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Fractions are perceived as one of the most difficult areas in school mathematics to learn and teach. The most frequently mentioned factors contributing to the complexity are fractions having five interrelated constructs (part-whole, ratio, operator, quotient and measure) and teachers' pedagogical approaches to address these constructs. The present study used the five interrelated constructs of fraction and the Knowledge Quartet Framework to investigate a teacher's teaching of fraction in a New Zealand Year 7 classroom. Video recordings and transcribed audio-recordings were analysed through the lenses of the five integrated constructs of fraction and the Knowledge Quartet Framework. The findings showed that students often initiated uses of fractions as quotient and as operator, drawing on part-whole understanding when solving fraction problems. The study showed that the teacher's focus in connecting the constructs and procedures of the lesson as well as shaping the classroom discussion when the lesson unfolded supported students' learning of the fraction constructs.

This paper considers the responses of 115 school students to two problems based on information provided in two-way tables. In each case the question asks if one of the variables involved depends on the other. Contextual knowledge might suggest a dependent relationship in both but in one problem the data show independence while in the other the data imply an inverse relationship. A wide range of solution strategies illustrates the cognitive complexity dealing with information in two-way tables.

The current COVID-19 situation has seen a plethora of data, statistics and predictions presented to society--some making incredible claims. The authors are concerned that statistics and probability still receive inadequate attention in the classroom, leaving students without the statistical literacy needed to make sense of the claims being made. COVID-19 has raised the visibility of data in society more than any other event in recent times. Although a crisis for society in both health and economic spheres, COVID-19 also provides a stimulus to examine the contribution that education, particularly statistics education and the "Australian Curriculum," can make to preparing society to understand the data and accompanying arguments presented to them. This article argues for more attention to be paid to Statistics by documenting the presence of data analysis across the curriculum and by presenting a selection of activities judged to contribute to building greater statistical understanding and literacy.

Explaining the complexity of mathematics teaching to others can be challenging. The Knowledge Quartet, a pedagogical content knowledge (PCK) framework from the UK, provides a fresh lens to reflect on mathematics classroom practice. Developed from videotaped mathematics lessons taught by final year pre-service teachers, the Knowledge Quartet describes four dimensions of teachers' knowledge for teaching mathematics: Foundation, Transformation, Connection, and Contingency. Each of these categories has several indicators that may be observed in the overall process of teaching, including during planning and preparation, classroom instruction, and reflection and evaluation phases.

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning. [ABSTRACT FROM AUTHOR]

The current expectation of teachers in Australia is that they are able to collect, interpret, and use data related to teaching and learning. Digital technologies in schools, such as electronic methods of record-keeping, offer enhanced opportunities for teachers to perform this skill, and its application has been growing steadily in education. The aim of this exploratory study was to examine fourth-year pre-service teachers' behaviour in record-keeping whilst on their final professional experience placement. Using Ajzen's (1992) theory of planned behavior, this study found that most pre-service teachers exhibited positive attitudes toward the behaviour of recording, using, and analysing classroom data. Despite this positive attitude, many pre-service teachers were unable to maintain any system of record-keeping whilst on placement. For many, this was due to a number of external influences or perceived external influences, which acted as a constraint to their behaviour. (Contains 4 tables and 1 figure.)

This paper examines the relationship between key predictors of middle school students' interest for statistical literacy, including their self-efficacy for statistical literacy and their prior mathematics achievement. Given the lack of research in the statistical literacy domain, a review of the mathematics education literature confirms the importance of self-efficacy and achievement in predicting interest and describes the nature of their influence on interest. The paper then reports on an empirical study involving a sample of 438 Australian middle school students from 8 schools across 3 states. Data from this study are used in a series of statistical models to confirm the influence of identified predictors and to study their inter-relationship. Results from the study suggest that the influence of prior achievement on interest is completely mediated by self-efficacy. In addition to this, the relationship between interest and self-efficacy appears to be quadratic, in that students with high levels of self-efficacy may not have corresponding high levels of interest. Implications for this research are then discussed. (Contains 2 tables and 3 footnotes.)

The following paper describes the development of an instrument designed to assess middle school students' interest in statistical literacy. The paper commences with a review of the literature as it relates to interest in this context and then proposes a theoretical model upon which the proposed instrument is based. The Rasch Rating Scale model is then applied to student responses to items in the instrument and fit statistics are analysed in order to assess the extent to which these responses conform to the requirements of the measurement model. The paper then presents evidence, including interview data, to support the validity of interpretations that can be made from the proposed instrument. The findings suggest that the proposed instrument provides a theoretically sound measure of middle school students' interest for statistical literacy that will be useful for the evaluation of interventions aimed at developing these students' statistical literacy. (Contains 6 tables and 2 figures.)

Since 2008, all Australian students in Years 3, 5, 7 and 9 have been assessed in literacy and numeracy through an annual National Assessment Program--Literacy and Numeracy (NAPLAN) test. In 2015, a team of mathematics education researchers across Australia conducted a nationwide research project to identify school practices and policies that were consistent between schools that showed growth and/or improvement in their NAPLAN numeracy results. This paper reports findings from three case study schools, using a school improvement framework to interpret evidence gathered from the schools' principals and school leaders. The study has particular implications for policy makers and school leaders who may be seeking ways to improve mathematical practices in their own jurisdictions and schools.

