Views of Coaching Practice Expressed by School-Based Coaches

This study reports on the views of coaching expressed by school-based coaches and coaching experts in response to observing the practice of a novice coach featured in a video. Researchers hypothesized that a coach participant\u27s observations about another coach\u27s practice would be a useful tool for examining participants\u27 beliefs about coaching. Researchers compared responses from school-based coaches to the responses of coaching experts and views expressed in leading coaching literature in order to examine the variation in school-based coaches\u27 views. Analysis of responses from both practicing coaches and coaching experts revealed eight themes that describe components of the videotaped coaching cycle: 1) coaching relationships; 2) the use of praise by the coach; 3) discussions of student learning; 4) how coaches respond to teachers\u27 questions; 5) how coaches prompt reflection; 6) how coaches address teacher knowledge and learning; 7) discussions of mathematics content; and, 8) facilitation of the coaching session. The analysis also revealed that these themes correspond to accepted domains of coaching knowledge reported in the coaching literature

The extinction and dust-to-gas structure of the planetary nebula NGC 7009 observed with MUSE

The large field and wavelength range of MUSE is well suited to mapping Galactic planetary nebulae (PN). The bright PN NGC 7009 was observed with MUSE on the VLT during the Science Verification of the instrument in seeing of 0.6". Emission line maps in hydrogen Balmer and Paschen lines were formed from analysis of the MUSE cubes. The measured electron temperature and density from the MUSE cube were employed to predict the theoretical hydrogen line ratios and map the extinction distribution across the nebula. After correction for the interstellar extinction to NGC 7009, the internal dust-to-gas ratio (A_V/N_H) has been mapped for the first time in a PN. The extinction map of NGC 7009 has considerable structure, broadly corresponding to the morphological features of the nebula. A large-scale feature in the extinction map, consisting of a crest and trough, occurs at the rim of the inner shell. The nature of this feature was investigated and instrumental and physical causes considered; no convincing mechanisms were identified to produce this feature, other than mass loss variations in the earlier asymptotic giant branch phase. The dust-to-gas ratio A_V/N_H increases from 0.7 times the interstellar value to >5 times from the centre towards the periphery of the ionized nebula. The integrated A_V/N_H is about 2 times the mean ISM value. It is demonstrated that extinction mapping with MUSE provides a powerful tool for studying the distribution of PN internal dust and the dust-to-gas ratio. (Abridged.)Comment: 10 pages, 7 figures. Accepted by A&

Sub-Gaussian short time asymptotics for measure metric Dirichlet spaces

This paper presents estimates for the distribution of the exit time from balls and short time asymptotics for measure metric Dirichlet spaces. The estimates cover the classical Gaussian case, the sub-diffusive case which can be observed on particular fractals and further less regular cases as well. The proof is based on a new chaining argument and it is free of volume growth assumptions

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica