Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Teaching and learning science and mathematics through technology practice

In this chapter we review studies of the engagement of students in design projects that emphasise integration of technology practice and the enabling sciences, which include physics and mathematics. We give special attention to affective and conceptual outcomes from innovative interventions of design projects. This is important work because of growing international concern that demand for professionals with technological expertise is increasing rapidly, while the supply of students willing to undertake the rigors of study in the enabling sciences is proportionally reducing (e.g., Barringtion, 2006; Hannover & Kessels, 2004; Yurtseven, 2002). The net effect is that the shortage in qualified workers is having a detrimental effect upon economic and social potential in Westernised countries (e.g., Department of Education, Science and Training [DEST], 2003; National Numeracy Review Panel and National Numeracy Review Secretarial, 2007; Yurtseven, 2002). Interestingly, this trend is reversed in developing economies including China and India (Anderson & Gilbride, 2003).

Analysing preservice teachers' potential for implementing engineering education in the middle school

Engineering is pivotal to any country's development. Yet there are insufficient engineers to take up available positions in many countries, including Australia (Engineers Australia, 2008). Engineering education is limited in Australia at the primary, middle and high school levels. One of the starting points for addressing this shortfall lies in preservice teacher education. This study explores second-year preservice teachers' potential to teach engineering in middle school, following their engagement with engineering concepts in their science curriculum unit and their teaching of engineering activities to Year 7 students. Using a literature-based pretest-posttest survey, items were categorised into four constructs (ie. personal professional attributes, student motivation, pedagogical knowledge and fused curricula). Results indicated that the preservice teachers' responses had not changed for instilling positive attitudes (88%) and accepting advice from colleagues (94%). However, there was statistical significance with 9 of the 25 survey items (p<0.05) after the preservice teachers' involvement in engineering activities. Fusing engineering education with other subjects, such as mathematics and science, is an essential first step in promoting preservice teachers' potential to implement engineering education in the middle school.

Theories of Mathematics Education : Seeking New Frontiers

This inaugural book in the new series Advances in Mathematics Education is the most up to date, comprehensive and avant garde treatment of Theories of Mathematics Education which use two highly acclaimed ZDM special issues on theories of mathematics education (issue 6/2005 and issue 1/2006), as a point of departure. Historically grounded in the Theories of Mathematics Education (TME group) revived by the book editors at the 29th Annual PME meeting in Melbourne and using the unique style of preface-chapter-commentary, this volume consist of contributions from leading thinkers in mathematics education who have worked on theory building. This book is as much summative and synthetic as well as forward-looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building. In addition a significant portion of the book includes newer developments in areas within mathematics education such as complexity theory, neurosciences, modeling, critical theory, feminist theory, social justice theory and networking theories. The 19 parts, 17 prefaces and 23 commentaries synergize the efforts of over 50 contributing authors scattered across the globe that are active in the ongoing work on theory development in mathematics education.

Surveying theories and philosophies of mathematics education

Any theory of thinking or teaching or learning rests on an underlying philosophy of knowledge. Mathematics education is situated at the nexus of two fields of inquiry, namely mathematics and education. However, numerous other disciplines interact with these two fields which compound the complexity of developing theories that define mathematics education. We first address the issue of clarifying a philosophy of mathematics education before attempting to answer whether theories of mathematics education are constructible? In doing so we draw on the foundational writings of Lincoln and Guba (1994), in which they clearly posit that any discipline within education, in our case mathematics education, needs to clarify for itself the following questions: (1) What is reality? Or what is the nature of the world around us? (2) How do we go about knowing the world around us? [the methodological question, which presents possibilities to various disciplines to develop methodological paradigms] and, (3) How can we be certain in the “truth” of what we know? [the epistemological question]

Politicizing mathematics education : has politics gone too far? Or not far enough?

In this chapter we tackle increasingly sensitive questions in mathematics education, those that have polarized the community into distinct schools of thought as well as impacted reform efforts.

A constraints-led perspective to understanding skill acquisition and game play: A basis for integration of motor learning theory and physical education praxis?

