Growing a greater understanding of multiplication through lesson study: Mathematics teacher educators’ professional development

Research cites the need for developing teachers’ mathematical knowledge for teaching (MKT) as well as for developing mathematics teacher educators’ (MTEs) mathematical knowledge for teaching teachers (MKTT). Using the framework of lesson study: formulating goals and researching, planning, implementing and observing, and reflecting (Lewis & Hurd, 2011), a group of MTEs designed and analyzed a lesson on multiplication for prospective elementary teachers. A qualitative analysis of MTE journal reflections and prospective teacher work showed a greater understanding of MTEs’ MKTT related to multiplication after completion of the lesson study. The authors recommend MTEs conduct lesson studies for other mathematics topics to further understand what MKTT MTEs need to develop to best support prospective teachers’ MKT

Similarity classes of 3x3 matrices over a local principal ideal ring

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.Comment: 14 pages, final version, to appear in Communications in Algebr

Understanding the small object argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.Comment: 42 pages; supersedes the earlier arXiv preprint math/0702290; v2: final journal version, minor corrections onl

Metastability in Two Dimensions and the Effective Potential

We study analytically and numerically the decay of a metastable phase in (2+1)-dimensional classical scalar field theory coupled to a heat bath, which is equivalent to two-dimensional Euclidean quantum field theory at zero temperature. By a numerical simulation we obtain the nucleation barrier as a function of the parameters of the potential, and compare it to the theoretical prediction from the bounce (critical bubble) calculation. We find the nucleation barrier to be accurately predicted by theory using the bounce configuration obtained from the tree-level (``classical'') effective action. Within the range of parameters probed, we found that using the bounce derived from the one-loop effective action requires an unnaturally large prefactor to match the lattice results. Deviations from the tree-level prediction are seen in the regime where loop corrections would be expected to become important.Comment: 13pp, LaTex with Postscript figs, CLNS 93/1202, DART-HEP-93/0

Expanding Bubbles in a Thermal Background

Real scalar field models incorporating asymmetric double well potentials will decay to the state of lowest energy. While the eventual nature of the system can be discerned, the determination of the dynamics of the bubble wall provides many difficulties. In the present study we investigate numerically the evolution of spherically symmetric expanding bubbles coupled to a thermal bath in 3+1 dimensions. A Markovian Langevin equation is employed to describe the interaction between bubble and bath. We find the shape and velocity of the wall to be independent of temperature, yet extremely sensitive to both asymmetry and viscosity.Comment: RevTeX, 9 pages (multicols), 9 figures, submitted to Phys. Rev.

Radiological Protection in Paediatric Diagnostic and Interventional Radiology

Paediatric patients have a higher average risk of developing cancer compared with adults receiving the same dose. The longer life expectancy in children allows more time for any harmful effects of radiation to manifest, and developing organs and tissues are more sensitive to the effects of radiation. This publication aims to provide guiding principles of radiological protection for referring clinicians and clinical staff performing diagnostic imaging and interventional procedures for paediatric patients. It begins with a brief description of the basic concepts of radiological protection, followed by the general aspects of radiological protection, including principles of justification and optimisation. Guidelines and suggestions for radiological protection in specific modalities - radiography and fluoroscopy, interventional radiology, and computed tomography - are subsequently covered in depth. The report concludes with a summary and recommendations. The importance of rigorous justification of radiological procedures is emphasised for every procedure involving ionising radiation, and the use of imaging modalities that are non-ionising should always be considered. The basic aim of optimisation of radiological protection is to adjust imaging parameters and institute protective measures such that the required image is obtained with the lowest possible dose of radiation, and that net benefit is maximised to maintain sufficient quality for diagnostic interpretation. Special consideration should be given to the availability of dose reduction measures when purchasing new imaging equipment for paediatric use. One of the unique aspects of paediatric imaging is with regards to the wide range in patient size (and weight), therefore requiring special attention to optimisation and modification of equipment, technique, and imaging parameters. Examples of good radiographic and fluoroscopic technique include attention to patient positioning, field size and adequate collimation, use of protective shielding, optimisation of exposure factors, use of pulsed fluoroscopy, limiting fluoroscopy time, etc. Major paediatric interventional procedures should be performed by experienced paediatric interventional operators, and a second, specific level of training in radiological protection is desirable (in some countries, this is mandatory). For computed tomography, dose reduction should be optimised by the adjustment of scan parameters (such as mA, kVp, and pitch) according to patient weight or age, region scanned, and study indication (e.g. images with greater noise should be accepted if they are of sufficient diagnostic quality). Other strategies include restricting multiphase examination protocols, avoiding overlapping of scan regions, and only scanning the area in question. Up-to-date dose reduction technology such as tube current modulation, organ-based dose modulation, auto kV technology, and iterative reconstruction should be utilised when appropriate. It is anticipated that this publication will assist institutions in encouraging the standardisation of procedures, and that it may help increase awareness and ultimately improve practices for the benefit of patients