Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Strategies for comparing decimal numbers with the same whole-number part

The strategies employed by 130 Grade 5 Brisbane students in comparing decimal numbers which have the same whole-number part were compared with those identified in similar studies conducted in the USA, France and Israel. Three new strategies were identified. Similar to USA results, the most common comparison errors stemmed from the incorrect whole-number strategy in which length is confused with size. The findings of this present study tend to support Resnick et al.’s (1989) hypothesis that the introduction of decimal-fraction recording before common-fraction recording seems to promote better comparison of decimal numbers.

Three primary school students’ cognition about 3D rotation in a virtual reality learning environment

This paper reports on three primary school students’ explorations of 3D rotation in a virtual reality learning environment (VRLE) named VRMath. When asked to investigate if you would face the same direction when you turn right 45 degrees first then roll up 45 degrees, or when you roll up 45 degrees first then turn right 45 degrees, the students found that the different order of the two turns ended up with different directions in the VRLE. This was contrary to the students’ prior predictions based on using pen, paper and body movements. The findings of this study showed the difficulty young children have in perceiving and understanding the non-commutative nature of 3D rotation and the power of the computational VRLE in giving students experiences that they rarely have in real life with 3D manipulations and 3D mental movements.

New stress and velocity fields for highly frictional granular materials

The idealised theory for the quasi-static flow of granular materials which satisfy the Coulomb-Mohr hypothesis is considered. This theory arises in the limit that the angle of internal friction approaches $\pi/2$, and accordingly these materials may be referred to as being `highly frictional'. In this limit, the stress field for both two-dimensional and axially symmetric flows may be formulated in terms of a single nonlinear second order partial differential equation for the stress angle. To obtain an accompanying velocity field, a flow rule must be employed. Assuming the non-dilatant double-shearing flow rule, a further partial differential equation may be derived in each case, this time for the streamfunction. Using Lie symmetry methods, a complete set of group-invariant solutions is derived for both systems, and through this process new exact solutions are constructed. Only a limited number of exact solutions for gravity driven granular flows are known, so these results are potentially important in many practical applications. The problem of mass flow through a two-dimensional wedge hopper is examined as an illustration.

Perturbation solutions for flow through symmetrical hoppers with inserts and asymmetrical wedge hoppers

Under certain circumstances, an industrial hopper which operates under the "funnel-flow" regime can be converted to the "mass-flow" regime with the addition of a flow-corrective insert. This paper is concerned with calculating granular flow patterns near the outlet of hoppers that incorporate a particular type of insert, the cone-in-cone insert. The flow is considered to be quasi-static, and governed by the Coulomb-Mohr yield condition together with the non-dilatant double-shearing theory. In two dimensions, the hoppers are wedge-shaped, and as such the formulation for the wedge-in-wedge hopper also includes the case of asymmetrical hoppers. A perturbation approach, valid for high angles of internal friction, is used for both two-dimensional and axially symmetric flows, with analytic results possible for both leading order and correction terms. This perturbation scheme is compared with numerical solutions to the governing equations, and is shown to work very well for angles of internal friction in excess of 45 degree.

Optimal fluid injection strategies for in situ mineral leaching in two-dimensions

This paper examines the ground-water flow problem associated with the injection and recovery of certain corrosive fluids into mineral bearing rock. The aim is to dissolve the minerals in situ, and then recover them in solution. In general, it is not possible to recover all the injected fluid, which is of concern economically and environmentally. However, a new strategy is proposed here, that allows all the leaching fluid to be recovered. A mathematical model of the situation is solved approximately using an asymptotic solution, and exactly using a boundary integral approach. Solutions are shown for two-dimensional flow, which is of some practical interest as it is achievable in old mine tunnels, for example.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Young children's understandings about "square" in 3D virtual reality microworlds

This paper reports an investigation of primary school children’s understandings about "square". 12 students participated in a small group teaching experiment session, where they were interviewed and guided to construct a square in a 3D virtual reality learning environment (VRLE). Main findings include mixed levels of "quasi" geometrical understandings, misconceptions about length and angles, and ambiguous uses of geometrical language for location, direction, and movement. These have implications for future teaching and learning about 2D shapes with particular reference to VRLE.

Energy and population policies in Australia

The Australian Government is about to release Australia’s first sustainable population policy. Sustainable population growth, among other things, implies sustainable energy demand. Current modelling of future energy demand both in Australia and by agencies such as the International Energy Agency sees population growth as one of the key drivers of energy demand. Simply increasing the demand for energy in response to population policy is sustainable only if there is a radical restructuring of the energy system away from energy sources associated with environmental degradation towards one more reliant on renewable fuels and less reliant on fossil fuels. Energy policy can also address the present nexus between energy consumption per person and population growth through an aggressive energy efficiency policy. The paper considers the link between population policies and energy policies and considers how the overall goal of sustainability can be achieved. The methods applied in this analysis draw on the literature of sustainable development to develop elements of an energy planning framework to support a sustainable population policy. Rather than simply accept that energy demand is a function of population increase moderated by an assumed rate of energy efficiency improvement, the focus is on considering what rate of energy efficiency improvement is necessary to significantly reduce the standard connections between population growth and growth in energy demand and what policies are necessary to achieve this situation. Energy efficiency policies can only moderate unsustainable aspects of energy demand and other policies are essential to restructure existing energy systems into on-going sustainable forms. Policies to achieve these objectives are considered. This analysis shows that energy policy, population policy and sustainable development policies are closely integrated. Present policy and planning agencies do not reflect this integration and energy and population policies in Australia have largely developed independently and whether the outcome is sustainable is largely a matter of chance. A genuinely sustainable population policy recognises the inter-dependence between population and energy policies and it is essential that this is reflected in integrated policy and planning agencies

