Towards a model for the analysis and description of mathematical knowledge and understanding

Mathematics education literature has called for an abandonment of ontological and epistemological ideologies that have often divided theory-based practice. Instead, a consilience of theories has been sought which would leverage the strengths of each learning theory and so positively impact upon contemporary educational practice. This research activity is based upon Popper’s notion of three knowledge worlds which differentiates the knowledge shared in a community from the personal knowledge of the individual, and Bereiter’s characterisation of understanding as the individual’s relationship to tool-like knowledge. Using these notions, a re-conceptualisation of knowledge and understanding and a subsequent re-consideration of learning theories are proposed as a way to address the challenge set by literature. Referred to as the alternative theoretical framework, the proposed theory accounts for the scaffolded transformation of each individual’s unique understanding, whilst acknowledging the existence of a body of domain knowledge shared amongst participants in a scientific community of practice. The alternative theoretical framework is embodied within an operational model that is accompanied by a visual nomenclature with which to describe consensually developed shared knowledge and personal understanding. This research activity has sought to iteratively evaluate this proposed theory through the practical application of the operational model and visual nomenclature to the domain of early-number counting, addition and subtraction. This domain of mathematical knowledge has been comprehensively analysed and described. Through this process, the viability of the proposed theory as a tool with which to discuss and thus improve the knowledge and understanding with the domain of mathematics has been validated. Putting of the proposed theory into practice has lead to the theory’s refinement and the subsequent achievement of a solid theoretical base for the future development of educational tools to support teaching and learning practice, including computer-mediated learning environments. Such future activity, using the proposed theory, will advance contemporary mathematics educational practice by bringing together the strengths of cognitivist, constructivist and post-constructivist learning theories.

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.

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.

Using a visually-based assignment to reinforce and assess design history knowledge and understanding

This paper presents a visual timeline-based assignment used in an undergraduate Industrial Design History, Theory and Critcism unit. The assignment was developed in order to find a better way of supporting design history learning than an exam or essay assessment. It was developed using constructive alignment and it allows design students to use their strong visual thinking skills to understand unfamiliar content, develop their visual literacy of design history, and think deeply about the links between the designs, styles, movements, events and people in their timeline. The task produced a variety of responses, from websites and electronic presentations to large paper timelines, scrolls and 3D models. These have been admired by peers and used for end of year shows and permanent displays. Questionnaires were issued to students to gain feedback about the assessment. Students stated that the visual nature of the assignment helped them to understand how different aspects of design history related to each other, assisted with retaining the information, and that it was more interesting and fun than a report or an exam. This paper explores the theories behind and the benefits of using such methods of assessment for design history courses.

Australian pre-service teachers' Knowledge and understanding of cyberbullying: Implications for school climate

The beliefs, attitudes and understandings of pre-service teachers towards bullying and more recently, cyberbullying remains unclear. Previous studies have found them to be generally lacking confidence to address bullying, which could impact negatively on school climate if, when they enter the profession, these beliefs undermine existing anti-bullying initiatives. This study explores Australian pre-service teachers' (N= 717) understanding and knowledge of traditional bullying and cyberbullying and their confidence and capacity to deal with it. Findings from self-report, anonymous questionnaires from students attending three universities in Australia indicated that two thirds (66%) of current pre-service teachers felt informed to very informed and capable to very capable (62%) of dealing with school bullying and 90% could discern cyber and traditional bullying behaviours from other online and offline aggressive acts. Gender and Year level differences were found. The potential impact of their knowledge and understanding of bullying and cyberbullying on school climate, and sustaining and maintaining anti-bullying interventions as they enter the profession is discussed.

Nursing students’ knowledge and practice of infection control precautions : an educational intervention

Aim. This paper is a report of the effectiveness of a purpose-designed education program in improving undergraduate nursing students’ understanding and practice of infection control precautions. Background. The severe acute respiratory syndrome outbreak in 2003 highlighted that healthcare workers were under-prepared for such an epidemic. While many in-service education sessions were arranged by institutions in response to the outbreak, preservice nursing education has overlooked preparation for handling such infectious disease epidemics. Method. A quasi-experimental design was used and a 16-hour, purpose-designed infection control education programme was implemented for preservice nursing students in southern Taiwan. Self-administered questionnaires were distributed at three time points during the period September 2005 to April 2006 to examine the sustainability and effectiveness of the intervention. Results. A total of 175 preservice nursing students participated in the study. Following the education programme, students in the intervention group showed a statistically significant improvement across time in their knowledge of these precautions [F(2, 180) = 13Æ53, P < 0Æ001] and confidence in resolving infectionrelated issues [F(1Æ79, 168Æ95) = 3Æ24] when compared with those in the control group. Conclusion. To improve nursing students’ capacity in responding to infectious epidemics, an educational programme that integrates the theme of infection precautions, learning theory and teaching strategies is recommended for all nursing institutes.

