Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Meeting under the “Omei” Tree in the Torres Strait Islands : networks and funds of knowledge of mathematical ideas

This paper focuses on the turning point experiences that worked to transform the researcher during a preliminary consultation process to seek permission to conduct of a small pilot project on one Torres Strait Island. The project aimed to learn from parents how they support their children in their mathematics learning. Drawing on a community research design, a consultative meeting was held with one Torres Strait Islander community to discuss the possibility of piloting a small project that focused on working with parents and children to learn about early mathematics processes. Preliminary data indicated that parents use networks in their community. It highlighted the funds of knowledge of mathematics that exist in the community and which are used to teach their children. Such knowledges are situated within a community’s unique histories, culture and the voices of the people. “Omei” tree means the Tree of Wisdom in the Island community.

Visualising mathematical knowledge and understanding

Contemporary mathematics education attempts to instil within learners the conceptualization of mathematics as a highly organized and inter-connected set of ideas. To support this, a means to graphically represent this organization of ideas is presented which reflects the cognitive mechanisms that shape a learner’s understanding. This organisation of information may then be analysed, with the view to informing the design of mathematics instruction in face-to-face and/or computer-mediated learning environments. However, this analysis requires significant work to develop both theory and practice.

Jabberwocky : the complexities of mathematical English

Mathematical English is a unique language based on ordinary English, with the addition of highly stylised formal symbol systems. Some words have a redefined status. Mathematical English has its own lexicon, syntax, semantics and literature. It is more difficult to understand than ordinary English. Ability in basic interpersonal communication does not necessarily result in proficiency in the use of mathematical English. The complex nature of mathematical English may impact upon the ability of students to succeed in mathematical and numeracy assessment. This article presents a review of the literature about the complexities of mathematical English. It includes examples of more than fifty language features that have been shown to add to the challenge of interpreting mathematical texts. Awareness of the complexities of mathematical English is an essential skill needed by mathematics teachers when teaching and when designing assessment tasks.

Get Into Vocational Education (GIVE) : motivating underperforming students

This study explores the effects of a vocational education-based program on academic motivation and engagement of primary school aged children. The Get Into Vocational Education (GIVE) program integrated ‘construction’ and the mathematics, English and science lessons of a Year 4 primary classroom. This paper focuses on investigating the components of the GIVE program that led to student changes in mathematical academic motivation and engagement resulting in outstanding gains in NAPLAN Numeracy results. The components proposed to have contributed to effectiveness of the GIVE program are: teacher and trainer expectations, task mastery and classroom relationships. These findings may be useful to researchers and educators who are interested in enhancing students’ mathematical academic motivation.

Students’ approaches to learning a new mathematical model

In this article, we report on the findings of an exploratory study into the experience of undergraduate students as they learn new mathematical models. Qualitative and quanti- tative data based around the students’ approaches to learning new mathematical models were collected. The data revealed that students actively adopt three approaches to under- standing a new mathematical model: gathering information for the task of understanding the model, practising with and using the model, and finding interrelationships between elements of the model. We found that the students appreciate mathematical models that have a real world application and that this can be used to engage students in higher level learning approaches.

Australian learning and teaching council projects in the mathematical sciences : a retrospective

Since 2004, the Australian Learning and Teaching Council (ALTC) and its predecessor, the Carrick Institute for Learning and Teaching in Higher Education, have funded numerous teaching and educational research-based projects in the Mathematical Sciences. In light of the Commonwealth Government’s decision to close the ALTC in 2011, it is appropriate to take account of the ALTCs input into the Mathe- matical Sciences in higher education. Here we present an overview of ALTC projects in the Mathematical Sciences, as well as report on the contributions they have made to the Discipline.

Repeating patterns : strategies to assist young students to generalise the mathematical structure

This paper focuses on very young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5-10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns.

