Theories of learning mathematics

According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. So, for healthy theories on development to occur, four Darwinian functions should function: (a) variation – avoid orthodoxy and encourage divergent thinking, (b) selection – submit all assumptions and innovations to rigorous testing, (c) diffusion – encourage the shareability of new and/or viable ways of thinking, and (d) accumulation – encourage the reuseability of viable aspects of productive innovations.

Cloud computing in the upstream oil & gas industry : a proposed way forward

Despite the compelling case for moving towards cloud computing, the upstream oil & gas industry faces several technical challenges—most notably, a pronounced emphasis on data security, a reliance on extremely large data sets, and significant legacy investments in information technology infrastructure—that make a full migration to the public cloud difficult at present. Private and hybrid cloud solutions have consequently emerged within the industry to yield as much benefit from cloud-based technologies as possible while working within these constraints. This paper argues, however, that the move to private and hybrid clouds will very likely prove only to be a temporary stepping stone in the industry's technological evolution. By presenting evidence from other market sectors that have faced similar challenges in their journey to the cloud, we propose that enabling technologies and conditions will probably fall into place in a way that makes the public cloud a far more attractive option for the upstream oil & gas industry in the years ahead. The paper concludes with a discussion about the implications of this projected shift towards the public cloud, and calls for more of the industry's services to be offered through cloud-based “apps.”

Evaluation of the "reconceptualising early mathematics learning" project

The Pattern and Structure Mathematics Awareness Project (PASMAP) has investigated the development of patterning and early algebraic reasoning among 4 to 8 year olds over a series of related studies. We assert that an awareness of mathematical pattern and structure enables mathematical thinking and simple forms of generalisation from an early age. The project aims to promote a strong foundation for mathematical development by focusing on critical, underlying features of mathematics learning. This paper provides an overview of key aspects of the assessment and intervention, and analyses of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in the evaluation throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ assessment and a Pattern and Structure Assessment (PASA) interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a relatively short period of time.

Modeling as a bridge between real world problems and school mathematics

This article argues for an interdisciplinary approach to mathematical problem solving at the elementary school, one that draws upon the engineering domain. A modeling approach, using engineering model eliciting activities, might provide a rich source of meaningful situations that capitalize on and extend students’ existing mathematical learning. The study reports on the developments of 48 twelve-year old students who worked on the Bridge Design activity. Results revealed that young students, even before formal instruction, have the capacity to deal with complex interdisciplinary problems. A number of students created quite appropriate models by developing the necessary mathematical constructs to solve the problem. Students’ difficulties in mathematizing the problem, and in revising and documenting their models are presented and analysed, followed by a discussion on the appropriateness of a modeling approach as a means for introducing complex problems to elementary school students.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

A comparison of Chinese and Australian early childhood mathematics

In the 21st century mathematics proficiency is synonymous with a numerate citizenry. In the past few decades young children’s ability to reason mathematically and develop mathematical proficiencies has been recognised. This paper explores the history of early childhood mathematics (ECME) that may explicate differences in Chinese and Australian contexts. Results of this review established that China and Australia are diametrically positioned in ECME. Influencing each countries philosophies and practices are their cultural beliefs. ECME in China and Australia must be culturally sustainable to achieve excellent outcomes for young children. Ongoing critique and review is necessary to ensure that ECME is meeting the needs of all teachers and children in their particular context. China and Australia with their rich contrasting philosophies can assist each other in their journeys to create exemplary ECME for the 21st century.

Mathematics learning in prep

Most teachers recognise the importance of mathematics teaching and learning in early years but there is not consensus on how and when this learning should occur. Young-Loveridge (cited in de Vries, Thomas, and Warren, 2010) suggests that quality early mathematical experiences are a key determinant to later achievement.