The development of science concepts emergent from science museum and post-visit activity experiences : students' construction of knowledge

This research investigated students' construction of knowledge about the topics of magnetism and electricity emergent from a visit to an interactive science centre and subsequent classroom-based activities linked to the science centre exhibits. The significance of this study is that it analyses critically an aspect of school visits to informal learning centres that has been neglected by researchers in the past, namely the influence of post-visit activities in the classroom on subsequent learning and knowledge construction. Employing an interpretive methodology, the study focused on three areas of endeavour. Firstly, the establishment of a set of principles for the development of post-visit activities, from a constructivist framework, to facilitate students' learning of science. Secondly, to describe and interpret students' scientific understandings : prior t o a visit t o a science museum; following a visit t o a science museum; and following post-visit activities that were related to their museum experiences. Finally, to describe and interpret the ways in which students constructed their understandings: prior to a visit to a science museum; following a visit to a science museum; and following post-visit activities directly related to their museum experiences. The study was designed and implemented in three stages: 1) identification and establishment of the principles for design and evaluation of post-visit activities; 2) a pilot study of specific post-visit activities and data gathering strategies related to student construction of knowledge; and 3) interpretation of students' construction of knowledge from a visit to a science museum and subsequent completion of post-visit activities, which constituted the main study. Twelve students were selected from a year 7 class to participate in the study. This study provides evidence that the series of post-visit activities, related to the museum experiences, resulted in students constructing and reconstructing their personal knowledge of science concepts and principles represented in the science museum exhibits, sometimes towards the accepted scientific understanding and sometimes in different and surprising ways. Findings demonstrate the interrelationships between learning that occurs at school, at home and in informal learning settings. The study also underscores for teachers and staff of science museums and similar centres the importance of planning pre- and post-visit activities, not only to support the development of scientific conceptions, but also to detect and respond to alternative conceptions that may be produced or strengthened during a visit to an informal learning centre. Consistent with contemporary views of constructivism, the study strongly supports the views that : 1) knowledge is uniquely structured by the individual; 2) the processes of knowledge construction are gradual, incremental, and assimilative in nature; 3) changes in conceptual understanding are can be interpreted in the light of prior knowledge and understanding; and 4) knowledge and understanding develop idiosyncratically, progressing and sometimes appearing to regress when compared with contemporary science. This study has implications for teachers, students, museum educators, and the science education community given the lack of research into the processes of knowledge construction in informal contexts and the roles that post-visit activities play in the overall process of learning.

Manipulating task constraints to improve tactical knowledge and collective decision-making in rugby union

In team sports such as rugby union, a myriad of decisions and actions occur within the boundaries that compose the performance perceptual- motor workspace. The way that these performance boundaries constrain decision making and action has recently interested researchers and has involved developing an understanding of the concept of constraints. Considering team sports as complex dynamical systems, signifies that they are composed of multiple, independent agents (i.e. individual players) whose interactions are highly integrated. This level of complexity is characterized by the multiple ways that players in a rugby field can interact. It affords the emergence of rich patterns of behaviour, such as rucks, mauls, and collective tactical actions that emerge due to players’ adjustments to dynamically varying competition environments. During performance, the decisions and actions of each player are constrained by multiple causes (e.g. technical and tactical skills, emotional states, plans, thoughts, etc.) that generate multiple effects (e.g. to run or pass, to move forward to tackle or maintain position and drive the opponent to the line), a prime feature in a complex systems approach to team games performance (Bar- Yam, 2004). To establish a bridge between the complexity sciences and learning design in team sports like rugby union, the aim of practice sessions is to prepare players to pick up and explore the information available in the multiple constraints (i.e. the causes) that influence performance. Therefore, learning design in training sessions should be soundly based on the interactions amongst players (i.e.teammates and opponents) that will occur in rugby matches. To improve individual and collective decision making in rugby union, Passos and colleagues proposed in previous work a performer- environment interaction- based approach rather than a traditional performer- based approach (Passos, Araújo, Davids & Shuttleworth, 2008).