Background: In order to design appropriate environments for performance and learning of movement skills, physical educators need a sound theoretical model of the learner and of processes of learning. In physical education, this type of modelling informs the organization of learning environments and effective and efficient use of practice time. An emerging theoretical framework in motor learning, relevant to physical education, advocates a constraints-led perspective for acquisition of movement skills and game play knowledge. This framework shows how physical educators could use task, performer and environmental constraints to channel acquisition of movement skills and decision making behaviours in learners. From this viewpoint, learners generate specific movement solutions to satisfy the unique combination of constraints imposed on them, a process which can be harnessed during physical education lessons. Purpose: In this paper the aim is to provide an overview of the motor learning approach emanating from the constraints-led perspective, and examine how it can substantiate a platform for a new pedagogical framework in physical education: nonlinear pedagogy. We aim to demonstrate that it is only through theoretically valid and objective empirical work of an applied nature that a conceptually sound nonlinear pedagogy model can continue to evolve and support research in physical education. We present some important implications for designing practices in games lessons, showing how a constraints-led perspective on motor learning could assist physical educators in understanding how to structure learning experiences for learners at different stages, with specific focus on understanding the design of games teaching programmes in physical education, using exemplars from Rugby Union and Cricket. Findings: Research evidence from recent studies examining movement models demonstrates that physical education teachers need a strong understanding of sport performance so that task constraints can be manipulated so that information-movement couplings are maintained in a learning environment that is representative of real performance situations. Physical educators should also understand that movement variability may not necessarily be detrimental to learning and could be an important phenomenon prior to the acquisition of a stable and functional movement pattern. We highlight how the nonlinear pedagogical approach is student-centred and empowers individuals to become active learners via a more hands-off approach to learning. Summary: A constraints-based perspective has the potential to provide physical educators with a framework for understanding how performer, task and environmental constraints shape each individual‟s physical education. Understanding the underlying neurobiological processes present in a constraints-led perspective to skill acquisition and game play can raise awareness of physical educators that teaching is a dynamic 'art' interwoven with the 'science' of motor learning theories.

Examination of social demographic status, disease status, health behaviours and quality of life in older Australians using the theory of planned behaviour model

The increase of life expectancy worldwide during the last three decades has increased age-related disability leading to the risk of loss of quality of life. How to improve quality of life including physical health and mental health for older people and optimize their life potential has become an important health issue. This study used the Theory of Planned Behaviour Model to examine factors influencing health behaviours, and the relationship with quality of life. A cross-sectional mailed survey of 1300 Australians over 50 years was conducted at the beginning of 2009, with 730 completed questionnaires returned (response rate 63%). Preliminary analysis reveals that physiological changes of old age, especially increasing waist circumference and co morbidity was closely related to health status, especially worse physical health summary score. Physical activity was the least adherent behaviour among the respondents compared to eating healthy food and taking medication regularly as prescribed. Increasing number of older people living alone with co morbidity of disease may be the barriers that influence their attitude and self control toward physical activity. A multidisciplinary and integrated approach including hospital and non hospital care is required to provide appropriate services and facilities toward older people.

Individual differences in learning from worked examples by senior secondary mathematics students