Cost-effectiveness of a telephone-delivered education programme to prevent early childhood caries in a disadvantaged area : a cohort study

Objectives Early childhood caries is a highly destructive dental disease which is compounded by the need for young children to be treated under general anaesthesia. In Australia, there are long waiting periods for treatment at public hospitals. In this paper, we examined the costs and patient outcomes of a prevention programme for early childhood caries to assess its value for government services. Design Cost-effectiveness analysis using a Markov model. Setting Public dental patients in a low socioeconomic, socially disadvantaged area in the State of Queensland, Australia. Participants Children aged 6 months to 6 years received either a telephone prevention programme or usual care. Primary and secondary outcome measures A mathematical model was used to assess caries incidence and public dental treatment costs for a cohort of children. Healthcare costs, treatment probabilities and caries incidence were modelled from 6 months to 6 years of age based on trial data from mothers and their children who received either a telephone prevention programme or usual care. Sensitivity analyses were used to assess the robustness of the findings to uncertainty in the model estimates. Results By age 6 years, the telephone intervention programme had prevented an estimated 43 carious teeth and saved £69 984 in healthcare costs per 100 children. The results were sensitive to the cost of general anaesthesia (cost-savings range £36 043–£97 298) and the incidence of caries in the prevention group (cost-savings range £59 496–£83 368) and usual care (cost-savings range £46 833–£93 328), but there were cost savings in all scenarios. Conclusions A telephone intervention that aims to prevent early childhood caries is likely to generate considerable and immediate patient benefits and cost savings to the public dental health service in disadvantaged communities.

Student identities in a mathematical community of practice : radical visible pedagogy and the teacher collective

This study investigated the practices of two teachers in a school that was successful in enabling the mathematical learning of students in Years 1 and 2, including those from backgrounds associated with low mathematical achievement. The study explained how the practices of the teachers constituted a radical visible pedagogy that enabled equitable outcomes. The study also showed that teachers’ practices have collective power to shape students’ mathematical identities. The role of the principal in the school was pivotal because she structured curriculum delivery so that students experienced the distinct practices of both teachers.

Rich and purposeful mathematical knowledge of mothers and children in a Torres Strait Islander community

This paper focuses on a pilot study that explored the situated mathematical knowledge of mothers and children in one Torres Strait Islander community in Australia. The community encouraged parental involvement in their children’s learning and schooling. The study explored parents’ understandings of mathematics and how their children came to learn about it on the island. A funds of knowledge approach was used in the study. This approach is based on the premise that people are competent and have knowledge that has been historically and culturally accumulated into a body of knowledge and skills essential for their functioning and well-being (Moll, 1992). The participants, three adults and one child are featured in this paper. Three separate events are described with epiphanic or illuminative moments analysed to ascertain the features that enabled an understanding of the nature of the mathematical events. The study found that Indigenous ways of knowing of mathematics were deeply embedded in rich cultural practices that were tied to the community. This finding has implications for teachers of children in the early years. Where school mathematics is often presented as disembodied and isolated facts with children seeing little relevance, learning a different perspective of mathematics that is tied to the resources and practices of children’s lives and facilitated through social relationships, may go a long way to improving the engagement of children and their parents in learning and schooling.

Applications of Spreadsheets in Education : The Amazing Power of a Simple Tool

This e-book is devoted to the use of spreadsheets in the service of education in a broad spectrum of disciplines: science, mathematics, engineering, business, and general education. The effort is aimed at collecting the works of prominent researchers and educators that make use of spreadsheets as a means to communicate concepts with high educational value. The e-book brings some of the most recent applications of spreadsheets in education and research to the fore. To offer the reader a broad overview of the diversity of applications, carefully chosen articles from engineering (power systems and control), mathematics (calculus, differential equations, and probability), science (physics and chemistry), and education are provided. Some of these applications make use of Visual Basic for Applications (VBA), a versatile computer language that further expands the functionality of spreadsheets. The material included in this e-book should inspire readers to devise their own applications and enhance their teaching and/or learning experience.

Clinical faculty : major contributors to the education of new CRNAs - Part 2

This study examines the important contributions of clinical faculty toward the education of the future workforce of Certified Registered Nurse Anesthetists (CRNAs). Differences in workload, work activities and income among clinical faculty, academic faculty and nonfaculty are examined. This is Part 2 of a 2-part column. Part 1, published in the April 2008 AANA Journal discussed salaries, recruitment, and retention of CRNA faculty.