Mathematical modelling of the refractive index reconstruction through global optimization

We examine a mathematical model of non-destructive testing of planar waveguides, based on numerical solution of a nonlinear integral equation. Such problem is ill-posed, and the method of Tikhonov regularization is applied. To minimize Tikhonov functional, and find the parameters of the waveguide, we use two new optimization methods: the cutting angle method of global optimization, and the discrete gradient method of nonsmooth local optimization. We examine how the noise in the experimental data influences the solution, and how the regularization parameter has to be chosen. We show that even with significant noise in the data, the numerical solution is of high accuracy, and the method can be used to process real experimental da.ta..

Developing formal mathematical assessment for 4-to-8-year-olds

The assessment of children in their years before school and their first years of school has been, traditionally, informal. Further, assessment of children's mathematical skills at this level has been infrequent compared to social, emotional and physical assessments. However, there are contexts where reliable, valid, standardised data from assessment in mathematics are required. This paper outlines the development of two assessment tools for mathematics that were originally developed for such contexts. Item Response Theory (IRT) analyses enabled the construction of assessment forms that address the range of abilities of 4- to 8-year-old children, and provided the scales used for constructing formative and summative reports of achievement. A description of the development of the assessment tools and the IRT analysis that provides the reporting formats are presented together with some research uses of the tools.

Rural parents, teenagers and alcohol: what are parents thinking?

INTRODUCTION: The fundamental role of alcohol in the lives of young Australians is mirrored in the level of drinking by adolescents. In 2001, more than one in four Australian adolescents aged 14-19 years consumed alcohol weekly. Teenagers in rural areas are more likely to consume alcohol than their metropolitan counterparts. Parents are key 'gatekeepers' of adolescent behaviour and as such are a salient group to consider in relation to adolescent alcohol use. The aim of this study was to explore parents' attitudes, beliefs, concerns and receptiveness to harm minimisation strategies with respect to teenage use of alcohol.

METHODS: A convenience sample of parents with adolescent children attended a series of focus groups across the north and north-western area of the State of Victoria, Australia. Schools were approached to advertise the project and invite parents to participate. Snowball sampling was used to enhance recruitment.

RESULTS: Parents described patterns of alcohol use such as 'drinking to get drunk' and the influence of both parents and peers on the consumption of alcohol by adolescents. Few parents were concerned about the long-term risks of alcohol use by teenagers; rather they were more concerned about the short-term harms, for example, road trauma and other accidents and risky behaviours such as binge drinking. Parents indicated that they perceived alcohol to be less harmful than other drugs and many indicated that alcohol was often not perceived to be a drug. A number of strategies were adopted by parents to negotiate teenagers' drinking and to minimise the risks associated with alcohol use. These included transporting teenagers to parties, providing teenagers with a mobile phone, setting clear guidelines about alcohol use and/or providing teenagers with a small amount of alcohol. These were seen by parents as strategies for reducing the risks associated with alcohol consumption. Many parents reported that they do not feel well informed about alcohol use and how and when to use harm reduction strategies.

CONCLUSIONS: Rural parents are unsure how to respond to teenagers' alcohol use and drunkenness. While some parental strategies for harm reduction (such as supplying adolescents with a small amount of alcohol) may have good face validity in reducing alcohol consumption among adolescents, these strategies are not supported by previous research findings.

On the EA-style integrated processing of self-contained mathematical texts

A sound and complete first-order goal-oriented sequent-type calculus is developed with ``large-block'' inference rules. In particular, the calculus contains formal analogues of such natural proof-search techniques as handling definitions and applying auxiliary propositions.

Evidence Algorithm and Processing Formalized Mathematical Texts

The goal of a research programme Evidence Algorithm is a development of an open system of automated proving that is able to accumulate mathematical knowledge and to prove theorems in a context of a self-contained mathematical text. By now, the first version of such a system called a System for Automated Deduction, SAD, is implemented in software. The system SAD possesses the following main features: mathematical texts are formalized using a specific formal language that is close to a natural language of mathematical publications; a proof search is based on special sequent-type calculi formalizing natural reasoning style, such as application of definitions and auxiliary propositions. These calculi also admit a separation of equality handling from deduction that gives an opportunity to integrate logical reasoning with symbolic calculation.

The graphic calculator as a thinking tool: perspectives from the classroom

For over a decade the graphic calculator has been promoted not only as a computational tool but also as a thinking tool — for example, as an aid to enhance conceptual understanding, as a problem-solving tool and as a means of enabling students to engage in meaningful investigations. However, research studies focusing on these aspects have shown mixed results and have mostly focused on graphs and functions. This paper reports on one aspect of a case study in a year 10 mathematics classroom — the role of the graphic calculator as a thinking tool. Data from observations of nine statistics lessons and interviews with the teacher and five students are analysed from three perspectives: the teacher’s intentions with respect to the use of the graphic calculator as a tool to promote conceptual understanding as opposed to procedural competence; the opportunities afforded during the lessons for student investigation; and students’ views of how the graphic calculator enhanced conceptual understanding. The results provide insights into ways in which students perceive the graphic calculator as promoting conceptual understanding, as well as some of the difficulties encountered in practice in a classroom.

Thinking multimedia

Children in our society have access to many information resources and communication options. As we witness the convergence of art, literacy and publishing, individuals need to learn how to make sense of information presented in many different forms, and how to construct their own communications in multiple media.
Thinking Multimedia is a program that has developed out of many projects that I have run in several school and some tertiary institutions over the past 12 years. It is an attempt to integrate skills and knowledge from different academic disciplines and to encourage students to understand learning processes and their own learning preferences. The course, offered at this stage at Year 10 level at St Catherine’s School in Melbourne, aims to provide background and basic skills in how to construct and deconstruct information in multiple media and to provide students with the opportunity to explore a ‘real need’ project of their own in a project-based team environment. The course is supported by an online resource and discussion component.
In this presentation I will explain the background to the Thinking Multimedia program and explore some of the work by the students involved.

Hops, steps and jumps : mathematical progress in the early years

While curriculum frameworks ale major influences on learning, teachers know that children progress at different rates. Sometimes this is evident within a particular topic. and at other times more obvious across different topics. In this paper. we present the hops, steps, and jumps of numeracy learning of some 3000 Australian children. All were assessed using I Can Do Maths, and their achievements mapped to provide a detailed picture of how children hop, step and jump on their numeracy journey. This mapping provides teachers with infonnation about key hurdles to numeracy learning for Australian children.

Mathematical and pedagogical understanding as situated cognition

One mathematics lesson was planned by two Grade 2 teachers together. TheirSeparate teaching of it was videotaped. and each teacher was interviewed before and after her lesson. The "same" lesson resulted in different sets of worthwhile learning outcomes. In this research report. the notion of situated cognition is used as a tool for analysis of how this divergence seemed to happen. It is agued that the teachers' development and uses of mathematical concepts were mediated by the social situations.