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Bridging research and practice is often at the core of collaborative research with schools. Longitudinal research designed to study teachers’ learning to teach mathematical inquiry grew from a small pilot into a large nationally funded project that has provided curricular resources for state implementation of the Australian Curriculum. The aim of the symposium is to present perspectives from key stakeholders for scaling up research. The symposium aims to tell a story of a partnership that worked to embed inquiry-based learning in mathematics into schools from the perspective of three key stakeholders: university researchers, primary teachers and curriculum writers from the state education department. Collectively, the papers explain the research development and outcomes, its classroom implementation, interactions between researchers and state department and creation of state-level curricular resources. Recommendations are given. In the second paper, Achievements and challenges encountered by classroom teachers involved in a research project: A reflection, teachers Allmond and Huntly share their initial apprehension, challenges and growing confidence in the project over more than six years. Recommendations from the teachers are given for researchers, teachers engaging with research projects and school leaders involving their schools in research.

Bridging research and practice is often at the core of collaborative research with schools. Longitudinal research designed to study teachers’ learning to teach mathematical inquiry grew from a small pilot into a large nationally funded project that has provided curricular resources for state implementation of the Australian Curriculum. The aim of the symposium is to present perspectives from key stakeholders for scaling up research. The symposium aims to tell a story of a partnership that worked to embed inquiry-based learning in mathematics into schools from the perspective of three key stakeholders: university researchers, primary teachers and curriculum writers from the state education department. Collectively, the papers explain the research development and outcomes, its classroom implementation, interactions between researchers and state department and creation of state-level curricular resources. Recommendations are given. Finally, Horne and Makar, authors of Building relationships between stakeholders and researchers: People, persistence and passion, close by discussing collaboration between the state education department and university researcher: the initiation and development of the partnership, challenges encountered, outcomes gained, and recommendations to stakeholders and researchers for building and maintaining productive partnerships.

The papers in this symposium outline the aims and design of a three year Teaching and Learning Research Initiative (TLRI) project involving mathematics education researchers and educator colleagues from Massey University and Victoria University of Wellington. The Learning the Work of Ambitious Mathematics Teaching project seeks to improve the mathematical experiences and educational outcomes for all students by looking to advance current practices for the preparation of mathematics teachers. In trialling new ways to support prospective teachers to not only ‘think’ like teachers, but also to put what they know into action, our project is focused on strengthening links between coursework and fieldwork. The second and third papers draw on our experiences within the first year of the three year project. Both papers concern our introduction of new Instructional Activities (IAs) to a range of different courses and programmes. In paper three, Averill, Drake, and Harvey draw on their classroom experiences of enacting the IAs through the rehearsal process. Key to the rehearsal is the public coaching by the teacher educators. An analysis of two groups of prospective teachers’ perceptions of the coaching process is used to further understand how teacher educators and prospective teachers might best engage in the public and interactive rehearsal process.

Worldwide interest in Japanese Lesson Study as a vehicle to improve mathematics teaching practice through professional learning has left largely unanswered questions about the extent to which it can be replicated elsewhere. The Implementing structured problem-solving mathematics lessons through Lesson Study project worked with teachers from three Melbourne primary schools to identify critical factors in adapting and implementing the structured problem-solving mathematics lessons that underpin Japanese Lesson Study; explore ways in which key elements of Japanese Lesson Study can be embedded in Australian practice; and establish the effect of participating in Japanese Lesson Study on teachers’ mathematics pedagogical content knowledge. Two Years 3 and 4 teachers, together with the curriculum director of numeracy coach from each of three schools and the network numeracy coach participated in the project. The teachers worked in two Lesson Study teams, with each team planning and teaching a research lesson in Term 3 and Term 4 in 2012. In this symposium, we report findings from the project and argue that the Japanese Lesson Study model for professional learning, together with the underpinning problem-solving structure of its research lessons, can build teacher capacity for effective mathematics teaching in Australian primary schools.

The symposium presents findings from a cross-country study, drawing attention to dimensions of students’ learning by analysing students’ mathematics sense making and assessment performance in high performing but culturally different contexts—Australia and Singapore. The symposium details the performance of Australian and Singaporean students on tasks sourced from each country’s respective national assessment instruments; identifies the approaches and strategies students from different cultures employ to solve mathematics tasks; and draws conclusions about the influence cultural and pedagogical practices have on students’ approaches to solving these tasks. Paper 4: Cross-country Comparisons of Student Sense Making: The Development of a Mathematics Processing Framework by Tom Lowrie. This paper identifies the strategies Singaporean and Australian students (n = 1187) 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 nongraphic 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 practices in the classroom.

In previous MERGA conferences, we have reported on our project to implement Polya-type problem solving in the main curriculum of one high achieving secondary school and the follow-up project to diffuse the innovation to a wider range of secondary schools. We used design experiment as the methodology in both of the projects. In this symposium, we report on various aspects of this design experiment while it is in progress. The three papers show the vital interactions between the researcher-designers and the teacher-implementers to fine-tune a workable design for each individual school.

In this symposium, we discuss our project Mathematical Progress and Value for Everyone (MProVE) which is about doing this hard research on helping low achievers at the Secondary levels improve in their learning of mathematics. We start with the recognition that a complex confluence of factors contribute to students’ low achievement. It is not restricted to their difficulties with the traditional ‘content’ domain – the subject matter of mathematics is viewed as difficult to grasp and they lack good foundations in mathematical knowledge. Other factors that contribute towards/hinder their learning include metacognitive dispositions, learning habits, threshold for frustration, feelings of confidence etc. It is our stance that efforts that can result in sustainable improvements in their achievement should take into consideration all these interacting factors that influence their learning of mathematics. In the first paper on “Four factors to consider in helping low achievers in mathematics", we propose and describe in some detail a framework for helping low achievers in mathematics that attends to the following areas: Mathematical content resources, Problem Solving disposition, Feelings towards the learning of mathematics, and Study habits.

Bridging research and practice is often at the core of collaborative research with schools. Longitudinal research designed to study teachers’ learning to teach mathematical inquiry grew from a small pilot into a large nationally funded project that has provided curricular resources for state implementation of the Australian Curriculum. The aim of the symposium is to present perspectives from key stakeholders for scaling up research. The symposium aims to tell a story of a partnership that worked to embed inquiry-based learning in mathematics into schools from the perspective of three key stakeholders: university researchers, primary teachers and curriculum writers from the state education department. Collectively, the papers explain the research development and outcomes, its classroom implementation, interactions between researchers and state department and creation of state-level curricular resources. Recommendations are given. In the third paper, Guided inquiry as a model for curricular resources in mathematics, Debritz and Horne describe the frameworks and adaptations made to inquiry in the project for statewide use.

In this symposium, we discuss our project Mathematical Progress and Value for Everyone (MProVE) which is about doing this hard research on helping low achievers at the Secondary levels improve in their learning of mathematics. We start with the recognition that a complex confluence of factors contribute to students’ low achievement. It is not restricted to their difficulties with the traditional ‘content’ domain – the subject matter of mathematics is viewed as difficult to grasp and they lack good foundations in mathematical knowledge. Other factors that contribute towards/hinder their learning include metacognitive dispositions, learning habits, threshold for frustration, feelings of confidence etc. It is our stance that efforts that can result in sustainable improvements in their achievement should take into consideration all these interacting factors that influence their learning of mathematics. In the third paper on “Helping low achievers develop a problem solving disposition”, we discuss another study with the same project school on helping students engage productively in mathematical problem solving. To this end, we envisage that such a graduating student when confronted with an unfamiliar mathematics question, instead of giving up early, he or she will try ways to tackle the problem productively, even independently of the teacher. We term such an approach towards mathematics problems as having a “Problem Solving Disposition” (PSD). This paper reports on our initial attempt to develop PSD in two classes of low-achieving students.

Worldwide interest in Japanese Lesson Study as a vehicle to improve mathematics teaching practice through professional learning has left largely unanswered questions about the extent to which it can be replicated elsewhere. The Implementing structured problem-solving mathematics lessons through Lesson Study project worked with teachers from three Melbourne primary schools to identify critical factors in adapting and implementing the structured problem-solving mathematics lessons that underpin Japanese Lesson Study; explore ways in which key elements of Japanese Lesson Study can be embedded in Australian practice; and establish the effect of participating in Japanese Lesson Study on teachers’ mathematics pedagogical content knowledge. Two Years 3 and 4 teachers, together with the curriculum director of numeracy coach from each of three schools and the network numeracy coach participated in the project. The teachers worked in two Lesson Study teams, with each team planning and teaching a research lesson in Term 3 and Term 4 in 2012. In this symposium, we report findings from the project and argue that the Japanese Lesson Study model for professional learning, together with the underpinning problem-solving structure of its research lessons, can build teacher capacity for effective mathematics teaching in Australian primary schools.

