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In Australia, all teachers are expected to explicitly attend to numeracy in the subjects they teach. Pre-service teachers, therefore, need to begin to develop effective numeracy teaching strategies but there is a limited research base to inform the design of courses that address this need. This paper reports on findings from a study investigating the impact of one such course. The paper draws on data from two pre-service teachers: two course tasks completed at the beginning of the course and interviews conducted at the end of the course. Data were analysed using the 21st Century Numeracy Model. Findings suggest there were changes to the pre-service teachers' understanding of numeracy and their capacity to embed numeracy in the subjects they will teach.

Engagement in mathematics education is an important factor in a successful student experience. This paper reports findings from a study of Year 5 student perceptions of engagement during a two-week, inquiry-based learning (IBL), problem-posing investigation. The study triangulated data from semistructured interviews, video observations and student work samples to understand the student's perspectives of IBL; however, this paper reports the interview data. The findings indicate that most of the students were behaviourally, emotionally, and cognitively engaged during the IBL investigation, and that the investigation provided an opportunity for students to experience levels of competence, relatedness, and autonomy need satisfaction.

Prospective secondary mathematics teachers in Australia are typically taught by mathematics educators and mathematicians who work in different faculties and seldom collaborate--a situation that can lead to conflicting views about how to teach mathematics. This paper reports on findings from semi-structured interviews with three mathematics teacher educators in a hybrid mathematics/mathematics education department. Valsiner's zone theory is used to analyse how their beliefs, institutional context and professional learning opportunities shape their identities as MTEs. Findings reveal that the MTEs had developed similar beliefs within supportive institutional context but drew on different sources of professional learning.

Mathematical reasoning is foundational to making sense of mathematics. Yet assessing mathematical reasoning can be challenging for teachers. This paper reports on a project where teachers taught two lessons with a specific focus on reasoning and came together with other teachers at their school to attempt to assess the reasoning of their students. Results, derived from an analysis of two post-lesson discussions about student work samples and associated completed assessment rubrics, provide insights into the challenges identified by 34 primary teachers at 4 Victorian government schools.

In this paper, we provide an overview of the "Maths Inside" project, funded by the Australian Maths and Science Partnership Program (AMSPP). The overall aim of the AMSPP is to improve uptake and participation of students in mathematics and science at secondary and tertiary levels. In this research project, we aim to improve student interest in mathematics and support mathematics teachers in their professional learning, through provision of rich and investigative learning resources, including video case studies of CSIRO scientists and mathematicians. Data collection on the outcomes of the project is ongoing and will be reported in subsequent papers.

Secondary mathematics pre-service teachers often have different experiences of mathematics and its teaching and learning during their initial teacher education. This paper documents the beliefs about mathematics, its teaching, and its learning, of mathematicians and mathematics educators who teach secondary mathematics pre-service teachers. The beliefs of the surveyed sample of eighty-two academics and differences between groups were characterised using descriptive statistics and one-way comparisons between groups ANOVA. Generally, respondents had a Problem-solving view of mathematics and those with education backgrounds were more in agreement with that method of teaching.

Pasifika cultures have a rich background of mathematics including a strong emphasis on patterns used within craft design (Finau & Stillman, 1995). However, there have been limited studies which have investigated the use of contextual Pasifika patterns in mathematics classrooms. The aim of this study was to explore how contextual Pasifika patterning tasks can potentially support young children to develop their understanding of growing patterns. Ten lessons using Pasifika and Maori patterns were undertaken with 27 Year 2 students (6-year-old). In this paper, analysis of one of the lessons is used to examine how a contextual task assisted these young students to generalise growing patterns.

This paper presents a new theoretical viewpoint blended from the perspectives that mathematical meaning is extracted (from objects falling under a particular concept) and that mathematical meaning is given (to objects that an individual interacts with). It is elaborated that neither uni-directional framing (whether involving extracting meaning or giving meaning) provides a comprehensive account of the complex emergence of evolving forms of meaning. It is argued for a framing that construes sense-making in mathematics as dialogical: where what meaning one extracts is a function of what meaning is given to, and vice versa.

This paper addresses the need for empirical research on the processes by which students create algorithms. I analyse the collaborative work of three high-school students on a contextualised graph theory task, in which they created an algorithm for maximising the happiness score of a seating arrangement. The group found an optimal arrangement but created an algorithm that did not fully account for this arrangement. The group's written algorithm reflected only the properties of their optimal arrangement that they explicitly noticed after creating the arrangement. And, these explicitly-noticed properties aligned with the group's predominant contextual considerations.

