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.

Mixing colours: An ICT tool based on a semiotic framework for mathematical meaning-makng 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.

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.

A comparison of international occupational therapy competencies : implications for Australian standards in the new millennium

Background/aim: A timely evaluation of the Australian Competency Standards for Entry-Level Occupational Therapists© (1994) was conducted. This thorough investigation comprised a literature review exploring the concept of competence and the applications of competency standards; systematic benchmarking of the Australian Occupational Therapy Competency Standards (OT AUSTRALIA, 1994) against other national and international competency standards and other affiliated documents, from occupational therapy and other cognate disciplines; and extensive nationwide consultation with the professional community. This paper explores and examines the similarities and disparities between occupational therapy competency standards documents available in English from Australia and other countries.----- Methods: An online search for national occupational therapy competency standards located 10 documents, including the Australian competencies.----- Results: Four 'frameworks' were created to categorise the documents according to their conceptual underpinnings: Technical-Prescriptive, Enabling, Educational and Meta-Cognitive. Other characteristics that appeared to impact the design, content and implementation of competency standards, including definitions of key concepts, authorship, national and cultural priorities, scope of services, intended use and review mechanisms, were revealed.----- Conclusion: The proposed 'frameworks' and identification of influential characteristics provided a 'lens' through which to understand and evaluate competency standards. While consistent application of and attention to some of these characteristics appear to consolidate and affirm the authority of competency standards, it is suggested that the national context should be a critical determinant of the design and content of the final document. The Australian Occupational Therapy Competency Standards (OT AUSTRALIA, 1994) are critiqued accordingly, and preliminary recommendations for revision are proposed.

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.

Mathematical modelling of muscle recruitment and function in the lumbar spine

Low back pain is an increasing problem in industrialised countries and although it is a major socio-economic problem in terms of medical costs and lost productivity, relatively little is known about the processes underlying the development of the condition. This is in part due to the complex interactions between bone, muscle, nerves and other soft tissues of the spine, and the fact that direct observation and/or measurement of the human spine is not possible using non-invasive techniques. Biomechanical models have been used extensively to estimate the forces and moments experienced by the spine. These models provide a means of estimating the internal parameters which can not be measured directly. However, application of most of the models currently available is restricted to tasks resembling those for which the model was designed due to the simplified representation of the anatomy. The aim of this research was to develop a biomechanical model to investigate the changes in forces and moments which are induced by muscle injury. In order to accurately simulate muscle injuries a detailed quasi-static three dimensional model representing the anatomy of the lumbar spine was developed. This model includes the nine major force generating muscles of the region (erector spinae, comprising the longissimus thoracis and iliocostalis lumborum; multifidus; quadratus lumborum; latissimus dorsi; transverse abdominis; internal oblique and external oblique), as well as the thoracolumbar fascia through which the transverse abdominis and parts of the internal oblique and latissimus dorsi muscles attach to the spine. The muscles included in the model have been represented using 170 muscle fascicles each having their own force generating characteristics and lines of action. Particular attention has been paid to ensuring the muscle lines of action are anatomically realistic, particularly for muscles which have broad attachments (e.g. internal and external obliques), muscles which attach to the spine via the thoracolumbar fascia (e.g. transverse abdominis), and muscles whose paths are altered by bony constraints such as the rib cage (e.g. iliocostalis lumborum pars thoracis and parts of the longissimus thoracis pars thoracis). In this endeavour, a separate sub-model which accounts for the shape of the torso by modelling it as a series of ellipses has been developed to model the lines of action of the oblique muscles. Likewise, a separate sub-model of the thoracolumbar fascia has also been developed which accounts for the middle and posterior layers of the fascia, and ensures that the line of action of the posterior layer is related to the size and shape of the erector spinae muscle. Published muscle activation data are used to enable the model to predict the maximum forces and moments that may be generated by the muscles. These predictions are validated against published experimental studies reporting maximum isometric moments for a variety of exertions. The model performs well for fiexion, extension and lateral bend exertions, but underpredicts the axial twist moments that may be developed. This discrepancy is most likely the result of differences between the experimental methodology and the modelled task. The application of the model is illustrated using examples of muscle injuries created by surgical procedures. The three examples used represent a posterior surgical approach to the spine, an anterior approach to the spine and uni-lateral total hip replacement surgery. Although the three examples simulate different muscle injuries, all demonstrate the production of significant asymmetrical moments and/or reduced joint compression following surgical intervention. This result has implications for patient rehabilitation and the potential for further injury to the spine. The development and application of the model has highlighted a number of areas where current knowledge is deficient. These include muscle activation levels for tasks in postures other than upright standing, changes in spinal kinematics following surgical procedures such as spinal fusion or fixation, and a general lack of understanding of how the body adjusts to muscle injuries with respect to muscle activation patterns and levels, rate of recovery from temporary injuries and compensatory actions by other muscles. Thus the comprehensive and innovative anatomical model which has been developed not only provides a tool to predict the forces and moments experienced by the intervertebral joints of the spine, but also highlights areas where further clinical research is required.