Bridging research and practice is often at the core of collaborative research with schools. Longitudinal research designed to study teachers’ learning to teach mathematical inquiry grew from a small pilot into a large nationally funded project that has provided curricular resources for state implementation of the Australian Curriculum. The aim of the symposium is to present perspectives from key stakeholders for scaling up research. The symposium aims to tell a story of a partnership that worked to embed inquiry-based learning in mathematics into schools from the perspective of three key stakeholders: university researchers, primary teachers and curriculum writers from the state education department. Collectively, the papers explain the research development and outcomes, its classroom implementation, interactions between researchers and state department and creation of state-level curricular resources. Recommendations are given. Makar and Dole outline the research project in Inquiry-based learning in mathematics: Designing collaborative research with schools. Recommendations from the researchers’ perspectives are given for designing collaborative research in schools.

In previous MERGA conferences, we have reported on our project to implement Polya-type problem solving in the main curriculum of one high achieving secondary school and the follow-up project to diffuse the innovation to a wider range of secondary schools. We used design experiment as the methodology in both of the projects. In this symposium, we report on various aspects of this design experiment while it is in progress. The three papers show the vital interactions between the researcher-designers and the teacher-implementers to fine-tune a workable design for each individual school. Paper 1: In a design experiment, the feedback from the teacher-implementer is crucial to the success of the innovation simply because the teacher is finally the one that brings the innovation to life in front of the students. We describe in this paper the changes and improvements we have made in our design of a problem solving module after observing and learning from the teachers of several schools who taught various versions of the module.

The papers in this symposium outline the aims and design of a three year Teaching and Learning Research Initiative (TLRI) project involving mathematics education researchers and educator colleagues from Massey University and Victoria University of Wellington. The Learning the Work of Ambitious Mathematics Teaching project seeks to improve the mathematical experiences and educational outcomes for all students by looking to advance current practices for the preparation of mathematics teachers. In trialling new ways to support prospective teachers to not only ‘think’ like teachers, but also to put what they know into action, our project is focused on strengthening links between coursework and fieldwork. The first (Anthony and Hunter) of three papers in this symposium outlines the project rationale, aims, and design. The paper provides a summary of seminal research studies in the U.S. informing our design. This includes a growing body of international research focused on ‘pedagogies of practice’ that involved providing prospective teachers with opportunities to rehearse—in class and in schools—teaching purposefully designed activities that highlight ambitious teaching practices.

Worldwide interest in Japanese Lesson Study as a vehicle to improve mathematics teaching practice through professional learning has left largely unanswered questions about the extent to which it can be replicated elsewhere. The Implementing structured problem-solving mathematics lessons through Lesson Study project worked with teachers from three Melbourne primary schools to identify critical factors in adapting and implementing the structured problem-solving mathematics lessons that underpin Japanese Lesson Study; explore ways in which key elements of Japanese Lesson Study can be embedded in Australian practice; and establish the effect of participating in Japanese Lesson Study on teachers’ mathematics pedagogical content knowledge. Two Years 3 and 4 teachers, together with the curriculum director of numeracy coach from each of three schools and the network numeracy coach participated in the project. The teachers worked in two Lesson Study teams, with each team planning and teaching a research lesson in Term 3 and Term 4 in 2012. In this symposium, we report findings from the project and argue that the Japanese Lesson Study model for professional learning, together with the underpinning problem-solving structure of its research lessons, can build teacher capacity for effective mathematics teaching in Australian primary schools.

This symposium is the initial outcome of a new collaborative relationship among early childhood mathematics researchers from Charles Sturt and Monash universities. This relationship is built around the researchers’ interests and research concerning mathematics learning and teaching at the time of children’s transition to primary school, including prior-to-school and the first years of school. In this symposium, we consider a variety of ways that we have used to ‘notice’ and ‘explore’ children’s mathematical learning and the ramifications of these for future research and practice. This work is particularly timely in Australia because of the recent introduction of the first national curriculum documents in both prior-to-school and school settings: The Early Years Learning Framework for Australia and the Australian Curriculum - Mathematics. The papers presented in the symposium will build on the new sociology of childhood by considering the strengths and agency of young children in accessing powerful mathematical thinking and then will consider complementary accounts of these strengths through the observations and analyses of mathematics education researchers, prior-to-school and school educators and young children themselves. Methodological approaches to gaining access to these accounts will be considered in the four papers to be presented.

In this symposium, we discuss our project Mathematical Progress and Value for Everyone (MProVE) which is about doing this hard research on helping low achievers at the Secondary levels improve in their learning of mathematics. We start with the recognition that a complex confluence of factors contribute to students’ low achievement. It is not restricted to their difficulties with the traditional ‘content’ domain – the subject matter of mathematics is viewed as difficult to grasp and they lack good foundations in mathematical knowledge. Other factors that contribute towards/hinder their learning include metacognitive dispositions, learning habits, threshold for frustration, feelings of confidence etc. It is our stance that efforts that can result in sustainable improvements in their achievement should take into consideration all these interacting factors that influence their learning of mathematics. In the second paper on “Positive feelings towards the learning of mathematics for low achievers”, we discuss an attempt to foreground one particular aspect of the framework in our work with the project school. A common area of difficulty and frustration for low achieving students is the operations involving negative numbers. The main objective of our innovations in this area with two classes of Year 7 low achieving students was to develop positive feelings towards mathematics, in particular of an appreciation for the ‘reasonableness’ and clear visualisation of the negative numbers ‘in action’.

This symposium is the initial outcome of a new collaborative relationship among early childhood mathematics researchers from Charles Sturt and Monash universities. This relationship is built around the researchers’ interests and research concerning mathematics learning and teaching at the time of children’s transition to primary school, including prior-to-school and the first years of school. In this symposium, we consider a variety of ways that we have used to ‘notice’ and ‘explore’ children’s mathematical learning and the ramifications of these for future research and practice. This work is particularly timely in Australia because of the recent introduction of the first national curriculum documents in both prior-to-school and school settings: The Early Years Learning Framework for Australia and the Australian Curriculum - Mathematics. The papers presented in the symposium will build on the new sociology of childhood by considering the strengths and agency of young children in accessing powerful mathematical thinking and then will consider complementary accounts of these strengths through the observations and analyses of mathematics education researchers, prior-to-school and school educators and young children themselves. Methodological approaches to gaining access to these accounts will be considered in the four papers to be presented.

Worldwide interest in Japanese Lesson Study as a vehicle to improve mathematics teaching practice through professional learning has left largely unanswered questions about the extent to which it can be replicated elsewhere. The Implementing structured problem-solving mathematics lessons through Lesson Study project worked with teachers from three Melbourne primary schools to identify critical factors in adapting and implementing the structured problem-solving mathematics lessons that underpin Japanese Lesson Study; explore ways in which key elements of Japanese Lesson Study can be embedded in Australian practice; and establish the effect of participating in Japanese Lesson Study on teachers’ mathematics pedagogical content knowledge. Two Years 3 and 4 teachers, together with the curriculum director of numeracy coach from each of three schools and the network numeracy coach participated in the project. The teachers worked in two Lesson Study teams, with each team planning and teaching a research lesson in Term 3 and Term 4 in 2012. In this symposium, we report findings from the project and argue that the Japanese Lesson Study model for professional learning, together with the underpinning problem-solving structure of its research lessons, can build teacher capacity for effective mathematics teaching in Australian primary schools.

This symposium is the initial outcome of a new collaborative relationship among early childhood mathematics researchers from Charles Sturt and Monash universities. This relationship is built around the researchers’ interests and research concerning mathematics learning and teaching at the time of children’s transition to primary school, including prior-to-school and the first years of school. In this symposium, we consider a variety of ways that we have used to ‘notice’ and ‘explore’ children’s mathematical learning and the ramifications of these for future research and practice. This work is particularly timely in Australia because of the recent introduction of the first national curriculum documents in both prior-to-school and school settings: The Early Years Learning Framework for Australia and the Australian Curriculum - Mathematics. The papers presented in the symposium will build on the new sociology of childhood by considering the strengths and agency of young children in accessing powerful mathematical thinking and then will consider complementary accounts of these strengths through the observations and analyses of mathematics education researchers, prior-to-school and school educators and young children themselves. Methodological approaches to gaining access to these accounts will be considered in the four papers to be presented.

In previous MERGA conferences, we have reported on our project to implement Polya-type problem solving in the main curriculum of one high achieving secondary school and the follow-up project to diffuse the innovation to a wider range of secondary schools. We used design experiment as the methodology in both of the projects. In this symposium, we report on various aspects of this design experiment while it is in progress. The three papers show the vital interactions between the researcher-designers and the teacher-implementers to fine-tune a workable design for each individual school. Paper 3: Problem solving task design is not only the design of a non-routine problem to be solved 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.

