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.

Customer-oriented deviance (COD) : a conceptualisation, development and validation of a behavioural measure

Frontline employee behaviours are recognised as vital for achieving a competitive advantage for service organisations. The services marketing literature has comprehensively examined ways to improve frontline employee behaviours in service delivery and recovery. However, limited attention has been paid to frontline employee behaviours that favour customers in ways that go against organisational norms or rules. This study examines these behaviours by introducing a behavioural concept of Customer-Oriented Deviance (COD). COD is defined as, “frontline employees exhibiting extra-role behaviours that they perceive to defy existing expectations or prescribed rules of higher authority through service adaptation, communication and use of resources to benefit customers during interpersonal service encounters.” This thesis develops a COD measure and examines the key determinants of these behaviours from a frontline employee perspective. Existing research on similar behaviours that has originated in the positive deviance and pro-social behaviour domains has limitations and is considered inadequate to examine COD in the services context. The absence of a well-developed body of knowledge on non-conforming service behaviours has implications for both theory and practice. The provision of ‘special favours’ increases customer satisfaction but the over-servicing of customers is also counterproductive for the service delivery and costly for the organisation. Despite these implications of non-conforming service behaviours, there is little understanding about the nature of these behaviours and its key drivers. This research builds on inadequacies in prior research on positive deviance, pro-social and pro-customer literature to develop the theoretical foundation of COD. The concept of positive deviance which has predominantly been used to study organisational behaviours is applied within a services marketing setting. Further, it addresses previous limitations in pro-social and pro-customer behavioural literature that has examined limited forms of behaviours with no clear understanding on the nature of these behaviours. Building upon these literature streams, this research adopts a holistic approach towards the conceptualisation of COD. It addresses previous shortcomings in the literature by providing a well bounded definition, developing a psychometrically sound measure of COD and a conceptually well-founded model of COD. The concept of COD was examined across three separate studies and based on the theoretical foundations of role theory and social identity theory. Study 1 was exploratory and based on in-depth interviews using the Critical Incident Technique (CIT). The aim of Study 1 was to understand the nature of COD and qualitatively identify its key drivers. Thematic analysis was conducted to analyse the data and the two potential dimensions of COD behaviours of Deviant Service Adaptation (DSA) and Deviant Service Communication (DSC) were revealed in the analysis. In addition, themes representing the potential influences of COD were broadly classified as individual factors, situational factors, and organisational factors. Study 2 was a scale development procedure that involved the generation and purification of items for the measure based on two student samples working in customer service roles (Pilot sample, N=278; Initial validation sample, N=231). The results for the reliability and Exploratory Factor Analyses (EFA) on the pilot sample suggested the scale had poor psychometric properties. As a result, major revisions were made in terms of item wordings and new items were developed based on the literature to reflect a new dimension, Deviant Use of Resources (DUR). The revised items were tested on the initial validation sample with the EFA analysis suggesting a four-factor structure of COD. The aim of Study 3 was to further purify the COD measure and test for nomological validity based on its theoretical relationships with key antecedents and similar constructs (key correlates). The theoretical model of COD consisting of nine hypotheses was tested on a retail and hospitality sample of frontline employees (Retail N=311; Hospitality N=305) of a market research panel using an online survey. The data was analysed using Structural Equation Modelling (SEM). The results provided support for a re-specified second-order three-factor model of COD which consists of 11 items. Overall, the COD measure was found to be reliable and valid, demonstrating convergent validity, discriminant validity and marginal partial invariance for the factor loadings. The results showed support for nomological validity, although the antecedents had differing impact on COD across samples. Specifically, empathy and perspective-taking, role conflict, and job autonomy significantly influenced COD in the retail sample, whereas empathy and perspective-taking, risk-taking propensity and role conflict were significant predictors in the hospitality sample. In addition, customer orientation-selling orientation, the altruistic dimension of organisational citizenship behaviours, workplace deviance, and social desirability responding were found to correlate with COD. This research makes several contributions to theory. First, the findings of this thesis extend the literature on positive deviance, pro-social and pro-customer behaviours. Second, the research provides an empirically tested model which describes the antecedents of COD. Third, this research contributes by providing a reliable and valid measure of COD. Finally, the research investigates the differential effects of the key antecedents in different service sectors on COD. The research findings also contribute to services marketing practice. Based on the research findings, service practitioners can better understand the phenomenon of COD and utilise the measurement tool to calibrate COD levels within their organisations. Knowledge on the key determinants of COD will help improve recruitment and training programs and drive internal initiatives within the firm.