As a contribution to honour the foresight of Ken Clements and John Foyster in founding MERGA [Mathematics Education Research Group of Australasia] so many years ago this paper is not a research paper in the usual sense. Rather it sets out to sample the context of Mathematics Education in Australasia and beyond (then and now) and to highlight some challenges as seen by this author. In this personal view I do not intend to expand in detail upon particular strands of research in which I have been involved, although for purposes of illustration examples will be drawn from time to time from this and other work. MERGA is about both people and scholarly activity, and so this paper will make reference to both--for history, culture, and challenge are essential components of the development of any organisation.

This paper describes how a new pre-service teacher engaged with mathematical content in order to learn it for teaching, during practicum. The results show that the PST learned mathematical content by initiating and carrying out a preparation phase prior to planning. This phase involved searching through internet sites and making notes that were then used to support her lesson planning and teaching. Implications for pre-service teacher education are also presented.

This paper explores how beliefs about mathematics may influence low-achieving adults' re-engagement with mathematics in the tertiary sector. Adult learners who have problematic mathematical histories often hold negative beliefs about the nature of mathematics and how it is learned. In New Zealand these adults are often required to re-engage in mathematical provision in the tertiary sector to gain qualifications for employment. The beliefs they hold about mathematics may negatively influence their approach to learning mathematics and their affective response to it. This paper explores several ways in which negative beliefs about mathematics may undermine adults' success.

This paper uses the hammer-and-nail metaphor to highlight the rigidity and impulsiveness that can be found in student mathematical behaviour. The hammer-and-nail phenomenon can be attributed to two plausible causes: (1) the way the human mind works; and (2) the way mathematics is traditionally taught in school. In this paper, the following aspects of the hammer-and-nail metaphor are discussed: (1) its origin and its usefulness; (2) its manifestations among learners of mathematics; (3) related literature; (4) pedagogical causes; and (5) suggestions for teaching.

Why have an organisation like MERGA? This question will be addressed from past, present and future perspectives (1976, 2012, and 2025). One focus of the paper will be the need to improve mathematics curricula, and to improve the teaching and learning of mathematics, at all levels. I shall argue that we have not done enough to make sure that MERGA has delivered, is still delivering, and will continue to deliver the goods on such basic curriculum/teaching/learning issues. Part of the difficulty is that we researchers have not reached agreement on what we mean by "improvement". That is as much a political issue as anything else, of course, but the MERGA community needs to do more to make sure that the responsibility for defining what improvement means, and how it is assessed, is not in the wrong hands. A second focus of the paper will be some reflections on what the "A" in MERGA might represent. This Conference is being held in Singapore, and the challenge is for a wider vision of MERGA's role in Asia to be formulated and implemented.

This paper reports on a project that aims to foster interdisciplinary collaboration between mathematicians and mathematics educators in pre-service teacher education. The project involves 23 investigators from six universities. Interviews were conducted with selected project participants to identify conditions that enable or hinder collaboration, and to identify learning mechanisms at the boundaries between disciplinary communities. A hybrid narrative constructed from the interviews is used to illustrate "transformation" as a learning mechanism that leads to new practices. [This project is funded by the Australian Government Office for Learning and Teaching (grant #MS13-3174).]

The purpose of this study was to explore the factorial structure of motivation and perception items from a student survey utilised as part of the Reframing Mathematical Futures II (RMFII) Project. Data was collected in 2017 from 442 students in Years 7 to 10 from various different States across Australia. An exploratory factor analysis identified four factors which were consistent with the studies the items were adapted from: Intrinsic and Cognitive Value of Mathematics, Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics. An analysis of variance (ANOVA) also revealed that there were statistically significant differences between Year Level and State for some of these factors.

To support teachers in their quest to incorporate reasoning as a mathematical proficiency as espoused in the Australian Curriculum: Mathematics, a professional learning research project using demonstration lessons was carried out. This paper reports on the impact of demonstration lessons on one participating teacher's pedagogical knowledge about reasoning. The growth in this teacher's knowledge was analysed using a phenomenographic framework established to evaluate teachers' development in mathematical reasoning. The results show that demonstration and subsequent trial lessons contributed to her growth.