The symposium presents findings from a cross-country study, drawing attention to dimensions of students’ learning by analysing students’ mathematics sense making and assessment performance in high performing but culturally different contexts—Australia and Singapore. The symposium details the performance of Australian and Singaporean students on tasks sourced from each country’s respective national assessment instruments; identifies the approaches and strategies students from different cultures employ to solve mathematics tasks; and draws conclusions about the influence cultural and pedagogical practices have on students’ approaches to solving these tasks. Paper 3: Students’ Performance on a Symmetry Task by Siew Yin Ho & Tracy Logan. This paper describes Singapore and Australian Grade 6 students (n=1187) performance on a Symmetry task in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia and Singapore’s national assessments, NAPLAN and PSLE. Only half the cohort solved the item successfully. It is possible that persistence of prototypical images of a vertical line of symmetry and reinforcements in the classrooms could have attributed to this low success rate.

The symposium presents findings from a cross-country study, drawing attention to dimensions of students’ learning by analysing students’ mathematics sense making and assessment performance in high performing but culturally different contexts—Australia and Singapore. The symposium details the performance of Australian and Singaporean students on tasks sourced from each country’s respective national assessment instruments; identifies the approaches and strategies students from different cultures employ to solve mathematics tasks; and draws conclusions about the influence cultural and pedagogical practices have on students’ approaches to solving these tasks. Paper 2: The Classic Word Problem: The Influence of Direct Teaching by Tracy Logan & Siew Yin Ho. Australian and Singaporean students have been exposed to different and alternate forms of teaching due to cultural differences in education. In each country, varying degrees of importance has been placed on the use of explicitly teaching problem-solving heuristics. This paper highlights the different strategies employed by students from each country when solving a word problem and the role direct teaching has played in the development of these strategies. Implications for classroom practice are also discussed.

The symposium presents findings from a cross-country study, drawing attention to dimensions of students’ learning by analysing students’ mathematics sense making and assessment performance in high performing but culturally different contexts—Australia and Singapore. The symposium details the performance of Australian and Singaporean students on tasks sourced from each country’s respective national assessment instruments; identifies the approaches and strategies students from different cultures employ to solve mathematics tasks; and draws conclusions about the influence cultural and pedagogical practices have on students’ approaches to solving these tasks. Paper 1: The Odd Couple: The Australian NAPLAN and Singaporean PSLE Jane Greenlees. The use of high-stakes assessment to measure students’ mathematical performance has become common place 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.

The papers in this symposium outline the aims and design of a three year Teaching and Learning Research Initiative (TLRI) project involving mathematics education researchers and educator colleagues from Massey University and Victoria University of Wellington. The Learning the Work of Ambitious Mathematics Teaching project seeks to improve the mathematical experiences and educational outcomes for all students by looking to advance current practices for the preparation of mathematics teachers. In trialling new ways to support prospective teachers to not only ‘think’ like teachers, but also to put what they know into action, our project is focused on strengthening links between coursework and fieldwork. The second and third papers draw on our experiences within the first year of the three year project. Both papers concern our introduction of new Instructional Activities (IAs) to a range of different courses and programmes. In paper two, Hunter, Hunter, and Anthony describe the design and role of the new IAs. IAs, used in a cyclic process of modelling, planning, guided rehearsal with feedback, enactment and reflection, are purposefully designed to support prospective teachers learn the practices, principles, and knowledge associated with ambitious mathematics teaching.

This symposium is the initial outcome of a new collaborative relationship among early childhood mathematics researchers from Charles Sturt and Monash universities. This relationship is built around the researchers’ interests and research concerning mathematics learning and teaching at the time of children’s transition to primary school, including prior-to-school and the first years of school. In this symposium, we consider a variety of ways that we have used to ‘notice’ and ‘explore’ children’s mathematical learning and the ramifications of these for future research and practice. This work is particularly timely in Australia because of the recent introduction of the first national curriculum documents in both prior-to-school and school settings: The Early Years Learning Framework for Australia and the Australian Curriculum - Mathematics. The papers presented in the symposium will build on the new sociology of childhood by considering the strengths and agency of young children in accessing powerful mathematical thinking and then will consider complementary accounts of these strengths through the observations and analyses of mathematics education researchers, prior-to-school and school educators and young children themselves. Methodological approaches to gaining access to these accounts will be considered in the four papers to be presented.

The study was designed to probe students’ thinking about which numerical values can be assigned to algebraic letters. The data from students in Year 7 (n=533), Year 8 (n=377) and Year 9 (n=172) was analysed using response patterns. The data confirmed that each year contained students with two misconceptions; Different Letter means Different Number and the Empty Box misconceptions. The findings provide support for the Steinle et al. (2009) hypothesis that a previously identified response pattern is a subset of the Empty Box misconception.

This study expands contemporary theorising about students’ conceptions of equality. A nationally representative sample of New Zealand students’ were asked to provide a spoken numerical response and an explanation as they solved an arithmetic additive missing number problem. Students’ responses were conceptualised as acts of communication and analysed according to their mathematical structure. Specifically, students’ spoken explanations were parsed and mapped using the properties of equality. These maps were classified according to their correspondence to the mathematical structure of the given problem. Students gave four different numerical responses and their explanations were interpreted as seven different conceptions of equality. These findings indicate that students’ conceptions of equality are more diverse and complex than previous accounts suggest.

This study, drawing on data from the Trends in International Mathematics and Science Study (TIMSS) 2011, examined whether mathematics teachers’ perceptions of their students’ mathematical competence were related to mathematics achievement, affect toward mathematics, and engagement in mathematics lessons among Grade 8 students in Singapore and Australia. Structural equation modelling (SEM) analyses revealed that mathematics teachers’ perceptions of their students’ mathematical competence were positively linked to eighth-graders’ mathematics achievement, affect toward mathematics, and engagement in mathematics lessons in both Singapore and Australia. Implications of the findings are briefly discussed.

Many primary schools in Australia are investing substantial funds introducing mobile technologies such as iPads to enhance teaching and learning. However, when new technologies are first introduced, teachers are often expected to integrate them into their practices without the support of appropriate professional development. This paper reports on a recent qualitative multiple case study that explored the pedagogical practices implemented by four primary teachers during the first six months of iPad use. Results of the study highlighted that although the iPads do have the potential to enhance teaching and learning of primary mathematics, appropriate professional development that addresses all aspects of technological and pedagogical content knowledge is required to ensure successful integration of new technologies into current teaching practices.

Teacher respect, important within culturally responsive practice, has seldom been explored in relation to mathematical pedagogy. Our study involving interviews, surveys, and lesson videos with Year 12 and 13 New Zealand mathematics students and teachers indicated specific pedagogical behaviours are important for demonstrating respect for students and their learning. Respectful teachers provide opportunities for mathematical decision-making, follow chains of reasoning with individuals, and sensitively guide next learning steps. Implications for practice include knowing individuals’ learning needs and prioritising one-to-one interactions.

A modified form of Lesson Study was used to deliver several lessons over two weeks to develop students’ relational thinking and to improve teachers’ knowledge of this thinking. Fifteen students in a multi-grade 4, 5 & 6 classroom were surveyed and interviewed using true/false sentence and open number problems involving one unknown number before and after the study. Students in all three grades increased their understanding of the role of equivalence and their capacity to use relational thinking. Questionnaires were also undertaken by three participating teachers before and after the study. Their knowledge about students’ relational thinking improved, and they demonstrated how they would integrate it into their future teaching of number and number operations.

Changes to students’ understanding of mathematical notation may be brought about by using technology within mathematics. Taking equality as a case study, the paper provides brief epistemological, historical, didactical, and computational reviews of its symbolic representation in pen-and-paper and technology-assisted mathematics, most especially in CAS. A multiplicity of special technology signs convey specific aspects of the broad meaning of the pen-and-paper sign. This provides a basis for new investigations into the effect on understanding of students’ doing mathematics with technology.

This paper reviews recent literature on teacher identity in order to propose an operational framework that can be used to investigate the formation and development of numeracy teacher identities. The proposed framework is based on Van Zoest and Bohl’s (2005) framework for mathematics teacher identity with a focus on those characteristics thought to be particularly important for numeracy teacher identity.

Statistical literacy has been argued to constitute a key aspect of students’ numeracy. In the present study we examine statistical literacy of two Year 7 students and their abilities to extract and interpret information from a familiar real-life context. Analysis of data indicate that while both of the students developed facility in extracting information and drawing graphs, the low-achieving student tended to process the graphical information somewhat superficially in comparison to the high-achiever. Implications for teaching and learning are discussed.

This paper reports part of a study of secondary mathematics teachers in Technology-Rich Teaching and Learning Environment (TRTLE’s). Three TRTLE’s, two year 11 and one year nine class and their teachers were the focus of the study. Seven Teaching Roles were identified as teachers acted to allow students to perceive and enact affordances of TRTLE’s appropriate to the learning of functions. Each role is important in allowing future independent perception and enactment of affordances by students.