Do board contacts matter? : an analysis of the relationship between boards of directors’ ties and the performance of Australia’s largest companies.

Boards of directors are thought to provide access to a wealth of knowledge and resources for the companies they serve, and are considered important to corporate governance. Under the Resource Based View (RBV) of the firm (Wernerfelt, 1984) boards are viewed as a strategic resource available to firms. As a consequence there has been a significant research effort aimed at establishing a link between board attributes and company performance. In this thesis I explore and extend the study of interlocking directorships (Mizruchi, 1996; Scott 1991a) by examining the links between directors’ opportunity networks and firm performance. Specifically, I use resource dependence theory (Pfeffer & Salancik, 1978) and social capital theory (Burt, 1980b; Coleman, 1988) as the basis for a new measure of a board’s opportunity network. I contend that both directors’ formal company ties and their social ties determine a director’s opportunity network through which they are able to access and mobilise resources for their firms. This approach is based on recent studies that suggest the measurement of interlocks at the director level, rather than at the firm level, may be a more reliable indicator of this phenomenon. This research uses publicly available data drawn from Australia’s top-105 listed companies and their directors in 1999. I employ Social Network Analysis (SNA) (Scott, 1991b) using the UCINET software to analyse the individual director’s formal and social networks. SNA is used to measure a the number of ties a director has to other directors in the top-105 company director network at both one and two degrees of separation, that is, direct ties and indirect (or ‘friend of a friend’) ties. These individual measures of director connectedness are aggregated to produce a board-level network metric for comparison with measures of a firm’s performance using multiple regression analysis. Performance is measured with accounting-based and market-based measures. Findings indicate that better-connected boards are associated with higher market-based company performance (measured by Tobin’s q). However, weaker and mostly unreliable associations were found for accounting-based performance measure ROA. Furthermore, formal (or corporate) network ties are a stronger predictor of market performance than total network ties (comprising social and corporate ties). Similarly, strong ties (connectedness at degree-1) are better predictors of performance than weak ties (connectedness at degree-2). My research makes four contributions to the literature on director interlocks. First, it extends a new way of measuring a board’s opportunity network based on the director rather than the company as the unit of interlock. Second, it establishes evidence of a relationship between market-based measures of firm performance and the connectedness of that firm’s board. Third, it establishes that director’s formal corporate ties matter more to market-based firm performance than their social ties. Fourth, it establishes that director’s strong direct ties are more important to market-based performance than weak ties. The thesis concludes with implications for research and practice, including a more speculative interpretation of these results. In particular, I raise the possibility of reverse causality – that is networked directors seek to join high-performing companies. Thus, the relationship may be a result of symbolic action by companies seeking to increase the legitimacy of their firms rather than a reflection of the social capital available to the companies. This is an important consideration worthy of future investigation.

An epidemic of pain in an Australian radiology practice

This paper presents findings from a study of an organisationally mandated assimilation process of an enterprise-wide information system in a radiology practice in Australia. A number of interviews with radiologists, radiographers and administrative staff are used to explore the impact of institutional structures on the assimilation process. The case study develops an argument that culture within and outside the Australian Radiology Practice (ARP), social structures within the ARP and organisational-level management mandates have impacted on the assimilation process. The study develops a theoretical framework that integrates elements of social actor theory (Lamb & Kling, 2003) to provide a more fine-grained analysis concentrating on the relationship among the radiology practitioners, the technology (an enterprise-wide Health Information System) and a larger social milieu surrounding its use. This study offers several theoretical and practical implications for technology assimilation in the health and radiology industry regarding the important roles social interactions, individual self-perceptions, organisational mandates and policies can play in assimilating new ICTs.

