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.

Incorporating design explanation within formal object-oriented method (FOOM)

Requirements engineering is a commencing phase in the development of either software applications or information systems. It is concerned with understanding and specifying the customer's requirements of the system to be delivered. Throughout the literature, this is agreed to be one of the most crucial and, unfortunately, problematic phases in development. Despite the diversity of research directions, approaches and methods, the question of process understanding and management is still limited. Among contemporary approaches to the improvement of the current practice of Requirements Engineering, Formal Object-Oriented Method (FOOM) has been introduced as a new promising solution. The FOOM approach to requirements engineering is based on a synthesis of socio-organisational theory, the object-oriented approach, and mathematical formal specification. The entire FOOM specification process is evolutionary and involves a large volume of changes in requirements. During this process, requirements evolve through various forms of informal, semi-formal, and formal while maintaining a semantic link between these forms and, most importantly, conforming to the customer's requirements. A deep understanding of the complexity of the requirements model and its dynamics is critical in improving requirements engineering process management. This thesis investigates the benefits of documenting both the evolution of the requirements model and the rationale for that evolution. Design explanation explains and justifies the deliberations of, and decisions made during, the design activity. In this thesis, design explanation is used to describe the requirements engineering process in order to improve understandability of, and traceability within, the evolving requirements specification. The design explanation recorded during this research project is also useful in assisting the researcher in gaining insights into the creativity and opportunistic characteristics of the requirements engineering process. This thesis offers an interpretive investigation into incorporating design explanation within FOOM in order to extend and advantage the method. The researcher's interpretation and analysis of collected data highlight an insight-driven and opportunistic process rather than a strictly and systematically predefined one. In fact, the process was not smoothly evolutionary, but involved occasional 'crisis' points at which the model was reconceptualised, simplified and restructured. Therefore, contributions of the thesis lie not only in an effective incorporation of design explanation within FOOM, but also a deep understanding of the dynamic process of requirements engineering. The new understanding of the complexity of the requirements model and its dynamics suggests new directions for future research and forms a basis for a new approach to process management.

Assessment of mathematical development in early childhood : some views of preschool practitioners in regional and rural Australia

As part of the project Mathematical Thinking of Preschool Children in Rural and Regional Australia: Research and Practice directors, teachers, and assistants in prior-to-school settings from regional and rural eastern Australia were interviewed to ascertain their beliefs and practices concerning early childhood mathematics. This paper reports the responses to  uestions about their assessment of children’s mathematical activity and development. The practitioners provided examples of both incidental and planned assessment activities, the different forms these took, methods of recording, and how the results were used.

Future teachers' professional knowledge on argumentation and proof : a case study from universities in three countries

In this paper, qualitative results of a case study about the professional knowledge in the area of argumentation and proof of future teachers from universities in three countries are described. Based on results of open questionnaires, data about the competencies these future teachers have in the areas of mathematical knowledge and knowledge of mathematics pedagogy are presented. The study shows that the majority of the future teachers at the participating universities situated in Germany, Hong Kong and Australia, were not able to execute formal proofs, requiring only lower secondary mathematical content, in an adequate and mathematically correct way. In contrast, in all samples there was evidence of at least average competencies of pedagogical content reflection about formal and pre-formal proving in mathematics teaching. However, it appears that possessing a mathematical background as mandated for teaching and having a high affinity with proving in mathematics teaching at the lower secondary level are not a sufficient preparation for teaching proof.

Planning with events and states

We present an overall planning system in which specifications can be described in terms of events and states. The underlying feature of this system is temporal logic, and its expressive power alloys one to deal with simultaneous actions and interacting actions. Moreover, one can represent both goal-oriented positive constraints and prevention-oriented negative constraints. The planning system can generate hierarchical plans and the overall model is capable of handling interacting agents.

How can the impact of cultural development work in local government be measured? Towards more effective planning and evaluation strategies

The field of cultural development in local government is relatively new, with most councils only having dedicated staff or teams within the last ten to fifteen years. Challenges are associated with that newness in the areas of planning, goal setting and formal evaluation of achievements in relation to goals. How can the best decisions be made about what is needed? How can the outcomes of that work be evaluated? What should be measured and how? This paper explores these challenges and presents some solutions. Program Logic is introduced as a methodology for effective planning and evaluation of cultural development work in local government. The need for both performance and outcome evaluation of cultural development work is discussed, as are the levels of evaluation required; considering the contribution of individual workers, departments, whole of council and the overall community outcomes. Factors beyond the influence of local government, which impact the outcomes of arts initiatives, are also considered in arguing that more sophisticated evaluation processes are required.

The role of contexts and teacher's questioning to enhance students' thinking

This paper discusses results from a design research in line with Realistic Mathematics Education (RME). Daily cycles of design, classroom experiments, and retrospective analysis are enacted in five days of working about division by fractions. Data consists of episodes of video classroom discussions, and samples of students’ work. The focus of discussion and analysis centres on the role of contexts and the role of teachers’ probing questions to elicit students’ thinking. Our findings suggest that contexts that are meaningful for and understandable by students bring out rich mathematical thinking and discussion amongst students. Meaningful contexts combined with teacher’s probing questions - highlighting big mathematical ideas - allow students to attain various approaches at different levels of formal mathematics.

Zonal banning and public order in urban Australia

In recent years, Australian governments of various ideological persuasions at local, state and territory and federal levels have introduced a range of zonal governing techniques to manage the flow of people in urban spaces. Zonal governance involves the identification and formal declaration of a specific urban geographic region to enable police and security personnel to deploy special powers and allied forms of surveillance technologies as a supplement to their conventional public order maintenance functions.

