Study abroad and complexity, accuracy and fluency (CAF) development: a longitudinal investigation of French and Chinese Learners of L2 English

This longitudinal study tracked third-level French (n=10) and Chinese (n=7) learners of English as a second language (L2) during an eight-month study abroad (SA) period at an Irish university. The investigation sought to determine whether there was a significant relationship between length of stay (LoS) abroad and gains in the learners' oral complexity, accuracy and fluency (CAF), what the relationship was between these three language constructs and whether the two learner groups would experience similar paths to development. Additionally, the study also investigated whether specific reported out-of-class contact with the L2 was implicated in oral CAF gains. Oral data were collected at three equidistant time points; at the beginning of SA (T1), midway through the SA sojourn (T2) and at the end (T3), allowing for a comparison of CAF gains arising during one semester abroad to those arising during a subsequent semester. Data were collected using Sociolinguistic Interviews (Labov, 1984) and adapted versions of the Language Contact Profile (Freed et al., 2004). Overall, the results point to LoS abroad as a highly influential variable in gains to be expected in oral CAF during SA. While one semester in the TL country was not enough to foster statistically significant improvement in any of the CAF measures employed, significant improvement was found during the second semester of SA. Significant differences were also revealed between the two learner groups. Finally, significant correlations, some positive, some negative, were found between gains in CAF and specific usage of the L2. All in all, the disaggregation of the group data clearly illustrates, in line with other recent enquiries (e.g. Wright and Cong, 2014) that each individual learner's path to CAF development was unique and highly individualised, thus providing strong evidence for the recent claim that SLA is "an individualized nonlinear endeavor" (Polat and Kim, 2014: 186).

The impact of learning context on the acquisition of sociopragmatic variation patterns in non-native speaker teachers of English

The study is a cross-linguistic, cross-sectional investigation of the impact of learning contexts on the acquisition of sociopragmatic variation patterns and the subsequent enactment of compound identities. The informants are 20 non-native speaker teachers of English from a range of 10 European countries. They are all primarily mono-contextual foreign language learners/users of English: however, they differ with respect to the length of time accumulated in a target language environment. This allows for three groups to be established – those who have accumulated 60 days or less; those with between 90 days and one year and the final group, all of whom have accumulated in excess of one year. In order to foster the dismantling of the monolith of learning context, both learning contexts under consideration – i.e. the foreign language context and submersion context are broken down into micro-contexts which I refer to as loci of learning. For the purpose of this study, two loci are considered: the institutional and the conversational locus. In order to make a correlation between the impact of learning contexts and loci of learning on the acquisition of sociopragmatic variation patterns, a two-fold study is conducted. The first stage is the completion of a highly detailed language contact profile (LCP) questionnaire. This provides extensive biographical information regarding language learning history and is a powerful tool in illuminating the intensity of contact with the L2 that learners experience in both contexts as well as shedding light on the loci of learning to which learners are exposed in both contexts. Following the completion of the LCP, the informants take part in two role plays which require the enactment of differential identities when engaged in a speech event of asking for advice. The enactment of identities then undergoes a strategic and linguistic analysis in order to investigate if and how differences in the enactment of compound identities are indexed in language. Results indicate that learning context has a considerable impact not only on how identity is indexed in language, but also on the nature of identities enacted. Informants with very low levels of crosscontextuality index identity through strategic means – i.e. levels of directness and conventionality; however greater degrees of cross-contextuality give rise to the indexing of differential identities linguistically by means of speaker/hearer orientation and (non-) solidary moves. When it comes to the nature of identity enacted, it seems that more time spent in intense contact with native speakers in a range of loci of learning allows learners to enact their core identity; whereas low levels of contact with over-exposure to the institutional locus of learning fosters the enactment of generic identities.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.