Energy and population policies in Australia

The Australian Government is about to release Australia’s first sustainable population policy. Sustainable population growth, among other things, implies sustainable energy demand. Current modelling of future energy demand both in Australia and by agencies such as the International Energy Agency sees population growth as one of the key drivers of energy demand. Simply increasing the demand for energy in response to population policy is sustainable only if there is a radical restructuring of the energy system away from energy sources associated with environmental degradation towards one more reliant on renewable fuels and less reliant on fossil fuels. Energy policy can also address the present nexus between energy consumption per person and population growth through an aggressive energy efficiency policy. The paper considers the link between population policies and energy policies and considers how the overall goal of sustainability can be achieved. The methods applied in this analysis draw on the literature of sustainable development to develop elements of an energy planning framework to support a sustainable population policy. Rather than simply accept that energy demand is a function of population increase moderated by an assumed rate of energy efficiency improvement, the focus is on considering what rate of energy efficiency improvement is necessary to significantly reduce the standard connections between population growth and growth in energy demand and what policies are necessary to achieve this situation. Energy efficiency policies can only moderate unsustainable aspects of energy demand and other policies are essential to restructure existing energy systems into on-going sustainable forms. Policies to achieve these objectives are considered. This analysis shows that energy policy, population policy and sustainable development policies are closely integrated. Present policy and planning agencies do not reflect this integration and energy and population policies in Australia have largely developed independently and whether the outcome is sustainable is largely a matter of chance. A genuinely sustainable population policy recognises the inter-dependence between population and energy policies and it is essential that this is reflected in integrated policy and planning agencies

Data modelling in the beginning school years

This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling. Some of the core components of data modelling are addressed. A selection of results from the first data modelling activity implemented during the second year (2010; second grade) of a current longitudinal study are reported. Data modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Reported here are children's abilities to identify diverse and complex attributes, sort and classify data in different ways, and create and interpret models to represent their data.

Free surface flow past topography : a beyond-all-orders approach

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck (2006), the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of incline. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

Young children’s metarepresentational competence in data modelling

longitudinal study of data modelling across grades 1-3. The activity engaged children in designing, implementing, and analysing a survey about their new playground. Data modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. The core components of data modelling addressed here are children’s structuring and representing of data, with a focus on their display of metarepresentational competence (diSessa, 2004). Such competence includes students’ abilities to invent or design a variety of new representations, explain their creations, understand the role they play, and critique and compare the adequacy of representations. Reported here are the ways in which the children structured and represented their data, the metarepresentational competence displayed, and links between their metarepresentational competence and conceptual competence.

Get Into Vocational Education (GIVE) : motivating underperforming students

This study explores the effects of a vocational education-based program on academic motivation and engagement of primary school aged children. The Get Into Vocational Education (GIVE) program integrated ‘construction’ and the mathematics, English and science lessons of a Year 4 primary classroom. This paper focuses on investigating the components of the GIVE program that led to student changes in mathematical academic motivation and engagement resulting in outstanding gains in NAPLAN Numeracy results. The components proposed to have contributed to effectiveness of the GIVE program are: teacher and trainer expectations, task mastery and classroom relationships. These findings may be useful to researchers and educators who are interested in enhancing students’ mathematical academic motivation.

Successful outcomes in vocational education and training courses : how pedagogy and expectations influence achievement

This paper reports on a four year Australian Research Council funded Linkage Project titled Skilling Indigenous Queensland, conducted in regional areas of Queensland, Australia from 2009 to 2013. The project sought to investigate vocational education, training (VET) and teaching, Indigenous learners’ needs, employer cultural and expectations and community culture and expectations to identify best practice in numeracy teaching for Indigenous VET learners. Specifically it focused on ways to enhance the teaching and learning of courses and the associated mathematics in such courses to benefit learners and increase their future opportunities of employment. To date thirty-nine teachers/trainers/teacher aides and two hundred and thirty-one students consented to participate in the project. Nine VET courses were nominated to be the focus on the study. This paper focuses on questionnaire and interview responses from four trainers, two teacher aides and six students. In recent years a considerable amount of funding has been allocated to increasing Indigenous Peoples’ participation in education and employment. This increased funding is predicated on the assumption that it will make a difference and contribute to closing the education gap between Indigenous and non-Indigenous Australians (Council of Australia Governments, 2009). The central tenet is that access to education for Indigenous People will create substantial social and economic benefits for regional and remote Indigenous People. The project’s aim is to address some of the issues associated with the gap. To achieve the aims, the project adopted a mixed methods design aimed at benefitting research participants and included: participatory collaborative action research (Kemmis & McTaggart, 1988) and, community research (Smith, 1999). Participatory collaborative action research refers to a is a “collective, self-reflective enquiry undertaken by participants in social situations in order to improve the rationality and justice of their own social and educational practices” (Kemmis et al., 1988, p. 5). Community research is described as an approach that “conveys a much more intimate, human and self-defined space” (p. 127). Community research relies on and validates the community’s own definitions. As the project is informed by the social at a community level, it is described as “community action research or emancipatory research” (Smith, 1999, p. 127). It seeks to demonstrate benefit to the community, making positive differences in the lives of Indigenous People and communities. The data collection techniques included survey questionnaires, video recording of teaching and learning processes, teacher reflective video analysis of teaching, observations, semi-structured interviews and student numeracy testing. As a result of these processes, the findings indicate that VET course teachers work hard to adopt contextualising strategies to their teaching, however this process is not always straight forward because of the perceptions of how mathematics has been taught and learned historically. Further teachers, trainers and students have high expectations of one another with the view to successful outcomes from the courses.