Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

Scaffolding techniques : a teacher training for cooperative learning in Thailand primary education

In general, the benefits of using cooperative learning include academic achievement, communication skills, problem-solving, social skills and student motivation. Yet cooperative learning as a Western educational concept may be ineffective in a different learning system. The study aims to investigate scaffolding techniques for cooperative learning in Thailand primary education. The program was designed to foster Thai primary school teachers’ cooperative learning implementation that includes the basic tenets of cooperative learning and socio-cognitive based learning. Two teachers were invited to participate in this experimental teacher training program for one and a half weeks. Then the teachers implemented a cooperative learning in their mathematics class for six weeks. The data from teacher interview and classroom observation indicated that the both teachers are able to utilise questions to scaffold their students’ engagement in cooperative learning. This initiative study showed that difficulty or failure of implementing cooperative learning in Thailand education may not be derived from cultural difference. The paper discussed the techniques the participant teachers applied with proactive scaffolding, reactive scaffolding and scaffolding questions that can be used to facilitate the implementation of cooperative learning in Thai school.

Elementary teacher’s conceptions of inquiry teaching : messages for teacher development

The advocacy for inquiry-based learning in contemporary curricula assumes the principle that students learn in their own way by drawing on direct experience fostered by the teacher. That students should be able to discover answers themselves through active engagement with new experiences was central to the thinking of eminent educators such as Pestalozzi, Dewey and Montessori. However, even after many years of research and practice, inquiry learning as a referent for teaching still struggles to find expression in the average teachers' pedagogy. This study drew on interview data from 20 elementary teachers. A phenomenographic analysis revealed three conceptions of teaching for inquiry learning in science in the elementary years of schooling: (a) The Experience- centred conception where teachers focused on providing interesting sensory experiences to students; (b) The Problem-centred conception where teachers focused on challenging students with engaging problems; and (c) The Question-centred conception where teachers focused on helping students to ask and answer their own questions. Understanding teachers' conceptions has implications for both the enactment of inquiry teaching in the classroom as well as the uptake of new teaching behaviours during professional development, with enhanced outcomes for engaging students in Science.

An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

Collaborative inquiries into literacy, place and identity in changing policy contexts : implications for teacher development

In this chapter we describe a history of collaboration between university-based literacy researchers and school-based teachers in teacher development programs and practitioner inquiries designed to improve literacy outcomes for students living in low-socioeconomic circumstances. We consider how an inquiry stance has informed teachers working for social justice through curriculum and pedagogy designed to connect children’s developing literacy repertoires with their changing material, social and linguistic contexts. We use examples from the practices of two of our long-term teacher-collaborators to show what has been possible to achieve, even in radically different policy contexts, because of teachers’ continued commitment to themes of place and belonging, and language and identity.

Analysing mentoring dialogues for developing a preservice teacher’s classroom management practices

Mentors (cooperating classroom teachers) have a shared responsibility with universities for developing preservice teachers’ pedagogical practices, particularly towards becoming reflective practitioners. Preservice teachers need to participate actively in their own learning, by reflecting and acting on the mentor’s constructive feedback provided during planning and feedback dialogue sessions. This case study uses feedback practices outlined within a five-factor mentoring model to analyse dialogue between a mentor and her respective mentee during different stages in their school-based programs (first practicum). This investigation uses multiple data sources such as video and audio-recorded interviews, archival documents from participants such as lesson plans, reflections and reports to examine preservice teacher’s reflections and implementations of practice as a result of her mentor’s feedback (e.g., establish expectations, review lesson plans, observe teaching then provide oral and written feedback, and evaluate progress). Findings indicated that reflective thinking was more apparent when the mentor did not dominate conversations but instead asked astute pedagogical knowledge questions to facilitate the mentee’s reflections on practice.

Practices for teaching moral values in the early years : a call for a pedagogy of participation

Schools have long been seen as institutions for preparing children for life, both academically and as moral agents in society. In order to become capable, moral citizens, children need to be provided with opportunities to learn moral values. However, little is known about how teachers enact social and moral values programs in the classroom. The aim of this paper is to investigate the practices that Australian early years teachers describe as important for teaching moral values. To investigate early years teachers’ understandings of moral pedagogy, 379 Australian teachers with experience teaching children in the early years were invited to participate in an on-line survey. This paper focuses on responses provided to an open-ended question relating to teaching practices for moral values. The responses were analysed using an interpretive methodology. The results indicate that the most prominent approaches to teaching moral values described by this group of Australian early years teachers were engaging children in moral activities. This was closely followed by teaching practices for transmitting moral values. Engaging children in building meaning and participatory learning for moral values were least often described.

