What does mathematics understanding look like?

The concept of mathematical understanding is central to curriculum development, classroom interaction, and the training of mathematics teachers In this paper, some models of the growth of understanding that the literature presents are outlined Some results of a study that documented four primary teachers’ mental models of, and beliefs about, different forms of understanding are reported It is proposed that linear models may restrict ways that these teachers plan lessons Questions for further research are raised

Teaching secondary school mathematics : research and practice for the 21st century

For the last decade or so, educational policy makers and researchers in many countries have been calling for significant changes to the way mathematics is taught in secondary schools. Australian mathematics curriculum documents now promote learning goals that go beyond mastery of a pre-determined body of knowledge and procedures - the traditional emphasis on facts, skills, formulae - to include mathematical reasoning and problem solving, communication, and real world applications. There is also pressure to move away from over-reliance on teacher-centred practices such as exposition and individual seatwork, towards activities that promote learners' involvement in constructing, applying, and evaluating mathematical ideas. Further impetus for reform comes from research recommending that if learners are to develop mathematically powerful forms of thinking and habits of mind, then classrooms should immerse them in the authentic practices of the discipline by supporting a culture of collaboration and sense-making. Teaching Secondary School Mathematics - incorporates recent developments in research and practice and applications to teaching mathematics in Australian secondary schools. Covering such areas as curriculum, pedagogy and assessment; teaching mathematical content; equity and diversity in the classroom; and professional and community engagement, it is an invaluable resource for all practising and pre-service mathematics teachers.

Year 3/4 children's forms of justification

Engaging children in justifying, forming conjectures and generalising is critical to develop their mathematical reasoning. Previous studies have revealed limited opportunities for primary school children to justify their thinking, form conjectures and generalise in mathematics lessons. Forms of justification of Year3/4 children from three schools in Victoria, Australia will be examined. Evidence from children's written explanations and their verbal explanations captured in video recordings revealed that some children employed sophisticated mathamatical ideas in their justifications. The value of making children's reasoning explicit through written explanations and verbal communications is highlighted.

Subject matters of science, technology, mathematics and engineering

This chapter offers a critical review of curriculum in the STEM disciplines, overviewing contemporary context and issues around the STEM acronym as a contested space, issues of inquiry, social justice, current concerns with student engagement and their history, theoretical perspectives on teaching and learning in these disciplines, and key forms and foci of inquiry into STEM curriculum.

Mathematics teacher education and technology

Various applications of electronic technology have contributed to the field of mathematics teacher education, but the array of tools now' available to teacher educators in developed countries can be somewhat daunting. In this chapter, the authors present an overview of some of these tools and the ways that they are being used. In addition, they address issues surrounding the development and applications of digital technology in teacher education programs in a number of countries. In particular, they present recent applications of videotapes, multimedia resources, internet-based communities, and tutorial conferencing facilities in pre-service and in-service education of mathematics teachers. As professionals who have used technology in teacher education, the authors blend their own insights into this review of technological tools and relevant issues.

Multiple forms of child abuse and neglect: adult retrospective reports

There are few research reports of the experience of multiple forms of child abuse and neglect (multi-type maltreatment). A critical review is presented of 29 studies in which adult retrospective reports of more than one form of child maltreatment (sexual abuse, physical abuse, psychological maltreatment, neglect, or witnessing family violence) are assessed. Empirical investigations of the extent and impact of multi-type maltreatment are identified. Data on the relationship between the different forms of maltreatment and the adjustment problems associated with each type of maltreatment are critiqued. As well as using dichotomous measures of maltreatment, an important methodological problem was the failure of researchers to assess all child maltreatment types, the relationship between multi-type maltreatment and adjustment, and the role of variables influencing the occurrence or impact of maltreatment. Considerable overlap was found in the occurrence of maltreatment types. The compounding or interacting effects of experiencing multi-type maltreatment were ignored in many studies. Where an assessment was made of the specific impact of multi-type maltreatment, it was associated with greater impairment than single forms of abuse or neglect. A multi-dimensional approach to prevention and treatment of child abuse and neglect is required in which children's vulnerability to multi-type maltreatment is addressed.

Pop art: making mathematics popular

Leicha Bragg shares an effective activity for integrating visual arts, technology and mathematics. Such an approach has the potential to enhance students' visual reasoning and spatial ability by using multiple perspectives of learning. 

