Modeling as a means for making powerful ideas accessible to children at an early age

The SimCalc Vision and Contributions Advances in Mathematics Education 2013, pp 419-436 Modeling as a Means for Making Powerful Ideas Accessible to Children at an Early Age Richard Lesh, Lyn English, Serife Sevis, Chanda Riggs … show all 4 hide » Look Inside » Get Access Abstract In modern societies in the 21st century, significant changes have been occurring in the kinds of “mathematical thinking” that are needed outside of school. Even in the case of primary school children (grades K-2), children not only encounter situations where numbers refer to sets of discrete objects that can be counted. Numbers also are used to describe situations that involve continuous quantities (inches, feet, pounds, etc.), signed quantities, quantities that have both magnitude and direction, locations (coordinates, or ordinal quantities), transformations (actions), accumulating quantities, continually changing quantities, and other kinds of mathematical objects. Furthermore, if we ask, what kind of situations can children use numbers to describe? rather than restricting attention to situations where children should be able to calculate correctly, then this study shows that average ability children in grades K-2 are (and need to be) able to productively mathematize situations that involve far more than simple counts. Similarly, whereas nearly the entire K-16 mathematics curriculum is restricted to situations that can be mathematized using a single input-output rule going in one direction, even the lives of primary school children are filled with situations that involve several interacting actions—and which involve feedback loops, second-order effects, and issues such as maximization, minimization, or stabilizations (which, many years ago, needed to be postponed until students had been introduced to calculus). …This brief paper demonstrates that, if children’s stories are used to introduce simulations of “real life” problem solving situations, then average ability primary school children are quite capable of dealing productively with 60-minute problems that involve (a) many kinds of quantities in addition to “counts,” (b) integrated collections of concepts associated with a variety of textbook topic areas, (c) interactions among several different actors, and (d) issues such as maximization, minimization, and stabilization.

Collective sampling and analysis of high order tensors for chatroom communications

This work investigates the accuracy and efficiency tradeoffs between centralized and collective (distributed) algorithms for (i) sampling, and (ii) n-way data analysis techniques in multidimensional stream data, such as Internet chatroom communications. Its contributions are threefold. First, we use the Kolmogorov-Smirnov goodness-of-fit test to show that statistical differences between real data obtained by collective sampling in time dimension from multiple servers and that of obtained from a single server are insignificant. Second, we show using the real data that collective data analysis of 3-way data arrays (users x keywords x time) known as high order tensors is more efficient than centralized algorithms with respect to both space and computational cost. Furthermore, we show that this gain is obtained without loss of accuracy. Third, we examine the sensitivity of collective constructions and analysis of high order data tensors to the choice of server selection and sampling window size. We construct 4-way tensors (users x keywords x time x servers) and analyze them to show the impact of server and window size selections on the results.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A framework for a dense matching algorithm : Results using matching metrics and order statistic filters

An algorithm for computing dense correspondences between images of a stereo pair or image sequence is presented. The algorithm can make use of both standard matching metrics and the rank and census filters, two filters based on order statistics which have been applied to the image matching problem. Their advantages include robustness to radiometric distortion and amenability to hardware implementation. Results obtained using both real stereo pairs and a synthetic stereo pair with ground truth were compared. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.

Order statistic filters for image matching

The rank and census are two filters based on order statistics which have been applied to the image matching problem for stereo pairs. Advantages of these filters include their robustness to radiometric distortion and small amounts of random noise, and their amenability to hardware implementation. In this paper, a new matching algorithm is presented, which provides an overall framework for matching, and is used to compare the rank and census techniques with standard matching metrics. The algorithm was tested using both real stereo pairs and a synthetic pair with ground truth. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.

Supreme Court Act 1995 (Qld) s 253 – whether refusal to stay execution of costs is an order “as to costs only”

In Virgtel Ltd v Zabusky [2009] QCA 92 the Queensland Court of Appeal considered the scope of an order “as to costs only” within the meaning of s 253 of the Supreme Court Act 1995 (Qld) (‘the Act”). The Court also declined to accept submissions from one of the parties after oral hearing, and made some useful comments which serve as a reminder to practitioners of their obligations in that regard.

