Problem solving for the 21st century

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

A problem-posing intervention in the development of problem-solving competence of underachieving, middle-year students

This study reported on the issues surrounding the acquisition of problem-solving competence of middle-year students who had been ascertained as above average in intelligence, but underachieving in problem-solving competence. In particular, it looked at the possible links between problem-posing skills development and improvements in problem-solving competence. A cohort of Year 7 students at a private, non-denominational, co-educational school was chosen as participants for the study, as they undertook a series of problem-posing sessions each week throughout a school term. The lessons were facilitated by the researcher in the students’ school setting. Two criteria were chosen to identify participants for this study. Firstly, each participant scored above the 60th percentile in the standardized Middle Years Ability Test (MYAT) (Australian Council for Educational Research, 2005) and secondly, the participants all scored below the cohort average for Criterion B (Problem-solving Criterion) in their school mathematics tests during the first semester of Year 7. Two mutually exclusive groups of participants were investigated with one constituting the Comparison Group and the other constituting the Intervention Group. The Comparison Group was chosen from a Year 7 cohort for whom no problem-posing intervention had occurred, while the Intervention Group was chosen from the Year 7 cohort of the following year. This second group received the problem-posing intervention in the form of a teaching experiment. That is, the Comparison Group were only pre-tested and post-tested, while the Intervention Group was involved in the teaching experiment and received the pre-testing and post-testing at the same time of the year, but in the following year, when the Comparison Group have moved on to the secondary part of the school. The groups were chosen from consecutive Year 7 cohorts to avoid cross-contamination of the data. A constructionist framework was adopted for this study that allowed the researcher to gain an “authentic understanding” of the changes that occurred in the development of problem-solving competence of the participants in the context of a classroom setting (Richardson, 1999). Qualitative and quantitative data were collected through a combination of methods including researcher observation and journal writing, video taping, student workbooks, informal student interviews, student surveys, and pre-testing and post-testing. This combination of methods was required to increase the validity of the study’s findings through triangulation of the data. The study findings showed that participation in problem-posing activities can facilitate the re-engagement of disengaged, middle-year mathematics students. In addition, participation in these activities can result in improved problem-solving competence and associated developmental learning changes. Some of the changes that were evident as a result of this study included improvements in self-regulation, increased integration of prior knowledge with new knowledge and increased and contextualised socialisation.

Technical note : modelling a complex micro-multileaf collimator using the standard BEAMnrc distribution

Purpose: The component modules in the standard BEAMnrc distribution may appear to be insufficient to model micro-multileaf collimators that have tri-faceted leaf ends and complex leaf profiles. This note indicates, however, that accurate Monte Carlo simulations of radiotherapy beams defined by a complex collimation device can be completed using BEAMnrc's standard VARMLC component module.---------- Methods: That this simple collimator model can produce spatially and dosimetrically accurate micro-collimated fields is illustrated using comparisons with ion chamber and film measurements of the dose deposited by square and irregular fields incident on planar, homogeneous water phantoms.---------- Results: Monte Carlo dose calculations for on- and off-axis fields are shown to produce good agreement with experimental values, even upon close examination of the penumbrae.--------- Conclusions: The use of a VARMLC model of the micro-multileaf collimator, along with a commissioned model of the associated linear accelerator, is therefore recommended as an alternative to the development or use of in-house or third-party component modules for simulating stereotactic radiotherapy and radiosurgery treatments. Simulation parameters for the VARMLC model are provided which should allow other researchers to adapt and use this model to study clinical stereotactic radiotherapy treatments.

Men's intimate partner abuse and control : reconciling paradoxical masculinities and social contradictions

