Mathematical thinking of preschool children in rural and regional Australia : research and practice

At what age do young children begin thinking mathematically? Can young children work on mathematical problems? How do early childhood educators ensure young children feel good about mathematics? Where do early childhood educators learn about suitable mathematics activities?

A good early childhood start in mathematics is critical for later mathematics success. Parents, carers and early childhood educators are teaching mathematics, either consciously or unconsciously, in any social interaction with a child.

Mathematical Thinking of Preschool Children in Rural and Regional Australia is an extension of a conference of Australian and New Zealand researchers that identified a number of important problems related to the mathematical learning of children prior to formal schooling. A project team of 11 researchers from top Australian universities sought to investigate how early childhood education can best have a positive influence on early mathematics learning.

The investigation complements and extends the work of Project Good Start by focusing attention on critical aspects of parents, carers and early childhood educators who care for young children. Early childhood educators from regional and rural New South Wales, Queensland and Victoria were interviewed, following a set of structured questions. The questions focused on: children’s mathematics learning; support for mathematics teaching; use of technology; attitudes to mathematics; and assessment and record keeping.

The researchers also reviewed research focusing on the mathematical capacities and potential foundations for further mathematical development in young children (0–5 years) published in the last decade and produced an annotated bibliography. This should provide a good basis for further research and reading.

Based upon the results of this investigation, the researchers make 11 recommendations for improving the practices of early childhood education centres in relation to young children’s mathematical thinking and development. The implications for policy and decision makers are outlined for teacher education, the provision of resources and further research.

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation : public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study (LPS) documented the voicing of mathematical ideas in public discussion and in teacher-student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher-student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation are central to the development of any theory of mathematics instruction.

Developing and using an S3R model to analyze reasoning in web-based cross-national exchanges on sustainability

Within the increasing body of research that examines students' reasoning on socioscientific issues, we consider in particular student reasoning concerning acute, open-ended questions that bring out the complexities and uncertainties embedded in ill-structured problems. In this paper, we propose a socioscientific sustainability reasoning (S3R) model to analyze students' reasoning exchanges on environmental socially acute questions (ESAQs). The paper describes the development of an epistemological analysis of how sustainability perspectives can be integrated into socioscientific reasoning, which emphasizes the need for S3R to be both grounded in context and collective. We argue the complexity of ESAQs requires a consideration of multiple dimensions that form the basis of our S3R analysis model: problematization, interactions, knowledge, uncertainties, values, and governance. For each dimension, in the model we have identified indicators of four levels of complexity. We investigated the usefulness of the model in identifying improvements in reasoning that flow from cross-national web-based exchanges between groups of French and Australian students, concerning a local and a global ESAQ. The S3R model successfully captured the nature of reasoning about socioscientific sustainability issues, with the collective negotiation of multiple forms of knowledge as a key characteristic in improving reasoning levels. The paper provides examples of collaborative argumentation in collective texts (wikis) to illustrate the various levels of reasoning in each dimension, and diagrammatic representation of the evolution of collective reflections. We observe that a staged process of construction and confrontation, involving groups representing to some extent different cultural and contextual stances, is powerful in eliciting reasoned argument of enhanced quality. © 2014 Wiley Periodicals, Inc.

Building fuzzy inference systems with similarity reasoning : NSGAII-based fuzzy rule selection and evidential functions

In our previous investigations, two Similarity Reasoning (SR)-based frameworks for tackling real-world problems have been proposed. In both frameworks, SR is used to deduce unknown fuzzy rules based on similarity of the given and unknown fuzzy rules for building a Fuzzy Inference System (FIS). In this paper, we further extend our previous findings by developing (1) a multi-objective evolutionary model for fuzzy rule selection; and (2) an evidential function to facilitate the use of both frameworks. The Non-Dominated Sorting Genetic Algorithms-p (NSGA-p) is adopted for fuzzy rule selection, in accordance with the Pareto optimal criterion. Besides that, two new evidential functions are developed, whereby given fuzzy rules are considered as evidence. Simulated and benchmark examples are included to demonstrate the applicability of these suggestions. Positive results were obtained.