Is domestic violence learned? The contribution of five forms of child maltreatment to men's violence and adjustment

On the basis of a learning-theory approach to the intergenerational transmission of violence, researchers have focused almost exclusively on violent men's childhood experiences of physical abuse and witnessing family violence. Little consideration has been given to the coexistence of other forms of child maltreatment or the role of family dysfunction in contributing to violence. This study shows the relationships between the level of child maltreatment (physical abuse, psychological maltreatment, sexual abuse, neglect, and witnessing family violence), childhood family characteristics, current alcohol abuse, trauma symptomatology, and the level of physical and psychological spouse abuse perpetrated by 36 men with a history of perpetrating domestic violence who had attended counseling. As hypothesized, a high degree of overlap between risk factors was found. Child maltreatment, low family cohesion and adaptability, and alcohol abuse was significantly associated with frequency of physical spouse abuse and trauma symptomatology scores, but not psychological spouse abuse. Rather than physical abuse or witnessing family violence, childhood neglect uniquely predicted the level of physical spouse abuse. Witnessing family violence (but not physical abuse) was found to have a unique association with psychological spouse abuse and trauma symptomatology. These results present a challenge to the understanding of domestic violence obtained from learning theory.

Correction of the copper transport defect of Menkes patient fibroblasts by expression of two forms of the sheep Wilson ATPase

The Wilson disease (WD) protein (ATP7B) is a copper-transporting P-type ATPase that is responsible for the efflux of hepatic copper into the bile, a process that is essential for copper homeostasis in mammals. Compared with other mammals, sheep have a variant copper phenotype and do not efficiently excrete copper via the bile, often resulting in excessive copper accumulation in the liver. To investigate the function of sheep ATP7B and its potential role in the copper-accumulation phenotype, cDNAs encoding the two forms of ovine ATP7B were transfected into immortalised fibroblast cell lines derived from a Menkes disease patient and a normal control. Both forms of ATP7B were able to correct the copper-retention phenotype of the Menkes cell line, demonstrating each to be functional copper-transporting molecules and suggesting that the accumulation of copper in the sheep liver is not due to a defect in the copper transport function of either form of sATP7B.

Teachers and computers for secondary mathematics

In the first stage of a three-year study in which the effects of using computers for the teaching and learning of mathematics are being explored, a questionnaire was developed and administered to teachers of students in grades 7–10 in a representative sample of co-educational post-primary schools in Victoria, Australia. Using open and closed response formats, the information sought included data on the teachers' professional backgrounds, computer ownership and use, and their beliefs and practices in using computers for the teaching of mathematics. In this article, findings related to ownership, professional development, perceptions of technological skills, beliefs about the efficacy of computer use in mathematics, and data on how teachers are using computers for teaching secondary mathematics are presented and discussed.

Psychometric data on the mathematics self-efficacy scale

Descriptive measurement reliability and validity data are reported on scores from the Mathematics Self-Efficacy Scale (MSES) in two contexts—class and test—for a sample of 302 Australian high school students. Summated scores on the MSES correlated r = .74, and together these items yielded one component that explained 49% of the variance. MSES scores demonstrated internal reliability for both class and test (Cronbach alphas = .86 and .90). Statistically significant correlations between MSES scores, past mathematics grades, and Marsh’s Self-Description Questionnaire III (Math) were used to indicate the convergent validity of scores for the MSES. Discriminant validity of the MSES scores was suggested by the lack of significant correlations between the MSES measures and students’ desired English grades. Statistically significant correlations between MSES scores and students’ desired mathematics grades indicated concurrent validity of the MSES measures. A statistically significant difference between class and test scores supported Bandura’s prescription for context specificity in efficacy research.

HRQOL and subjective well-being: noncomplementary forms of outcome measurement

This review considers some of the broad principles that concern quality of life assessment. These are discussed in relation to health-related quality of life (HRQOL) and the measurement of subjective well-being. It is argued that there are serious logical and methodological issues concerning HRQOL measurement, to the extent that the instruments may not be regarded as valid measures of life quality as this term is generally understood. It is recommended that HRQOL measurement be abandoned in favor of three separate forms of measurement as medical symptoms, subjective well-being and specific dimensions of psychological ill-being.

Organizing forms in change management: the role of structures, process and boundaries in a longitudinal case analysis

This paper examines change management at William Angliss Institute of Technical and Further Education (TAFE) against the three organizational dimensions of structures, processes and boundaries identified by the INNFORM Study. Its experience confirms that even when an organization adopts a systemic approach and implements change across each design dimension, optimal performance benefits depend on mutually reinforcing and complementary changes. Furthermore, improvement to processes, particularly communications and human resources practices, plays a pivotal role, as complementary change across all dimensions depends ultimately on the contribution and commitment of organization members. Case findings also highlight the need for ambidextrous forms of organizing that combine 'controllability' with 'responsiveness'. The conceptual notion of organizing dualities has been employed to provide a practical interpretation of the ostensibly competing imperatives implied by ambidexterity. This case explores the dualities that can be demonstrated for the INNFORM triumvirate of structures, processes and boundaries. The dualities interpretation emphasizes an acceptance of texture and the simultaneous presence of what are conventionally viewed as incompatible organizing forms. This was considered a useful conceptual vehicle in the analysis of a case study covering nearly ten years of serious change interventions, where one theoretical view can be misleading in understanding the subtleties and complexities of the actual changes that occurred.