Mathematics, mastery and metacognition: how adding a creative approach can support children in maths

Background: Children who hold an incremental view of ability show greater perseverance, improved help-seeking skills and are better able to cope with unexpected challenges. Classroom instruction can influence how children view themselves as learners. Aim: To explore how mastery-orientated classroom instruction, collaborative learning and metacognitive reflection can foster learners’ attitudes to their task performance. We hypothesised that using a mastery-oriented approach within a mathematics curriculum encourages metacognition, improves motivation and helps children achieve an underlying understanding of mathematical concepts thus improving mathematics performance. Method: This paper reports an 11-week project aiming to embed problem-solving strategies within a mastery-oriented whole-class environment. Children completed pre- and post-task semi-structured interviews and maths problems in addition to the 11-week collaborative maths project. Participants were 24 children from a rural primary school in East Sussex, 12 boys and 12 girls (mean age 8 years and 9 months). The interviews are presented qualitatively and a repeated measures analysis of variance on mathematics motivation and performance was conducted. Findings: The learners showed increased metacognitive reflection on learning strategies as well as increases in girls’ motivation for mathematics. Limitations: This is a small sample size and, being conducted within a typical everyday classroom, there were several uncontrolled variables. Although change was evident in both attitude and maths scores, it was difficult to apportion added value to the different variables contributing to the change in maths scores. Conclusions: Challenging children’s perceptions of mathematics encouraged greater self-reflection and increased motivation for girls

Magnetic cloaking by a paramagnet/superconductor cylindrical tube in the critical state

Cloaking of static magnetic fields by a finite thickness type-II superconductor tube being in the full critical state and surrounded by a coaxial paramagnet shell is studied. On the basis of exact solutions to the Maxwell equations, it is shown that, additionally to previous studies assuming the Meissner state of the superconductor constituent, perfect cloaking is still realizable at fields higher than the field of full flux penetration into the superconductor and for arbitrary geometrical parameters of both constituents. It is also proven that simultaneously the structure is fully undetectable under the cloaking conditions. Differently from the case of the Meissner state the cloaking properties in the application relevant critical state are realized, however, only at a certain field magnitude.Comment: 5 pages, 4 figures; to be published in Applied Physics Letters. arXiv admin note: substantial text overlap with arXiv:1401.356

Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity

A stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state $P=\rho$. A wide class of self-similar solutions turn out to be unstable against kink mode perturbation. According to the criterion, the Evans-Coleman stiff-fluid solution is unstable and cannot be a critical solution for the spherical collapse of a stiff fluid if we allow sufficiently small discontinuity in the density gradient field in the initial data sets. The self-similar scalar-field solution, which was recently found numerically by Brady {\it et al.} (2002 {\it Class. Quantum. Grav.} {\bf 19} 6359), is also unstable. Both the flat Friedmann universe with a scalar field and that with a stiff fluid suffer from kink instability at the particle horizon scale.Comment: 15 pages, accepted for publication in Classical and Quantum Gravity, typos correcte

Multiple imaging by gravitational waves

Gravitational waves act like lenses for the light propagating through them. This phenomenon is described using the vector formalism employed for ordinary gravitational lenses, which was proved to be applicable also to a non-stationary spacetime, with the appropriate modifications. In order to have multiple imaging, an approximate condition analogous to that for ordinary gravitational lenses must be satisfied. Certain astrophysical sources of gravitational waves satisfy this condition, while the gravitational wave background, on average, does not. Multiple imaging by gravitational waves is, in principle, possible, but the probability of observing such a phenomenon is extremely low.Comment: 23 pages, LaTeX, no figures, to appear in Int. J. Mod. Phys.

Cell proliferation and oncogene expression after bile duct ligation in the rat: Evidence of a specific growth effect on bile duct cells

The proliferative response of the rat liver was measured after temporary or permanent total biliary obstruction (BDO) and in different regions after selective ligation of the lobar ducts draining the right 60% of the hepatic mass. The results were compared with those after 70% partial hepatectomy (PH). Cell proliferation was assessed globally by measuring DNA synthesis and stratified to the separate cell populations with cytostaining techniques that allowed distinction of hepatocytes, duct cells, and nonparenchymal cells (NPCs). In selected experimental groups, gene expression was determined of transforming growth factor-β1 (TGFβ-1), prothrombin, c-erb-B2, transforming growth factor alpha (TGFα), human Cyclophilin (CyP), and 28S ribosomal RNA. The stimulation of a proliferative response to total BDO required obstruction for longer than 24 hours, but after this deligation did not switch off regeneration. In the first week after permanent BDO, there was progressive infiltration of NPCs, fibrous linkage of some portal areas, and a crescendo of DNA synthesis that was obvious at 24 hours, maximal at 48 hours, and back nearly to baseline at 6 days. At the 2-day mark, the bile duct cells had a 17-fold increase in proliferation, accompanied by a threefold to fourfold increase in hepatocyte renewal. Little or no increase in expression of TGFα or the hepatocyte-specific prothrombin gene was detectable in the first 48 hours, whereas levels of the oncogene c-erb-B2 that is associated with cholangiocarcinoma were expressed from 48 to 96 hours. Livers subjected to regional BDO with or without immunosuppressive treatment with FK 506 and cyclosporine had an inflammatory reaction only on the side with ligated ducts. DNA synthesis increased in both the obstructed and freely draining lobes to approximately half the level that occurred after total BDO. The proliferation of the obstructed side was similar to the mixed duct cell/hepatocyte response after total BDO, but this almost exclusively involved duct cells on the freely draining side. In contrast to the findings after BDO, livers after PH regenerated maximally at 24 hours rather than 48 hours, had a predominantly noninflammatory hepatocyte as opposed to duct cell response, and had marked expression of the prothrombin and TGFα genes but only weakly and late of c-erb-B2 messenger RNA. The results show that the liver responds as a whole and in a biologically intelligent way to the nature of the injury inflicted on any part of it. It further implies the presence of humoral communications and control networks that assure organ homeostasis and relate this to total body homeostasis. © 1995

