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.

Implementing a Pattern and Structure Mathematics Awareness Program (PASMAP) in kindergarten

This paper provides an interim report of a large empirical evaluation study in progress. An intervention was implemented to evaluate the effectiveness of the Pattern and Structure Mathematical Awareness Program (PASMAP) on Kindergarten students’ mathematical development. Four large schools (two from Sydney and two from Brisbane), 16 teachers and their 316 students participated in the first phase of a 2-year longitudinal study. Eight of 16 classes implemented the PASMAP program over three school terms. This paper provides an overview of key aspects of the intervention, and preliminary analysis of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas.

Promoting student understanding through complex learning

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s learning potential, as increasingly complex learners capable of dealing with cognitively demanding tasks.

Bayesian multinomial logit: Theory and route choice example

Statisticians along with other scientists have made significant computational advances that enable the estimation of formerly complex statistical models. The Bayesian inference framework combined with Markov chain Monte Carlo estimation methods such as the Gibbs sampler enable the estimation of discrete choice models such as the multinomial logit (MNL) model. MNL models are frequently applied in transportation research to model choice outcomes such as mode, destination, or route choices or to model categorical outcomes such as crash outcomes. Recent developments allow for the modification of the potentially limiting assumptions of MNL such as the independence from irrelevant alternatives (IIA) property. However, relatively little transportation-related research has focused on Bayesian MNL models, the tractability of which is of great value to researchers and practitioners alike. This paper addresses MNL model specification issues in the Bayesian framework, such as the value of including prior information on parameters, allowing for nonlinear covariate effects, and extensions to random parameter models, so changing the usual limiting IIA assumption. This paper also provides an example that demonstrates, using route-choice data, the considerable potential of the Bayesian MNL approach with many transportation applications. This paper then concludes with a discussion of the pros and cons of this Bayesian approach and identifies when its application is worthwhile

Modeling the stepped potential discharge of primary alkaline battery cathodes

A novel model for the potentiostatic discharge of primary alkaline battery cathodes is presented. The model is used to simulate discharges resulting from the stepped potential electrochemical spectroscopy (SPECS) of primary alkaline battery cathodes cathodes, and the results are validated with experimental data. We show that a model based on a single (or mean) reaction framework can be used to simulate multi-reaction discharge behaviour and we develop a consistent functional modification to the kinetic equation of the model that allows for this to occur. The model is used to investigate the effects that the initial exchange current density, i00, and the diffusion coefficient for protons in electrolytic manganese dioxide (EMD), DH+, have on SPECS discharge. The behaviour observed is consistent with the idea that individual reduction reactions, within the multi-reaction, reduction behaviour of EMD, have distinct i00 and DH+ values.

The role of bricolage and resource constraints in high potential sustainability ventures

Sustainability decisions and their impacts may be among the greatest challenges facing the world in the 21st century (Davos 2000). Apart from adaptation on the part of established organizations these challenges are arguably going to require solutions developed by new actors However, young ventures have only recently begun generating research interest within sustainability literature (Shepherd et al. 2009). In particular, little is known about resource behaviours of these ventures and how they adapt to substantial resource constraints. One promising theory that has been identified as a way that some entrepreneurs manage constraints is bricolage: a construct defined as “making do by applying combinations of the resources at hand to new problems and opportunities” (Baker and Nelson 2005: 333). Bricolage may be critical as the means of continued venture success as these ventures are frequently developed in severe resource constraint, owing to higher levels of technical sophistication (Rothaermel and Deeds 2006). Further, they are often developed by entrepreneurs committed to personal and social goals of resourcefulness, including values that focus on conservation rather than consumption of resources (Shepherd et al. 2009). In this paper, using seven novel cases of high potential sustainability firms from CAUSEE we consider how constraints impact resource behaviours and further illustrate and extend bricolage domains previously developed by Baker and Nelson (2005) with recommendations for theory and practice provided.