This paper reports on an intervention program, "Prepare 2 Learn", that was designed taking into account a range of components from other successful intervention programs. The program is focussed on year 6 students from a school in Melbourne, Australia, who are falling approximately 6 months behind with the hope that extra help at an early stage may result in them reaching the required standard and realising their potential. While the students' academic results moved substantially a more pleasing result was the noticeable improvement in the students' approaches to their learning.

Self-belief can directly predict students' academic motivation and achievement. Research indicates that mathematical self-belief often decreases during the middle years of schooling. This study explored the mathematical self-belief development of 15 Year 7 students. Data were gathered from a survey, a mathematics achievement test and interviews. Results were analysed and interpreted from a multilevel perspective. Findings indicate that student-level characteristics, such as persistence, were the most influential on mathematical self-belief. While class-level contexts, such as ability grouping, were less influential, interpersonal relationships with teachers played a major role. [The research reported in this paper was part of a larger project supported by an Australian Research Council Linkage grant.]

This design-based research project investigated the development of functional thinking in algebra for the upper primary years of schooling. Ten teachers and their students were involved in a sequence of five cycles of collaborative planning, team-teaching, evaluating and revising five lessons on functional thinking for their students over one year. This paper focuses on two aspects of the study related to developing students' functional thinking by visualising the structure of a growing pattern in different ways. An appendix presents the assessment task used at the beginning of the lesson sequence.

This paper reports teachers' beliefs about the extent to which expertise in one-to-one teaching can be transferred to classroom teaching. The study involved 21 mathematics intervention specialists. Data collection involved a structured questionnaire with six openended questions. Participants were found to be very positive towards transferring strategies developed in one-to-one teaching to their classroom teaching. The strategies included using material settings, using particular questioning techniques, incorporating assessment into teaching, focusing on dimensions of mathematisation, valuing students' responses, teaching at the 'cutting edge', and using mathematical language.

Mathematical reasoning features in curriculum documents around the world, but is understood and enacted poorly by teachers in classrooms. We explore teachers' noticing of reasoning during observed lessons. Two teams of primary teachers in Canada and Australia worked to plan, deliver, and observe lessons intended to include reasoning. They observed each other teaching a lesson that was planned with the assistance of a researcher, and later, a researcher observed each post-lesson discussion. Given the reported benefits of teachers' noticing of reasoning during peer-observed lessons, targeted professional learning support is required to further enact teachers' peer discourse to facilitate mathematical reasoning.

Problem solving task design is not only the design of a non-routine problem to be solves by the students. Our task design also requires a supporting document, the practical worksheet, which would act as a cognitive scaffold for the students in the initial stages of the problem solving process before they can internalize the metacognitive strategies and automate the use of these strategies when faced with a new problem. A further enhancement of the scaffolding that can be provided by the teacher as she facilitates forty or more students working on the practical worksheet is a set of scaffolding cards. In this paper, we describe the cards and the preliminary use of these cards to facilitate problem solving for teachers in a professional development workshop.

This paper identifies the strategies Singaporean and Australian students (n = 1,187) employed to solve a 24-item mathematics test. A mathematics-processing framework is proposed, which describes the way primary-aged students successfully process graphic and non-graphic mathematics tasks. There were distinct differences in the way in which the students from the respective countries approached the tasks with the Singaporean students more likely to employ strategies that were explicitly taught and practiced in the classroom, whereas the Australian students tended to employ a more diverse range of approaches.

The learning and teaching of mathematics are key elements for primary school teachers, and various approaches for teaching mathematics to pre-service teachers are evident in mathematics education. This paper reports on a project to develop a critical approach to using mathematical subject knowledge in choosing learning examples and examines data from United Kingdom pre-service teachers' lessons. The data suggests pre-service teachers have no structured method for choosing examples, which impacts on the quality of the learning experiences of students.

The use of high-stakes assessment to measure students' mathematical performance has become commonplace in schools all over the world. Such assessment instruments provide national or international comparisons of student (and potentially teacher performance). Each form of assessment is specialised in nature and is characteristic of the culture and intent of the governing bodies. The purpose of this paper is to highlight differences and similarities between two national high-stakes assessments and the possible implications to students' sense making.

