Group metacognition during mathematical problem solving

Although various studies have shown that groups are more productive than individuals in complex mathematical problem solving, not all groups work together cooperatively. This review highlights that addressing organisational and cognitive factors to help scaffold group mathematical problem solving is necessary but not sufficient. Successful group problem solving also needs to incorporate metacognitive factors in order for groups to reflect on the organisational and cognitive factors influencing their group mathematical problem solving.

Implementation of an integrated, technology-based, discovery mode assessment item involving an incubation period to enhance learning outcomes for engineering maths students

In this paper we discuss our current efforts to develop and implement an exploratory, discovery mode assessment item into the total learning and assessment profile for a target group of about 100 second level engineering mathematics students. The assessment item under development is composed of 2 parts, namely, a set of "pre-lab" homework problems (which focus on relevant prior mathematical knowledge, concepts and skills), and complementary computing laboratory exercises which are undertaken within a fixed (1 hour) time frame. In particular, the computing exercises exploit the algebraic manipulation and visualisation capabilities of the symbolic algebra package MAPLE, with the aim of promoting understanding of certain mathematical concepts and skills via visual and intuitive reasoning, rather than a formal or rigorous approach. The assessment task we are developing is aimed at providing students with a significant learning experience, in addition to providing feedback on their individual knowledge and skills. To this end, a noteworthy feature of the scheme is that marks awarded for the laboratory work are primarily based on the extent to which reflective, critical thinking is demonstrated, rather than the amount of CBE-style tasks completed by the student within the allowed time. With regard to student learning outcomes, a novel and potentially critical feature of our scheme is that the assessment task is designed to be intimately linked to the overall course content, in that it aims to introduce important concepts and skills (via individual student exploration) which will be revisited somewhat later in the pedagogically more restrictive formal lecture component of the course (typically a large group plenary format). Furthermore, the time delay involved, or "incubation period", is also a deliberate design feature: it is intended to allow students the opportunity to undergo potentially important internal re-adjustments in their understanding, before being exposed to lectures on related course content which are invariably delivered in a more condensed, formal and mathematically rigorous manner. In our presentation, we will discuss in more detail our motivation and rationale for trailing such a scheme for the targeted student group. Some of the advantages and disadvantages of our approach (as we perceived them at the initial stages) will also be enumerated. In a companion paper, the theoretical framework for our approach will be more fully elaborated, and measures of student learning outcomes (as obtained from eg. student provided feedback) will be discussed.

Promoting interdisciplinarity through mathematical modelling

This article presents one approach to addressing the important issue of interdisciplinarity in the primary school mathematics curriculum, namely, through realistic mathematical modelling problems. Such problems draw upon other disciplines for their contexts and data. The article initially considers the nature of modelling with complex systems and discusses how such experiences differ from existing problem-solving activities in the primary mathematics curriculum. Principles for designing interdisciplinary modelling problems are then addressed, with reference to two mathematical modelling problems— one based in the scientific domain and the other in the literary domain. Examples of the models children have created in solving these problems follow. A reflection on the differences in the diversity and sophistication of these models raises issues regarding the design of interdisciplinary modelling problems. The article concludes with suggested opportunities for generating multidisciplinary projects within the regular mathematics curriculum.