Multi-device key management using visual side channels in pervasive computing environments

In the modern connected world, pervasive computing has become reality. Thanks to the ubiquity of mobile computing devices and emerging cloud-based services, the users permanently stay connected to their data. This introduces a slew of new security challenges, including the problem of multi-device key management and single-sign-on architectures. One solution to this problem is the utilization of secure side-channels for authentication, including the visual channel as vicinity proof. However, existing approaches often assume confidentiality of the visual channel, or provide only insufficient means of mitigating a man-in-the-middle attack. In this work, we introduce QR-Auth, a two-step, 2D barcode based authentication scheme for mobile devices which aims specifically at key management and key sharing across devices in a pervasive environment. It requires minimal user interaction and therefore provides better usability than most existing schemes, without compromising its security. We show how our approach fits in existing authorization delegation and one-time-password generation schemes, and that it is resilient to man-in-the-middle attacks.

Street Computing : Urban Informatics and City Interfaces

This book develops tools and techniques that will help urban residents gain access to urban computing. Metaphorically speaking, it is taking computing to the street by giving the general public – rather than just researchers and professionals – the power to leverage available city infrastructure and create solutions tailored to their individual needs. It brings together five chapters that are based on presentations given at the Street Computing Workshop held on 24 November 2009 in Melbourne in conjunction with the Australian Computer-Human Interaction Conference (OZCHI 2009). This book focuses on applying urban informatics, urban and community sensing and open application programming interfaces (APIs) to the public space through the delivery of online services, on demand and in real time. It then offers a case study of how the city of Singapore has harnessed the potential of an online infrastructure so that residents and visitors can access services electronically. This book was published as a special issue of the Journal of Urban Technology, 19(2), 2012.

Assessing the potential of mathematics textbooks to promote deep learning

Curriculum documents for mathematics emphasise the importance of promoting depth of knowledge rather than shallow coverage of the curriculum. In this paper, we report on a study that explored the analysis of junior secondary mathematics textbooks to assess their potential to assist in teaching and learning aimed at building and applying deep mathematical knowledge. The method of analysis involved the establishment of a set of specific curriculum goals and associated indicators, based on research into the teaching and learning of a particular field within the mathematics curriculum, namely proportion and proportional reasoning. Topic selection was due to its pervasive nature throughout the school mathematics curriculum at this level. As a result of this study, it was found that the five textbook series examined provided limited support for the development of multiplicative structures required for proportional reasoning, and hence would not serve well the development of deep learning of mathematics. The study demonstrated a method that could be applied to the analysis of junior secondary mathematics in many parts of the world.

Learning with FIRST LEGO League

Robotics is a valuable tool for engaging students in the hands-on application of science, technology, engineering, and mathematics (STEM) concepts. Robotics competitions such as FIRST LEGO League (FLL) can increase students’ interest in the STEM subjects and can foster their problem solving and teamwork skills. This paper reports on a study investigating students’ perceptions on the influence of participating in a FLL competition on their learning. The students completed questionnaires regarding their perceptions of their learning during the FLL challenge and were also interviewed to gain a deeper understanding of their questionnaire responses. The results show that the students were engaged with the FLL challenge and held positive views regarding their experience. The results also suggest that students involved with the FLL challenge improved their learning about real-world applications, problem solving, engagement, communication, and the application of the technology/engineering cycle.

Managing memory and reducing I/O cost for correlation matrix calculation in bioinformatics

The generation of a correlation matrix from a large set of long gene sequences is a common requirement in many bioinformatics problems such as phylogenetic analysis. The generation is not only computationally intensive but also requires significant memory resources as, typically, few gene sequences can be simultaneously stored in primary memory. The standard practice in such computation is to use frequent input/output (I/O) operations. Therefore, minimizing the number of these operations will yield much faster run-times. This paper develops an approach for the faster and scalable computing of large-size correlation matrices through the full use of available memory and a reduced number of I/O operations. The approach is scalable in the sense that the same algorithms can be executed on different computing platforms with different amounts of memory and can be applied to different problems with different correlation matrix sizes. The significant performance improvement of the approach over the existing approaches is demonstrated through benchmark examples.

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.