Intimate partner abuse and control is one of the most common forms of violence against women, and is considered an international problem of social, political, legal and human rights significance. Yet few studies have attempted to understand this problem from the perspective of male perpetrators. This gap is addressed by conducting in-depth interviews with 16 able-bodied men of white European ancestry born and educated in New Zealand or Australia, who have been physically violent and/or emotionally, intellectually, sexually or financially controlling of a live-in female partner. This thesis extends and deepens the dominant ways of thinking about men’s intimate partner abuse by utilising a new theoretical framework compatible with contemporary feminist scholarship. A synthesis of Connell’s theory of masculinities and Bourdieu’s field theory is utilised for the purpose of exploring more nuanced, complex understandings of manliness and men’s relationships with men, women and social structures. Through such an analysis, this thesis finds that men’s perpetration of power and control over women is driven by a need to avoid the stigma of appearing weak. As a consequence, their desire and ability to show love, care and empathy is suppressed in favour of a presumed honourable manliness, and their female partners are used as weapons in the pursuit of symbolic capital in the form of recognition, prestige and acceptance from real and/or imagined men. This research also uncovers the complex interplay between masculine practices and particular social contexts. For example, the norms of practice encountered from those in authority, such as teachers, sports coaches, police, court judges and workplace management, influences the decision making of the men in this study, to use, or not to use, physical violence, psychological abuse and structural control. The principal conclusion is that there is a repertoire of paradoxical masculinities and contradictory social messages available to the men in this study. But gender policing by other men, complicit women and those in authority provides little room for legitimate complexity in masculine practices. Perpetrators in this study reconcile these conflicts of interest by generally avoiding subordinated masculinity and possible ostracism, and instead practicing more heroic hegemonic masculinities by abusing and controlling women and particular other men. This thesis concludes that for intimate partner abuse and control to cease, changes in power structures have to occur at all levels of society.

Metaheuristic approaches to the hybrid flow shop scheduling problem with a cost-related criterion

In the paper, the flow-shop scheduling problem with parallel machines at each stage (machine center) is studied. For each job its release and due date as well as a processing time for its each operation are given. The scheduling criterion consists of three parts: the total weighted earliness, the total weighted tardiness and the total weighted waiting time. The criterion takes into account the costs of storing semi-manufactured products in the course of production and ready-made products as well as penalties for not meeting the deadlines stated in the conditions of the contract with customer. To solve the problem, three constructive algorithms and three metaheuristics (based one Tabu Search and Simulated Annealing techniques) are developed and experimentally analyzed. All the proposed algorithms operate on the notion of so-called operation processing order, i.e. the order of operations on each machine. We show that the problem of schedule construction on the base of a given operation processing order can be reduced to the linear programming task. We also propose some approximation algorithm for schedule construction and show the conditions of its optimality.

Short-termism : challenges and resolutions

The global financial crisis has highlighted the vulnerability of our economy to systemic risks. While its causes are numerous and relate to complex problems deeply embedded in capital markets, ‘short-termism’ (excessive focus on short-term outcomes at the expense of long-term wealth creation and sustainability) has frequently been flagged as a major contributor to this crisis. Although short-termism is not new, the global financial crisis has highlighted the presence of short-termism among institutional investors, and the failure of global markets and regulators to deal with such perverse and destructive behaviour (Guyatt 2009). Solutions are clearly needed. Although there is a body of research that provides evidence of the presence of short-termism in capital markets and the consequences of short-term decision-making on the financial wellbeing of both individuals and organisations, there is no consensus on mitigating solutions to short-termism. What emerges from the literature is the need to take a broad interdisciplinary perspective in seeking solutions to the problem.

Interdisciplinarity vs disciplinarity : developing knowledge for problem solving in middle years of schooling

Interdisciplinary studies are fundamental to the signature practices for the middle years of schooling. Middle years researchers claim that interdisciplinarity in teaching appropriately meets the needs of early adolescents by tying concepts together, providing frameworks for the relevance of knowledge, and demonstrating the linking of disparate information for solution of novel problems. Cognitive research is not wholeheartedly supportive of this position. Learning theorists assert that application of knowledge in novel situations for the solution of problems is actually dependent on deep discipline based understandings. The present research contrasts the capabilities of early adolescent students from discipline based and interdisciplinary based curriculum schooling contexts to successfully solve multifaceted real world problems. This will inform the development of effective management of middle years of schooling curriculum.

Service driven asset management

Local governments are service driven rather than asset driven. Understanding this distinction is critical to ensuring that community needs are appropriately addressed. Translating community needs and desires into infrastructure is a complex yet little understood process. In this paper, we look at two case studies that explore the interface between service outcomes and the specification of performance requirements for the assets. The two case studies we look at are: a public health issue resulting from inadequate public amenities in a beach resort and the prioritisation of maintenance work in a world of increasing service demands and declining funding. The case studies all use the same investment logic mapping framework to establish clear drivers as to the problem that councils are responding to in delivering their services. The key to the framework is the separation of concern between service management and asset management.