A framework for network RTK data processing based on grid computing

Real-Time Kinematic (RTK) positioning is a technique used to provide precise positioning services at centimetre accuracy level in the context of Global Navigation Satellite Systems (GNSS). While a Network-based RTK (N-RTK) system involves multiple continuously operating reference stations (CORS), the simplest form of a NRTK system is a single-base RTK. In Australia there are several NRTK services operating in different states and over 1000 single-base RTK systems to support precise positioning applications for surveying, mining, agriculture, and civil construction in regional areas. Additionally, future generation GNSS constellations, including modernised GPS, Galileo, GLONASS, and Compass, with multiple frequencies have been either developed or will become fully operational in the next decade. A trend of future development of RTK systems is to make use of various isolated operating network and single-base RTK systems and multiple GNSS constellations for extended service coverage and improved performance. Several computational challenges have been identified for future NRTK services including: • Multiple GNSS constellations and multiple frequencies • Large scale, wide area NRTK services with a network of networks • Complex computation algorithms and processes • Greater part of positioning processes shifting from user end to network centre with the ability to cope with hundreds of simultaneous users’ requests (reverse RTK) There are two major requirements for NRTK data processing based on the four challenges faced by future NRTK systems, expandable computing power and scalable data sharing/transferring capability. This research explores new approaches to address these future NRTK challenges and requirements using the Grid Computing facility, in particular for large data processing burdens and complex computation algorithms. A Grid Computing based NRTK framework is proposed in this research, which is a layered framework consisting of: 1) Client layer with the form of Grid portal; 2) Service layer; 3) Execution layer. The user’s request is passed through these layers, and scheduled to different Grid nodes in the network infrastructure. A proof-of-concept demonstration for the proposed framework is performed in a five-node Grid environment at QUT and also Grid Australia. The Networked Transport of RTCM via Internet Protocol (Ntrip) open source software is adopted to download real-time RTCM data from multiple reference stations through the Internet, followed by job scheduling and simplified RTK computing. The system performance has been analysed and the results have preliminarily demonstrated the concepts and functionality of the new NRTK framework based on Grid Computing, whilst some aspects of the performance of the system are yet to be improved in future work.

Scaffolding students’ thinking in mathematical investigations

This article describes some of the issues that teachers might encounter when scaffolding students’ thinking during mathematical investigations. It describes four episodes where a teacher’s scaffolding failed to support students’ mathematical thinking and explores the reasons why the scaffolding was ineffective. Understanding what is ineffective and why is one way to improve pedagogical practice. As a background to these episodes, we first provide an overview of the mathematical investigation. Our paper concludes with some recommendations for judicious scaffolding during investigations.

Mathematical models for describing the shape of the in vitro unstretched human crystalline lens

We developed orthogonal least-squares techniques for fitting crystalline lens shapes, and used the bootstrap method to determine uncertainties associated with the estimated vertex radii of curvature and asphericities of five different models. Three existing models were investigated including one that uses two separate conics for the anterior and posterior surfaces, and two whole lens models based on a modulated hyperbolic cosine function and on a generalized conic function. Two new models were proposed including one that uses two interdependent conics and a polynomial based whole lens model. The models were used to describe the in vitro shape for a data set of twenty human lenses with ages 7–82 years. The two-conic-surface model (7 mm zone diameter) and the interdependent surfaces model had significantly lower merit functions than the other three models for the data set, indicating that most likely they can describe human lens shape over a wide age range better than the other models (although with the two-conic-surfaces model being unable to describe the lens equatorial region). Considerable differences were found between some models regarding estimates of radii of curvature and surface asphericities. The hyperbolic cosine model and the new polynomial based whole lens model had the best precision in determining the radii of curvature and surface asphericities across the five considered models. Most models found significant increase in anterior, but not posterior, radius of curvature with age. Most models found a wide scatter of asphericities, but with the asphericities usually being positive and not significantly related to age. As the interdependent surfaces model had lower merit function than three whole lens models, there is further scope to develop an accurate model of the complete shape of human lenses of all ages. The results highlight the continued difficulty in selecting an appropriate model for the crystalline lens shape.

Elicitator : an expert elicitation tool for regression in ecology

Expert elicitation is the process of retrieving and quantifying expert knowledge in a particular domain. Such information is of particular value when the empirical data is expensive, limited, or unreliable. This paper describes a new software tool, called Elicitator, which assists in quantifying expert knowledge in a form suitable for use as a prior model in Bayesian regression. Potential environmental domains for applying this elicitation tool include habitat modeling, assessing detectability or eradication, ecological condition assessments, risk analysis, and quantifying inputs to complex models of ecological processes. The tool has been developed to be user-friendly, extensible, and facilitate consistent and repeatable elicitation of expert knowledge across these various domains. We demonstrate its application to elicitation for logistic regression in a geographically based ecological context. The underlying statistical methodology is also novel, utilizing an indirect elicitation approach to target expert knowledge on a case-by-case basis. For several elicitation sites (or cases), experts are asked simply to quantify their estimated ecological response (e.g. probability of presence), and its range of plausible values, after inspecting (habitat) covariates via GIS.