In this article, data from a study of the mental computation competence of students in grades 3 to 10 are presented. Students responded to mental computation items, presented orally, that included operations applied to fractions, decimals and percents. The data were analysed using Rasch modelling techniques, and a six-level hierarchy of part-whole computation was identified. This hierarchy is described in terms of the three different representations of part-whole reasoning--fraction, decimal, and percent--and is elaborated by a consideration of the likely cognitive demands of the items. Discussion includes reasons for the relative difficulties of the items, performance across grades and directions for future research. (Contains 2 tables and 3 figures.)

Using mathematical gossip in the classroom allows teachers to expand their students' horizons, and provide pathways to improvement of understanding. The expansion of a simple idea into another mathematical context can enrich a student's learning. In particular it may help to bridge the gap between purely procedural approaches and a conceptual understanding of the underlying mathematics. Having both procedural and conceptual understanding provides a springboard for strategic use of mathematics in both mathematical and social contexts. The connections built through making use of mathematical gossip allow a "big picture" view of mathematics to be taken, and may improve the ability to use mathematics strategically.

Interest has long been regarded as an important motivational construct in the learning of mathematics. It has been contended that the development of interest is directed by two control systems: an emotional and a cognitive. Under the former, students are attracted to activities that are enjoyable, whereas under the latter they consciously engage in tasks that might satisfy, for example, later goals. Younger students are more likely to associate emotional responses to interest-related stimuli, whereas older students tend to associate cognitive responses, suggesting qualitative developmental differences for students' interest in mathematics. Yet there are differences between the classroom motivational environments experienced by primary/elementary vs. secondary school students, and the mathematical interest and enthusiasm likely felt and expressed by primary vs. secondary teachers, which may influence dimensions of student interest. Of concern in this study is the extent to which teacher enthusiasm impacts the classroom motivational environments (mastery vs. performance goal orientation) perceived by students and reported by teachers, and thereby dimensions of students' interest for mathematics. Using survey data from 471 students in Grades 3-10 and 44 of their mathematics teachers, we find that students' perceptions of their teachers' enthusiasm for teaching mathematics positively predict their perceptions of a classroom mastery environment, which in turn predicts their interest. However, at the class-level, teachers' enthusiasm for the subject of mathematics negatively predicts the emotional interest of their students. Implications of these results are discussed.

School autonomy has been identified as having an impact on a school's performance, yet less has been reported about the effect this has on students' goal orientations and engagement with mathematics. In a national study conducted in schools across Australia, measures of school autonomy were collected from teachers and school leaders, along with students' perceptions of the mastery and performance goal orientations of their classrooms and personally using surveys. Schools were identified as having high or low levels of autonomy on the basis of school leaders' responses. For the study discussed in this paper, a subset of 14 schools for which matched student and teacher data were available provided students' responses to a variety of variables including goal orientations. The findings suggested students in high-autonomy schools were less likely to hold a personal performance approach and avoidance goals than their peers in low-autonomy schools. Fifty-five case studies conducted in 52 schools provided evidence of some of the practical aspects of these findings, which have implications for systems, schools and teachers.

Statistical literacy increasingly is considered an important outcome of schooling. There is little information, however, about appropriate expectations of students at different stages of schooling. Some progress towards this goal was made by Watson and Callingham (2005), who identified an empirical 6-level hierarchy of statistical literacy and the distribution of middle school students across the levels, using archived data from 1993-2000. There is interest in reconsidering these outcomes a decade later, during which statistics and probability has become a recognised strand of the Australian mathematics curriculum. Using a new data-set of over 7000 student responses from middle-years students in different parts of Australia during the period 2007-2009, the nature of the hierarchy was confirmed. Longitudinal analysis identified how students performed across time against the hierarchy. Suggestions are made for systems and teachers about realistic expectations for middle-years students, and possible curriculum challenges.

The Best Practice in Mathematics Education project was funded by the Australian Office of the Chief Scientist, to examine promotion of students' learning, engagement and aspirations in this core learning domain. We draw upon cross-sectional survey data from 551 students in grades three to nine to examine how students' mathematics engagement relates to key dimensions of their learning climate (mastery or performance focused classrooms), teacher enthusiasm, and school caring. Engagement is known to be associated with positive school outcomes and influenced by environmental factors. Less known is whether, and the extent to which, students have different profiles of engagement across component dimensions (Fredricks, Blumenfeld, & Paris, 2004); and, how profiles may differ according to experienced environment dimensions. We first develop profiles of adolescents' behavioral, emotional, and cognitive engagement using multilevel latent class analysis, educing three profiles of "engaged", "compliant", and "disengaged" students, who systematically differed on experienced environmental factors. Mastery focused classrooms, enthusiastic teachers and caring school environment were experienced most by engaged, and least by disengaged students; performance focused classrooms were unrelated to engagement profiles. Identified patterns will be of particular use to teachers in understanding how class, teacher, and school influences together shape students' own engagement in mathematics.