On the Classification of Recursive Languages

A one-sided classifier for a given class of languages converges to 1 on every language from the class and outputs 0 infinitely often on languages outside the class. A two-sided classifier, on the other hand, converges to 1 on languages from the class and converges to 0 on languages outside the class. The present paper investigates one-sided and two-sided classification for classes of recursive languages. Theorems are presented that help assess the classifiability of natural classes. The relationships of classification to inductive learning theory and to structural complexity theory in terms of Turing degrees are studied. Furthermore, the special case of classification from only positive data is also investigated.

The stewardship paradigm : an enquiry into the ethical obligation associated with being in control of resorces

The resource allocation and utilization discourse is dominated by debates about rights particularly individual property rights and ownership. This is due largely to the philosophic foundations provided by Hobbes and Locke and adopted by Bentham. In our community, though, resources come not merely with rights embedded but also obligations. The relevant laws and equitable principles which give shape to our shared rights and obligations with respect to resources take cognizance not merely of the title to the resource (the proprietary right) but the particular context in which the right is exercised. Moral philosophy regarding resource utilisation has from ancient times taken cognizance of obligations but with ascendance of modernity, the agenda of moral philosophy regarding resources, has been dominated, at least since John Locke, by a preoccupation with property rights; the ethical obligations associated with resource management have been largely ignored. The particular social context has also been ignored. Exploring this applied ethical terrain regarding resource utilisation, this thesis: (1) Revisits the justifications for modem property rights (and in that the exclusion of obligations); (2) Identifies major deficiencies in these justifications and reasons for this; (3) Traces the concept of stewardship as understood in classical Greek writing and in the New Testament, and considers its application in the Patristic period and by Medieval and reformist writers, before turning to investigate its influence on legal and equitable concepts through to the current day; 4) Discusses the nature of the stewardship obligation,maps it and offers a schematic for applying the Stewardship Paradigm to problems arising in daily life; and, (5) Discusses the way in which the Stewardship Paradigm may be applied by, and assists in resolving issues arising from within four dominant philosophic world views: (a) Rawls' social contract theory; (b) Utilitarianism as discussed by Peter Singer; (c) Christianity with particular focus on the theology of Douglas Hall; (d) Feminism particularly as expressed in the ethics of care of Carol Gilligan; and, offers some more general comments about stewardship in the context of an ethically plural community.

On the data consumption benefits of accepting increased uncertainty

In the context of learning paradigms of identification in the limit, we address the question: why is uncertainty sometimes desirable? We use mind change bounds on the output hypotheses as a measure of uncertainty, and interpret ‘desirable’ as reduction in data memorization, also defined in terms of mind change bounds. The resulting model is closely related to iterative learning with bounded mind change complexity, but the dual use of mind change bounds — for hypotheses and for data — is a key distinctive feature of our approach. We show that situations exists where the more mind changes the learner is willing to accept, the lesser the amount of data it needs to remember in order to converge to the correct hypothesis. We also investigate relationships between our model and learning from good examples, set-driven, monotonic and strong-monotonic learners, as well as class-comprising versus class-preserving learnability.

