Body fractions: A physical approach to fraction learning

Many students experience great difficulty understanding the meaning of fractions (Anthony & Walshaw, 2007; Behr, Lesh, Post & Silver, 1983; Davis, Hunting & Pearn, 1993; Lamon, 2007; Verschaffel, Greer & Torbeyns, 2006; Young-Loveridge, Taylor, Hawera & Sharma, 2007). For many students who have spent their early mathematics lessons focusing on counting (whole) numbers, recognising that there are many numbers between those whole numbers called fractional numbers, is quite revolutionary. The foundation of understanding fractions is the idea that they are parts of a whole. The fact that one whole object can be divided into many equal parts, with each part having a name relative to the original whole, opens up a whole new realm of number understanding for the students

Deepening students' understanding of multiplication and division by exploring divisibility by nine

This paper explores how a focus on understanding divisibility rules can be used to help deepen students’ understanding of multiplication and division with whole numbers. It is based on research with seven Year 7–8 teachers who were observed teaching a group of students a rule for divisibility by nine. As part of the lesson, students were shown a way of proving why the divisibility rule for nine works, using materials grouped in tens and hundreds. After the lesson, students’ understanding of multiplication and division was considerably deepened

The mathematical content knowledge and attitudes of New Zealand pre-service primary teachers

This paper presents data on the mathematical content knowledge and attitudes of pre-service primary teacher education students. The assessment consisted of nine tasks, including 2-digit computations and proportional reasoning. Students rated their liking for mathematics at three time points: primary, secondary, .and when assessed. Fewer than half the students liked mathematics, currently. Those with positive attitudes tended to perform well on mathematics tasks, but some low scorers were positive and some high' scorers were negative about mathematics. Most students used algorithmic procedures to solve problems and several consistent misconceptions were identified. Performance was noticeably poor on adding common fractions and converting fractions to percentages using knowledge of common factors. The implications of these findings for Initial Teacher Education (ITE) providers are presented

Teachers’ reflecting on professional knowledge in the numeracy (mathematics) classroom

New Zealand primary school teachers are expected to regularly reflect on their teaching practice in order to consider the implications of past teaching on future planning. Aligned to teachers’ ongoing reflection, the New Zealand Curriculum (Ministry of Education, 2007) contains a section on effective pedagogy—teacher actions promoting student learning, which includes a Teaching as Inquiry Cycle (pp. 34–35). Embedded within their inquiry, teachers consider the teaching-learning relationship and often turn to frameworks of knowledge for guidance. This article shares the implications of using a framework of teacher knowledge in research. While the framework used contained much detail for the researcher, it overlapped categories and at the same time lacked acknowledgement of some important concepts for teachers in classroom practice. Findings from using a framework in this research were combined with findings from previous research to formulate the Wheel of Professional Knowledge, which was developed for mathematics teachers to use when reflecting on their practice

Investigating the Professional Knowledge of New Zealand Primary School Teachers when Teaching Mathematics for Numeracy

This research investigated the relationship between teachers’ espoused professional knowledge, professional knowledge in practice, and student learning, when teaching ‘mathematics for numeracy’ in the New Zealand primary school classroom. The focus was on teaching within the multiplicative and proportional domains, as research at the time the study commenced indicated that these areas of mathematics were problematic for many teachers. The purpose of the research was to identify strengths, weaknesses, and inconsistencies in teachers’ practice; links between espoused views and actions; and to consider the usefulness of a framework for investigating teacher knowledge in practice. This study is intended to inform teacher reflection and professional development, and contribute to improvements in teaching practice and student achievement. A multiple-case study design, underpinned by an interpretivist paradigm, was used, which included four case-study teachers from two primary schools: School A, was a central city full primary school (Years 1 to 8) and School B, a rural town primary school (Years 1 to 6). The study aligned with a social constructivist perspective on teaching and learning. The data were obtained through four main sources: (1) pre-unit and post-unit student assessment tasks; (2) recorded observations; (3) semi-structured interviews); and (4) teacher questionnaires. Comparison between students’ initial and final assessment data showed little progress in understanding of multiplication and division, with a more noticeable improvement in fractional understanding. Classroom observations were analysed under three broad categories: content knowledge, pedagogical knowledge, and pedagogical content knowledge, and highlighted important issues relating to the professional knowledge of teachers and the contribution this made to student learning. Results indicated that the mathematical content knowledge of the teachers was stronger than their content knowledge in a pedagogical context. While teaching for conceptual understanding frequently challenged the teachers, they recognised the importance of conceptual understanding prior to procedural learning for their students. They struggled with on-the-spot identification of the next steps of learning for individual students and there was little evidence of focus on questioning that extended students’ thinking that might have assisted in overcoming misconceptions and confusions with concepts. There were times when the teachers’ espoused theories differed from their theory-in-practice, while at times they were similar to each other. The research concluded that in teaching practice the many facets of PCK, within the broader construct of professional knowledge, were more than topic-specific. Instead, they were person-specific and lesson-specific, with different categories coming to the fore in different proportions, for different reasons, including: lesson structure, context, problem types, the opportunities afforded students for conversation, and use of manipulatives. While not all categories of professional knowledge were evident in every lesson, they combined over a period to underscore the complexities of teaching and ultimately have an effect on student learning. An outcome of the study was a Wheel of Knowledge designed for teachers, identifying key areas of knowledge to be addressed in mathematics teaching. Alongside this, a more detailed Professional Knowledge Framework was created for researchers, based on categories identified from this research as important in identifying teacher professional knowledge in classroom practice. Both models have the potential to identify areas of teacher professional knowledge required to improve students’ mathematical understanding

