Tools or crutches? Apparatus as a sense-making aid in mathematics teaching with children with moderate learning difficulties

This paper challenges a view of concrete materials as abstracts used within a rigid instructional sequence that particular children are perceived to require or not, as the case may be. Focussing on mathematics teaching, it contends that it is more useful to consider the function of these materials as tools, artefacts used flexibly and selectively by pupils to make sense of mathematics, rather than as crutches, devices which may support procedural competency in mathematics but with no guarantees of understanding

High-Resolution Simulations of Cosmic Microwave Background non-Gaussian Maps in Spherical Coordinates

We describe a new numerical algorithm to obtain high-resolution simulated maps of the Cosmic Microwave Background (CMB), for a broad class of non-Gaussian models. The kind of non-Gaussianity we account for is based on the simple idea that the primordial gravitational potential is obtained by a non-linear but local mapping from an underlying Gaussian random field, as resulting from a variety of inflationary models. Our technique, which is based on a direct realization of the potential in spherical coordinates and fully accounts for the radiation transfer function, allows to simulate non-Gaussian CMB maps down to the Planck resolution ($\ell_{\rm max} \sim 3,000$), with reasonable memory storage and computational time.Comment: 9 pages, 5 figures. Submitted to ApJ. A version with higher quality figures is available at http://www.pd.infn.it/~liguori/content.htm

'I like it instead of maths' : how pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems

This study shows how a group of 24 children in three Scottish primary schools for pupils with moderate learning difficulties responded to word problems following their teachers' introduction to the principles of Cognitively Guided Instruction (CGI). CGI is a professional development programme in mathematics instruction based on constructivist principles developed at the University of Wisconsin-Madison. The study found that the sample group of pupils were able to develop their understanding of mathematical concepts through actively engaging in word problems without prior explicit instruction and with minimal teacher adjustments. The pupils' conceptual understandings demonstrated by their solution strategies within CGI activities were generally not consistent with classroom records of assessment. The results were encouraging in illustrating the capacity of the sample group of pupils with moderate learning difficulties to reveal their mathematical thinking and considers the importance of this insight for instructional decision making

The jeely nursery, letting the children lead: final report to the robertson trust

This is the final report written at the conclusion of a three year Robertson Trust funded project at the Jeely Nursery in Castlemilk, Glasgow, 2007 to 2010. The project purpose was to meet the particular needs of children vulnerable to highly adverse social and economic circumstances, including those living with parental addiction. The aim was to develop a collaborative strategy which would, by involving children, nursery staff and parents together, help to build enduring resources for the emotional resilience needed by children to overcome adversity and improve their chances of achieving educational success. The well validated premise underpinning the child-led pedagogy, Special Playtime, is that early negative attachment experiences can be transformed through direct positive experience with trained staff. The report examines the project using a three dimensional conceptual framework located in the literature on attachment, resilience and child-led pedagogy and focuses on the manner in which the several and differing relationships within the project interacted with and sustained each other

Kurtosis and Large--Scale Structure

We discuss the non--linear growth of the excess kurtosis parameter of the smoothed density fluctuation field $\delta$, S_4\equiv[\lan\delta^{\,4}\ran-3\lan\delta^{\,2}\ran^2]/ \lan\delta^{\,2}\ran^3 in an Einstein--de Sitter universe. We assume Gaussian primordial density fluctuations with scale--free power spectrum $P(k)\propto k^{\,n}$ and analyze the dependence of $S_4$ on primordial spectral index $n$, after smoothing with a Gaussian filter. As already known for the skewness ratio $S_3$, the kurtosis parameter is a {\it decreasing function} of $n$, both in exact perturbative theory and in the Zel'dovich approximation. The parameter $S_4$ provides a powerful statistics to test different cosmological scenarios.Comment: 11 pages in Latex (plus 1 figure), SISSA 127/93/