Complex modelling of open system design for sustainable architecture

This paper argues a model of complex system design for sustainable architecture within a framework of entropy evolution. The spectrum of sustainable architecture consists of the efficient use of energy and material resource in life-cycle of buildings, the active involvement of the occupants in micro-climate control within buildings, and the natural environmental context. The interactions of the parameters compose a complex system of sustainable architectural design, of which the conventional linear and fragmented design technologies are insufficient to indicate holistic and ongoing environmental performance. The complexity theory of dissipative structure states a microscopic formulation of open system evolution, which provides a system design framework for the evolution of building environmental performance towards an optimization of sustainability in architecture.

Philanthropy for Indigenous causes : more than a 'cup of tea'?

In the face of improved First Nation outcomes in many western nations, Australia is still dealing with a seemingly intractable gap between the quality of life of its Aboriginal and Torres Strait Islander peoples and the non-Aboriginal and Torres Strait Islander population. Philanthropy in Australia provides a smaller proportion of funding for community projects than is the case in other countries and Aboriginal and Torres Strait Islander causes have been significantly under - represented as recipients. This paper reports on a qualitative study aimed at understanding the issues affecting the decisions and actions of grantmaking organisations and individuals who wish to support Aboriginal and Torres Strait Islander causes in the current Australian context. The aims were to build on the limited research in this arena, add to the future research agenda and contribute to practice and policy insights for Australia and beyond. The study found that while government funding programs are perceived as output driven, inflexible and dogmatic - ‘a cup of tea mob’- participants see the Australian philanthropic sector as capable of addressing the complex Aboriginal and Torres Strait Islander ‘problem’ with more innovative and independent thinking. From the point of view of contextual impacts, success criteria, barriers, structural imposts and emotional involvement, the practical experience in grantmaking for Indigenous causes of participants in this study reflects that found elsewhere. However the focus of many grantmakers on organisational rather than community capacity and the potentially elitist emphasis on established relationships continues to hamper Aboriginal and Torres Strait Islander access to philanthropic funding in Australia. Further, if the strategic changes currently visible in the sector are unsupported by a depth of policy and a proactive transfer and distribution of skill and knowledge they may be unsustainable.

Building a successful partnership to increase affordable rental housing supply in Queensland

Accessibility to housing for low to moderate income groups in Australia has been experiencing a severe decline since 2001. On the supply side, the public sector has been reducing its commitment to the direct provision of public housing. Despite high demand for affordable housing, there has been limited supply generated by non-government housing providers. One possible solution to promote an increase in affordable housing supply, like other infrastructure, is through the development of multi-stakeholder partnerships and private financing. This research aims to identify current issues underlying decision-making criteria for building multi-stakeholder partnerships to deliver affordable housing projects. It also investigates strategies for minimising risk and ensuring the financial outcomes of these partnership arrangements. A mix of qualitative in-depth interviews and quantitative surveys has been used as the main method to explore stakeholder experiences regarding their involvement in partnership arrangements in the affordable housing sector in Queensland. Two sets of interviews were conducted following an exploratory pilot study: one set in 2003-2004 and the other in 2007-2008. There were nineteen respondents representing government, private and not-for-profit organisations in the first stage interviews and surveys. The second stage interviews were focussed on twenty-two housing providers in South East Queensland. Initial analyses have been conducted using thematic and statistical analyses. This study extends the use of existing decision making tools and combines the use of a Soft System Framework to analyse the ideal state questionnaires using qualitative thematic analysis. Soft System Methodology (SSM) has been used to analyse this unstructured complex problem by using systematic thinking to develop a conceptual model and carrying it to the real world situations to solve the problem. This research found that the diversity of stakeholder capability and their level of risk acceptance will allow partnerships to develop the best synergies and a degree of collaboration which achieves the required financial return within acceptable risk parameters. However, some of the negativity attached to future commitment to such partnerships has been found to be the anticipation of a worse outcome than that expected from independent action. Many interviewees agree that housing providers' fear of financial risk and community rejection has been central to dampening their enthusiasm for entering such investment projects. The creation of a mixed-use development structure will mitigate both risk and return as the commercial income will subsidise the affordable housing development and will normalise concentration of marginalised low-income people who live in a prime location with an award winning design. In addition, tenant support schemes and rent-to-buy incentive programs will encourage them to secure their tenancies and significantly reduce the risk of rent arrears and property damage. There is also a breakthrough investment vehicle offered by the social developer which sells the non-physical but financial product to individual and institutional investors to mitigate further financial risk. Finally, this study recommends modification of the current value-for-money framework in favour of broader partnership arrangements which are more closely aligned with risk minimisation strategies.

