Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Perception of affordances in multi-scale dynamics as an alternative explanation for equivalence of analogical and inferential reasoning in animals and humans

Recent claims of equivalence of animal and human reasoning are evaluated and a study of avian cognition serves as an exemplar of weaknesses in these arguments. It is argued that current research into neurobiological cognition lacks theoretical breadth to substantiate comparative analyses of cognitive function. Evaluation of a greater range of theoretical explanations is needed to verify claims of equivalence in animal and human cognition. We conclude by exemplifying how the notion of affordances in multi-scale dynamics can capture behavior attributed to processes of analogical and inferential reasoning in animals and humans.

Systematic variations in polymer gel dosimeter calibration due to container influence and deviations from water equivalence

There are a number of gel dosimeter calibration methods in contemporary usage. The present study is a detailed Monte Carlo investigation into the accuracy of several calibration techniques. Results show that for most arrangements the dose to gel accurately reflects the dose to water, with the most accurate method involving the use of a large diameter flask of gel into which multiple small fields of varying dose are directed. The least accurate method was found to be that of a long test tube in a water phantom, coaxial with the beam. The large flask method is also the most straightforward and least likely to introduce errors during setup, though, to its detriment, the volume of gel required is much more than other methods.

A comparative study of the effect of calibration conditions on the water equivalence of a range of gel dosimeters

Gel dosimeters are of increasing interest in the field of radiation oncology as the only truly three-dimensional integrating radiation dosimeter. There are a range of ferrous-sulphate and polymer gel dosimeters. To be of use, they must be water-equivalent. On their own, this relates to their radiological properties as determined by their composition. In the context of calibration of gel dosimeters, there is the added complexity of the calibration geometry; the presence of containment vessels may influence the dose absorbed. Five such methods of calibration are modelled here using the Monte Carlo method. It is found that the Fricke gel best matches water for most of the calibration methods, and that the best calibration method involves the use of a large tub into which multiple fields of different dose are directed. The least accurate calibration method involves the use of a long test tube along which a depth dose curve yields multiple calibration points.

A comparative study of the effect of calibration conditions on the water equivalence of a range of gel dosimeters

Gel dosimeters are of increasing interest in the field of radiation oncology as the only truly three-dimensional integrating radiation dosimeter. There are a range of ferrous-sulphate and polymer gel dosimeters. To be of use, they must be water-equivalent. On their own, this relates to their radiological properties as determined by their composition. In the context of calibration of gel dosimeters, there is the added complexity of the calibration geometry; the presence of containment vessels may influence the dose absorbed. Five such methods of calibration are modelled here using the Monte Carlo method. It is found that the Fricke gel best matches water for most of the calibration methods, and that the best calibration method involves the use of a large tub into which multiple fields of different dose are directed. The least accurate calibration method involves the use of a long test tube along which a depth dose curve yields multiple calibration points.

Offsetting methane emissions : an alternative to emission equivalence metrics

It is widely recognised that defining trade-offs between greenhouse gas emissions using ‘emission equivalence’ based on global warming potentials (GWPs) referenced to carbon dioxide produces anomalous results when applied to methane. The short atmospheric lifetime of methane, compared to the timescales of CO2 uptake, leads to the greenhouse warming depending strongly on the temporal pattern of emission substitution. We argue that a more appropriate way to consider the relationship between the warming effects of methane and carbon dioxide is to define a ‘mixed metric’ that compares ongoing methane emissions (or reductions) to one-off emissions (or reductions) of carbon dioxide. Quantifying this approach, we propose that a one-off sequestration of 1 t of carbon would offset an ongoing methane emission in the range 0.90–1.05 kg CH4 per year. We present an example of how our approach would apply to rangeland cattle production, and consider the broader context of mitigation of climate change, noting the reverse trade-off would raise significant challenges in managing the risk of non-compliance. Our analysis is consistent with other approaches to addressing the criticisms of GWP-based emission equivalence, but provides a simpler and more robust approach while still achieving close equivalence of climate mitigation outcomes ranging over decadal to multi-century timescales.

Dispositional resistance to change : measurement equivalence and the link to personal values across 17 nations

The concept of dispositional resistance to change has been introduced in a series of exploratory and confirmatory analyses through which the validity of the Resistance to Change (RTC) Scale has been established (S. Oreg, 2003). However, the vast majority of participants with whom the scale was validated were from the United States. The purpose of the present work was to examine the meaningfulness of the construct and the validity of the scale across nations. Measurement equivalence analyses of data from 17 countries, representing 13 languages and 4 continents, confirmed the cross-national validity of the scale. Equivalent patterns of relationships between personal values and RTC across samples extend the nomological net of the construct and provide further evidence that dispositional resistance to change holds equivalent meanings across nations.

Measurement equivalence of the dispositional resistance to change scale

Intended to bridge the gap between the latest methodological developments and cross-cultural research, this interdisciplinary resource presents the latest strategies for analyzing cross-cultural data. Techniques are demonstrated through the use of applications that employ cross national data sets such as the latest European Social Survey. With an emphasis on the generalized latent variable approach, internationally?prominent researchers from a variety of fields explain how the methods work, how to apply them, and how they relate to other methods presented in the book. Syntax and graphical and verbal explanations of the techniques are included. [from publisher's website]

New light on the optical equivalence theorem and a new type of discrete diagonal coherent state representation

In many instances we find it advantageous to display a quantum optical density matrix as a generalized statistical ensemble of coherent wave fields. The weight functions involved in these constructions turn out to belong to a family of distributions, not always smooth functions. In this paper we investigate this question anew and show how it is related to the problem of expanding an arbitrary state in terms of an overcomplete subfamily of the overcomplete set of coherent states. This provides a relatively transparent derivation of the optical equivalence theorem. An interesting by-product is the discovery of a new class of discrete diagonal representations.

Equivalence conditions for classes of linear and non-linear distributed parameter systems

Equivalence of certain classes of second-order non-linear distributed parameter systems and corresponding linear third-order systems is established through a differential transformation technique. As linear systems are amenable to analysis through existing techniques, this study is expected to offer a method of tackling certain classes of non-linear problems which may otherwise prove to be formidable in nature.

A note on the equivalence of shock manifold equations

The shock manifold equation is a first order nonlinear partial differential equation, which describes the kinematics of a shockfront in an ideal gas with constant specific heats. However, it was found that there was more than one of these shock manifold equations, and the shock surface could be embedded in a one parameter family of surfaces, obtained as a solution of any of these shock manifold equations. Associated with each shock manifold equation is a set of characteristic curves called lsquoshock raysrsquo. This paper investigates the nature of various associated shock ray equations.