Despite the impetus towards open flows or movement within sovereign territories or larger territorial groupings, such as the European Union, considerable governmental effort has been directed towards the use of new forms of criminal law to re-territorialize urban space through new administrative, property law and regulatory measures. These low-level spatial demarcations introduce various supplementary police powers and discretionary procedures that enhance surveillance within a declared area to increase the level of contemporary urban security. Of particular concern is the legal right to ban or exclude “undesirable” individuals and groups from entering or using certain designated urban zones, to prevent antisocial or violent behavior usually associated with alcohol consumption.

To date, most discussion of the impact of banning and related surveillance measures focuses on illegal migration through ports of entry into sovereign nations and the commensurate burdens this creates for both citizens and non-citizens to authenticate their movements at national geographic borders. This logic is permeating more localized forms of regulation adopted by Australian local and mid-tier state and territory governments to control the movement of people in and out of major event sites and in the urban night-time economy.

A survey of recent reforms in the state of Victoria reveals how this new logic of mass-surveillance aims to promote greater levels of urban security while reshaping the conventional order maintenance functions of both the public and private police. This chapter describes these procedures and their impact in sanctioning the efficient screening of people to promote order in specific zones within the contemporary Australian urban environment, at the expense of more progressive and inclusive crime prevention initiatives. We focus on two exemplars of the intensification of surveillance through zonal governance techniques: ‘major events’ and ‘designated alcohol zones’.

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.

Children's perspectives on mathematics and game playing

This paper reports on data collected from a larger study investigating the effect of game playing on students' mathematical learning and motivation. The study was conducted in three schools with 240 students participating in four different treatment groups. This paper reports some of the results of interviews with students to gather data on their attitudes towards mathematics and game playing. The interview data collected from a sample of students in the· three game playing groups pertaining to attitudes are presented and discussed below.

Rhetorical theory of public relations: opening the door to semiotic and pragmatism approaches

The 2001 Handbook of Public Relations edited by Robert Heath contains a prominent article advocating the use of rhetorical theory or ‘rhetorical enactment rational’ as a fruitful way of advancing theoretical understandings of public relations. In 2004 Heath and Dan Millar edited: Responding to Crisis: A Rhetorical Approach to Crisis Communication. These are the latest excursions into a perspective on public relations reflecting the extensive study of rhetoric in North America. Other examples are Public Relations Inquiry as Rhetorical Criticism (Elwood, 1995); Rhetorical and Critical Approaches to Public Relations (Toth and Heath, 1992); and a chapter Public Relations? No, Relations with Publics: A Rhetorical-Organisational Approach to Contemporary Corporate Communication (Cheney and Dionisopoulos, in Botan and Hazleton (Eds.) 1989).

The conventional notion of rhetoric is argumentation and persuasion stemming from the ancient Greek sophists, such as Aristotle, and from the Romans, particularly Cicero and Quintillion. Rhetoric became a fundamental plank of the trivium of ancient and medieval education: grammar, logic and rhetoric. Then in the 20th century Kenneth Burke, Stephen Toulmin and Chaim Perelman with Lucie Olbrechts-Tyteca extended Aristotle’s suggestion that: “Rhetoric is the counterpart of dialectic” Aristotle (trans. 1991). To use the rhetorical approach to argue that rational discourse cannot describe the world on its own. Instead living, enculturated human beings have to perceive ‘their’ truths. They take a perceptual ‘position’ on reason.

Public relations, is an industry for influencing perceptual ‘positions’. But the study of perception and attempts to influence perception cannot be claimed by rhetorical scholars alone. Semioticians and linguists who take the perspective of linguistic pragmatics also claim this field. This paper takes the example of ‘public relations’ as a focus for the confluence of rhetorical, semiotic and pragmatism approaches to the ‘problematic’ of understanding and truth.

Increasing access to mathematical thinking

This article is a contribution to understanding of teacher actions that can contribute to a successful mathematics learning experience, defined as one that engages all students, especially those who may sometimes feel alienated from mathematics and schooling, in productive and successful mathematical thinking and learning. We offer an example of a task that can form the basis of such a learning experience. The key elements are that the task is open-ended, that the teacher offers specific pedagogical prompts to support student leaning, that the teacher builds a sense of community by ensuring that there are some common experiences, and the teacher prepare prompts that can be used to support students who are experiencing difficulty, or to extend those students who complete the
basic task readily.

Introductory computer programming as a means for extending spatial and temporal understanding

Should computer programming be taught within schools of architecture?

Incorporating even low-level computer programming within architectural education curricula is a matter of debate but we have found it useful to do so for two reasons: as an introduction or at least a consolidation of the realm of descriptive geometry and in providing an environment for experimenting in morphological time-based change.

Mathematics and descriptive geometry formed a significant proportion of architectural education until the end of the 19th century. This proportion has declined in contemporary curricula, possibly at some cost for despite major advances in automated manufacture, Cartesian measurement is still the principal ‘language’ with which to describe building for construction purposes. When computer programming is used as a platform for instruction in logic and spatial representation, the waning interest in mathematics as a basis for spatial description can be readdressed using a left-field approach. Students gain insights into topology, Cartesian space and morphology through programmatic form finding, as opposed to through direct manipulation.

In this context, it matters to the architect-programmer how the program operates more than what it does. This paper describes an assignment where students are given a figurative conceptual space comprising the three Cartesian axes with a cube at its centre. Six Phileban solids mark the Cartesian axial limits to the space. Any point in this space represents a hybrid of one, two or three transformations from the central cube towards the various Phileban solids. Students are asked to predict the topological and morphological outcomes of the operations. Through programming, they become aware of morphogenesis and hybridisation. Here we articulate the hypothesis above and report on the outcome from a student group, whose work reveals wider learning opportunities for architecture students in computer programming than conventionally assumed.