Population Monte Carlo Algorithm in High Dimensions

The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix for the importance function can be estimated in special cases.

Visualisation of groundwater flow using texture based visualisation techniques

Texture based techniques for visualisation of unsteady vector fields have been applied for the visualisation of a Finite volume model for variably saturated groundwater flow through porous media. This model has been developed by staff in the School of Mathematical Sciences QUT for the study of salt water intrusion into coastal aquifers. This presentation discusses the implementation and effectiveness of the IBFV algorithm in the context of visualisation of the groundwater simulation outputs.

Mixing colours : an ICT tool based on a semiotic framework for mathematical meaning-making about ratio and fractions

This paper reports on the research and development of an ICT tool to facilitate the learning of ratio and fractions by adult prisoners. The design of the ICT tool was informed by a semiotic framework for mathematical meaning-making. The ICT tool thus employed multiple semiotic resources including topological, typological, and social-actional resources. The results showed that individual semiotic resource could only represent part of the mathematical concept, while at the same time it might signify something else to create a misconception. When multiple semiotic resources were utilised the mathematical ideas could be better learnt.

Street computing workshop 2009

The Street Computing workshop, held in conjunction with OZCHI 2009, solicits papers discussing new research directions, early research results, works-in-progress and critical surveys of prior research work in the areas of ubiquitous computing and interaction design for urban environments. Urban spaces have unique characteristics. Typically, they are densely populated, buzzing with life twenty-four hours a day, seven days a week. These traits afford many opportunities, but they also present many challenges: traffic jams, smog and pollution, stress placed on public services, and more. Computing technology, particularly the kind that can be placed in the hands of citizens, holds much promise in combating some of these challenges. Yet, computation is not merely a tool for overcoming challenges; rather, when embedded appropriately in our everyday lives, it becomes a tool of opportunity, for shaping how our cities evolve, for enabling us to interact with our city and its people in new ways, and for uncovering useful, but hidden relationships and correlations between elements of the city. The increasing availability of an urban computing infrastructure has lead to new and exciting ways inhabitants can interact with their city. This includes interaction with a wide range of services (e.g. public transport, public services), conceptual representations of the city (e.g. local weather and traffic conditions), the availability of a variety of shared and personal displays (e.g. public, ambient, mobile) and the use of different interaction modes (e.g. tangible, gesture-based, token-based). This workshop solicits papers that address the above themes in some way. We encourage researchers to submit work that deals with challenges and possibilities that the availability of urban computing infrastructure such as sensors and middleware for sensor networks pose. This includes new and innovative ways of interacting with and within urban environments; user experience design and participatory design approaches for urban environments; social aspects of urban computing; and other related areas.

Road network extraction with new vectorization and pruning from high-resolution RS images

With the increasing resolution of remote sensing images, road network can be displayed as continuous and homogeneity regions with a certain width rather than traditional thin lines. Therefore, road network extraction from large scale images refers to reliable road surface detection instead of road line extraction. In this paper, a novel automatic road network detection approach based on the combination of homogram segmentation and mathematical morphology is proposed, which includes three main steps: (i) the image is classified based on homogram segmentation to roughly identify the road network regions; (ii) the morphological opening and closing is employed to fill tiny holes and filter out small road branches; and (iii) the extracted road surface is further thinned by a thinning approach, pruned by a proposed method and finally simplified with Douglas-Peucker algorithm. Lastly, the results from some QuickBird images and aerial photos demonstrate the correctness and efficiency of the proposed process.