Theories of learning mathematics

According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. So, for healthy theories on development to occur, four Darwinian functions should function: (a) variation – avoid orthodoxy and encourage divergent thinking, (b) selection – submit all assumptions and innovations to rigorous testing, (c) diffusion – encourage the shareability of new and/or viable ways of thinking, and (d) accumulation – encourage the reuseability of viable aspects of productive innovations.

Evaluation of the "reconceptualising early mathematics learning" project

The Pattern and Structure Mathematics Awareness Project (PASMAP) has investigated the development of patterning and early algebraic reasoning among 4 to 8 year olds over a series of related studies. We assert that an awareness of mathematical pattern and structure enables mathematical thinking and simple forms of generalisation from an early age. The project aims to promote a strong foundation for mathematical development by focusing on critical, underlying features of mathematics learning. This paper provides an overview of key aspects of the assessment and intervention, and analyses of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in the evaluation throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ assessment and a Pattern and Structure Assessment (PASA) interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a relatively short period of time.

Power system sensitivity analysis considering induction motor loads

Power system operation and planning are facing increasing uncertainties especially with the deregulation process and increasing demand for power. Probabilistic power system stability assessment and probabilistic power system planning have been identified by EPRI as one of the important trends in power system operations and planning. Probabilistic small signal stability assessment studies the impact of system parameter uncertainties on system small disturbance stability characteristics. Researches in this area have covered many uncertainties factors such as controller parameter uncertainties and generation uncertainties. One of the most important factors in power system stability assessment is load dynamics. In this paper, composite load model is used to consider the uncertainties from load parameter uncertainties impact on system small signal stability characteristics. The results provide useful insight into the significant stability impact brought to the system by load dynamics. They can be used to help system operators in system operation and planning analysis.

Modeling as a bridge between real world problems and school mathematics

This article argues for an interdisciplinary approach to mathematical problem solving at the elementary school, one that draws upon the engineering domain. A modeling approach, using engineering model eliciting activities, might provide a rich source of meaningful situations that capitalize on and extend students’ existing mathematical learning. The study reports on the developments of 48 twelve-year old students who worked on the Bridge Design activity. Results revealed that young students, even before formal instruction, have the capacity to deal with complex interdisciplinary problems. A number of students created quite appropriate models by developing the necessary mathematical constructs to solve the problem. Students’ difficulties in mathematizing the problem, and in revising and documenting their models are presented and analysed, followed by a discussion on the appropriateness of a modeling approach as a means for introducing complex problems to elementary school students.

A comparison of Chinese and Australian early childhood mathematics

In the 21st century mathematics proficiency is synonymous with a numerate citizenry. In the past few decades young children’s ability to reason mathematically and develop mathematical proficiencies has been recognised. This paper explores the history of early childhood mathematics (ECME) that may explicate differences in Chinese and Australian contexts. Results of this review established that China and Australia are diametrically positioned in ECME. Influencing each countries philosophies and practices are their cultural beliefs. ECME in China and Australia must be culturally sustainable to achieve excellent outcomes for young children. Ongoing critique and review is necessary to ensure that ECME is meeting the needs of all teachers and children in their particular context. China and Australia with their rich contrasting philosophies can assist each other in their journeys to create exemplary ECME for the 21st century.

Mathematics learning in prep

Most teachers recognise the importance of mathematics teaching and learning in early years but there is not consensus on how and when this learning should occur. Young-Loveridge (cited in de Vries, Thomas, and Warren, 2010) suggests that quality early mathematical experiences are a key determinant to later achievement.

Beginning teachers’ achievements and challenges : implications for induction and mentoring

It is recognised worldwide that beginning teachers require more support as reasons for high attrition rates (e.g., lack of appreciation from colleagues, unsatisfying working conditions, inadequate teacher preparation) indicate current systems are failing them. One way of addressing their specific needs is to understand their achievements and challenges during their first year of teaching. This qualitative study tracks 10 beginning primary teachers’ achievements and challenges at two points (April and September) during their first year of teaching in Australian public schools. Findings showed that building relationships and behaviour management were considered achievements at these two points, yet behaviour management was also considered a challenge. Other challenges included: learning differentiation, working with parents, and negotiating a life-work balance. Induction into the school culture and infrastructure continued to be important, especially developing skills on handling difficult parents and creating a life-work balance. Simultaneously, they required mentoring for effective teaching in classroom management and differentiation. A two-prong approach of induction into the school culture and infrastructure and mentoring for effective teaching needs to continue throughout the first year of teaching, and possibly beyond.

Assessing the potential of mathematics textbooks to promote deep learning

Curriculum documents for mathematics emphasise the importance of promoting depth of knowledge rather than shallow coverage of the curriculum. In this paper, we report on a study that explored the analysis of junior secondary mathematics textbooks to assess their potential to assist in teaching and learning aimed at building and applying deep mathematical knowledge. The method of analysis involved the establishment of a set of specific curriculum goals and associated indicators, based on research into the teaching and learning of a particular field within the mathematics curriculum, namely proportion and proportional reasoning. Topic selection was due to its pervasive nature throughout the school mathematics curriculum at this level. As a result of this study, it was found that the five textbook series examined provided limited support for the development of multiplicative structures required for proportional reasoning, and hence would not serve well the development of deep learning of mathematics. The study demonstrated a method that could be applied to the analysis of junior secondary mathematics in many parts of the world.