A review of the research literature indicated that there were probable orders in which students develop understandings and skills for calculating with percentages. Such calculations might include using models to represent percentages, knowing fraction equivalents, selection of strategies to solve problems and determination of percentage change. To further describe the hierarchies, an assessment instrument was constructed and piloted before being refined and trialled with lower secondary students. Rasch model analysis was applied to the results.

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.

With reports of declining enrolments in mathematics related degrees and low female participation rates in these degrees, the issue of gender differences in mathematics remains relevant. Results of recent studies suggest gender differences in mathematics are nuanced and that small differences in the early years can manifest as larger differences in later years. This study explores differences in teachers’ ratings of children’s achievement across a number of mathematical content domains. It is based on observations from the K-cohort of the Longitudinal Study of Australian Children in 2006, when the children were aged between six and seven, and in 2008, when they were aged between eight and nine. Gender differences in achievement are analysed using the Mantel-Haenszel procedure associated with the implementation of the Rasch model. Results indicate that teachers rate girls higher on tasks related to data, whereas they rate boys higher on tasks related to place-value and computation. Implications of these findings are discussed.

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.

This paper reports the results of a pencil-and-paper test developed to assess young children’s understanding of mass measurement. The innovative element of the test was its use of photographs. We found many children of the 295 6-8 year-old children tested could “read” the photographs and diagrams and recognise the images as representations of their classroom experiences. While the test had its limitations, it also required explanation, deductive reasoning, and justification of thinking through the open response questions. We have demonstrated that it is possible to develop pencil-and-paper tests that use photographs and diagrams to closely connect written assessment to classroom experiences of young children. Such assessment tools can reveal a range of children’s thinking and can be a useful addition to the various authentic assessment practices.

The findings discussed here are a small part of a larger study entitled, Encouraging Persistence Maintaining Challenge. The paper reports five teachers’ observations of the implementation of a task which was new to them. The teachers were asked to identify aspects of the task which they perceived as challenging for the Year 6 students. The teachers’ responses are discussed using a framework of features of challenging tasks proposed by Sullivan et al. (2011). The findings show that teachers identified the challenges involved as demanding mathematical reasoning, interpreting complex mathematics and in expecting students to create their own solution pathways.

In this study, ways in which problem posing activities aid our understanding of children’s learning of addition of unlike fractions and product of proper fractions was examined. In particular, how a simple problem posing activity helps teachers take a second, deeper look at children’s understanding of fraction concepts will be discussed. The problems posed by the students were explored and insights into the students’ understanding of fractions were identified.

This paper presents a framework for examining the Pedagogical Content Knowledge (PCK) required of mathematics teacher educators as they endeavour to build the PCK for teaching mathematics that is required of the pre-service teachers with whom they work. The framework builds on existing research into PCK, and provides a series of filters through which to examine the complexity of the work of teacher education. The usefulness of the framework is trialled by using it to study the PCK used by the first author in working with pre-service teachers to build understanding about ways of interacting with students.

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.

Teacher mathematical noticing is a key component of mathematics teaching expertise and has been a focus of recent professional development efforts. In this paper, I propose and describe explicitly the notion of productive mathematical noticing, which surfaces from a case study involving a group of seven mathematics teachers who collaborated as part of a lesson study team at a primary school in Singapore. Two vignettes—one that happened during the planning stage, and the other taken from the actual lesson—are discussed to illustrate the notion of productive mathematical noticing.

This study aims to design learning situations and tasks that promote and assess the capacity of primary school children to transfer mathematical knowledge to new contexts. We discuss previous studies investigating mathematical transfer, and particularly the strengths and limitations of tasks used to assess transfer in these studies. We describe some pilot tasks that were used with upper primary children and provide some responses to teacher prompts. We describe some design principles for the construction of tasks and an associated categorisation of prompts that might be used as the basis of further research into mathematical transfer.

The Accelerating the Mathematics Learning of Low Socio-Economic Status Junior Secondary Students project aims to address the issues faced by very underperforming mathematics students as they enter high school. Its aim is to accelerate learning of mathematics through a vertical curriculum to enable students to access Year 10 mathematics subjects, thus improving life chances. This paper reports upon the theory underpinning this project and illustrates it with examples of the curriculum that has been designed to achieve acceleration.

Teacher educators need to identify those aspects of preservice teacher (PST) mathematical content knowledge (MCK) that need developing. A methodology that unpacks the MCK that PSTs use in their teaching is presented in this paper. MCK in the teaching acts themselves and in the PST reflections on those acts is categorised and evaluated. The process is illustrated with a lesson excerpt from a secondary mathematics PST. The benefits and limitations associated with this methodology are also discussed.

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.

Research suggests that teachers’ beliefs about teaching are strongly influenced by their personal experiences with mathematics. This study aimed to explore Pacific Island pre-service secondary mathematics teacher’s perceptions about good and bad mathematics teachers. Thirty pre-service teachers, enrolled in a mathematics teaching methods course during their third year of University study were asked to write reflections on their personal mathematical memories. Results indicate that pre-service teachers rate good mathematics teachers using a varied combination of characteristics.

Counting strategies initially used by young children to perform simple addition are often replaced by more efficient counting strategies, decomposition strategies and rule-based strategies until most answers are encoded in memory and can be directly retrieved. Practice is thought to be the key to developing fluent retrieval of addition facts. This study examines the errors made by five children in Year 3 as they perform simple addition and illustrates why practice does not assist all children to develop retrieval strategies.

This paper reports on 13 Grade 3 students’ approaches to partitive and quotitive division word problems. Of particular interest was the extent to which students drew on their knowledge of multiplication to solve division problems. The findings suggest that developing a relationship between multiplication and division is more significant than differentiating the types of division.

New Zealand and Australian curricula require students to learn about weight/mass for at least six years. However, little research identifies what should be taught. This study reports cognitive interviews with 17 Year 9 students who were asked how heavy is my rock? Only one student demonstrated some understanding of how to use analogue kitchen scales, most had multiple errors. Results suggest that teachers would benefit from better guidance about teaching the skill set needed for such a task.

Slope is a mathematical concept that is both fundamental to the study of advanced calculus and commonly perceived in everyday life. The measurement of steepness of terrain as a ratio is an example of an everyday application of the concept of slope. In this study, a group of pre-service teachers were tested for their capacity to mathematize the measurement of steepness. Findings suggest that accuracy in the measurement of steepness may be related to success in linear algebra as they are both related to the perception of slope as a ratio.

Effective mathematics teaching for Indigenous language speaking students, currently the lowest achieving group in Australia, needs to be based on fair expectations of both students and teachers. Teacher interviews in a small Northern Territory school, conducted within an ethnographic study, showed that teachers’ decisions regarding level of mathematics curriculum taught were informed by students’ prior learning and by the language dynamic in their classrooms. The need and pressure to teach Standard Australian English also affected how mathematics was taught.

This paper addresses one of the foundational components of beginning inference, namely variation, with 5 classes of Year 4 students undertaking a measurement activity using scaled instruments in two contexts: all students measuring one person’s arm span and recording the values obtained, and each student having his/her own arm span measured and recorded. The results included documentation of students’ explicit appreciation of the variety of ways in which variation can occur, including outliers, and their ability to create and describe valid representations of their data.

The case study reported here examined three scaffolding practices employed by two teachers in two Year 5 and 6 mathematics classrooms. One scaffolding practice was the use of discussion. This paper describes the use of whole class discussions as scaffolding, drawing upon observations of six mathematics lessons in each classroom and the responses of four low-attaining target students. Implications for teachers on the use of whole class discussions as scaffolding for low-attaining students in mathematics are also explored.

Argumentation in mathematics teaching has potential to move students beyond tacit understanding of mathematical concepts and procedures towards articulation and justification of their ideas; a practice in which evidence is central. Design-based research was used to examine the nature of evidence used by a class of primary students through levels of argument and explanation. Results of this exploratory study indicate that evidence put forward became increasingly sophisticated as students’ conceptions became public and therefore open to increased potential challenge.

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.

Teachers of inquiry mathematics face many challenges as understandings of how children learn in this environment are still developing. Children learn as they hold onto the moment of not knowing what to do. This paper operationalises learning for a small group of Year Three students exploring the usefulness of maps. As part of a PhD study using Design Research, the teacher-as-researcher identified perturbations generated by the students in her class and applied an operational definition of learning to track student development of mathematical knowledge.

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.

This paper reports research on student produced screencasts to support learning. Participants in a Mathematics for Teachers course were asked to create and peer critique screencasts to explain concepts (year 4 to 9 level). They were also asked about their experience with screencasting and its impact on their own teaching and learning. This paper will discuss preliminary results of a pre-survey and highlight features of initial screencasts and their critiques. The paper concludes with an outline of future directions.

