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.

Meeting under the “Omei” Tree in the Torres Strait Islands : networks and funds of knowledge of mathematical ideas

This paper focuses on the turning point experiences that worked to transform the researcher during a preliminary consultation process to seek permission to conduct of a small pilot project on one Torres Strait Island. The project aimed to learn from parents how they support their children in their mathematics learning. Drawing on a community research design, a consultative meeting was held with one Torres Strait Islander community to discuss the possibility of piloting a small project that focused on working with parents and children to learn about early mathematics processes. Preliminary data indicated that parents use networks in their community. It highlighted the funds of knowledge of mathematics that exist in the community and which are used to teach their children. Such knowledges are situated within a community’s unique histories, culture and the voices of the people. “Omei” tree means the Tree of Wisdom in the Island community.

Visualising mathematical knowledge and understanding

Contemporary mathematics education attempts to instil within learners the conceptualization of mathematics as a highly organized and inter-connected set of ideas. To support this, a means to graphically represent this organization of ideas is presented which reflects the cognitive mechanisms that shape a learner’s understanding. This organisation of information may then be analysed, with the view to informing the design of mathematics instruction in face-to-face and/or computer-mediated learning environments. However, this analysis requires significant work to develop both theory and practice.

Rich and purposeful mathematical knowledge of mothers and children in a Torres Strait Islander community

This paper focuses on a pilot study that explored the situated mathematical knowledge of mothers and children in one Torres Strait Islander community in Australia. The community encouraged parental involvement in their children’s learning and schooling. The study explored parents’ understandings of mathematics and how their children came to learn about it on the island. A funds of knowledge approach was used in the study. This approach is based on the premise that people are competent and have knowledge that has been historically and culturally accumulated into a body of knowledge and skills essential for their functioning and well-being (Moll, 1992). The participants, three adults and one child are featured in this paper. Three separate events are described with epiphanic or illuminative moments analysed to ascertain the features that enabled an understanding of the nature of the mathematical events. The study found that Indigenous ways of knowing of mathematics were deeply embedded in rich cultural practices that were tied to the community. This finding has implications for teachers of children in the early years. Where school mathematics is often presented as disembodied and isolated facts with children seeing little relevance, learning a different perspective of mathematics that is tied to the resources and practices of children’s lives and facilitated through social relationships, may go a long way to improving the engagement of children and their parents in learning and schooling.

Shifting the Focus. Incorporating knowledge about Aboriginal engineering into main stream content

BACKGROUND OR CONTEXT The concept of 'Aboriginal engineering' has had little exposure in conventional engineering education programs, despite more than 40,000 years of active human engagement with the diverse Australian environment. The work reported in this paper began with the premise that Indigenous Student Support Through Indigenous Perspectives Embedded in Engineering Curricula (Goldfinch, et al 2013) would provide a clear and replicable means of encouraging Aboriginal teenagers to consider a career in engineering. Although that remains a key outcome of this OLT project, the direction taken by the research had led to additional insights and perspectives that have wide implications for engineering education more generally. There has only been passing reference to the achievements of Aboriginal engineering in current texts, and the very absence of such references was a prompt to explore further as our work developed. PURPOSE OR GOAL Project goals focused on curriculum-based change, including development of a model for inclusive teaching spaces, and study units employing key features of the model. As work progressed we found we needed to understand more about the principles and practices informing the development of pre-contact Aboriginal engineering strategies for sustaining life and society within the landscape of this often harsh continent. We also found ourselves being asked 'what engineering did Aboriginal cultures have?' Finding that there are no easy-to- access answers, we began researching the question, while continuing to engage with specific curriculum trials. APPROACH Stakeholders in the project had been identified as engineering educators, potential Aboriginal students and Aboriginal communities local to Universities involved in the project. We realised, early on, that at least one more group was involved - all the non-Aboriginal students in engineering classes. This realisation, coupled with recognition of the need to understand Aboriginal engineering as a set of viable, long term practices, altered the focus of our efforts. Rather than focusing primarily on finding ways to attract Aboriginal engineering students, the shift has been towards evolving ways of including knowledge about Aboriginal practices and principles in relevant engineering content. DISCUSSION This paper introduces the model resulting from the work of this project, explores its potential influence on engineering curriculum development and reports on implementation strategies. The model is a static representation of a dynamic and cyclic approach to engaging with Aboriginal engineering through contact with local communities in regard to building knowledge about the social beliefs underlying Aboriginal engineering principles and practices. Ways to engage engineering educators, students and the wider community are evolving through the continuing work of the project team and will be reported in more detail in the paper. RECOMMENDATIONS/IMPLICATIONS/CONCLUSION While engineering may be considered by some to be agnostic in regard to culture and social issues, the work of this project is drawing attention to the importance of including such issues into curriculum materials at a number of levels of complexity. The paper will introduce and explore the central concepts of the research completed to date, as well as suggesting ways in which engineering educators can extend their knowledge and understanding of Aboriginal engineering principles in the context of their own specialisations.