The internetalisation of information, knowledge and interaction components of the firm’s internationalisation process

Transcending traditional national borders, the Internet is an evolving technology that has opened up many new international market opportunities. However, ambiguity remains, with limited research and understanding of how the Internet influences the firm’s internationalisation process components. As a consequence, there has been a call for further investigation of the phenomenon. Thus, the purpose of this study was to investigate the Internet’s impact on the internationalisation process components, specifically, information availability, information usage, interactive communication and international market growth. Analysis was undertaken using structural equation modelling. Findings highlight the mediating impact of the Internet on information and knowledge transference in the internationalisation process. Contributions of the study test conceptualisations and give statistical validation of interrelationships, while illuminating the Internet’s impact on firm internationalisation.

Indigenous knowledge and IP food security : a review of concepts and cases

Dáwat, Pamahándí, Tawíd, Ságda, Lampísa, Ibabások, Lapát, Panedlák: for most of us gathered here, these are words that we don’t usually use in our daily lives. Others may consider them as exotic, alien, funny and even backward. However, for indigenous kindred among us, these words denote an intimate identity and deep understanding of the world around them. It constitutes a broader knowledge system, be written or otherwise, which guides them in the management of resources within their ancestral land. This paper will provide a brief theoretical framework of the concepts of indigenous knowledge systems—hereinafter called IKS, and indigenous peoples food security, and hopefully a deeper or continued appreciation in the study of both concepts in general.

Understanding the impact of legislative changes on child car seating and restraint practices in regional Queensland

Road crashes contribute to a significant amount of child mortality and morbidity in Australia. In fact, passenger injuries contribute to the majority of child crash road trauma. A number of factors contribute to child injury and death in motor vehicles, including inappropriate seating position, inappropriate choice of restraint, and incorrect installation and use of child restraints. Prior to March 2010, child restraint legislation in Queensland only required children twelve months and younger to be seated in a properly adjusted and fastened child restraint. This legislation left older infants and young children potentially suboptimally protected. From March 2010, new legislation specified seating position and type of child restraint required, depending on the age of the child. This research was underpinned by the Health Belief Model (HBM), which explores health related behaviour, behaviour change, environmental factors influencing behaviour change (including legislative changes) and is flexible enough to be used in relation to parents' health practices for their children, rather than parent health directly. This thesis investigates the extent to which the changes to child restraint legislation have led parents in regional areas of Queensland to use appropriate restraint practices for their children and determines the extent to which the constructs of the HBM, parental perceptions, barriers and environmental factors contribute to the appropriateness of child seating and restraint use. Study One included three sets of observations taken in two regional cities of Queensland prior to the legislative amendment, during an educative period of six months, and after the enactment of the legislation. Each child's seating position and restraint type were recorded. Results showed that the proportion of children observed occupying the front seat decreased by 15.6 per cent with the announcement the legislation. There was no decrease in front seat use at the enactment of the legislation. The proportion of children observed using dedicated child restraints increased by 8.8 per cent with the announcement of the legislation when there was one child in the vehicle. Further, there was a 10.1 per cent increase in the proportion of children observed using a seat belt that fit with the announcement when there was one child in the vehicle and with the enactment of the legislation regardless of the number of children in the vehicle (21.8 per cent for one child, 39.7 per cent for two children and 40.2 per cent for three or more children). Study Two comprised initial intercept interviews, later followed up by telephone, with parents with children aged eight years and younger at the announcement and telephone interviews at the enactment of the legislation in one regional city in Queensland. Parents reported their child restraint practices, and opinions, knowledge and understanding of the requirements of the new legislation. Parent responses were analysed in terms of the constructs in the HBM. When asked which seating position their child 'usually' used, parents reported child front seat use was nil (0.0 per cent) and did not change with the enactment of the legislative amendment. However, when parents were asked whether they allowed children to use the front seat at some point within the six months prior to the interview, reported child front seat use was 7 (5.4 per cent) children at T2 and 10 (9.6 per cent) at T3. Reported use of age-appropriate child restraints did not increase with the enactment of the legislation (p = 0.77, ns). Parents reported restraint practices were classed as either appropriate or inappropriate. Parents who reported appropriate restraint practices were those whose children were sitting in optimal restraints and seating positions for their age according to the requirements of the legislation. Parents who reported inappropriate restraint practices were those who had one or more children who were suboptimally restrained or seated for their age according to the requirements of the legislation. Neither parents' perceptions about their susceptibility of being in a crash nor the likelihood of severity of child injury if involved in a crash yielded significant differences in the appropriateness of reported parent restraint practices over time with the enactment of the legislation. A trend in the data suggested parents perceived a benefit to using appropriate restraint practices was to avoid fines and demerit points. Over 75 per cent of parents who agreed that child restraints provide better protection for children than an adult seat belt reported appropriately seating and restraining their children (2 (1) = 8.093, p<.05). The self-efficacy measure regarding parents' confidence in installing a child restraint showed a significant association with appropriate parental restraint practices (2 (1) = 7.036, p<.05). Results suggested that some parents may have misinterpreted the announcement of the legislative amendment as the announcement of the enforcement of the legislation instead. Some parents who correctly reported details of the legislation did not report appropriate child restraint practices. This finding shows that parents' knowledge of the legislative amendment does not necessarily have an impact on their behaviour to appropriately seat and restrain children. The results of these studies have important implications for road safety and the prevention of road-related injury and death to children in Queensland. Firstly, parents reported feeling unsure of how to install restraints, which suggests that there may be children travelling in restraints that have not been installed correctly, putting them at risk. Interventions to alert and encourage parents to seek advice when unsure about the correct installation of child restraints could be considered. Secondly, some parents in this study although they were using the most appropriate restraint for their children, reported using a type that was not the most appropriate restraint for the child's age according to the legislation. This suggests that intervention may be effective in helping parents make a more accurate choice of the most appropriate type of restraint to use with children, especially as the child ages and child restraint requirements change. Further research could be conducted to ascertain the most effective methods of informing and motivating parents to use the most appropriate restraints and seating positions for their children, as these results show a concerning disparity between reported restraint practices and those that were observed.