Numeracy is a fundamental component of the Australian National Curriculum as a General Capability identified in each F-10 subject. In this paper we report on an aspect of a larger project aimed at investigating how effective cross-curricular numeracy practice can be implemented within the context of the Australian National Curriculum. Specifically, we draw on a numeracy audit of the English curriculum and on data collected via a semi-structured teacher interview to explore the numeracy demands and opportunities that exist in the teaching of English. Despite limited guidance for teachers in how to take advantage of numeracy skills within English, our investigation reveals that multiple opportunities to do so abound.

The introduction of the Early Years Learning Framework and the Australian Curriculum – Mathematics in Australian preschools and primary schools has caused early childhood educators to reconsider what may be appropriate levels of mathematics knowledge to expect from children as they start school. This paper reports on initial data from an extensive evaluation of an early mathematics intervention and asks whether the expectations of young children recorded in the Australian Curriculum – Mathematics are realistic

The longitudinal progress of 42 Grade 1 students who participated in a 10-20 week Extending Mathematical Understanding (EMU) intervention program was examined to evaluate the effectiveness of the program for enhancing and accelerating mathematics learning. Overall the students made accelerated progress during Grade 1 and their learning was maintained after the six-week summer break and also when they were assessed 12 months later. However, the rate of progress for many students was less during Grade 2 when they participated only in regular classroom lessons.

This study describes the construction of a questionnaire instrument to measure mathematics teacher educators’ knowledge for technology integrated mathematics teaching. The study was founded on a reconceptualisation of the generic Technological Pedagogical Content Knowledge framework in the specific context of mathematics teaching. Steps in the development of the questionnaire were; consideration of the context in which the questionnaire would be used, comparison of proposed items with and existing instrument, expert review, and pilot testing. The process described provides a model for other researchers interested in adapting generic tools for mathematics specific use.

This paper addresses data arising from initial discussions with school principals concerning the implementation of a doctoral project in their schools. The doctoral project involved the prep teacher working with the preschool teacher to support children’s numeracy practises as they made the transition to school. The findings presented in this paper suggest that numeracy might not be a key priority for schools as children make the transition from preschool to primary school, despite government policy, frameworks and curriculum documentation advocating the otherwise.

In this paper we will examine mathematics education using practice theory. We outline the theoretical and philosophical ideas that have been developed, and in particular, we discuss the ‘sayings’, ‘doings’, and ‘relatings’ inherent in the teaching and learning practices of mathematics education. This theorising is drawn from an empirical study that focused on the broader practices of education in schools. We exemplify these ideas with a small excerpt of data. Understanding mathematics education as a practice highlights the site-based nature and also ecological arrangement of practices, and we conclude by outlining some implications that emerge from this perspective.

Increasingly iPadsTM are being used in schools and prior-to-school settings, with a plethora of Apps available for mathematics learning. Despite the growing number of Apps available in the iTunes App Store, there has been limited systematic analysis of the pedagogic design of Apps designed for mathematics learning. This paper describes a content analysis of Apps that are currently available as ‘educational’ content in the iTunes App Store and highlights the limited range of pedagogic designs available for mathematics learning.

This paper reports results from a study exploring classroom modelling as an approach to teacher education. In particular, it presents analysis of responses on an observation proforma completed by primary school teachers to indicate what teacher actions they observed when watching modelled lessons and which of those actions they intended to implement in their classrooms as a result of the observation. One hundred and sixty two teachers participated in observing at least one modelled lesson in a variety of classroom contexts over a period of twelve months. The results of the observation proformas revealed that teachers focussed on desirable pedagogies when observing modelled lessons, suggesting that classroom modelling may be a powerful form of teacher learning.

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.

In this paper, the mathematical engagement of Colin and Robyn is compared. Through this comparison, and informed by longitudinal research into the mathematical journeys of a group of students in New Zealand, a set of engagement skills emerged. Both students had high levels of engagement in mathematics. However, Colin was a thriving mathematics student with effective engagement skills, whereas Robyn had ineffective engagement skills and was, over time, vulnerable to disengagement, negative feelings and non-participation in the subject of mathematics.

Drawing on survey data from over 2000 parents, this paper explores the possibility of early-years swimming to add mathematical capital to young children. Using developmental milestones as the basis, it was found that parents reported significantly earlier achievement on many of these milestones. Such data suggest that the early years swim environment may offer enhanced opportunities for learning skills that help transition young children into formal schooling. This paper explores those milestones that are related to early mathematics.

This study, drawing on data from the Programme for International Student Assessment (PISA) 2009, examined the relationships of out-of-school-time mathematics lessons to mathematical literacy in Singapore and Australia. Results of two-level hierarchical linear modelling (HLM) analyses revealed that out-of-school-time enrichment lessons in mathematics were not significantly associated with mathematical literacy in Singapore and Australia. Out-of-school-time remedial lessons in mathematics were negatively associated with mathematical literacy in Australia, while such remedial lessons in mathematics were not significantly related to mathematical literacy in Singapore. Learning time in out-of-school-time lessons in mathematics was significantly negatively linked to mathematical literacy in Singapore and Australia. Implications of the findings are discussed.

Applications (apps) for hand-held devices, such as IPads and smart phones, are in great supply. Many of these focus on mathematics. A recent search revealed more than 4,000 apps for mathematics education. The ease of access and the fact that they are generally low cost, often free, means that they are readily available to the general population but this raises questions as to their quality and what is being learned through the use of these apps. Using two quantitative measures and one qualitative measure, this article evaluates 142 apps which met initial search criteria, and recommends 34 mathematics apps for further evaluation and trial with primary school teachers and students.

This paper describes the use of transactional distance theory (TDT) as a conceptual framework to underpin the design and delivery of a fully online first-year mathematics education methods course to pre-service teachers. It identifies key issues evident in the mathematical literature concerning mathematics education in general and pre-service teacher mathematics education in particular. It describes the use of a course design process based on data collection and reflection, course redesign, and implementation of design changes in a cyclical process to achieve course objectives.

The mathematics content knowledge of pre-service teachers is a growing area of inquiry. This topic requires further theoretical development due to the limited applicability of current cognitive and practice-oriented frameworks of mathematics content knowledge to beginning pre-service teachers. Foundation content knowledge is an integrated, growth-oriented concept of mathematics content knowledge specifically for beginning pre-service primary teachers. While we acknowledge that our proposal is preliminary and incomplete we also maintain that it addresses a number of important issues faced by pre-service teachers, their initial teacher education providers, and the mathematics education community.

Proportional reasoning is important for informed decisions in proportional problem situations. This paper reports on mathematical content knowledge related to proportional reasoning of second-year, pre-service teachers. Responses to two ratio items provide insights into their correct method of solutions and common misconceptions. Anchor Points (Tatto, Peck, Schwille, Bankov, et al., 2012) were used to score the items. Some pre-service teachers had difficult with a scale ratio item but many more were unable to correctly respond to a proportional item.

This paper examines correspondences and disjunctions within a national curriculum and between various aspects of its delivery, and how these align with the mathematical needs of the workplace. This is investigated in the context of the New Zealand school mathematics curriculum; the Numeracy Development Project; the senior school assessment régime, and the numeracy requirements of toolmaking. In theory, the numeracy aspects of the various curricula and the workplace should form a logical learning progression in numeracy. The content was consistent across the various curricula, however, the assessment régime was not particularly congruent with gaining the thorough knowledge (mastery) required for workplace practice.

Little is known about how Australian teachers interpret, enact and assess reasoning. This paper reports on primary teachers’ perceptions of reasoning prior to observation and subsequent trialling of demonstration lessons in a primary school. The findings indicate that while some teachers were able to articulate what reasoning means, others were unsure. It is argued that to facilitate curricular change and reform, teachers need support in understanding mathematical reasoning and how they can further develop this proficiency in their primary classrooms.

Effective teachers have good pedagogical content knowledge (PCK). Pedagogical content knowledge is the intersection of discipline specific content knowledge and pedagogical knowledge. How effectively are pre-service teachers helped to develop good PCK? In this project we asked our pre-service teachers how they would respond to a particular student misconception before and after teaching three topics, to determine if there had been any growth in their PCK. Although the pre-service teachers had deepened their knowledge on teaching specific mathematics content, few changed their answer to the question or showed a deeper understanding of what the student had understood. This then has implications for our teaching - we need to make our thinking explicit so that pre-service teachers can see the complexity of these issues.

Research affirms that pattern and structure underlie the development of a broad range of mathematical concepts. However, the concept of pattern also occurs in other fields. This theoretical paper explores pattern recognition, a neurological construct based on the work of Goldberg (2005), and pattern as defined in the field of mathematics to highlight what is intrinsically similar about the concept in these domains. An emerging model of patterning is proposed to describe this relationship.