Rhetoric about the importance of students being equipped to apply mathematics to relevant problems arising in their lives, individually, as citizens, and in the workplace has never been matched by serious policy or curricular support. This paper identifies and elaborates authenticity implications for addressing this issue, and describes aspects of a modelling challenge in which students were mentored to engage in problem solving located in real world settings. Characteristics of the approach and selected student responses to the challenge are provided.

"Make It Count" was a large scale, government-funded, project aimed at improving the mathematics learning of Indigenous students. NAPLAN Numeracy test results were used as one measure of the effect of the program. In this paper we report on the performance on these tests of Indigenous students in schools involved in the project. Group data and, where available, longitudinal data for individual students are reported.

What types of performance do we encourage in our mathematics students? How do these performances reflect students' identity constructions? This paper uses a performance metaphor for identity to analyse interviews with teachers and students in their first year of secondary school. "Perseverance" is valued by teachers, yet many mathematics classrooms appear not to provide the opportunities to persevere and students do not perform identities of perseverance. Rather, "quickness" and "just knowing" the answer are considered part of a successful mathematics identity.

Students develop understanding of mathematics when they translate between and within different mathematical representations. This paper explores a student-generated story and content descriptors from the Australian Curriculum: Mathematics to highlight how primary school students can represent mathematical concepts through exploring the links between everyday physical objects, pictures, oral/written language, models and mathematical symbols. This active experience enhances the students' capacity to represent mathematical concepts and ideas, symbolise these, and eventually learn to abstract and generalise.

A current concern is student learning outcomes and these are largely a function of teachers' knowledge and their practice. This position paper is premised on the notion that certain knowledge is required for the teaching of mathematics. An exploration of literature demonstrates that such professional knowledge development can be supported by Learning Trajectories (LT). We propose to use LT as theoretical lens to examine pre-service teachers' Content and Pedagogical Content knowledge and advance a research design.

This paper presents the results from an initial lesson in a series of design experiments focusing on young Indigenous students' understandings of growing patterns. Indigenous students in Year 2 and 3 (n = 16) participated in pre lesson activities and a 45 minute lesson on growing patterns. Tentative findings from this study suggest that; (a) Year 2 and 3 Indigenous students are capable of working with growing patterns; (b) contextual artefacts assisted with communication; and (c) gesture played an important two-fold role in the lessons and communication of the mathematics experienced.

In this paper, the RBC+C framework (Hershkowitz, Schwartz, & Dreyfus, 2001) is used to analyse and describe construction and consolidation of mathematical knowledge by primary pupils in a whole-class setting. I describe a lesson that concerned what is commonly termed the Handshakes problem. One pupil spontaneously established a connection with a related problem in which the class had engaged a month previously. There followed a conversation in which an older construct was consolidated while a new construct emerged--the nature of this intertwined construction and consolidation is discussed.

In this study we examine the use of cumulative and exploratory talk types in a year 5 computer supported collaborative learning environment. The focus for students in this environment was to participate in mathematical problem solving, with the intention of developing the proficiencies of problem solving and reasoning. Findings suggest that students engaged in exploratory talk may more regularly attempt the use of technical (tier 3) mathematical vocabulary.

Mathematics anxiety affects primary pre-service teachers' engagement with and future teaching of mathematics. The study aimed to assess the level and range of mathematics anxiety in first year pre-service teachers entering their teacher education course, and to investigate the sources of this anxiety as perceived and identified by them. Data collection methods included the RMARS survey, and Critical Incident Technique. The results indicate that the most common negative impacts on pre-service teacher mathematical self-concept involved experiences with teachers. However, their current mathematics anxiety is most commonly aroused under testing or evaluation situations.

This study investigated potential gender differences in a sample of 807 Year 6 Singaporean students in relation to two variables: spatial visualisation ability and cognitive style. In contrast to the general trend, overall there were no significant gender differences on spatial visualisation ability. However, gender differences were prevalent among students who possessed high spatial visualisation ability, in favour of boys. In terms of cognitive style, there were significant gender differences in the spatial imagery and verbal information processing dimensions. Boys gave higher ratings to their spatial-imagery encoding and processing preferences than their verbal information processing preferences. Some of these findings are in contrast to studies undertaken in the educational-psychology literature. Implications are drawn regarding pedagogical practices in Singaporean schools.