Imago veritas falsa: for a (post-)Schmittian decisionist theory of law, legal reasoning, and judging

For decades, while approaching the ‘normativism/pragmatism’ divide and discussing the legitimacy of (and opportunity for) the judge to act as a ‘social engineer’, socio-legal scholars have tried to ascertain whether the jurist should also consider the impact of his/her activity on society at large, and if so, why and to what extent. The present contribution understands instead the law in terms of a structurally incomplete image (imago veritas falsa) which always needs the decisive intervention of the legal interpreter to exercise its performative instances. In particular, by adopting an unconventional theoretico-philosophical approach that transcends the classic boundaries of foundationalist metaphysics as expressed by the dichotomy of Western logic, this paper argues for the necessity of a tertium comparationis capable of explaining that the real essence of law, legal reasoning, and judging is neither that of normativism, nor of pragmatism, but rather of (post-)Schmittian decisionism.

A universal optimization strategy for ant colony optimization algorithms based on the Physarum-inspired mathematical model

Ant colony optimization (ACO) algorithms often fall into the local optimal solution and have lower search efficiency for solving the travelling salesman problem (TSP). According to these shortcomings, this paper proposes a universal optimization strategy for updating the pheromone matrix in the ACO algorithms. The new optimization strategy takes advantages of the unique feature of critical paths reserved in the process of evolving adaptive networks of the Physarum-inspired mathematical model (PMM). The optimized algorithms, denoted as PMACO algorithms, can enhance the amount of pheromone in the critical paths and promote the exploitation of the optimal solution. Experimental results in synthetic and real networks show that the PMACO algorithms are more efficient and robust than the traditional ACO algorithms, which are adaptable to solve the TSP with single or multiple objectives. Meanwhile, we further analyse the influence of parameters on the performance of the PMACO algorithms. Based on these analyses, the best values of these parameters are worked out for the TSP.

A case study on developing a teacher's capacity in mathematical modelling

Research has indicated that for teachers to facilitate mathematical modelling activities in the mathematics classroom, they need to be familiar with the process of mathematical modelling. As such, it is imperative that teachers experience the whole mathematical modelling process. This paper reports on a Multi-tiered Teaching Experiment designed to help a teacher develop his capacity in the domain of mathematical modelling. Drawing on part of a larger case-based study conducted using Design Research phases situated within the Multi-tiered Teaching Experiment framework, the purpose of this paper is to exemplify how the research design fostered growth in teacher capacity through the natural development of critical moments of learning by the teacher during interactions between the researchers and the teacher-modeller himself. The potential of the Multi-tiered Teaching Experiment as a useful non-prescriptive teacher development approach building upon the existing repertoire of individual teachers will be discussed.

A multi-objective ant colony optimization algorithm based on the physarum-inspired mathematical model

Multi-objective traveling salesman problem (MOTSP) is an important field in operations research, which has wide applications in the real world. Multi-objective ant colony optimization (MOACO) as one of the most effective algorithms has gained popularity for solving a MOTSP. However, there exists the problem of premature convergence in most of MOACO algorithms. With this observation in mind, an improved multiobjective network ant colony optimization, denoted as PMMONACO, is proposed, which employs the unique feature of critical tubes reserved in the network evolution process of the Physarum-inspired mathematical model (PMM). By considering both pheromones deposited by ants and flowing in the Physarum network, PM-MONACO uses an optimized pheromone matrix updating strategy. Experimental results in benchmark networks show that PM-MONACO can achieve a better compromise solution than the original MOACO algorithm for solving MOTSPs.

Developing teacher competencies through videos for facilitation of mathematical modelling in Singapore primary schools

Mathematical modelling tasks which are situated in real-world contexts encourage students to draw connections between school-based mathematics and the real-world, enhancing their engagement in learning. Such tasks often require varied interpretations of the real-world problem context resulting in multiple pathways of solutions. Although mathematical modelling has been introduced in the Singapore mathematics curriculum since 2007, its incorporation in schools has been limited. One reason for this could be that teachers are challenged by how best to facilitate for rich student mathematisation processes during such tasks. This chapter reports how a multi-tiered teaching experiment using design research methodology was conducted to build teachers’ capacity in designing, facilitating, and evaluating student mathematisation during mathematical modelling tasks with an intact class of Primary 5 students (aged 10-11). The use of videos was critical because grounded images helped capture the dynamics and complexity of authentic classroom interactions. This chapter highlights how video recordings of teacher-student interactions during a modelling task were harnessed during design methodology cycles, particularly during the Retrospective Analysis phase, to activate critical moments of learning for the teacher towards developing her competencies in facilitating students’ mathematisation processes.

