Conceptualising and teaching discursive and performative reflection in higher education

The importance of reflection in higher education, and across disciplinary fields is widely recognised. It is generally embedded in university graduate attributes, professional standards and course objectives. Furthermore, reflection is commonly included in assessment requirements in higher education subjects, often without necessary scaffolding or clear expectations for students. It is essential that academic staff have substantive knowledge and clear expectations about the aims of reflective activities, the most effective mode of representation, and appropriate teaching strategies to support students in deep, critical reflection. The paper argues the case for reflection to be represented in different modes, using discursive (language) or performative (symbolic practice) forms of expression according to disciplinary context and individual communicative strengths. It introduces key discursive and expressive elements that constitute different modes of representation in reflective tasks. This functional analysis of textual elements provides explicit knowledge for teaching and assessing multiple modes of reflection in higher education.

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation

Stochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have been proposed in the literature by Stirk, Molina-París and van den Berg (2008). A stochastic modelling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probability density function (PDF) approaches can be expensive but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by the Finite State Projection and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise in some parameter regimes. Time-dependent propensities naturally arise in immunological processes due to, for example, age-dependent effects. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology.

Stochastic chemical kinetics and the quasi-steady-state assumption : application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.

Reflection of My Digital Mind

This artwork visualised my digital mind which is built on my technology experience. The contemporary technologies make available many small pieces of factual information that have affected the formation of my identity. This results that multiple identities can exist in a rhizomatic form through a vertical gene transfer. This does not refer to schizophrenia, but an ongoing transformation of digital mind.

An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular 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 fractional partial differential equations.

Developing tools to facilitate integrated reflection

Much has been said and documented about the key role that reflection can play in the ongoing development of e-portfolios, particularly e-portfolios utilised for teaching and learning. A review of e-portfolio platforms reveals that a designated space for documenting and collating personal reflections is a typical design feature of both open source and commercial off-the-shelf software. Further investigation of tools within e-portfolio systems for facilitating reflection reveals that, apart from enabling personal journalism through blogs or other writing, scaffolding tools that encourage the actual process of reflection are under-developed. Investigation of a number of prominent e-portfolio projects also reveals that reflection, while presented as critically important, is often viewed as an activity that takes place after a learning activity or experience and not intrinsic to it. This paper assumes an alternative, richer conception of reflection: a process integral to a wide range of activities associated with learning, such as inquiry, communication, editing, analysis and evaluation. Such a conception is consistent with the literature associated with ‘communities of practice’, which is replete with insight into ‘learning through doing’, and with a ‘whole minded’ approach to inquiry. Thus, graduates who are ‘reflective practitioners’ who integrate reflection into their learning will have more to offer a prospective employer than graduates who have adopted an episodic approach to reflection. So, what kinds of tools might facilitate integrated reflection? This paper outlines a number of possibilities for consideration and development. Such tools do not have to be embedded within e-portfolio systems, although there are benefits in doing so. In order to inform future design of e-portfolio systems this paper presents a faceted model of knowledge creation that depicts an ‘ecology of knowing’ in which interaction with, and the production of, learning content is deepened through the construction of well-formed questions of that content. In particular, questions that are initiated by ‘why’ are explored because they are distinguished from the other ‘journalist’ questions (who, what, when, where, and where) in that answers to them demand explanative, as opposed to descriptive, content. They require a rationale. Although why questions do not belong to any one genre and are not simple to classify — responses can contain motivational, conditional, causal, and/or existential content — they do make a difference in the acquisition of understanding. The development of scaffolding that builds on why-questioning to enrich learning is the motivation behind the research that has informed this paper.

Scaffolding for integrated reflection

This paper presents some theoretical perspectives that might inform the design and development of information and communications technology (ICT) tools to support integrated (in-session) reflection and deep learning during e-learning. The role of why questioning provides the focus of discussion and is informed by the literature on critical thinking, sense-making, and reflective practice, as well as recent developments in knowledge management, computational linguistics and automated question generation. It is argued that there exists enormous scope for the development of ICT scaffolding targeted at supporting reflective practice during e-learning. The first generations of e-Portfolio tools provide some evidence for the significance of the benefits of integrating reflection into the design of ICT systems; however, following the review of a number of such systems, as well as a range of ICT applications and services designed to support e-learning, it is argued that the scope of implementation is limited.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.