Mathematical modelling in the early school years

In this article we explore young children's development of mathematical knowledge and reasoning processes as they worked two modelling problems (the Butter Beans Problem and the Airplane Problem). The problems involve authentic situations that need to be interpreted and described in mathematical ways. Both problems include tables of data, together with background information containing specific criteria to be considered in the solution process. Four classes of third-graders (8 years of age) and their teachers participated in the 6-month program, which included preparatory modelling activities along with professional development for the teachers. In discussing our findings we address: (a) Ways in which the children applied their informal, personal knowledge to the problems; (b) How the children interpreted the tables of data, including difficulties they experienced; (c) How the children operated on the data, including aggregating and comparing data, and looking for trends and patterns; (c) How the children developed important mathematical ideas; and (d) Ways in which the children represented their mathematical understandings.

Curriculum design innovation in flexible science teaching

In this paper you will be introduced to a number of principles which can be used to inform good teaching practice and rigorous curriculum design. Principles relate to: * Application of a common sequence of events for how learners learn; * Accommodating different learning styles; * Adopting a purposeful approach to teaching and learning; * Using assessment as a central driving force in the curriculum and as an organising structure leading to coherence of teaching and learning approach; and * The increasing emphasis that is being placed on the development of generic graduate competencies over and above discipline content knowledge. The principles are particularly significant in relation to adult learning. The paper will use three specific applications as illustrations to help you to learn how these principles can be applied. The illustrations are taken from a second year subject in supercomputing that uses scientific case studies. The subject has been developed (with support from Silicon Graphics Inc. and Intel) to be taught entirely via the Internet.

Vision-based navigational competencies for a car-like vehicle

If mobile robots are to perform useful tasks in the real-world they will require a catalog of fundamental navigation competencies and a means to select between them. In this paper we describe our work on strongly vision-based competencies: road-following, person or vehicle following, pose and position stabilization. Results from experiments on an outdoor autonomous tractor, a car-like vehicle, are presented.

Predicting active material utilisation in LiFePO4 electrodes using a multi-scale mathematical model

A mathematical model is developed to simulate the discharge of a LiFePO4 cathode. This model contains 3 size scales, which match with experimental observations present in the literature on the multi-scale nature of LiFePO4 material. A shrinking-core is used on the smallest scale to represent the phase-transition of LiFePO4 during discharge. The model is then validated against existing experimental data and this validated model is then used to investigate parameters that influence active material utilisation. Specifically the size and composition of agglomerates of LiFePO4 crystals is discussed, and we investigate and quantify the relative effects that the ionic and electronic conductivities within the oxide have on oxide utilisation. We find that agglomerates of crystals can be tolerated under low discharge rates. The role of the electrolyte in limiting (cathodic) discharge is also discussed, and we show that electrolyte transport does limit performance at high discharge rates, confirming the conclusions of recent literature.

Evolution of vocational rehabilitation competencies in Australia

Over the past decade, there has been growth in the delivery of vocational rehabilitation services globally, as countries seek to control disability-related expenditure, yet there has been minimal research outside the United States on competencies required to work in this area. This study reports on research conducted in Australia to determine current job function and knowledge areas in terms of their importance and frequency of use in the provision of vocational rehabilitation. A survey comprising items from the Rehabilitation Skills Inventory-Amended and International Survey of Disability Management was completed by 149 rehabilitation counselors and items submitted to factor analysis. T-tests and analyses of variance were used to determine differences between scores of importance and frequency and differences in scores based on work setting and professional training. Six factors were identified as important and frequently used: (i) vocational counseling, (ii) professional practice, (iii) personal counseling, (iv) rehabilitation case management, (v) workplace disability case management, and (vi) workplace intervention and program management. Vocational counseling, professional practice and personal counseling were significantly more important and performed more frequently by respondents in vocational rehabilitation settings than those in compensation settings. These same three factors were rated significantly higher in importance and frequency by those with rehabilitation counselor training when compared with those with other training. In conclusion, although ‘traditional’ knowledge and skill areas such as vocational counseling, professional practice, and personal counseling were identified as central to vocational rehabilitation practice in Australian rehabilitation agencies, mean ratings suggest a growing emphasis on knowledge and skills associated with disability management practice.