Core components of communication of clinical reasoning: a qualitative study with experienced Australian physiotherapists

Communication is an important area in health professional education curricula, however it has been dealt with as discrete skills that can be learned and taught separate to the underlying thinking. Communication of clinical reasoning is a phenomenon that has largely been ignored in the literature. This research sought to examine how experienced physiotherapists communicate their clinical reasoning and to identify the core processes of this communication. A hermeneutic phenomenological research study was conducted using multiple methods of text construction including repeated semi-structured interviews, observation and written exercises. Hermeneutic analysis of texts involved iterative reading and interpretation of texts with the development of themes and sub-themes. Communication of clinical reasoning was perceived to be complex, dynamic and largely automatic. A key finding was that articulating reasoning (particularly during research) does not completely represent actual reasoning processes but represents a (re)construction of the more complex, rapid and multi-layered processes that operate in practice. These communications are constructed in ways that are perceived as being most relevant to the audience, context and purpose of the communication. Five core components of communicating clinical reasoning were identified: active listening, framing and presenting the message, matching the co-communicator, metacognitive aspects of communication and clinical reasoning abilities. We propose that communication of clinical reasoning is both an inherent part of reasoning as well as an essential and complementary skill based on the contextual demands of the task and situation. In this way clinical reasoning and its communication are intertwined, providing evidence for the argument that they should be learned (and explicitly taught) in synergy and in context.

Promoting clinical reasoning in general practice trainees: role of the clinical teacher

BACKGROUND: Clinical reasoning requires knowledge, cognition and metacognition, and is contextually bound. Clinical teachers can and should play a key role in explicitly promoting clinical reasoning.

CONTEXT: The aim of this article is to relate the clinical reasoning literature to the general practice or family medicine context, and to provide clinical teachers with strategies to promote clinical reasoning.

INNOVATION: It is important that the clinical teacher teaches trainees the specific skills sets of the expert general practitioner (e.g. synthesising skills, recognising prototypes, focusing on cues and clues, using community resources and dealing with uncertainty) in order to promote clinical reasoning in the context of general practice or family medicine. Clinical teachers need to understand their own reasoning processes as well as be able to convey that knowledge to their trainees. They also need to understand the developmental stages of clinical reasoning and be able to nurture each trainee's own expertise. Strategies for facilitating effective clinical reasoning in trainees include adequate exposure to patients, offering the trainees opportunity for reflection and feedback, and coaching on the techniques of reasoning in the general practice context.

IMPLICATIONS: The journey to expertise in clinical reasoning is unique to each clinician, with different skills developing at different rates, depending on content, context and past experience. Doctors enter into general practice training with the building blocks of biomedical and clinical knowledge and a desire to learn how to be a general practitioner. Clinical teachers are integral in the process of helping trainees learn how to 'think like a general practitioner'.

Clinical reasoning sessions: back to the patient

BACKGROUND: Problem-based learning (PBL) was developed as a facilitated small group learning process based around a clinical problem. Originally designed for pre-clinical years of medical education, its application across all years poses a number of difficulties, including the risk of reducing patient contact, providing a learning process that is skewed towards an understanding of pathophysiological processes, which may not be well understood in all areas of medicine, and failing to provide exposure to clinically relevant reasoning skills. CONTEXT: Curriculum review identified dissatisfaction with PBLs in the clinical years of the Sydney Medical School's Graduate Medical Program, from both staff and students. A new model was designed and implemented in the Psychiatry and Addiction Medicine rotation, and is currently being evaluated. INNOVATION: We describe an innovative model of small-group, student-generated, case-based learning in psychiatry - clinical reasoning sessions (CRS) - led by expert facilitators. IMPLICATIONS: The CRS format returns the student to the patient, emphasises clinical assessment skills and considers treatment in the real-world context of the patient. Students practise a more sophisticated reasoning process with real patients modelled upon that of their expert tutor. This has increased student engagement compared with the previous PBL programme.

Using hermeneutic phenomenology to investigate how experienced practitioners learn to communicate clinical reasoning

This paper is primarily targeted at doctoral students and other researchers considering using hermeneutic phenomenology as a research strategy. We present interpretive paradigm research designed to investigate how experienced practitioners learn to communicate their clinical reasoning in professional practice. Twelve experienced physiotherapy practitioners participated in this research. Using hermeneutic phenomenology enabled access to a phenomenon that is often subconscious and provided a means of interpreting participants’ experiences of personal learning journeys. Within the philosophy underpinning hermeneutic phenomenology, researchers need to design a research strategy that flows directly from the research question and goals of the research project. This paper explores such a strategy.