EXPLICIT WRITTEN CORRECTIVE FEEDBACK AND LANGUAGE APTITUDE IN SLA: IMPLICATIONS FOR IMPROVEMENT OF LINGUISTIC ACCURACY

Most second language researchers agree that there is a role for corrective feedback in second language writing classes. However, many unanswered questions remain concerning which linguistic features to target and the type and amount of feedback to offer. This study examined two new pieces of writing by 151 learners of English as a Second Language (ESL), in order to investigate the effect of direct and metalinguistic written feedback on errors with the simple past tense, the present perfect tense, dropped pronouns, and pronominal duplication. This inquiry also considered the extent to which learner differences in language-analytic ability (LAA), as measured by the LLAMA F, mediated the effects of these two types of explicit written corrective feedback. Learners in the feedback groups were provided with corrective feedback on two essays, after which learners in all three groups completed two additional writing tasks to determine whether or not the provision of corrective feedback led to greater gains in accuracy compared to no feedback. Both treatment groups, direct and metalinguistic, performed better than the comparison group on new pieces of writing immediately following the treatment sessions, yet direct feedback was more durable than metalinguistic feedback for one structure, the simple past tense. Participants with greater LAA proved more likely to achieve gains in the direct feedback group than in the metalinguistic group, whereas learners with lower LAA benefited more from metalinguistic feedback. Overall, the findings of the present study confirm the results of prior studies that have found a positive role for written corrective feedback in instructed second language acquisition.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.