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This book presents a wide range of methodological perspectives on researching what teachers think and do in language teaching. It contains chapters by the editors and a leading expert in teacher cognition, as well as eight case studies by new researchers, accompanied by commentaries by internationally known researchers.

We investigate the relative efficiency of the empirical “tail median” versus “tail mean” as estimators of location when the data can be modeled by an exponential power distribution (EPD), a flexible family of light-tailed densities. By considering appropriate probabilities so that the quantile of the untruncated EPD (tail median) and mean of the left-truncated EPD (tail mean) coincide, limiting results are established concerning the ratio of asymptotic variances of the corresponding estimators. The most remarkable finding is that in the limit of the right tail, the asymptotic variance of the tail median estimate is approximately 36% larger than that of the tail mean, irrespective of the EPD shape parameter. This discovery has important repercussions for quantitative risk management practice, where the tail median and tail mean correspond to value-at-risk and expected shortfall, respectively. To this effect, a methodology for choosing between the two risk measures that maximizes the precision of the estimate is proposed. From an extreme value theory perspective, analogous results and procedures are discussed also for the case when the data appear to be heavy-tailed.

The dynamical systems of trigonometric functions are explored, with a focus on sz=sin(z) and the fractal image created by iterating the Newton map, Fs(z), of s(z). The basins of attraction created from iterating Fs(z) are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x and y-axis symmetry of the Newton map as well as some interesting results on periodic points on the real axis.

The dynamical systems of trigonometric functions are explored, with a focus on tz=tanz and the fractal image created by iterating the Newton map, Ftz, of tz. The basins of attraction created from iterating Ftz are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x- and y-axis symmetry of the Newton map and explore the nature of the fractal images.

We prove some versions of Brannan’s Conjecture on Taylor coefficients of the ratio of two binomials of the form (1+zx)α/(1−x)β(1+zx)^\\alpha /(1-x)^\\beta and discuss some related inequalities for trigonometric sums.

Motivated by an engineering pullout test applied to a steel strip embedded in earth, we show how the resulting linearly decreasing force leads naturally to a new distribution, if the force under constant stress is modeled via a three-parameter Weibull. We term this the LDSWeibull distribution, and show that inference on the parameters of the underlying Weibull can be made upon collection of data from such pullout tests. Various classical finite-sample and asymptotic properties of the LDSWeibull are studied, including existence of moments, distribution of extremes, and maximum likelihood based inference under different regimes. The LDSWeibull is shown to have many similarities with the Weibull, but does not suffer from the problem of having an unbounded likelihood function under certain parameter configurations. We demonstrate that the quality of its fit can also be very competitive with that of the Weibull in certain applications.

This text contains case studies relating the experience of bilingual children in various settings in New Zealand primary schools. The contexts include a Maori immersion school, a Samoan bilingual unit, and mainstream classrooms which cater for immigrant and deaf children.

We extend recently obtained sharp bounds for ratios of zero-balanced hypergeometric functions to the general k k -balanced case, k ∈ N kın N . We also discuss the absolute monotonicity of generalizations of previously studied functions involving generalized complete elliptic integrals.

We extend recently obtained sharp bounds for ratios of zero-balanced hypergeometric functions to the general k-balanced case, k ∈ ℕ. We also discuss the absolute monotonicity of generalizations of previously studied functions involving generalized complete elliptic integrals.

This text contains case studies relating the experience of bilingual children in various settings in New Zealand primary schools. The contexts include a Maori immersion school, a Samoan bilingual unit, and mainstream classrooms which cater for immigrant and deaf children. Suggestions for educational policy, teacher development and research are made.