MobileMums: A randomized controlled trial of an SMS-based physical activity intervention

Background Postnatal women (<12 months postpartum) are at increased risk of physical inactivity. Purpose To evaluate the efficacy and feasibility of a theory-based physical activity (PA) intervention delivered to postnatal women primarily via mobile telephone short message service (SMS). Methods Eighty-eight women were randomized to the intervention (n=45) or minimal contact control (n=43) condition. The 12-week intervention consisted of a face-to-face PA goal-setting consultation, a goal-setting magnet, three to five personally tailored SMS/week and a nominated support person who received two SMS per week. SMS content targeted constructs of social cognitive theory. Frequency (days/week) and duration (min/week) of PA participation and walking for exercise were assessed via self-report at baseline, 6 and 13 weeks. Results Intervention participants increased PA frequency by 1.82 days/week (SE±0.18) by 13 weeks (F(2,85)=4.46, p=0.038) and walking for exercise frequency by 1.08 days/ week (SE±0.24) by 13 weeks (F(2,85)=5.38, p=0.02). Positive trends were observed for duration (min/week) of PA and walking for exercise. Conclusions Intervention exposure resulted in increased frequency of PA and walking for exercise in postnatal women.

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 Stochastic View of Optimal Regret through Minimax Duality

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional--the minimizer over the player's actions of expected loss--defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary. Peter L. Bartlett, Alexander Rakhlin

Physical activity and nutrition program for seniors (PANS) : protocol of a randomized controlled trial

Background Along with reduced levels of physical activity, older Australian's mean energy consumption has increased. Now over 60% of older Australians are considered overweight or obese. This study aims to confirm if a low-cost, accessible physical activity and nutrition program can improve levels of physical activity and diet of insufficiently active 60-70 year-olds. Methods/Design This 12-month home-based randomised controlled trial (RCT) will consist of a nutrition and physical activity intervention for insufficiently active people aged 60 to 70 years from low to medium socio-economic areas. Six-hundred participants will be recruited from the Australian Federal Electoral Role and randomly assigned to the intervention (n = 300) and control (n = 300) groups. The study is based on the Social Cognitive Theory and Precede-Proceed Model, incorporating voluntary cooperation and self-efficacy. The intervention includes a specially designed booklet that provides participants with information and encourages dietary and physical activity goal setting. The booklet will be supported by an exercise chart, calendar, bi-monthly newsletters, resistance bands and pedometers, along with phone and email contact. Data will be collected over three time points: pre-intervention, immediately post-intervention and 6-months post-study. Discussion This trial will provide valuable information for community-based strategies to improve older adults' physical activity and dietary intake. The project will provide guidelines for appropriate sample recruitment, and the development, implementation and evaluation of a minimal intervention program, as well as information on minimising barriers to participation in similar programs.

Living well with diabetes: a randomized controlled trial of a telephone-delivered intervention for maintenance of weight loss, physical activity and glycaemic control in adults with type 2 diabetes

Background By 2025, it is estimated that approximately 1.8 million Australian adults (approximately 8.4% of the adult population) will have diabetes, with the majority having type 2 diabetes. Weight management via improved physical activity and diet is the cornerstone of type 2 diabetes management. However, the majority of weight loss trials in diabetes have evaluated short-term, intensive clinic-based interventions that, while producing short-term outcomes, have failed to address issues of maintenance and broad population reach. Telephone-delivered interventions have the potential to address these gaps. Methods/Design Using a two-arm randomised controlled design, this study will evaluate an 18-month, telephone-delivered, behavioural weight loss intervention focussing on physical activity, diet and behavioural therapy, versus usual care, with follow-up at 24 months. Three-hundred adult participants, aged 20-75 years, with type 2 diabetes, will be recruited from 10 general practices via electronic medical records search. The Social-Cognitive Theory driven intervention involves a six-month intensive phase (4 weekly calls and 11 fortnightly calls) and a 12-month maintenance phase (one call per month). Primary outcomes, assessed at 6, 18 and 24 months, are: weight loss, physical activity, and glycaemic control (HbA1c), with weight loss and physical activity also measured at 12 months. Incremental cost-effectiveness will also be examined. Study recruitment began in February 2009, with final data collection expected by February 2013. Discussion This is the first study to evaluate the telephone as the primary method of delivering a behavioural weight loss intervention in type 2 diabetes. The evaluation of maintenance outcomes (6 months following the end of intervention), the use of accelerometers to objectively measure physical activity, and the inclusion of a cost-effectiveness analysis will advance the science of broad reach approaches to weight control and health behaviour change, and will build the evidence base needed to advocate for the translation of this work into population health practice.