Making multiplication meaningful: Teaching for conceptual understanding

The term numeracy is used widely in schools today and brings with it the expectation that students will be taught both how to do the mathematics, alongside an understanding of the concepts associated with the procedural application. One issue, which has arisen with the terminology ‘numeracy classroom’, is how to best support teachers to enhance their teaching of mathematics to allow this understanding to occur. This article stems from a larger research study that analysed the professional knowledge of teachers when teaching numeracy, and the impact their mathematics knowledge and procedural application had on children’s learning. This article presents observations of three teachers teaching a multiplication lesson (the first in a series of lessons over a six-week period) as they developed their students’ understanding of the mathematical concepts associated with the interpretation of the multiplication symbol. An analysis of the findings shows when the teachers used manipulatives, related word problems to the children’s lives, and promoted discussion in groups, a greater understanding of multiplication was apparent

Teachers as researchers

Research in educational settings has often been undertaken by a third-person (a term used by Ball, 2000), who sits on the edge of the classroom, maybe moves among the students from time to time, and looks ‘at’ the teaching taking place. The assumption was that an outside observer may see and hear things taking place that the teacher takes for granted,and may notice things occurring that might not have been planned for by the teacher. This approach harks back to research objectivity and a value-free research agenda (Denzin & Lincoln, 2005). However, while an outsider seeks to understand what goes on in the lessons, he/she does not necessarily appreciate or understand some of the relational, historical and affective factors beyond that gaze,and that combine in the interactions, classroom climate and situation ofthat moment. The first-person approach to research in educational settings means the teacher is in a position to probe beneath the observed actions associated with their practice and focus on issues arising through the questioning, evidence gathering and analysing of their actions alongside their students’ learning. First-person research by teachers brings to scholarly research and writing,a perspective direct from the classroom practitioner. Practitioner research, or the first-person approach to research, is referred to as ‘working in the inside’(Ball, 2000). Ball advocated that this form of research provides a valuable contribution to the improvement of teaching as teachers examine in depth some of the puzzles associated with classroom practice

Introduction to papers in this issue

The overall theme for this issue is the sharing of good ideas and support to look at alternatives - look at what might be familiar in alternative ways, and from alternative perspectives. This issue has three sections offering our professional learning community opportunities to explore pedagogy from early childhood to tertiary, from New Zealand and international authors. One thread in this issue is mathematics education. Another is tertiary education, with articles from the perspectives of the teachers and learners. The four think pieces are designed to provoke and challenge your thinking while providing something useful

Prisoners’ Families’ Research: Developments, Debates and Directions

After many years of relative obscurity, research on prisoners’ families has gained significant momentum. It has expanded from case-oriented descriptive analyses of family experiences to longitudinal studies of child and family development and even macro analyses of the effects on communities in societies of mass incarceration. Now the field engages multi-disciplinary and international interest although it arguably still remains on the periphery of mainstream criminological, psychological and sociological research agendas. This chapter discusses developments in prisoners’ families’ research and its positioning in academia and practice. It does not aim to provide an all-encompassing review of the literature rather it will offer some reflections on how and why the field has developed as it has and on its future directions. The chapter is divided into three parts. The first discusses reasons for the historically small body of research on prisoners’ families and for the growth in research interest over the past two decades. The second analyses patterns and shifts in the focus of research studies and considers how the field has been shaped by intersecting disciplinary interests of psychology, sociology, criminology and socio-legal studies. The final part reflects on substantive and ethical issues that are likely to shape the direction of prisoners’ families’ research in the future

New genetic loci link adipose and insulin biology to body fat distribution.

Body fat distribution is a heritable trait and a well-established predictor of adverse metabolic outcomes, independent of overall adiposity. To increase our understanding of the genetic basis of body fat distribution and its molecular links to cardiometabolic traits, here we conduct genome-wide association meta-analyses of traits related to waist and hip circumferences in up to 224,459 individuals. We identify 49 loci (33 new) associated with waist-to-hip ratio adjusted for body mass index (BMI), and an additional 19 loci newly associated with related waist and hip circumference measures (P < 5 × 10(-8)). In total, 20 of the 49 waist-to-hip ratio adjusted for BMI loci show significant sexual dimorphism, 19 of which display a stronger effect in women. The identified loci were enriched for genes expressed in adipose tissue and for putative regulatory elements in adipocytes. Pathway analyses implicated adipogenesis, angiogenesis, transcriptional regulation and insulin resistance as processes affecting fat distribution, providing insight into potential pathophysiological mechanisms