Disclosure obligations on ex parte application for freezing order - duty of practitioners to the court - failure to comply with duty - serious dereliction of duty to court - supplemental order for costs against solicitors personally

The decision of Applegarth J in Heartwood Architectural & Joinery Pty Ltd v Redchip Lawyers [2009] QSC 195 (27 July 2009) involved a costs order against solicitors personally. This decision is but one of several recent decisions in which the court has been persuaded that the circumstances justified costs orders against legal practitioners on the indemnity basis. These decisions serve as a reminder to practitioners of their disclosure obligations when seeking any interlocutory relief in an ex parte application. These obligations are now clearly set out in r 14.4 of the Legal Profession (Solicitors) Rule 2007 and r 25 of 2007 Barristers Rule. Inexperience or ignorance will not excuse breaches of the duties owed to the court.

Analysis of higher-order aberrations in a large clinical population

Purpose: To use a large wavefront database of a clinical population to investigate relationships between refractions and higher order aberrations and between aberrations of right and left eyes. Methods: Third and fourth-order aberration coefficients and higher-order root-mean-squared aberrations (HO RMS), scaled to a pupil size of 4.5 mm diameter, were analysed in a population of about 24,000 patients from Carl Zeiss Vision's European wavefront database. Correlations were determined between the aberrations and the variables of refraction, near addition and cylinder. Results: Most aberration coefficients were significantly dependent upon these variables, but the proportions of aberrations that could be explained by these factors were less than 2% except for spherical aberration (12%), horizontal coma (9%) and HO RMS (7%). Near addition was the major contributor for horizontal coma (8.5% out of 9.5%) and spherical equivalent was the major contributor for spherical aberration (7.7% out of 11.6%). Interocular correlations were highly significant for all aberration coefficients, varying between 0.16 and 0.81. Anisometropia was a variable of significance for three aberrations (vertical coma, secondary astigmatism and tetrafoil), but little importance can be placed on this because of the small proportions of aberrations that can be explained by refraction (all less than 1.0 %). Conclusions: Most third- and fourth-order aberration coefficients were significantly dependent upon spherical equivalent, near addition and cylinder, but only horizontal coma (9%) and spherical aberration (12%) showed dependencies of greater than 2%. Interocular correlations were highly significant for all aberration coefficients, but anisometropia had little influence on aberration coefficients.

Critical discourse analysis and political economy of communication : understanding the new corporate order

This article uses critical discourse analysis to analyse material shifts in the political economy of communications. It examines texts of major corporations to describe four key changes in political economy: (1) the separation of ownership from control; (2) the separation of business from industry; (3) the separation of accountability from responsibility; and (4) the subjugation of ‘going concerns’ by overriding concerns. The authors argue that this amounts to a political economic shift from traditional concepts of ‘capitalism’ to a new ‘corporatism’ in which the relationships between public and private, state and individual interests have become redefined and obscured through new discourse strategies. They conclude that the present financial and regulatory ‘crisis’ cannot be adequately resolved without a new analytic framework for examining the relationships between corporation, discourse and political economy.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

A new first-order decision-making model for the procurement of public sector infrastructure: Procedures and Testing

Given global demand for new infrastructure, governments face substantial challenges in funding new infrastructure and delivering Value for Money (VfM). As part of the background to this challenge, a critique is given of current practice in the selection of the approach to procure major public sector infrastructure in Australia and which is akin to the Multi-Attribute Utility Approach (MAUA). To contribute towards addressing the key weaknesses of MAUA, a new first-order procurement decision-making model is presented. The model addresses the make-or-buy decision (risk allocation); the bundling decision (property rights incentives), as well as the exchange relationship decision (relational to arms-length exchange) in its novel approach to articulating a procurement strategy designed to yield superior VfM across the whole life of the asset. The aim of this paper is report on the development of this decisionmaking model in terms of the procedural tasks to be followed and the method being used to test the model. The planned approach to testing the model uses a sample of 87 Australian major infrastructure projects in the sum of AUD32 billion and deploys a key proxy for VfM comprising expressions of interest, as an indicator of competition.