Search engine content analysis

Search engines have forever changed the way people access and discover knowledge, allowing information about almost any subject to be quickly and easily retrieved within seconds. As increasingly more material becomes available electronically the influence of search engines on our lives will continue to grow. This presents the problem of how to find what information is contained in each search engine, what bias a search engine may have, and how to select the best search engine for a particular information need. This research introduces a new method, search engine content analysis, in order to solve the above problem. Search engine content analysis is a new development of traditional information retrieval field called collection selection, which deals with general information repositories. Current research in collection selection relies on full access to the collection or estimations of the size of the collections. Also collection descriptions are often represented as term occurrence statistics. An automatic ontology learning method is developed for the search engine content analysis, which trains an ontology with world knowledge of hundreds of different subjects in a multilevel taxonomy. This ontology is then mined to find important classification rules, and these rules are used to perform an extensive analysis of the content of the largest general purpose Internet search engines in use today. Instead of representing collections as a set of terms, which commonly occurs in collection selection, they are represented as a set of subjects, leading to a more robust representation of information and a decrease of synonymy. The ontology based method was compared with ReDDE (Relevant Document Distribution Estimation method for resource selection) using the standard R-value metric, with encouraging results. ReDDE is the current state of the art collection selection method which relies on collection size estimation. The method was also used to analyse the content of the most popular search engines in use today, including Google and Yahoo. In addition several specialist search engines such as Pubmed and the U.S. Department of Agriculture were analysed. In conclusion, this research shows that the ontology based method mitigates the need for collection size estimation.

Research Monograph: The role of community and lifestyle in the making of a knowledge city

Our brief is to investigate the role of community and lifestyle in the making of a globally successful knowledge city region. Our approach is essentially pragmatic. We start by broadly examining knowledge-based urban development from a number of different perspectives. The first view is historical. In this context knowledge work and knowledge workers are seen as vital parts of a new emergent mode of production reliant on the continual production of abstract knowledge. We briefly develop this perspective to encompass the work of Richard Florida who has, notedly, claimed: “Wherever talent goes, innovation, creativity, and economic growth are sure to follow.” Our next perspective examines concepts of knowledge and modes of its production to discover knowledge is not an unchanging object but a human activity that changes in form and content through history. The suggestion emerges that not only is the production of contemporary ‘knowledge’ organised in a specific (and new) manner but also the output of this networked production is a particular type of knowledge (i.e. techné). The third perspective locates knowledge production and its workers in the contemporary urban context. As such, it co-ordinates the knowledge city in the increasingly global structure of cities and develops a typology of different groups of knowledge workers in their preferred urban environment(s). We see emerging here a distinctive geography of knowledge production. It is an urban phenomenon. There is, in short, something about the nature of cities that knowledge workers find particularly attractive. In the next, essentially anthropological, perspective we start to explore the needs and desires of the individual knowledge worker. Beyond the needs basic to any modern human household an attempt is made to deduce, from a base understanding of knowledge work as mental labour, the compensatory cultural needs of the knowledge worker when not at work - and the expression of these needs in the urban fabric. Our final perspective consists of two case studies. In a review of the experiences of Austin, Texas and Singapore’s one-north precinct we collect empirical data on, respectively, a knowledge city that has sustained itself for over 50 years and an urban precinct newly launched into the global market for knowledge work and knowledge workers. Interwoven The Role of Community and Lifestyle in the Making of a Knowledge City Urban Research Program 8 through all perspectives, in the form of apposite citation, is that of ‘expert opinion’ gathered in a rudimentary poll of academic and industry sources. This opinion appears in text boxes while details of the survey can be found in Appendix A. In the conclusion of the report we interpret the wide range of evidence gathered above in a policy frame. It is our hope this report will leave the reader with a clearer picture of the decisive organisational, infrastructural, aesthetic and social dimensions of a knowledge precinct.