Assessing students in senior science: an analysis of questions in contextualised chemistry exams

This study explores the development of a coding system for analysing test questions in two context-based chemistry exams. We describe our unique analytical procedures before contrasting the data from both tests. Our findings indicate that when a new curriculum is developed such as a context-based curriculum, teachers are required to combine the previously separate domains of context and concept to develop contextualised assessment. We argue that constructing contextualised assessment items requires teachers to view concepts and context as interconnected rather than as separate entities that may polarise scientific endeavour. Implications for practice, curriculum and assessment-development in context-based courses are proposed.

Peer partnership to enhance scholarship of teaching : a case study

This paper reports on the outcomes of a peer partnership program trialled at the Queensland University of Technology (QUT), Australia. The program was designed based on a community of practice methodology to bring together academic staff for the purpose of advancing teaching practice. The program encouraged professional and supportive environments for the purpose of critical reflection and personal development. The belief was that quality teaching is core business and vital to university organisational goals. Peer partnership programs support improvement in teaching and learning. Participants in the program reported the program enhanced their commitment and insight into teaching and that there is willingness to be involved if supported by colleagues and an organisation. Feedback from participants in the program was positive and outcomes arising from the QUT Peer Partnership Project were the development of an online peer partner tool-kit, staff development training, an instructional DVD and integration of the project goals within QUT staff development programs.

Connecting online with external students : do we really know how to?

Students are often time poor and find it difficult to manage their time in relation to study and other external factors including work. Online study is no exception to this and in many cases where the student is studying in an online only environment, they are also working in full time employment. Higher Education institutions are now offering an abundance of courses online to attract more under-graduate and post-graduate students. It is in this sense that there is an ever-increasing need to understand the student of today and find ways to connect with them and support them in their studies. This paper will report on a small-scale case study of an undergraduate online-only group of first year education students and their associated online experiences in developing a sense of community whilst interacting with a learning management system and its associated tools. Further the paper will explore the mis-conceptions that are widely held by course designers and lecturers involved with online courses.

Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Converting theory to practice : university-school collaboration on devising strategies for mentoring pedagogical knowledge

There appears no shortage of theorists for preservice teacher education; however many ideas are abandoned without practical applications. Indeed, it can take years for theories to materialise into practice, if they materialise at all. The quality of preservice teacher education is central for enhancing an education system, and mentors’ roles can assist to shape preservice teachers’ development within the school context. Yet mentoring can be haphazard without being underpinned by a theoretical framework. A mentoring model (personal attributes, system requirements, pedagogical knowledge, modelling, and feedback) has emerged from research and the literature to guide mentors’ practices. This qualitative study investigates mentors’ pedagogical knowledge as one factor crucial to the mentoring process. More specifically, this study involves a questionnaire and audio-recorded focus group meetings with experienced mentors (n=14) who deliberated on devising practical applications for mentoring pedagogical knowledge. Findings revealed that these experienced mentors pinpointed practical applications around a mentor’s role for providing pedagogical knowledge to the mentee. These strategies were varied and demonstrated that any one mentoring practice may be approached from a number of different angles. Nevertheless, there were core mentoring practices in pedagogical knowledge such as showing the mentee how to plan for teaching, articulating classroom management approaches, and talking about how to connect learning to assessment. Mentors may require education on current mentoring practices with practical strategies that are linked to theoretical underpinnings.

Partners in education : the teacher education done differently (TEDD) project

There has been an abundance of education reform recommendations for teaching and teacher education as a result of national and international reviews. A major criticism in education is the lack of connection between theory and practice (or praxis), that is, how the learning at university informs practical applications for teaching in the classroom. This paper presents the Teacher Education Done Differently (TEDD) project, funded by the Department of Education, Employment and Workplace Relations (DEEWR). It outlines how it has re-structured its offering of coursework in a Bachelor of Education (BEd) held at an Australian university campus to embrace praxis. Establishing partnerships was crucial to the development of this project. TEDD initially gathered a reference group of educators, which included university staff, school executives, and other key stakeholders, who formed an Advisory Group and Steering Committee. These groups formed a collective vision for TEDD and aimed to motivate others, foster team work, and create leadership roles that would benefit all stakeholders. The paper presents how university units changed to include a stronger praxis development for preservice teachers. Preservice teachers take their learning into schools within lead-up programs such as Ed Start for practicum I, III, and IV; Science in Schools, and Studies of Society and its Environment (SOSE). Findings showed that opportunities for undertaking additional real-world experiences were perceived to assist the preservice teachers’ praxis development. Additional school-based experiences as lead-up days for field experiences and as avenues for exploring the teaching of specific subject areas presented as an opportunity for enhancing education for all.