Higher order thinking in chemistry curriculum and its assessment

Higher-order thinking has featured persistently in the reform agenda for science education. The intended curriculum in various countries sets out aspirational statements for the levels of higher-order thinking to be attained by students. This study reports the extent to which chemistry examinations from four Australian states align and facilitate the intended higher-order thinking skills stipulated in curriculum documents. Through content analysis, the curriculum goals were identified for each state and compared to the nature of question items in the corresponding examinations. Categories of higher-order thinking were adapted from the OECD’s PISA Science test to analyze question items. There was considerable variation in the extent to which the examinations from the states supported the curriculum intent of developing and assessing higher-order thinking. Generally, examinations that used a marks-based system tended to emphasize lower-order thinking, with a greater distribution of marks allocated for lower-order thinking questions. Examinations associated with a criterion-referenced examination tended to award greater credit for higher-order thinking questions. The level of complexity of chemistry was another factor that limited the extent to which examination questions supported higher-order thinking. Implications from these findings are drawn for the authorities responsible for designing curriculum and assessment procedures and for teachers.

Robust image hash functions using higher order spectra

Robust hashing is an emerging field that can be used to hash certain data types in applications unsuitable for traditional cryptographic hashing methods. Traditional hashing functions have been used extensively for data/message integrity, data/message authentication, efficient file identification and password verification. These applications are possible because the hashing process is compressive, allowing for efficient comparisons in the hash domain but non-invertible meaning hashes can be used without revealing the original data. These techniques were developed with deterministic (non-changing) inputs such as files and passwords. For such data types a 1-bit or one character change can be significant, as a result the hashing process is sensitive to any change in the input. Unfortunately, there are certain applications where input data are not perfectly deterministic and minor changes cannot be avoided. Digital images and biometric features are two types of data where such changes exist but do not alter the meaning or appearance of the input. For such data types cryptographic hash functions cannot be usefully applied. In light of this, robust hashing has been developed as an alternative to cryptographic hashing and is designed to be robust to minor changes in the input. Although similar in name, robust hashing is fundamentally different from cryptographic hashing. Current robust hashing techniques are not based on cryptographic methods, but instead on pattern recognition techniques. Modern robust hashing algorithms consist of feature extraction followed by a randomization stage that introduces non-invertibility and compression, followed by quantization and binary encoding to produce a binary hash output. In order to preserve robustness of the extracted features, most randomization methods are linear and this is detrimental to the security aspects required of hash functions. Furthermore, the quantization and encoding stages used to binarize real-valued features requires the learning of appropriate quantization thresholds. How these thresholds are learnt has an important effect on hashing accuracy and the mere presence of such thresholds are a source of information leakage that can reduce hashing security. This dissertation outlines a systematic investigation of the quantization and encoding stages of robust hash functions. While existing literature has focused on the importance of quantization scheme, this research is the first to emphasise the importance of the quantizer training on both hashing accuracy and hashing security. The quantizer training process is presented in a statistical framework which allows a theoretical analysis of the effects of quantizer training on hashing performance. This is experimentally verified using a number of baseline robust image hashing algorithms over a large database of real world images. This dissertation also proposes a new randomization method for robust image hashing based on Higher Order Spectra (HOS) and Radon projections. The method is non-linear and this is an essential requirement for non-invertibility. The method is also designed to produce features more suited for quantization and encoding. The system can operate without the need for quantizer training, is more easily encoded and displays improved hashing performance when compared to existing robust image hashing algorithms. The dissertation also shows how the HOS method can be adapted to work with biometric features obtained from 2D and 3D face images.