The implementation of a Post Entrance Numeracy Assessment (PENA) program at the University of Notre Dame Australia (UNDA) Fremantle campus, demonstrated that many Health Science students had inadequate mathematical skills on entry. The development of a support course, offered as a ‘strongly encouraged’ option, resulted in only 4% of students (not achieving the PENA benchmark) actually engaging. Whilst recognising that Health Science students at this institution have a history of limited engagement with support, this poor level of engagement demanded particular attention. It was clearly linked to mathematics per se, where student negativity to the subject is almost palpable in many undergraduate courses. In attempting to explain this engagement reluctance, the clear need to improve student skills with mathematics demanded a review of how such a support program may be offered.

The ubiquitous practice of providing worked solutions to exercises in mathematics education has been under-researched. Little is known about what elements of a worked solution are valued by students. This exploratory study sought in-depth feedback from six undergraduate students who experienced a range of worked solutions designed to encourage engagement. Elements that were valued included detail, explanation, and choice. Elements of worked solutions with which students did not engage were extension and reflection tasks.

This paper reports an exploration into the use of a combination of semiotic resources when teaching the part- whole model of fractions. The study involved a single case study of one class teacher and six students in an Australian primary classroom. Using video as the predominate research tool it was possible to describe how gesture and language were combined with two and three dimensional representations of folding paper, fraction walls and number lines to build images that appeared to enhance students understanding of the part-whole model of fractions. I conjecture that the variety semiotic resources including gesture should have more prominence in teacher’s planning documents.

The data from national tests such as the National Assessment Program Literacy and Numeracy (NAPLAN) and its precursor Victorian Achievement Improvement Monitor (AIM) are an important resource. The 2006 Year 3 AIM assessment included two subtraction items that are similar in content, and which were presented without text or images. The detailed, novel analysis of the children’s responses presented here provides insight into children's fluency and understanding of these items.

A recent Google search for ‘Help with maths’ produced 57 600 000 results, indicating that there are literally millions of online resources claiming to provide assistance with mathematics. As mathematics educators, however, we remain largely uninformed about students’ use of such resources, particularly when they are self-initiated and often accessed in an out of classroom environment. This paper reports on a study that investigated the resources Grade 8 and 9 students accessed when requiring support with understanding mathematical concepts. The study found that while friends and teachers were often students’ preferred options, they did access online sites, particularly in the later years of schooling. The study has implications for students and teachers including the potential for online resources to both complement and challenge the traditional role of the teacher and contemporary classroom practices.

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.

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.

This research study investigates how pre-service teachers integrate statistical content, students’ thinking, and pedagogy as they examine how 11- to 12-year olds develop mathematically. The findings provide insights into: a) how pre-service teachers identify some of the difficulties that students commonly have, and b) what pedagogical approaches pre-service teachers use to address students’ difficulties and enhance students’ learning. These findings have implications for the design and delivery of professional development that improves teachers’ knowledge, understanding, and skills in teaching statistics.

While previous work in the domain of proportional reasoning has primarily focused on the coordination of integer quantities, this study investigates how students coordinate fractional quantities. Fine-grained analysis of two seventh graders’ responses to a set of systematically designed proportional tasks, shows how their knowledge of multiplication and division from the domain of integers affords them the necessary resources to coordinate the fractional quantities. Further, it succinctly shows how the numeric feature of the fractional quantities cues and constrains them.

Measures of centre (the mean, median and mode) are fundamental to the discipline of statistics. Yet previous research shows that students may not have a thorough conceptual understanding of these measures, even though these statistics are easy to calculate. This study describes the findings of a study of pre-service teachers’ ideas of measures of centre. The results indicate that while some participants had ideas about these statistics that were valid; a substantial proportion displayed little understanding of these measures.

An understanding of conditional probability is essential for students of inferential statistics as it is used in Null Hypothesis Tests. Conditional probability is also used in Bayes’ theorem, in the interpretation of medical screening tests and in quality control procedures. This study examines the understanding of conditional probability of students entering an introductory applied statistics unit at an Australian university. These students answered questions that tested their ability to interpret conditional statements in two-way tables and their ability to discern the difference between a conditional statement and its inverse in written form. They also answered questions to determine if they held the time-axis fallacy.

Tablet PCs were used interactively in a Discrete Mathematics course in the first year of a Computing degree. The main benefit expected was an improvement in student engagement, but peer instruction was very evident and the ability to display many student answers led to very effective and immediate feedback, particularly when incorrect answers were displayed to the class. We discuss our experiences and the student reactions, and in particular, what they reported as the main advantages.

As we move into the 21st century, educationalists are exploring the myriad of possibilities associated with Computer Based Assessment (CBA). At first glance this mode of assessment seems to provide many exciting opportunities in the mathematics domain, yet one must question the validity of CBA and whether our school systems, students and teachers are ready to harness this form of assessment. The most obvious advantages of CBA are the speed and accuracy of accessing results and the opportunities for innovative item development. This paper will aim to highlight how several factors can obstruct the validity and reliability of this assessment mode, particularly at an item level. These threats to validity must be carefully considered by test designers to ensure CBA is used effectively in primary school mathematics classrooms.

Queensland rural and remote schools have difficulty in attracting experienced, in-field mathematics teachers. Thus, when such teachers arrive, much is expected of them to increase the mathematics knowledge of students. This paper looks at one such teacher who, against the high expectations placed upon him as an in-field teacher, experienced difficulties in teaching mathematics to underperforming Indigenous Australian students. The paper discusses pedagogical adjustments taken to overcome the disadvantage of being an in-field mathematics teacher in an underperforming classroom.

Everyday financial dilemmas require us to draw on social, interdisciplinary, and mathematical understandings simultaneously and in synergy if we are to make informed financial decisions. Financial literacy is enjoying an elevated status across the Australian Curriculum. This paper reviews some of the literature on financial literacy, and describes aspects of a research project in which Year 6 students were presented with financial dilemmas to gain insights into their social and mathematical understandings and financial decision making. The findings suggest that teaching that provides meaningful contexts in which mathematical concepts are situated is critical to student learning.

The concept of mobile technologies is now an emergent theme in educational research, yet the playing of these edutainment applications and their impact on early childhood learning needs to be fully explored. This study highlights current research and explores how iPads improve student learning. It also examines how the introduction of iPads, affects children’s motivation and self-efficacy towards numeracy learning. These findings contribute to the positive use of iPads to foster children’s development in numeracy.

While there are many considerations for effective mathematics teachers, one key factor is the development of a classroom culture that supports the desired form of learning. In examining the opportunities and constraints associated with posing challenging tasks, we are exploring ways that teachers might influence classroom culture positively. The data presented below suggest that it is possible to foster a classroom culture in which teachers pose tasks that challenge students and encourage them to persist when working on those tasks. The key elements seem to be the ways the tasks are posed, the interactive support for students when engaged in the tasks, collaborative reviews of class explorations and assessment against criteria.

At some stage when planning, teachers make decisions about the mathematics tasks they will pose and how they will structure lessons. It seems, though, that these decisions are complex, and that this complexity has been underestimated by curriculum developers and teacher educators. The following is a report of data collection that simulated some of these planning decisions. The results suggest that teachers may need support in matching tasks to curriculum content statements, in articulating the purposes of tasks, and in considering how tasks might be used to address differences in student readiness.

Forty seven Aboriginal students in the Kimberley were interviewed in English and Kimberley Kriol to investigate their understanding of ‘everyday’ words used within mathematics classrooms. The results showed that some of the Kindergarten and Pre-primary students had difficulty with both the Kriol and English words, indicating they need to learn concepts associated with these words. The research also showed that many of the Year 3 students understood most of the Kriol words, but still needed to learn some of the commonly used English words and phrases.

This paper considers the responses of 26 teachers to items exploring their pedagogical content knowledge (PCK) about the concept of average. The items explored teachers’ knowledge of average, their planning of a unit on average, and their understanding of students as learners in devising remediation for two student responses to a problem. Results indicated a wide range of performance and a wide range of ability in relation to a hierarchical statistical PCK scale. Suggestions are made about developing teachers’ PCK.

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 video-stimulated interview study of problem-solving activity of a high performing Grade 6 girl who displayed confidence in her mathematical ability, provides a microanalysis of tensions she encountered when her findings using concrete aides did not match her rule application. It highlights her disinclination to explore these inconsistencies. This study points to the problematic nature of pedagogical approaches that develop only instrumental understandings and emphasises the need to explicitly value what policies promote; creative and innovative thinking.

The ability of primary (elementary) pre-service teachers to engage effectively in mathematics units in a rigorous program is vital to producing citizens who are able to use mathematics effectively in their lives. Mathematics anxiety affects pre-service primary teachers’ engagement with and future teaching of mathematics. The study measured the range of mathematics anxiety in two hundred and nineteen pre-service teachers starting a teacher education course in an Australian university. They completed the Revised Mathematics Anxiety Scale (RMARS) and a set of demographic questions. Age differences in anxiety were found to be significant, and this has implications for university retention of mature age pre-service primary teachers.

