A mathematical model for the spread of streptococcus pneumoniae with transmission due to sequence type

This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter. We start off with a discussion of Streptococcus pneumoniae and a review of previous work. We propose a simple mathematical model with two sequence types and then perform an equilibrium and (global) stability analysis on the model. We show that in general there are only three equilibria, the carriage-free equilibrium and two carriage equilibria. If the effective reproduction number Re is less than or equal to one, then the carriage will die out. If Re > 1, then the carriage will tend to the carriage equilibrium corresponding to the multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus sequence types have the same transmission parameter then there is a line of carriage equilibria. Provided that carriage is initially present then as time progresses the carriage will approach a point on this line. The results generalize to many competing sequence types. Simulations with realistic parameter values confirm the analytical results.

Adapting the task : what preservice teachers notice when adapting mathematical tasks

This paper reports on an in-depth study that explores preservice teachers’ pedagogical adaptations to a rich mathematical task. Data were collected from six elementary preservice teachers working in pairs to first solve a mathematics problem and then design adaptations to make the problem more accessible and more challenging for diverse learners. Results indicate that preservice teachers are able to draw upon a range of strategies to vary the mathematical content, the context, and the question asked. However, they also did not notice or attend to how their adaptations changed the mathematical structure of the problem. This study provides insights into what is involved in learning to adapt classroom mathematical tasks as an important pedagogical practice.

Opportunities to promote mathematical content knowledge for primary teaching

Understanding the development of pre-service teachers’ mathematical content knowledge (MCK) is important for improving primary mathematics’ teacher education. This paper reports on a case study, Rose , and her opportunities to develop MCK during the four years of her program. Program opportunities to promote MCK when planning and practicing primary teaching included: coursework experiences and responding to assessment requirements. Discussion includes the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. By fourth-year, Rose demonstrated development of different categories of MCK when practicing her teaching because of her program experiences.

A framework for teachers’ knowledge of mathematical reasoning

Exploring and developing primary teachers’ understanding of mathematical reasoning was the focus of the Mathematical Reasoning Professional Learning Research Program. Twenty-four primary teachers were interviewed after engagement in the first stage of the program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts exploring variations in primary teachers’ perceptions of mathematical reasoning revealed seven categories of description based on four dimensions of variation, establishing a framework to evaluate development in understanding of reasoning.

A primary teacher's developing understanding of mathematical reasoning

To support teachers in their quest to incorporate reasoning as a mathematical proficiency as espoused in the Australian Curriculum: Mathematics, a professional learning research project using demonstration lessons was carried out. This paper reports on the impact of demonstration lessons on one participating teacher’s pedagogical knowledge about reasoning. The growth in this teacher’s knowledge was analysed using a phenomenographic framework established to evaluate teachers’ development in mathematical reasoning. The results show that demonstration and subsequent trial lessons contributed to her growth.

Mathematical thinking of preschool children in rural and regional Australia : research and practice

At what age do young children begin thinking mathematically? Can young children work on mathematical problems? How do early childhood educators ensure young children feel good about mathematics? Where do early childhood educators learn about suitable mathematics activities?<br /><br />A good early childhood start in mathematics is critical for later mathematics success. Parents, carers and early childhood educators are teaching mathematics, either consciously or unconsciously, in any social interaction with a child.<br /><br />Mathematical Thinking of Preschool Children in Rural and Regional Australia is an extension of a conference of Australian and New Zealand researchers that identified a number of important problems related to the mathematical learning of children prior to formal schooling. A project team of 11 researchers from top Australian universities sought to investigate how early childhood education can best have a positive influence on early mathematics learning.<br /><br />The investigation complements and extends the work of Project Good Start by focusing attention on critical aspects of parents, carers and early childhood educators who care for young children. Early childhood educators from regional and rural New South Wales, Queensland and Victoria were interviewed, following a set of structured questions. The questions focused on: children&rsquo;s mathematics learning; support for mathematics teaching; use of technology; attitudes to mathematics; and assessment and record keeping.<br /><br />The researchers also reviewed research focusing on the mathematical capacities and potential foundations for further mathematical development in young children (0&ndash;5 years) published in the last decade and produced an annotated bibliography. This should provide a good basis for further research and reading.<br /><br />Based upon the results of this investigation, the researchers make 11 recommendations for improving the practices of early childhood education centres in relation to young children&rsquo;s mathematical thinking and development. The implications for policy and decision makers are outlined for teacher education, the provision of resources and further research.