This paper presents the findings of a small-scale study that investigated the issues and challenges of teaching and learning about quantitative reasoning (QR) within a project-based learning (PjBL) context. Students and teachers were surveyed and interviewed about their experiences of learning and teaching QR in that context in contrast to teaching and learning mathematics in more traditional settings. The grade 9-12 student participants were characterised by a history of disengagement with mathematics and school in general, and the teacher participants were non-mathematics specialist teachers. Both students and teachers were new to the PjBL situation, which resulted in the teaching/learning relationship being a reciprocal one. The findings indicated that students and teachers viewed QR positively, particularly when compared with traditional mathematics teaching, yet tensions were identified for aspects such as implementation of curriculum and integration of relevant mathematics into projects. Both sets of participants identified situations where learning QR was particularly successful, along with concerns or difficulties about integrating QR into project work. The findings have implications for educators, who may need to examine their own approaches to mathematics teaching, particularly in terms of facilitating student engagement with the subject.

The increased importance of developing statistical understanding in school education is recognised in curriculum documents across the world. The role of technology in enhancing the teaching of statistics is emphasised in these documents and the emergence of quality computer software and websites provides teachers with access to unprecedented resources for teaching statistics to young students. Assessment processes, however, have not kept pace with the advances in technology. This paper highlights some emerging and existing issues in the assessment of statistical understanding at the school level, and includes discussion of the implications for teachers and researchers.

Statistics is an increasingly important component of the mathematics curriculum. "StatSmart" was a project intended to influence middle-years students' learning outcomes in statistics through the provision of appropriate professional learning opportunities and technology to teachers. Participating students in grade 5/6 to grade 9 undertook three tests, a pre-test, a post-test and a longitudinal retention test over a period of 2 years. Their teachers completed a survey that included items measuring pedagogical content knowledge (PCK) for teaching statistics. Despite the development of valid instruments to measure both student and teacher content knowledge and teachers' PCK, linking teachers' knowledge directly to students' learning outcomes has proved elusive. Multilevel modelling of results from 789 students for whom there were 3 completed tests and measures from their teachers indicated that students' outcomes were influenced positively by their initial teacher's PCK. Extended participation of teachers in the project also appeared to reduce negative effects of changing teachers.

This article presents a case study of one particular lesson in which a teacher uses the features of an interactive whiteboard (IWB) to teach mental computation strategies to a Grade 1 class. The framework of Technological Pedagogical Content Knowledge (TPACK) is used to evaluate the teacher's use of the IWB to teach mathematical concepts to young children. The findings demonstrate that the TPACK framework can be useful in considering the mathematics knowledge required for teaching in the early years and helpful for describing teaching practices that incorporate technology. The study has implications for teachers and teacher educators who may use the framework to understand teaching approaches using technology with the aim of enhancing early years' classroom practice.

In Ethiopia, primary school teachers of science and mathematics are encouraged to integrate Information and Communication Technology (ICT) into their teaching as a means to improve the quality of education. However, there has not been the same emphasis placed on providing professional learning opportunities for teachers on how to use ICT in their teaching. The present study investigated how a group of practising primary school science and mathematics teachers developed the skills needed to integrate ICT into their teaching. The study employed a combination of qualitative and quantitative research methods within an action research approach. The teachers took part in activities from a Technological Pedagogical and Content Knowledge (TPACK) based professional learning workshop, including designing lessons, classroom instruction, and reflection activities in teams. A lesson evaluation sheet, questionnaire, observation checklist, and logbook were used to gather data. The results showed that the teachers acquired an improved competency to integrate available ICT into their teaching through the intervention activities.

The nature of numeracy implies that both mathematics and social context are involved. In the Australian curriculum, numeracy is included as one of seven general capabilities that it is the responsibility of every teacher to develop in their students. Many teachers, however, dislike or actively avoid mathematics. This situation tends to lead to opportunities for numeracy development in school students to be overlooked or ignored. A key issue is then to help teachers to recognise the appropriate mathematical ideas in subjects other than mathematics, and then to suggest ways in which these ideas can be acknowledged and developed in context. Three different approaches to working with both pre- and in-service high school teachers are described and the outcomes discussed in the context of building professional capital. Findings suggest that teachers do not recognise the complexity of numeracy, and benefit from supportive professional learning developing professional capital with respect to numeracy.

The gender-related challenges facing males entering the primary-school teaching profession have been well documented in the academic literature over recent decades. The majority of these data have come about through qualitative reports. Whilst qualitative methods provide important perspectives into these issues, the use of valid and reliable quantitative survey tools has received less attention. This paper discusses the construction, piloting, and subsequent tests for reliability and validity involved in developing a robust survey tool to measure the gender-related challenges faced by pre-service male primary teachers during their university study and professional experience in schools. Utilizing Senocak's four stages of survey development and Rasch modelling techniques, data analysis showed high levels of validity and reliability for the survey tool.