Student learning and performance in mathematics may be associated with issues of connectivity related to curriculum content, acting together with a larger and more complex set of issues, such as those related to teacher quality and socioeconomic influences. Complex and non-linear connectivity of mathematics and other concepts, in particular, may underpin the development of mathematics expertise, and student failures may be related to inadequate development of the networks that connect these concepts. This paper investigates the scope and influence of spatiotemporal networks that may link concepts in the learning of mathematics.

Concept image is proposed by Vinner and Tall (1981) to differentiate the elements and relationships that a learner constructs about a concept from formal mathematical definition for the same concept. The answers of 119 University students to an examination question are analysed to establish the concept images they have for circle and ellipse. The results show the difficulty the students have in reasoning with properties at the higher levels of Van Hiele’s (2004) model of geometric thought.

With the advancement of new technologies, this author has in 2010 started to engineer an online learning environment for investigating the nature and development of spatial abilities, and the teaching and learning of geometry. This paper documents how this new digital learning environment can afford the opportunity to integrate the learning about 3D shapes with direction, location and movement, and how young children can mentally and visually construct virtual 3D shapes using movements in both egocentric and fixed frames of reference (FOR). Findings suggest that year 4 (aged 9) children can develop the capacity to construct a cube using egocentric FOR only, fixed FOR only or a combination of both FOR. However, these young participants were unable to articulate the effect of individual or combined FOR movements. Directions for future research are proposed.

This paper explores the decline in boys’ participation in post-compulsory rigorous mathematics using the perspectives of eight experienced teachers at an independent, boys’ College located in Brisbane, Queensland. This study coincides with concerns regarding the decline in suitably qualified tertiary graduates with requisite mathematical skills and abilities to meet increasing employment demands and opportunities in science, technology, engineering, and mathematics (STEM) careers. Individual interviews and a focus-group interview with teachers in various curriculum leadership, careers/counselling, and mathematics teaching roles revealed.

As part of a longitudinal study on mathematics identity formation and senior subject selection, responses from five top streamed classes of Year 8 students, to the open-ended question “Am I a maths type of person?” have been thematically analysed through examination of key words. Consideration is given to the type of mathematical identity these top streamed students are constructing and how this is related to their intended mathematics pathway in Years 11 and 12.

Indonesian government policy stipulating English as the language of mathematics instruction has created dilemmas for mathematics teachers since they are themselves not proficient in English communication, or with the English mathematics register. The question thus arose as to how mathematics online learning resources (in English) could support the development of learners’ “English Maths” proficiency. Would the language in which mathematical ideas are communicated deny learners’ access to mathematics learning and constrain teachers’ capacity to develop rich mathematical talk? Both questions will be discussed through critical incidents from video data analysis of one teacher, teaching fractions in a secondary school.

What is acceleration and how do we achieve it? Effective classroom pedagogy occurs in classrooms where the teacher has evidence of accelerating the progress of priority group learners. Accelerated Learning in Mathematics (ALiM) is a national intervention introduced in New Zealand in 2010 aimed at accelerating the learning of those students below and well-below national expectations. It focuses on the expertise within the school to evaluate the effectiveness of current practices that support accelerated mathematics learning and to closely monitor the impact of a 10 -15 week intervention for a small group of students. The attention is on supporting teachers and schools to inquire into how an effective teacher provides a short and intensive supplementary programme alongside their classroom programme to accelerate progress. The key themes for teaching are accelerated learning; pedagogical response to individual learning strengths and needs; carefully designed mathematics task in response to identity, language and culture; genuine engagement with parents and family; collaborative inquiry; and high levels of teacher reflective practice. In this round table presentation we will present findings from schools who participated in this intervention in 2012. We will examine the main focus for teaching these students and the impact this intervention had on the rest of the school. We will look at how these schools engaged the parent/family and what effect this had on the rate of acceleration. Finally we will analyse to what extent the teachers were engaged into inquiring into their own teaching practice and to what extent this impacted on the learning of the students.

The New Zealand Curriculum advocates a reflective strategy termed ‘teaching as inquiry’, which encourages teachers to plan for their learners, then continually reflect and respond to their learners’ needs (Ministry of Education, 2007). The February 2013 ERO report, Mathematics in Years 4 to 8: Developing a Responsive Curriculum (Education Review Office, 2013), has questioned the ability of some schools to be able to provide a responsive mathematics curriculum, particularly for students who are under-achieving. One of the factors which may be having a negative impact on teacher ability to foster ‘teaching as inquiry’ is the practice of streaming which has become quite common in recent years in New Zealand primary school mathematics (Years 1-8). This paper looks at the background of streaming in classrooms, and explores the connections that can be made with current research in regards to effective classroom practice for all learners of mathematics.

Developing quality teachers of mathematics is a global concern and research into mathematics teaching, early career primary teacher identity and teacher self-efficacy often focused on teachers’ beliefs and the relationship between beliefs and teaching practice. While some studies have looked at early career teachers and mathematics, none have focused solely on pre-service teacher beliefs about their teaching identity as teachers of primary mathematics as it is constructed over the duration of the practicum. The proposed longitudinal case study aims to track the impact on self-efficacy and identity of pre-service primary teachers as they participate in their practice teaching.

Increasing numbers of students enrolled in primary pre-service teacher Education degrees in Australia enter university with insufficient mathematical content knowledge (Livy & Vale, 2011) and low confidence levels about their ability to teach and do the mathematics required for their intended role as classroom teachers (Wilson, 2009). Teachers need to have the knowledge and teaching skills to improve student outcomes in the mathematics field (Beswick, 2012). This research project explores the development of a mentoring model aimed at increasing the confidence and competence of pre-service primary teachers by matching them with well trained, highly capable, confident and supportive primary mathematics teachers as mentors.

Mathematics identity, as a specific type of identity, may be considered through a variety of paradigms. If identity is defined as a narrative, analysis of the formation of mathematics identity may be undertaken through narrative inquiry. A narrative approach allows the researcher to consider both personal understandings and meanings relating to mathematics identity, in the participants’ spatial and temporal location. Narrative inquiry allows the consideration of the “why” behind participants’ statements and actions, within their particular context, over a period of time.

Research indicates that students are ‘switching-off’ mathematics from as early as Year 5. This presentation reports on an intervention study aimed at improving middle year students’ engagement in mathematics. Twenty middle year teachers and their students (N=339) from seven schools were involved in a year-long professional development program. Student motivation and engagement levels in mathematics were assessed prior to and at the completion of the intervention. Comparison of student data with those from a similar cohort not involved in the intervention indicates that it is possible to reduce, and even reverse, the downward shift in student engagement levels in mathematics.

In this presentation findings from part of a PhD study on students’ learning preferences and their use of advanced calculators such as graphics calculators and CAS calculators will be shared. Students’ responses to a question asking for their preferred method of learning how to use the calculators to solve mathematics problems will be shared. Amongst the different methods, the highest percentage of students indicated that they most preferred to try out the calculator steps while receiving instructions such as observing a demonstration, listening to an explanation, or reading the instructions. The implications of the findings will be discussed.

This short communication outlines a three-year study with 15 Aboriginal Community Children’s Services across New South Wales and the Australian Capital Territory. The project engaged more than 60 early childhood educators and approximately 240 children aged 4 to 5 years. Following an Early Mathematical Patterning Assessment (Papic, in press; Papic, Mulligan, & Mitchelmore, 2011) the project implemented an early patterning framework that developed young children's mathematical thinking and problem-solving skills. Follow up interviews with kindergarten teachers, supported by data from Best Start assessments (NSW Department of Education & Training, 2009), provides evidence of the potential impact of this program on children’s mathematics learning. A key finding is the increased confidence and pedagogical content knowledge of early childhood educators.

This paper discusses the differences of students’ achievement between using open-source dynamic mathematics software (GeoGebra) and Geometer’s Sketchpad in learning geometry. There were 43 participants taken from two secondary school classes in Indonesia. The GeoGebra group consists of 21 students, and the Geometer’s Sketchpad group consists of 22 students. The findings show that the use dynamic mathematics software can have positive effect on students’ achievement. However, findings do not show any significant difference between the two groups.

Although there has been considerable research into the importance of teaching numeracy and being numerate, little is reported on how numeracy is regarded in the secondary school setting by non-mathematics teachers. This paper reports on a preliminary study into the prominence of numeracy in Australian curriculum documentation and teacher perceptions of numeracy in their daily practice. Results indicate that secondary teachers have a narrow view of numeracy and have limited access to professional learning in that area.