The aim of this study was to determine the role of semiotics in assisting young Indigenous students to engage with and identify the general structure of growing patterns. The theoretical perspective of semiotics underpinned the study. Data are drawn from two Year 3 students, including analysis of pretest questions and two conjecture-driven lessons. Results indicate that particular semiotic signs (iconic signs) contribute to how young Indigenous students attend to, and identify the structure of growing patterns.

According to Stenhouse (1984), "research is systematic 'inquiry' made public". By bringing "inquiry" into teaching practice we promote learning in three layers: learning of mathematics; learning of the "teaching" of mathematics; and learning of the processes through which mathematics teaching and learning develop. Through examples of developmental practice from school-based research in the UK and Norway, and research in university-based mathematics teaching in the UK, I discuss ways in which mathematics teachers and mathematics educators can form communities of inquiry to promote development in learning and teaching and look critically at some of the issues involved. These issues raise challenges for promoting development within a philosophy of educational practice and at scale, with reference to the wider dimensions of society, system and culture.

Ill-structured tasks presented in an inquiry learning environment have the potential to affect students' beliefs and attitudes towards mathematics. This empirical research followed a Design Experiment approach to explore how aspects of using ill-structured tasks may have affected students' beliefs and attitudes. Results showed this task type and learning environment created situations that exposed and challenged students' beliefs and attitudes and required them to defend their position. Insights regarding factors that may influence students' beliefs and attitudes are discussed.

Declining participation rates in advanced mathematics courses and STEM-related occupations has been an issue in Australia for some time, particularly for females. As students continue to disengage with mathematics and complain about its usefulness, it is important to explore what we can do to stem the tide of departing students. One area worthy of investigation is students' interest in mathematics including whether they are able to name a mathematical role model in their lives. Forty-three students in Years 7 to 9 from three schools were asked to name people they knew who were interested in mathematics. There was a strong bias towards male figures (44 to 17), particularly fathers and male peers.

This study investigates the relationship between students ability to answer reduced language dependency mathematical questions with their overall numeracy level. It investigates whether a student's success at reduced language mathematical questions translates into better overall numeracy scores. It was found, students have up to two years advancement if able to correctly answer reduced language dependency questions. This phenomenon was clearly apparent in the overall findings, but was most pronounced at the Year 3 level test, and for female students.

As part of an intervention project to encourage exploratory talk with young children in mathematics, it was found that, although the children did not engage fully in reasoning, the intervention had supported some children in developing more cohesive discourse. The cohesion was evidenced through the children's use of deictic words, in particular the word "that". Examples of dialogue are contrasted to illustrate the changes in this use of deixis and are related to children's meaning-making in mathematics.

A 3-year longitudinal study "Transforming Children's Mathematical and Scientific Development" integrates, through data modelling, a pedagogical approach focused on mathematical patterns and structural relationships with learning in science. As part of this study, a purposive sample of 21 highly able Grade 1 students was engaged in an innovative data modelling program. In the majority of students, representational development was observed. Their complex graphs depicting categorical and continuous data revealed a high level of structure and enabled identification of structural features critical to this development.

The purpose of the study was to investigate relationship between mathematics anxiety and attitude towards mathematics among secondary school students in South India. Data were collected from 112 secondary school students in a private school. Demographic information such as gender and age of the students were also collected. Structural equation modelling was used to test the hypothesised relationships between mathematics anxiety and attitude to mathematics variables. Independent-samples t-tests were used to examine the differences in the measured variables based on gender and age.

The processes of mathematisation, the use of mathematical models and representations of real world contexts, and contextualisation, the embedding of mathematical ideas into a meaningful context, are key aspects of students' mathematical learning. We present a conceptual framework for thinking about mathematising and contextualising developed as part of the "Make it Count," a national project that seeks to develop an evidence base of practices that improve Indigenous students' learning in mathematics. We suggest that an intentional focus on mathematisation and contextualisation helps to make mathematics meaningful, particularly for Indigenous students. In particular we suggest that such a focus has the potential to enhance the mathematical resilience of Aboriginal students.

Many key aspects of modern theory relating to classroom discourse in mathematics education, especially the theory of didactical situations, have not progressed much further than theories put forward in the 1840s by the North American educator David Page. Aspects of Brousseau's (1997) description of what he called the Topaze effect, are remarkably similar to Page's idea of "drawing out"--a mode of questioning that was likely to remove the cognitive challenge for learners engaged in non-trivial mathematical tasks.