examined not only in terms of the mediating role of the calculator, but also within the broader context of the classroom community, the teachers' beliefs and intentions, and the classroom norms and the division of labour. Use of this model in a post hoc situation suggests that activity theory can play a significant role in the planning of future classroom research.

Using spreadsheets in chemical education to avoid symbolic mathematics.

Traditionally, quantum theory has traditionally relied heavily on the use of  mathematics. However, there is a significant cohort of students who are  weak in mathematics, for example, students who are majoring in   biochemistry, biological sciences, etc. This paper reports on the use of  spreadsheets to generate approximate numerical solutions and visual  (graphical) descriptions as a method of avoiding or minimizing symbolic  manipulations, mathematical derivations and numerical computation. A  specific example from quantum theory is provided. Some aspects of  educational pedagogy of spreadsheet usage are discussed.

Calculators, mathematics and young children : a study of six children using calculators as part of the mathematics curriculum during their first two years of school

The thesis investigates the role a calculator can play in the developing number knowledge of three girls and three boys as part of their mathematics program, during their first two years at primary school. Random sampling was used initially to select six girls and six boys from the twenty-four children entering a 1993 prep class. These twelve children were interviewed on entrance to school and based on the performance of the twelve children on the initial interview, a girl and a boy were chosen from the higher, middle and lower achievers to take part in the full study. The class teachers involved were previously participants in the ‘Calculators in Primary Mathematics’ research program and were committed to the use of calculators in their mathematics program. A case study approach using qualitative methods within the activity theory framework is used to collect relevant data and information, an analysis of five interviews with each child and observations of the children in forty-one classroom lessons provides comprehensive data on the children's developing number knowledge during the two years. The analysis questionnaires establishes each teacher's perceptions of the children's number learning at the beginning and end of each year, compares teacher expectations with children's actual performance for the year and compares curriculum expectations with children's actual performance. A teacher interview established reasons for changes in teaching style; teacher expectations; children's number learning; and was used to confirm my research findings. An activity theory framework provides an appropriate means of co-coordinating perspectives within this research to enable a description of the child's number learning within a social environment. This framework allows for highlighting the mediation offered by the calculator supporting the children's number learning in the classroom. Levels of children's developing number knowledge reached when working with a calculator and as a result of calculator use are mapped against the levels recommended in ‘Mathematics in the National Curriculum’ (National Curriculum Council, December 1988), and the Curriculum and Standards Framework: Mathematics (Board of Studies 2000). Findings from this comparison illustrate that the six children's performance in number was enhanced when using a calculator and indicate that on-going development and understanding of number concepts occurred at levels of performance at least two years in advance of curriculum recommendations for the first two years of school.

Implementation of student active learning in primary mathematics in Indonesia

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.

Spectral shape descriptor using spherical harmonics

In this paper, we propose a spectral descriptor for shapes of objects. The method relies on transforming the 2D objects into 3D space; distance transform and scale space theory is used to transform objects into 3D space. Spherical harmonics of the voxel grid are used to obtain shape descriptors. The proposed methods are compared against two existing methods which use spherical harmonics for shape based retrieval of images. Comparison is done based on ranking of images which is articulated in recall-precision curves. MPEG-7 Still Images Content Set is used for performing experiments. Experimental results show that the performance of the proposed descriptor is significantly better than other methods in the same category.

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation : public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study (LPS) documented the voicing of mathematical ideas in public discussion and in teacher-student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher-student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation are central to the development of any theory of mathematics instruction.

Online professional learning for rural teachers of mathematics and science

Professional learning is crucial for the ongoing development of teachers and the improvement of student outcomes. Professional learning in mathematics and science education has the potential to address concerns about shortages in these areas and their impact on economic growth. However, attendance at face-to-face professional learning is problematic for many teachers located in rural areas. This study explores the utilisation of online professional learning in mathematics and science education by teachers in rural areas, and canvasses teachers’ requirements for this form of professional learning. An activity system, using cultural-historical activity theory, is developed for online professional learning for teachers of mathematics and/or science. Qualitative interpretive analysis of transcripts of 14 semi-structured individual interviews with three different groups of people suggests that teachers of mathematics/science require professional learning programs which are not only flexibly delivered but also provide professional learning content that has high utility value. By better understanding how teachers respond to the notion and practice of online professional learning, informed decisions can be made about how best to support teachers and thus build capacity in schools for success in mathematics and science.

A collaborative development of quality mathematics assessment tasks to promote preservice teachers’ professional eye

The Mathematics Teacher Education Collective (MTEC) — a self-study group based at the University of British Columbia, Canada — collaborates to enhance our pedagogical practice in mathematics teacher education through analysing, constructing, and reflecting on variations to assessment tasks. The theoretical framework underlying the establishment of the MTEC is based on Lave and Wenger’s (1991) view of learning through a Community of Practice (CoP). “Communities of practice are groups of people who share a concern or a passion for something they do and learn how to do it better as they interact regularly” (Wenger-Trayner, 2011, http://wenger-trayner.com/theory/). Three characteristics are viewed as crucial to the CoP: domain, community, and practice. The shared domain for MTEC is a commitment to gaining a deeper understanding of practice as mathematics teacher educators. In particular, as members we are interested in better understanding the role and development of tasks for learning to teach mathematics. As a community MTEC engages “in joint activities and discussions, [that] help each other, and share information” (Wenger-Trayner, 2011, http://wenger-trayner.com/theory/), with the goal to build relationships that provide members with opportunities to learn from one other. MTEC members are practitioners in the field. The success of the MTEC is based on the sharing, evaluation and critical reflection of assessment tasks and pedagogical approaches refined by the CoP.

Synthesis, structure and spectral properties of dithiocarbamato bridged dirhenium(III,II) complexes: a combined experimental and theoretical study

Sodium salts of dimethyldithiocarbamate, diethyldithiocarbamate and pyrrolidinedithiocarbamate react with the multiply bonded paramagnetic dirhenium(III,II) complex Re₂(μ-O₂CCH₃)Cl₄(μ-dppm)₂, 1 (dppm = Ph₂PCH₂PPh₂) in refluxing ethanol to afford the paramagnetic substitution products of the type Re₂(η²-S,S)₂(μ-S,S)(μ-Cl)₂(μ-dppm), where S,S represents the dithiocarbamato ligands [S,S = S₂CNMe₂, 4(L_Me); S₂CNEt₂, 4(L_Et) and S₂CN(CH₂)₄, 4(L_Pyr)]. These are the first examples of dirhenium complexes that contain bridging dithiocarbamato ligand along with the dppm ligand. These complexes have very similar spectral (UV-Vis, IR, EPR) and electrochemical properties which are also reported. The identity of 4(L_Et) has been established by single-crystal X-ray structure determination (Re-Re distance 2.6385 (9) Å) and is shown to have edge-shared bioctahedral structure. The electronic structure and the absorption spectra of the complexes are scrutinized by the density functional theory (DFT) and time-dependent density functional theory (TD-DFT) analyses.

Tail conditional expectation for multivariate distributions : a game theory approach

This paper proposes using the Shapley values in allocating the total tail conditional expectation (TCE) to each business line (X j, j = 1, ... , n) when there are n correlated business lines. The joint distributions of X j and S (S = X1 + X2 + ⋯ + X n) are needed in the existing methods, but they are not required in the proposed method.