Lensing dispersion of supernova flux: a probe of nonlinear structure growth

The scatter in the apparent magnitude of type Ia supernovae induced by stochastic gravitational lensing is highly dependent on the nonlinear growth of cosmological structure. In this paper, we show that such a dependence can potentially be employed to gain significant information about the mass clustering at small scales. While the mass clustering ultimately hinges on cosmology, here we demonstrate that, upon obtaining more precise observational measurements through future cosmological surveys, the lensing dispersion can very effectively be used to gain information on the poorly understood astrophysical aspects of structure formation, such as the clumpiness of dark matter halos and the importance of gas physics and star formation into shaping the large-scale structure. In order to illustrate this point we verify that even the tentative current measurements of the lensing dispersion performed on the Supernova Legacy Survey sample favor a scenario where virialized structures are somewhat less compact than predicted by $n-$body cosmological simulations. Moreover, we are also able to put lower limits on the slope of the concentration-mass relation. By artificially reducing the statistical observational error we argue that with forthcoming data the stochastic lensing dispersion will allow one to importantly improve constraints on the baryonic physics at work during the assembly of cosmological structure.Comment: 13 pages, 6 figures. Accepted for publication by MNRA

Letting the children lead: the jeely nursery - a first interim report to the robertson trust

This is the first interim report undertaken for the Robertson Trust (RT) of an ongoing project developed by the Jeely Nursery (JN) in Castlemilk Glasgow 2007 – 2010. The Robertson Trust funding has enabled the JN to initiate a radical multi faceted programme of development focussed on children living in highly adverse socioeconomic conditions who may in addition be subject to the negative effects of living in families coping with substance abuse. They represent one of the most vulnerable groups of children in contemporary society. The task ahead for the nursery staff is complex, challenging and long term, requiring a high degree of consistency and commitment to both professional and personal growth. There is no question that the challenge is understood and accepted at all levels; that commitment to the project is well established and that a high level of motivation is sustained in spite of a number of difficult circumstances occurring over the year, unrelated to the project

Constraining Primordial Magnetic Fields with Future Cosmic Shear Surveys

The origin of astrophysical magnetic fields observed in galaxies and clusters of galaxies is still unclear. One possibility is that primordial magnetic fields generated in the early Universe provide seeds that grow through compression and turbulence during structure formation. A cosmological magnetic field present prior to recombination would produce substantial matter clustering at intermediate/small scales, on top of the standard inflationary power spectrum. In this work we study the effect of this alteration on one particular cosmological observable, cosmic shear. We adopt the semi-analytic halo model in order to describe the non-linear clustering of matter, and feed it with the altered mass variance induced by primordial magnetic fields. We find that the convergence power spectrum is, as expected, substantially enhanced at intermediate/small angular scales, with the exact amplitude of the enhancement depending on the magnitude and power-law index of the magnetic field power spectrum. We use the predicted statistical errors for a future wide-field cosmic shear survey, on the model of the ESA Cosmic Vision mission \emph{Euclid}, in order to forecast constraints on the amplitude of primordial magnetic fields as a function of the spectral index. We find that the amplitude will be constrained at the level of $\sim 0.1$ nG for $n_B\sim -3$, and at the level of $\sim 10^{-7}$ nG for $n_B\sim 3$. The latter is at the same level of lower bounds coming from the secondary emission of gamma-ray sources, implying that for high spectral indices \emph{Euclid} will certainly be able to detect primordial magnetic fields, if they exist. The present study shows how large-scale structure surveys can be used for both understanding the origins of astrophysical magnetic fields and shedding new light on the physics of the pre-recombination Universe. (abridged)Comment: 24 pages, 9 figures. To appear on JCA

CoMaLit - II. The scaling relation between mass and Sunyaev-Zel'dovich signal for Planck selected galaxy clusters

We discuss the scaling relation between mass and integrated Compton parameter of a sample of galaxy clusters from the all-sky {\it Planck} Sunyaev-Zel'dovich catalogue. Masses were measured with either weak lensing, caustics techniques, or assuming hydrostatic equilibrium. The retrieved $Y_{500}$-$M_{500}$ relation does not strongly depend on the calibration sample. We found a slope of 1.4-1.9, in agreement with self-similar predictions, with an intrinsic scatter of $20\pm10$ per cent. The absolute calibration of the relation can not be ascertained due to systematic differences of $\sim$20-40 per cent in mass estimates reported by distinct groups. Due to the scatter, the slope of the conditional scaling relation, to be used in cosmological studies of number counts, is shallower, $\sim$1.1-1.6. The regression methods employed account for intrinsic scatter in the mass measurements too. We found that Planck mass estimates suffer from a mass dependent bias.Comment: 14 pages, 7 figures; v2: 17 pages, 11 figures; MNRAS in press, results unchanged; extended discussion of the Planck calibration sample; added discussion of conditional vs symmetric scaling relations and of mixture of Gaussian functions as distribution of the independent variable; products from the CoMaLit series at http://pico.bo.astro.it/~sereno/CoMaLi