Exploring the experiences of embedding Indigenous knowledge and perspectives on teaching practicum through interpretive phenomenology

In this paper, we discuss interpretive/hermeneutic phenomenology as a theoretical approach to explore the experiences of three stakeholder groups in embedding Indigenous knowledge and perspectives on teaching practicum, a project sponsored by ALTC. We begin by asking the phenomenological question ‘what is your experience of practice teaching?’ An open, explorative, phenomenological framework seeks the meanings of experiences, not truths, from the participants’ words themselves. Interpretive phenomenology is particularly suitable to explore educational experiences (Grumet, 1992; M. van Manen, 1990), as it provides rich ground for listening to the stakeholders’ lived experience and documenting it for interpretation. In an interpretive process, perspectives on lifeworlds, worldview and lenses get highlighted (Cunningham & Stanley, 2003). We establish how through various project stages, interpretive phenomenology gets to the essence of practice teaching experience creating a pedagogical ‘understanding’ of the essential nature of shared experience as lived by the participants (M van Manen, 2002). Thereby, it foregrounds voices of agency, dissent, acceptance and resistance. We consider how our research study focuses on the pedagogic voice of Indigenous pre-service teachers and the recognition of complex pedagogic fields in Indigenous education. We explain how this study seeks insights into their evaluation of pedagogic relations with two other education stakeholders – their practicum supervising teachers at schools and university staff involved practicum experience. As such, our study aims to support and develop long term, future-oriented opportunities for Indigenous pre-service teachers to embed Indigenous knowledge in the curricula. We conclude with some projections into the discourse on how Indigenous knowledge (IK) and perspectives might be diversely exemplified in pre-service teachers’ professional works (particularly E-portfolios). We speculate how this change could in turn maximise opportunities for Indigenous pre-service teachers, their supervising teachers and university staff to demonstrate leadership in their field through the creation of future tangible products such as units of work, resources, assessment and reflection tools. The processes contextualising the cultural interface of competing knowledge systems (Nakata, 2007) provide important analytical tools for understanding issues affecting student-teacher-mentor relationships occurring on practicum.

Knowledge and the City : Concepts, Applications and Trends of Knowledge-Based Urban Development

This book underlines the growing importance of knowledge for the competitiveness of cities and their regions. Examining the role of knowledge - in its economic, socio-cultural, spatial and institutional forms - for urban and regional development, identifying the preconditions for innovative use of urban and regional knowledge assets and resources, and developing new methods to evaluate the performance and potential of knowledge-based urban and regional development, the book provides an in-depth and comprehensive understanding of both theoretical and practical aspects of knowledge-based development and its implications and prospects for cities and regions.