Induction of antioxidative Nrf2 gene transcription by coffee in humans : depending on genotype?

The Nrf2/ARE pathway is a major cellular defense mechanism that prevents damage by reactive oxygen species through induction of antioxidative phase II enzymes. However, the activity of the Nrf2/ARE system is not uniform with variability in response presumed to be dependent on the Nrf2 genotype. We recently completed a pilot human coffee intervention trial with healthy humans, where large interindividual differences in the antioxidative response to the study coffee were examined. Here, we address the question whether differences in the modulation of Nrf2 gene transcription, assessed as an induction of Nrf2 gene transcription by Q-PCR, might be correlated with specific Nrf2 genotypes. To date, nine single nucleotide polymorphisms (SNPs) have been identified in the Nrf2 (NFE2L2) gene. Two of these, the -617C/A and -651G/A SNPs are located within the promoter region and have previously been reported to influence the activity of the Nrf2/ARE pathway by reducing Nrf2 transcriptional activity. Sequencing of the critical Nrf2 gene promoter region not only confirmed the existence of these SNPs within the participants of the trial at the expected frequency (33% carrying the -617C/A, 17% the -651G/A and 56% the -653A/G SNP) but also indicated reduced Nrf2 gene transcription associated with a normal diet if the SNPs at position -617, -651 or -653 were present. Of note, the data also indicated the study coffee increased Nrf2 gene transcription even in SNP carriers. This further highlights the relevance of genotype-dependent induction of Nrf2 gene transcription that appears to be largely influenced by dietary factors.

Induction of multiple reinstatements of ethanol- and sucrose-seeking behavior in Long–Evans rats by the α-2 adrenoreceptor antagonist yohimbine

Rationale Developing models to efficiently explore the mechanisms by which stress can mediate reinstatement of drug-seeking behavior is crucial to the development of new pharmacotherapies for alcohol use disorders. Objectives We examined the effects of multiple reinstatement sessions using the pharmacological stressor, yohimbine, in ethanol- and sucrose-seeking rats in order to develop a more efficient model of stress-induced reinstatement. Methods Long–Evans rats were trained to self-administer 10% ethanol with a sucrose-fading procedure, 20% ethanol without a sucrose-fading procedure, or 5% sucrose in 30-min operant self-administration sessions, followed by extinction training. After reaching extinction criteria, the animals were tested once per week with yohimbine vehicle and yohimbine (2 mg/kg), respectively, 30 min prior to the reinstatement sessions or blood collection. Levels of reinstatement and plasma corticosterone (CORT) were determined each week for four consecutive weeks. Results Yohimbine induced reinstatement of ethanol- and sucrose-seeking in each of the 4 weeks. Interestingly, the magnitude of the reinstatement decreased for the 10% ethanol group after the first reinstatement session but remained stable for the 20% ethanol group trained without sucrose. Plasma CORT levels in response to injection of both vehicle and yohimbine were significantly higher in the ethanol-trained animals compared to sucrose controls. Conclusions The stable reinstatement in the 20% ethanol group supports the use of this training procedure in studies using within-subject designs with multiple yohimbine reinstatement test sessions. Additionally, these results indicate that the hormonal response to stressors can be altered following extinction from self-administration of relatively modest amounts of ethanol.

Data and measurement in Year 4 of the national curriculum : Mathematics

The activities introduced here were used in association with a research project in four Year 4 classrooms and are suggested as a motivating way to address several criteria for Measurement and Data in the Australian Curriculum: Mathematics. The activities involve measuring the arm span of one student in a class many times and then of all students once.

Perspectives on Reconceptualizing Early Mathematics Learning

This introductory section provides an overview of the different perspectives on reconceptualizing early mathematics learning. The chapters provide a broad scope in their topics and approaches to advancing young children’s mathematical learning. They incorporate studies that highlight the importance of pattern and structure across the curriculum, studies that target particular content such as statistics, early algebra, and beginning number, and studies that consider how technology and other tools can facilitate early mathematical development. Reconceptualizing the professional learning of teachers in promoting young children’s mathematics, including a consideration of the role of play, is also addressed. Although these themes are diffused throughout the chapters, we restrict our introduction to the core focus of each of the chapters.

Reconceptualizing Early Mathematics Learning : The Fundamental Role of Pattern and Structure

The Pattern and Structure Mathematics Awareness Program (PASMAP) was developed concurrently with the studies of AMPS and the development of the Pattern and Structure Assessment (PASA) interview. We summarize some early classroom-based teaching studies and describe the PASMAP that resulted. A large-scale two-year longitudinal study, Reconceptualizing Early Mathematics Learning (REML) resulted. We provide an overview of the REML study and discuss the consequences for our view of early mathematics learning. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in an evaluation of the PASMAP intervention throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ standardized assessment and the PASA interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students. Implications for pedagogy and curriculum are discussed.