The primary purpose of this research was to examine individual differences in learning from worked examples. By integrating cognitive style theory and cognitive load theory, it was hypothesised that an interaction existed between individual cognitive style and the structure and presentation of worked examples in their effect upon subsequent student problem solving. In particular, it was hypothesised that Analytic-Verbalisers, Analytic-Imagers, and Wholist-lmagers would perform better on a posttest after learning from structured-pictorial worked examples than after learning from unstructured worked examples. For Analytic-Verbalisers it was reasoned that the cognitive effort required to impose structure on unstructured worked examples would hinder learning. Alternatively, it was expected that Wholist-Verbalisers would display superior performances after learning from unstructured worked examples than after learning from structured-pictorial worked examples. The images of the structured-pictorial format, incongruent with the Wholist-Verbaliser style, would be expected to split attention between the text and the diagrams. The information contained in the images would also be a source of redundancy and not easily ignored in the integrated structured-pictorial format. Despite a number of authors having emphasised the need to include individual differences as a fundamental component of problem solving within domainspecific subjects such as mathematics, few studies have attempted to investigate a relationship between mathematical or science instructional method, cognitive style, and problem solving. Cognitive style theory proposes that the structure and presentation of learning material is likely to affect each of the four cognitive styles differently. No study could be found which has used Riding's (1997) model of cognitive style as a framework for examining the interaction between the structural presentation of worked examples and an individual's cognitive style. 269 Year 12 Mathematics B students from five urban and rural secondary schools in Queensland, Australia participated in the main study. A factorial (three treatments by four cognitive styles) between-subjects multivariate analysis of variance indicated a statistically significant interaction. As the difficulty of the posttest components increased, the empirical evidence supporting the research hypotheses became more pronounced. The rigour of the study's theoretical framework was further tested by the construction of a measure of instructional efficiency, based on an index of cognitive load, and the construction of a measure of problem-solving efficiency, based on problem-solving time. The consistent empirical evidence within this study that learning from worked examples is affected by an interaction of cognitive style and the structure and presentation of the worked examples emphasises the need to consider individual differences among senior secondary mathematics students to enhance educational opportunities. Implications for teaching and learning are discussed and recommendations for further research are outlined.

Are you a mindful driver? A review of the potential explanatory value of mindfulness in predicting speeding behaviour

Mindfulness is a concept which has been widely used in studies on consciousness, but has recently been applied to the understanding of behaviours in other areas, including clinical psychology, meditation, physical activity, education and business. It has been suggested that mindfulness can also be applied to road safety, though this has not yet been researched. A standard definition of mindfulness is “paying attention in a particular way, on purpose in the present moment and non-judgemental to the unfolding of experience moment by moment” [1]. Scales have been developed to measure mindfulness; however, there are different views in the literature on the nature of the mindfulness construct. This paper reviews the issues raised in the literature and arrives at an operational definition of mindfulness considered relevant to road safety. It is further proposed that mindfulness is best construed as operating together with other psychosocial factors to influence road safety behaviours. The specific case of speeding behaviour is outlined, where the psychosocial variables in the Theory of Planned Behaviour (TPB) have been demonstrated to predict both intention to speed and actual speeding behaviour. A role is proposed for mindfulness in enhancing the explanatory and predictive powers of the TPB concerning speeding. The implications of mindfulness for speeding countermeasures are discussed and a program of future research is outlined.

A grounded theory study of technology appropriation in anaesthesia

The human-technology nexus is a strong focus of Information Systems (IS) research; however, very few studies have explored this phenomenon in anaesthesia. Anaesthesia has a long history of adoption of technological artifacts, ranging from early apparatus to present-day information systems such as electronic monitoring and pulse oximetry. This prevalence of technology in modern anaesthesia and the rich human-technology relationship provides a fertile empirical setting for IS research. This study employed a grounded theory approach that began with a broad initial guiding question and, through simultaneous data collection and analysis, uncovered a core category of technology appropriation. This emergent basic social process captures a central activity of anaesthestists and is supported by three major concepts: knowledge-directed medicine, complementary artifacts and culture of anaesthesia. The outcomes of this study are: (1) a substantive theory that integrates the aforementioned concepts and pertains to the research setting of anaesthesia and (2) a formal theory, which further develops the core category of appropriation from anaesthesia-specific to a broader, more general perspective. These outcomes fulfill the objective of a grounded theory study, being the formation of theory that describes and explains observed patterns in the empirical field. In generalizing the notion of appropriation, the formal theory is developed using the theories of Karl Marx. This Marxian model of technology appropriation is a three-tiered theoretical lens that examines appropriation behaviours at a highly abstract level, connecting the stages of natural, species and social being to the transition of a technology-as-artifact to a technology-in-use via the processes of perception, orientation and realization. The contributions of this research are two-fold: (1) the substantive model contributes to practice by providing a model that describes and explains the human-technology nexus in anaesthesia, and thereby offers potential predictive capabilities for designers and administrators to optimize future appropriations of new anaesthetic technological artifacts; and (2) the formal model contributes to research by drawing attention to the philosophical foundations of appropriation in the work of Marx, and subsequently expanding the current understanding of contemporary IS theories of adoption and appropriation.