Consumer misbehaviour during professional service encounters

To date, consumer behaviour research is still over-focused on the functional rather than the dysfunctional. Both empirical and anecdotal evidence suggest that service organisations are burdened with the concept of consumer sovereignty, while consumers freely flout the ‘rules’ of social exchange and behave in deviant and dysfunctional ways. Further, the current scope of consumer misbehaviour research suggests that the phenomenon has principally been studied in the context of economically-focused exchange. This limits our current understanding of consumer misbehaviour to service encounters that are more transactional than relational in nature. Consequently, this thesis takes a Social Exchange approach to consumer misbehaviour and reports a three-stage multi-method study that examined the nature and antecedents of consumer misbehaviour in professional services. It addresses the following broad research question: What is the nature of consumer misbehaviour during professional service encounters? Study One initially explored the nature of consumer misbehaviour in professional service encounters using critical incident technique (CIT) within 38 semi-structured in-depth interviews. The study was designed to develop a better understanding of what constitutes consumer misbehaviour from a service provider’s perspective. Once the nature of consumer misbehaviour had been qualified, Study Two focused on developing and refining calibrated items that formed Guttman-like scales for two consumer misbehaviour constructs: one for the most theoretically-central type of consumer misbehaviour identified in Study One (i.e. refusal to participate) and one for the most well-theorised and salient type of consumer misbehaviour (i.e. verbal abuse) identified in Study One to afford a comparison. This study used Rasch modelling to investigate whether it was possible to calibrate the escalating severity of a series of decontextualised behavioural descriptors in a valid and reliable manner. Creating scales of calibrated items that capture the variation in severity of different types of consumer misbehaviour identified in Study One allowed for a more valid and reliable investigation of the antecedents of such behaviour. Lastly, Study Three utilised an experimental design to investigate three key antecedents of consumer misbehaviour: (1) the perceived quality of the service encounter [drawn from Fullerton and Punj’s (1993) model of aberrant consumer behaviour], (2) the violation of consumers’ perceptions of justice and equity [drawn from Rousseau’s (1989) Psychological Contract Theory], and (3) consumers’ affective responses to exchange [drawn from Weiss and Cropanzano’s (1996) Affective Events Theory]. Investigating three key antecedents of consumer misbehaviour confirmed the newly-developed understanding of the nature of consumer misbehaviour during professional service encounters. Combined, the results of the three studies suggest that consumer misbehaviour is characteristically different within professional services. The most salient and theoretically-central behaviours can be measured using increasingly severe decontextualised behavioural descriptors. Further, increasingly severe forms of consumer misbehaviour are likely to occur as a response to consumer anger at low levels of interpersonal service quality. These findings have a range of key implications for both marketing theory and practice.

The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network

Sample complexity results from computational learning theory, when applied to neural network learning for pattern classification problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classification problem and the learning algorithm finds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feedforward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A3 √((log n)/m) (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training examples is considerably smaller than the number of weights. It also supports heuristics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples.

Bounded parameter Markov Decision Processes with average reward criterion

Bounded parameter Markov Decision Processes (BMDPs) address the issue of dealing with uncertainty in the parameters of a Markov Decision Process (MDP). Unlike the case of an MDP, the notion of an optimal policy for a BMDP is not entirely straightforward. We consider two notions of optimality based on optimistic and pessimistic criteria. These have been analyzed for discounted BMDPs. Here we provide results for average reward BMDPs. We establish a fundamental relationship between the discounted and the average reward problems, prove the existence of Blackwell optimal policies and, for both notions of optimality, derive algorithms that converge to the optimal value function.