An Asymptotic Analysis of Space Charge Layers in a Mathematical Model of a Solid Electrolyte

We review a model for a solid electrolyte derived under thermodynamics principles. We non-dimensionalise and scale the model to identify small parameters, where we identify a scaling that controls the width of the space-charge layer in the electrolyte. We present asymptotic analyses and numerical solutions for the one dimensional zero charge flux equilibrium problem. We introduce an auxiliary variable to remove singularities from the domain in order to facilitate robust numerical simulations. From the asymptotics we identify three distinct regions: the bulk, boundary layers, and intermediate layers. The boundary and intermediate layers form the space charge layer of the solid electrolyte, which we can further distinguish as strong and weak space-charge-layers respectively. The weak space-charge-layer is characterised by a length, $\lambda$, which is equivalent to the Debye length of a standard liquid electrolyte. The strong space-charge-layer is characterised by a scaled Debye length, which is larger than $\lambda$. We find that both layers exhibit distinct behaviour, we see quadratic behaviour in the strong space-charge-layer and exponential behaviour in the weak space-charge-layer. We find that matching between these two asymptotic regimes is not standard and we implement a pseudo-matching approach to facilitate the transition between the quadratic and exponential behaviours. We demonstrate excellent agreement between asymptotics and simulation.Comment: 24 pages plus 14 page supplementary materials, 19 figures total (counting subfigures) Submitted to SIAM Journal on Applied Mathematic

Asymptotic reduction of a porous electrode model for lithium-ion batteries

We present a porous electrode model for lithium-ion batteries using Butler--Volmer reaction kinetics. We model lithium concentration in both the solid and fluid phase along with solid and liquid electric potential. Through asymptotic reduction, we show that the electric potentials are spatially homogeneous which decouples the problem into a series of time-dependent problems. These problems can be solved on three distinguished time scales, an early time scale where capacitance effects in the electrode dominate, a mid-range time scale where a spatial concentration gradient forms in the electrolyte, and a long-time scale where each of the electrodes saturate and deplete with lithium respectively. The solid-phase concentration profiles are linear functions of time and the electrolyte potential is everywhere zero, which allows the model to be reduced to a system of two uncoupled ordinary differential equations. Analytic and numerical results are compared with full numerical simulations and experimental discharge curves demonstrating excellent agreement.Comment: Accepted in SIAM Journal on Applied Mathematic

From exam to education: the math exam/educational resources

peer-reviewedThe Math Exam/Education Resources (MER) is an open online learning resource hosted at The University of British Columbia (UBC), aimed at providing mathematics education resources for students and instructors at UBC. In this paper, there will be a discussion of the motivation for creating this resource on the MediaWiki platform, key features of the implementation that support student learning (including the evolution of the MER wiki from an exam database to more general learning resource), data on student use and response, potential for future development, and a brief description of how the project was implemented. Preliminary correlation data between wiki usage and exam performance are shared along with some preliminary data from an ongoing impact study.peer-reviewe

Outdoor play as a mean to achieve educational goals: a case study in a Portuguese day-care group

This study aims to explore the pedagogical potential of outdoor play situations, considering the educational goals established in the Portuguese Pedagogical Guidelines for 0–3-year-old children (ongoing work), namely development of a sense of security and positive selfesteem; development of curiosity and exploratory impetus; and development of social and communication skills. Following a qualitative methodology, during a period of 9 months, the outdoor play experiences of 14 children from a day-care centre were observed and monitored and the professionals were interviewed. The data shows that outdoor experiences can contribute to the achievement of the educational goals, enlarging opportunities for creative play, problem solving, experimentation, inquiry, dialogues and collaboration. The achievement of the educational goals is facilitated through supportive and positive relationships with adults and peers, provision for proper equipment, and contact with nature. This study offers a deep insight into how educational policies can be put into practice in the outdoor area.publishe

Challenges of Early Years leadership preparation: a comparison between early and experienced Early Years practitioners in England

Leadership has been under-researched in the Early Years (EY) sector of primary schools in England, especially in leading change for professional development. The aim of this paper is to theorise what the leadership culture for EY practitioners looks like, and how Initial Teacher Training providers and schools are preparing practitioners for leadership. Using case studies of EY practitioners in different stages of their career in primary schools, we offer an insight into their preparedness for leadership in EY, the implication being that leadership training requires an understanding and embedding of the EY culture and context. Interviews with both sample groups allowed for deeper insight into the lived world. Interviews were also conducted with the head teachers to gain an overview of the leadership preparation they provided. The main findings suggest that newer EY practitioners are better prepared for leadership from their university training in comparison to more experienced EY practitioners

Career changers and fast-track induction: teacher perspectives on their early professional development

In this article, the early professional development of mature, early career teachers who entered the profession via an employment-based route to teaching in England is presented and explored from the teachers’ own perspectives. From a larger sample in a longitudinal study, the development of four career changers is traced in detail, using a model of professional learning which highlights teaching and professional skills. There is some evidence that those who undertake the programme experience a smoother induction into teaching than those from more traditional routes, although the internship year itself is challenging and demands a high level of commitment and resilience on the part of the pre-service teachers. Contextual factors, such as school support and learning from experienced teachers, are vital in enabling early professional development and the absence of these can hinder learning and self-confidence. Findings are analysed with reference to the model, as well as research on teachers’ development, within a socio-cultural learning framework. The study makes a valuable contribution to knowledge about the professional development of mature entrants to teaching

Pedagogy: the silent partner in early years learning

This paper sets out to look critically at the influences on pedagogy in early years education, at the ways in which it is enacted in practice and the pedagogical perspectives held by practitioners. The aim of the paper is to explore the current state of understanding and suggest areas to be included in an agenda for future research. The factors that influence practitioners’ actions are reviewed and the consensus around child-centredness and play is challenged. Findings from two studies of pedagogy in action in the early years are presented and examined in terms of socio-cultural theory and its implications for practice. The first of these studies draws attention to the varied nature of pedagogical interactions that support learning while the second looks at the experience of pedagogical innovation. The need for researchers and practitioners to find effective ways of communicating and working together is stressed throughout the paper