Forecasting time-varying value-at-risk

In this thesis we are interested in financial risk and the instrument we want to use is Value-at-Risk (VaR). VaR is the maximum loss over a given period of time at a given confidence level. Many definitions of VaR exist and some will be introduced throughout this thesis. There two main ways to measure risk and VaR: through volatility and through percentiles. Large volatility in financial returns implies greater probability of large losses, but also larger probability of large profits. Percentiles describe tail behaviour. The estimation of VaR is a complex task. It is important to know the main characteristics of financial data to choose the best model. The existing literature is very wide, maybe controversial, but helpful in drawing a picture of the problem. It is commonly recognised that financial data are characterised by heavy tails, time-varying volatility, asymmetric response to bad and good news, and skewness. Ignoring any of these features can lead to underestimating VaR with a possible ultimate consequence being the default of the protagonist (firm, bank or investor). In recent years, skewness has attracted special attention. An open problem is the detection and modelling of time-varying skewness. Is skewness constant or there is some significant variability which in turn can affect the estimation of VaR? This thesis aims to answer this question and to open the way to a new approach to model simultaneously time-varying volatility (conditional variance) and skewness. The new tools are modifications of the Generalised Lambda Distributions (GLDs). They are four-parameter distributions, which allow the first four moments to be modelled nearly independently: in particular we are interested in what we will call para-moments, i.e., mean, variance, skewness and kurtosis. The GLDs will be used in two different ways. Firstly, semi-parametrically, we consider a moving window to estimate the parameters and calculate the percentiles of the GLDs. Secondly, parametrically, we attempt to extend the GLDs to include time-varying dependence in the parameters. We used the local linear regression to estimate semi-parametrically conditional mean and conditional variance. The method is not efficient enough to capture all the dependence structure in the three indices —ASX 200, S&P 500 and FT 30—, however it provides an idea of the DGP underlying the process and helps choosing a good technique to model the data. We find that GLDs suggest that moments up to the fourth order do not always exist, there existence appears to vary over time. This is a very important finding, considering that past papers (see for example Bali et al., 2008; Hashmi and Tay, 2007; Lanne and Pentti, 2007) modelled time-varying skewness, implicitly assuming the existence of the third moment. However, the GLDs suggest that mean, variance, skewness and in general the conditional distribution vary over time, as already suggested by the existing literature. The GLDs give good results in estimating VaR on three real indices, ASX 200, S&P 500 and FT 30, with results very similar to the results provided by historical simulation.

Investigating users’ intuitive interaction with complex artefacts

This paper examines the role of intuition in the way that people operate unfamiliar devices. Intuition is a type of cognitive processing that is often non-conscious and utilises stored experiential knowledge. Intuitive interaction involves the use of knowledge gained from other products and/or experiences. Two initial experimental studies revealed that prior exposure to products employing similar features helped participants to complete set tasks more quickly and intuitively, and that familiar features were intuitively used more often than unfamiliar ones. A third experiment confirmed that performance is affected by a person's level of familiarity with similar technologies, and also revealed that appearance (shape, size and labelling of features) seems to be the variable that most affects time spent on a task and intuitive uses during that time. Age also seems to have an effect. These results and their implications are discussed.

Representation and learning of visual information for pose recognition

Recovering position from sensor information is an important problem in mobile robotics, known as localisation. Localisation requires a map or some other description of the environment to provide the robot with a context to interpret sensor data. The mobile robot system under discussion is using an artificial neural representation of position. Building a geometrical map of the environment with a single camera and artificial neural networks is difficult. Instead it would be simpler to learn position as a function of the visual input. Usually when learning images, an intermediate representation is employed. An appropriate starting point for biologically plausible image representation is the complex cells of the visual cortex, which have invariance properties that appear useful for localisation. The effectiveness for localisation of two different complex cell models are evaluated. Finally the ability of a simple neural network with single shot learning to recognise these representations and localise a robot is examined.