Exploring the influence of a science content course incorporating explicit nature of science and argumentation instruction on preservice primary teachers' views of nature of science

There exists a general consensus in the science education literature around the goal of enhancing students. and teachers. views of nature of science (NOS). An emerging area of research in science education explores NOS and argumentation, and the aim of this study was to explore the effectiveness of a science content course incorporating explicit NOS and argumentation instruction on preservice primary teachers. views of NOS. A constructivist perspective guided the study, and the research strategy employed was case study research. Five preservice primary teachers were selected for intensive investigation in the study, which incorporated explicit NOS and argumentation instruction, and utilised scientific and socioscientific contexts for argumentation to provide opportunities for participants to apply their NOS understandings to their arguments. Four primary sources of data were used to provide evidence for the interpretations, recommendations, and implications that emerged from the study. These data sources included questionnaires and surveys, interviews, audio- and video-taped class sessions, and written artefacts. Data analysis involved the formation of various assertions that informed the major findings of the study, and a variety of validity and ethical protocols were considered during the analysis to ensure the findings and interpretations emerging from the data were valid. Results indicated that the science content course was effective in enabling four of the five participants. views of NOS to be changed. All of the participants expressed predominantly limited views of the majority of the examined NOS aspects at the commencement of the study. Many positive changes were evident at the end of the study with four of the five participants expressing partially informed and/or informed views of the majority of the examined NOS aspects. A critical analysis of the effectiveness of the various course components designed to facilitate the development of participants‟ views of NOS in the study, led to the identification of three factors that mediated the development of participants‟ NOS views: (a) contextual factors (including context of argumentation, and mode of argumentation), (b) task-specific factors (including argumentation scaffolds, epistemological probes, and consideration of alternative data and explanations), and (c) personal factors (including perceived previous knowledge about NOS, appreciation of the importance and utility value of NOS, and durability and persistence of pre-existing beliefs). A consideration of the above factors informs recommendations for future studies that seek to incorporate explicit NOS and argumentation instruction as a context for learning about NOS.

An instrument for assessing primary students’ knowledge of information graphics in mathematics

Information graphics have become increasingly important in representing, organising and analysing information in a technological age. In classroom contexts, information graphics are typically associated with graphs, maps and number lines. However, all students need to become competent with the broad range of graphics that they will encounter in mathematical situations. This paper provides a rationale for creating a test to measure students’ knowledge of graphics. This instrument can be used in mass testing and individual (in-depth) situations. Our analysis of the utility of this instrument informs policy and practice. The results provide an appreciation of the relative difficulty of different information graphics; and provide the capacity to benchmark information about students’ knowledge of graphics. The implications for practice include the need to support the development of students’ knowledge of graphics, the existence of gender differences, the role of cross-curriculum applications in learning about graphics, and the need to explicate the links among graphics.

Human resource management and innovation: what are knowledge-intensive firms doing?

An undeniable shift in focus from traditional production companies to Knowledge-Intensive Firms (KIFs) poses challenges for academics and practioners alike. In particular, effective management of an organization's human resources has become a critical issue for ensuring sustained innovation capacity. The relationship between Human Resource Management (HRM) in KIFs is however still a largely unexplored arena. The objective of this paper is to explore this relationship in an effort to identify HRM practices that support innovation. To this end, the paper includes reviews of the literature relevant to HRM and innovation in KIFs and four case studies from companies in Denmark and Australia that have been recognized for excellence in innovation. On the basis of content analyses conducted on the case data, some preliminary conclusions are posited regarding the role of HRM in KIFs. More specifically, the findings from this study suggest that while there are commonalities between HRM practices in traditional manufacturing companies and KIFs, there are also important differences, especially in terms of staffing practices. The paper contributes by offering recommendations for management of HRM in innovative KIFs and potential avenues for research to further develop our understanding of how HRM can support innovation in KIFs.

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.

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.