Walk a mile in my shoes: A case study of the everyday lives and work experiences of a group of Irish primary school Principals

While much has been written about theories and practices of management and leadership in education in recent years, what school Principals actually do on a daily basis is relatively unresearched in Ireland. Moreover, how Principals experience the job personally goes largely unnoticed. To investigate such questions, the researcher adopts a case study approach to gathering data from a group of 31 Irish primary school Principals. Researcher-driven diaries offer opportunities for self-observation by recording Principals’ personal reflections on management and leadership activities as they arise during the day. On completion, all of the diaries are collated into a single bound volume and a copy of the booklet is returned to each participant. Principals report that it is both interesting and worthwhile to read the entries of others and to gain insights into the daily work practices and lived experiences of colleagues. 21 of the 31 Principals are available in the following weeks for a second round of data gathering in recorded interviews. They comment about their own experiences of keeping the diaries and about their impressions of the experiences of other school leaders also. The diary and interview data are then coded and queried using QSR’s NVivo computer application and an organised framework of thematic flowcharts. The results are presented in a case study report with supporting empirical evidence and 12 different aspects of journeying in the Principal’s shoes through a myriad of daily work practices are explored. A narrative account explores Principals’ engagement with the internal and external school environments. It demonstrates evidence of positive work ethic and time management issues. It details Principals’ involvement with Boards of Management. Principals are vocal about their emotional investment in their roles and about the many positives and negatives that they encounter. Conclusions are drawn about career progression and about the sustainability of certain practices within the current system. The collated Principals’ diaries are available in the Appendices. They offer the opportunity to readers in different contexts to draw relevant and meaningful conclusions of their own

Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems

Finite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)¿ 0

Reflection-on-action in qualitative research processes: deconstructing research and developing an honest critical self-appraisal rubric.

In this paper, four critical friends meet to discuss qualitative research practices. Together they put one of their own case studies under the knife and deconstruct it to investigate the possibilities that knowledge work is complicated not only by the dynamics of socially constructed enterprises and the actors involved therein, but by the positioning of the researcher. The case describes an evaluative study of a university programme where students engaged in directed experiential learning in group integrated learning settings. The researcher was also the course lead-tutor and this gave rise to some concern, on later reflection and in discussions among critical friends, when issues of researcher positioning were considered. Together, through questioning the topic, the literature, the research experience and the role of the researcher, we developed a reflection-on-action rubric. In a research arena where subjective, interpretative and messy examples abound, as they should, this paper offers an example of our own work, an honest self-appraisal, a rubric for readers’ consideration and a discussion that adds to the perpetual flux of knowledge work

An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

A convergence analysis is given for the Grünwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald–Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away from t=0. Numerical results for a test example show that these theoretical results are sharp