As a teacher educator in primary mathematics, I am intrigued by the number of ‘worksheets’ used; this is the antithesis of my philosophy of how primary mathematics should be implemented. My proposed research is informed by a study by Marcia L. Tate (2009). My study will compare the engagement of students, concept retention and practical application of numeracy skills of children using worksheets to those involved in practical hands on activities. The children will be taught using a range of activities. Their learning behaviours will be monitored, they will be interviewed regarding their attitudes and formative assessment will be administered.

Within a larger design research project, we developed an instructional framework for multiplication and division, to be refined through a teaching experiment with low-attaining primary students. We describe the instruction at multiple scales, from the broad organization into domains and phases, through the sequencing of small topics, to the details of specific instructional activities. We also map the multiple dimensions of mathematisation involved: progressions toward larger numbers, more abstract settings, more formal notations, more sophisticated strategies, and so on. The framework contributes to research on arithmetic instruction; and also to our developing notions of frameworks, learning trajectories, and instructional design.

The aim of this preliminary study is to explore the effects of using different types of display and pattern rules on achievement in pattern recognition among pre-schoolers in Malaysia. A total of one hundred and fifty six pre-schoolers were involved in this study. The instrument used was adapted from Gadzichowski (2012). It contains 25 patterns which were divided into five different groups based on display; colour, shape, object, letter and number. Each group comprised five different patterns with rules of increasing difficulty. Each child was interviewed individually. A correct answer was given 1, otherwise zero. Descriptive statistics and a two factor ANOVA for correlated measures were conducted. Results show that the overall achievement of the children was rather low. Children find certain rules easier than others. The different displays had no significant impact on the achievement of pattern recognition amongst these children.

One of the most basic tasks in statistics is to represent data graphically and this suggests that teachers need to possess graphical competence. We explore the statistical graph comprehension of one pre-service and one in-service secondary school mathematics teachers through a series of video recorded interviews. Initially, both teachers claimed strong self-efficacy towards teaching statistical graphs conceptually. However, thinking processes deployed by them to selected statistical tasks revealed procedural knowledge rather than the claimed conceptual knowledge. These consequences suggest that the focus should be on developing the necessary competencies of teachers to work with statistical graphs effectively.

The incorporation of contextualised tasks has been highly recommended by reform documents and curricula. Nevertheless, the role that task context plays in assessments is an unsolved matter because there are arguments relate to whether it makes a task easier or harder for students. This study represents an attempt to scrutinise to what extent the nature of demand of the second-order use of context may affect students’ performance on literacy tasks. It is anticipated that this study can provide a deeper understanding of how task context impacts students’ performance, thereby contributing to the improvement of contextualised assessments among teachers, policy makers, and assessment writers.

In the traditional Mathematics classroom, usually a small fraction of students are able to form or recognise patterns which are core to solving problems, however, many other students never get as far. Pattern-based learning is a new developing strategy that aims to promote effective teaching/learning of Mathematics by enabling all students to recognise patterns. The technique involves presentation of the solutions to standard well-known problems through software. Once the student has gained experience in solving multiple problems using a clear pattern of solution, he/she can then independently apply the technique. In this poster, Pattern-based learning is applied to Linear Algebra.

This study aims to analyse student’s thinking process in geometric argumentation from geometric examples and counter-examples between Grade 3, 5 and 7 students in Victoria, Australia and Taiwan. There are two experiments in this study. The first will test and compare cognitive frameworks of geometric argumentation. The second will analyse student’s thinking process though geometric examples. It is anticipated that this study can provide a better understanding of thinking process in geometric activities and assist students enhance their abilities in geometry.

The Australian Curriculum: Mathematics (Australian Curriculum, Assessment and Reporting Authority, 2012), with its specification of content for year levels, represents a break from stage based curricula which have become the norm in Australian educational jurisdictions in recent decades. It thus provides an opportunity to rethink the appropriateness of developmental approaches to mathematics teaching and the concept of readiness that underpins the widely accepted tenet of teaching from where students are at (Anderson, 2010). Such an approach has the risk of students who fall behind their peers remaining behind even when they make progress (Capraro, Young, Lewis, Yetkiner, & Woods, 2009). This is exacerbated in mathematics because of a prevailing belief that mathematics, to a greater extent than other school subjects, is inherently hierarchical and hence must be taught in a linear fashion that precludes access to advanced content (e.g., algebra) until more basic topics (e.g., arithmetic) have been mastered. A year level based mathematics curriculum has the potential to contribute to solving at least two major problems that currently characterise mathematics learning particularly in the middle and secondary years of schooling. These are 1) persistent gaps in attainment between various disadvantaged groups and a majority of their year level peers, and 2) impoverished curriculum offerings for low attaining students who struggle to master 'basic' content. This Roundtable will provide a forum for discussion of these propositions and the opportunity afforded by the implementation of the year level based Australian Curriculum: Mathematics. Stimulus in the form of evidence that challenges the hierarchical and linear nature of mathematics learning will presented and ways that these ideas might contribute to closing attainment gaps discussed.

This roundtable will report and seek participants’ feedback on progress towards the project Developing a shared understanding of assessment criteria and standards for undergraduate mathematics, funded by the Office of Learning and Teaching. The project seeks to engage the higher education mathematics community in a conversation around assessment standards which builds upon the Learning and Teaching Academic Standards project outcomes for the sciences (Yates, Jones & Kelder, 2011), and their contextualisation within the mathematics discipline. It aims to influence assessment practices in mathematics departments, to move away from idiosyncratic marking and grading approaches that favour procedural mastery towards practices that measure the quality of all aspects of student work against external anchors, ensuring comparability of standards within and across mathematics departments. The project will result in a reference framework and toolkit to support tertiary educators in the development of quality assessment standards and criteria. The project approach incorporates the four essential elements that, according to Sadler (2009), are required to convey and apply achievement standards: (i) exemplars of different levels of achievement invoking the criteria relevant to the judgment made, each of them with an (ii) explanation of how the judgment was made; a (iii) conversation about the exemplars and their corresponding judgments to establish a common vocabulary; and (iv) the sharing of what has been tacit knowledge within the discipline community.

The recent publication of the Trends in International Mathematics and Science Study (TIMSS) 2011, has raised much debate in the public and political arena in New Zealand. Analysis of the data indicates that New Zealand students performed less well than most developed countries, and performance of ten year olds has declined since 2001. The question being asked is, ‘Why are New Zealand’s ten year olds not performing as well as those in other developed countries?’ In 2010, New Zealand introduced National Standards in mathematics, reading and writing. The mathematics standards rely on teachers making judgments about a student’s overall learning from a wide range of relevant evidence. Other countries such as Australia, England and the United States of America have introduced national testing. The notion that New Zealand students aren’t practised in taking tests in this manner has been offered as an explanation for the decline. How does a student’s prior test-taking skills and experience impact on results in such a high stakes activity? Do international tests like TIMSS provide an accurate measurement of a student’s mathematical understanding and ability to solve complex problems? Should teachers be investing some time in practising the techniques for tests of this type? This round table forum presents a small-scale study investigating the impact of practiced skills involved in test taking in relation to mathematics standard data. Discussion will focus on high stakes testing versus overall teacher judgments in assessing mathematical competence.

In the 21st century algebra continues to be seen as a “gatekeeper course” for mathematics (Rand Mathematics Study Panel, 2003). Many future career opportunities are lost to students who do not have a good understanding of algebra at some level. The Australian Curriculum, in its strand Number and algebra, introduces formal algebra to students at an earlier stage than has been the case in most Australian states in the past. In this round table presentation we will firstly examine some aspects of what the curriculum says about early algebraic ideas and reasoning. We will then examine three lesson plans designed to introduce students to foundational algebraic concepts, also discussing some current doctoral research. This research asks: What strategies do teachers use when teaching algebra in the transition years, and how do these choices reflect their beliefs about mathematics teaching and learning? Participants will have the opportunity to discuss the results of some current research on teachers’ beliefs and practices in this area of teaching. They will also review some research findings in current literature, and what this says about the teaching and learning of early algebraic concepts.

National Standards, introduced into New Zealand schools in 2010, require teachers in years one to eight to make overall teacher judgments in mathematics. This new assessment policy asks teachers to use the standards and exemplars to make defensible and dependable holistic judgments about whether a student is above, at, below or well below their year standard. The centrality, complexity and nature of teacher judgment practice in mathematics in such a policy context need to be understood. My study drew from principals’ and teachers’ perspectives about how teachers approach and make overall teacher judgments in mathematics and was gathered using semi-structured interviews and from document analysis. Participants included four principals and seven teachers of students in years three to six. A range of approaches to judgment making emerged from exploring the beliefs, understandings and judgment practices teachers adopted. Teachers utilised both explicit and tacit knowledge in the decision making process and valued their relationship with and knowledge of their students, giving attention to features other than those specified in the mathematics standards. This round table forum will begin with a short presentation of findings to initiate discussion regarding influences that could be considered to ensure teacher judgments in mathematics are dependable and whether exemplars and standards are sufficient to inform professional judgments in mathematics.