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&rsquo;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 &lsquo;the Asian classroom&rsquo; 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 &ldquo;public oral interactivity&rdquo; (the number of utterances in whole class and teacher-student interactions in each lesson) and &ldquo;mathematical orality&rdquo; (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.

A universal optimization strategy for ant colony optimization algorithms based on the Physarum-inspired mathematical model

Ant colony optimization (ACO) algorithms often fall into the local optimal solution and have lower search efficiency for solving the travelling salesman problem (TSP). According to these shortcomings, this paper proposes a universal optimization strategy for updating the pheromone matrix in the ACO algorithms. The new optimization strategy takes advantages of the unique feature of critical paths reserved in the process of evolving adaptive networks of the Physarum-inspired mathematical model (PMM). The optimized algorithms, denoted as PMACO algorithms, can enhance the amount of pheromone in the critical paths and promote the exploitation of the optimal solution. Experimental results in synthetic and real networks show that the PMACO algorithms are more efficient and robust than the traditional ACO algorithms, which are adaptable to solve the TSP with single or multiple objectives. Meanwhile, we further analyse the influence of parameters on the performance of the PMACO algorithms. Based on these analyses, the best values of these parameters are worked out for the TSP.

A case study on developing a teacher's capacity in mathematical modelling

Research has indicated that for teachers to facilitate mathematical modelling activities in the mathematics classroom, they need to be familiar with the process of mathematical modelling. As such, it is imperative that teachers experience the whole mathematical modelling process. This paper reports on a Multi-tiered Teaching Experiment designed to help a teacher develop his capacity in the domain of mathematical modelling. Drawing on part of a larger case-based study conducted using Design Research phases situated within the Multi-tiered Teaching Experiment framework, the purpose of this paper is to exemplify how the research design fostered growth in teacher capacity through the natural development of critical moments of learning by the teacher during interactions between the researchers and the teacher-modeller himself. The potential of the Multi-tiered Teaching Experiment as a useful non-prescriptive teacher development approach building upon the existing repertoire of individual teachers will be discussed.

A multi-objective ant colony optimization algorithm based on the physarum-inspired mathematical model

Multi-objective traveling salesman problem (MOTSP) is an important field in operations research, which has wide applications in the real world. Multi-objective ant colony optimization (MOACO) as one of the most effective algorithms has gained popularity for solving a MOTSP. However, there exists the problem of premature convergence in most of MOACO algorithms. With this observation in mind, an improved multiobjective network ant colony optimization, denoted as PMMONACO, is proposed, which employs the unique feature of critical tubes reserved in the network evolution process of the Physarum-inspired mathematical model (PMM). By considering both pheromones deposited by ants and flowing in the Physarum network, PM-MONACO uses an optimized pheromone matrix updating strategy. Experimental results in benchmark networks show that PM-MONACO can achieve a better compromise solution than the original MOACO algorithm for solving MOTSPs.

Developing teacher competencies through videos for facilitation of mathematical modelling in Singapore primary schools

Mathematical modelling tasks which are situated in real-world contexts encourage students to draw connections between school-based mathematics and the real-world, enhancing their engagement in learning. Such tasks often require varied interpretations of the real-world problem context resulting in multiple pathways of solutions. Although mathematical modelling has been introduced in the Singapore mathematics curriculum since 2007, its incorporation in schools has been limited. One reason for this could be that teachers are challenged by how best to facilitate for rich student mathematisation processes during such tasks. This chapter reports how a multi-tiered teaching experiment using design research methodology was conducted to build teachers&rsquo; capacity in designing, facilitating, and evaluating student mathematisation during mathematical modelling tasks with an intact class of Primary 5 students (aged 10-11). The use of videos was critical because grounded images helped capture the dynamics and complexity of authentic classroom interactions. This chapter highlights how video recordings of teacher-student interactions during a modelling task were harnessed during design methodology cycles, particularly during the Retrospective Analysis phase, to activate critical moments of learning for the teacher towards developing her competencies in facilitating students&rsquo; mathematisation processes.