Deep subcutaneous application of poly-L-lactic acid as a filler for facial lipoatrophy in HIV-infected patients

Introduction: Facial lipoatrophy is a crucial problem of HIV-infected patients undergoing highly active antiretroviral therapy (HAART). Poly-L-lactic acid (PLA), provided as New-Fill(R)/Sculptra(TM), is known as one possible treatment option. In 2004 PLA was approved by the FDA as Sculptra(TM) for the treatment of lipoatrophy of the face in HIV-infected patients. While the first trials demonstrated relevant efficacy, this was to some extent linked to unwanted effects. As the depth of injection was considered relevant in this context, the application modalities of the preparation were changed. The preparation was to be injected more deeply into subcutaneous tissue, after increased dilution. Material and Methods: To test this approach we performed a pilot study following the new recommendations in 14 patients. Results: While the efficacy turned out to be about the same, tolerability was markedly improved. The increase in facial dermal thickness was particularly obvious in those patients who had suffered from lipoatrophy for a comparatively small period of time. Conclusion: With the new recommendations to dilute PLA powder and to inject it into the deeper subcutaneous tissue nodule formation is a minor problem. However, good treatment results can only be achieved if lipoatrophy is not too intense; treatment intervals should be about 2 - 3 weeks. Copyright (C) 2005 S. Karger AG, Basel

Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

We address the issue of mobility of localized modes in two-dimensional nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes e.g. discrete spatial solitons in a tight-binding approximation of two-dimensional optical waveguide arrays made from photorefractive crystals. We discuss numerically obtained exact stationary solutions and their stability, focussing on three different solution families with peaks at one, two, and four neighboring sites, respectively. When varying the power, there is a repeated exchange of stability between these three solutions, with symmetry-broken families of connecting intermediate stationary solutions appearing at the bifurcation points. When the nonlinearity parameter is not too large, we observe good mobility, and a well defined Peierls-Nabarro barrier measuring the minimum energy necessary for rendering a stable stationary solution mobile.Comment: 19 pages, 4 figure

A complete classification of spherically symmetric perfect fluid similarity solutions

We classify all spherically symmetric perfect fluid solutions of Einstein's equations with equation of state p/mu=a which are self-similar in the sense that all dimensionless variables depend only upon z=r/t. For a given value of a, such solutions are described by two parameters and they can be classified in terms of their behaviour at large and small distances from the origin; this usually corresponds to large and small values of z but (due to a coordinate anomaly) it may also correspond to finite z. We base our analysis on the demonstration that all similarity solutions must be asymptotic to solutions which depend on either powers of z or powers of lnz. We show that there are only three similarity solutions which have an exact power-law dependence on z: the flat Friedmann solution, a static solution and a Kantowski-Sachs solution (although the latter is probably only physical for a1/5, there are also two families of solutions which are asymptotically (but not exactly) Minkowski: the first is asymptotically Minkowski as z tends to infinity and is described by one parameter; the second is asymptotically Minkowski at a finite value of z and is described by two parameters. A complete analysis of the dust solutions is given, since these can be written down explicitly and elucidate the link between the z>0 and z<0 solutions. Solutions with pressure are then discussed in detail; these share many of the characteristics of the dust solutions but they also exhibit new features.Comment: 63 pages. To appear in Physical Review

Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media

We review the latest progress and properties of the families of bright and dark one-dimensional periodic waves propagating in saturable Kerr-type and quadratic nonlinear media. We show how saturation of the nonlinear response results in appearance of stability (instability) bands in focusing (defocusing) medium, which is in sharp contrast with the properties of periodic waves in Kerr media. One of the key results discovered is the stabilization of multicolor periodic waves in quadratic media. In particular, dark-type waves are shown to be metastable, while bright-type waves are completely stable in a broad range of energy flows and material parameters. This yields the first known example of completely stable periodic wave patterns propagating in conservative uniform media supporting bright solitons. Such results open the way to the experimental observation of the corresponding self-sustained periodic wave patterns.Comment: 29 pages, 10 figure

Tunable tunneling: An application of stationary states of Bose-Einstein condensates in traps of finite depth

The fundamental question of how Bose-Einstein condensates tunnel into a barrier is addressed. The cubic nonlinear Schrodinger equation with a finite square well potential, which models a Bose-Einstein condensate in a quasi-one-dimensional trap of finite depth, is solved for the complete set of localized and partially localized stationary states, which the former evolve into when the nonlinearity is increased. An immediate application of these different solution types is tunable tunneling. Magnetically tunable Feshbach resonances can change the scattering length of certain Bose-condensed atoms, such as $^{85}$Rb, by several orders of magnitude, including the sign, and thereby also change the mean field nonlinearity term of the equation and the tunneling of the wavefunction. We find both linear-type localized solutions and uniquely nonlinear partially localized solutions where the tails of the wavefunction become nonzero at infinity when the nonlinearity increases. The tunneling of the wavefunction into the non-classical regime and thus its localization therefore becomes an external experimentally controllable parameter.Comment: 11 pages, 5 figure