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Typically, more primary students qualify for mathematics intervention support than schools have the capacity to assist. This highlights the importance of every primary mathematics teacher having the expertise to design inclusive and responsive curricula and instruction for students who may experience difficulty. Our study addresses this issue through exploring how specialist intervention teachers can support the mathematics professional learning of teachers using co-teaching cycles. The findings provide insight about the challenges experienced by classroom teachers when teaching students who experience difficulty with mathematics, and the actions of the specialist intervention teachers that assisted the teachers' professional growth.

This paper reports on a key representation, a triple number line, designed as part of the first author's doctoral study. The study sought ways to represent multiple constructs of fractions in the context of merging music and mathematics to support learners' understanding of fractions. A problem scenario was designed guided by Realistic Mathematics Education principles. Findings shared in this paper are based on the process of designing and implementing the tasks around the triple number line. Data for this qualitative, participatory dual-design experiment in task design were collected via formal and informal interviews in two micro-Communities of Practice. We conclude that the key representation of the triple number line can be a powerful tool for supporting learners in their fraction understanding.

Big Ideas in school mathematics can be seen as overarching concepts that occur in various mathematical topics in a syllabus. For teachers, this knowledge can be used to help students develop a better understanding of mathematics by making visible the central ideas, and connection across topics and across levels. For students, this knowledge can further be helpful affectively by engendering an appreciation of mathematics as a subject that is coherent and comprehensible. Although there has been much interest recently in the understanding of Big Ideas, there is little research done in the assessment of Big Ideas thinking. In this paper, we discuss our development of an instrument to measure the Big Ideas of equivalence and proportionality. Our analysis of some pilot items suggests that Big Idea thinking is a multidimensional construct within most school environments.

This paper is a brief summary of a large historic research project in Papua New Guinea (PNG). The project aimed to document and analyse the nature of mathematics education from tens of thousands of years ago to the present. Data sources varied from first contact and later records, archaeology, oral histories, language analyses, lived experiences, memoirs, government documents, field studies, and previous research especially doctoral studies. The impacts of colonisation, post-colonial aid and globalisation on mathematics education have been analysed and an understanding of the current status of mathematics education established as neocolonial. Managing neocolonial education policies may minimise the loss of cultural ways of thinking.

We explored primary pre-service teachers' beliefs about challenging mathematical tasks and the role they perceived their initial teacher education played in influencing those aspects. Fifty-seven pre-service teachers completed an online questionnaire, and four participants were individually interviewed. Results showed that most participants recognised the importance of teaching with challenging mathematical tasks even prior to exposure to such content in their teacher education program. The teacher education program was perceived to have positively impacted final year pre-service teachers' perspectives about challenging tasks. Implications for teacher education regarding challenging tasks are discussed.

In this study, we aim to investigate the role of language in small group peer collaboration. Based on the notion of symmetrical scaffolding, we draw on theories of dialogic space and functional linguistics to analyse a transcript of three six-year-old students as they explore a shared understanding of 'two more than'. Using Martin and White's (2005) engagement framework as an analytical tool, we identify how the three students used language resources in contractive and expansive ways to move each other's learning forward. These findings provide a perspective of symmetrical scaffolding closely focused on students' language choices that support engagement with each other's thinking. We suggest that a focus on use of language with young students is valuable in identifying the type of mathematical discourse that will support peer collaboration in problem solving and mathematical reasoning.

This paper explores use of a set of diagrammatic tools for representation and analysis of the moves a teacher makes implementing a mathematical modelling task. The focus here is on identifying what the teacher did so we can subsequently interrogate this, as to the why. Data include pre and post lesson teacher interviews and transcripts of a video and audio-recorded task implementation. The analytical tools developed, with one teacher and one task early in a three-year project were particularly useful in ascertaining what the teacher moves were as we subsequently sought to determine reasons for these.

This qualitative study investigates remote Australian primary school parents' attitudes towards their children's learning of mathematics and the role of technology. Using semi-structured interviews and thematic analysis, results revealed that parents aimed to provide the best mathematics learning opportunities for their children, yet they felt obsolete and isolated from their children's learning. Additionally, they valued the role of technology, but did not think school was providing adequate digital literacy skills. Discussion about what changes to the educational community are suggested to support parents in their children's learning.

In this paper, the SOLO taxonomy is used to identify different levels of student understanding of the statistical concepts associated with sampling distribution. This study was part of a research project investigating students' conceptual understanding of concepts of hypothesis testing taught with the support of simulation learning activities. The study involved eight students enrolled in a first-year tertiary introductory statistics unit and the examination of their written responses to three questions about sampling distribution concepts. The SOLO taxonomy categorisations revealed that some students had only pre- and unistructural understanding of sampling distribution, and none providing responses at the extended abstract level.

Place value is one of the 'big ideas' in number and plays a critical role in helping students develop their number sense, problem solving and computation skills. Yet, the elegant simplicity of our place value system belies the abstract nature of the construct. This paper presents data from 606 Year 3-6 students (ages 8-12) from two metropolitan Melbourne primary schools who completed the Place Value Assessment Tool (PVAT). Each student's place value knowledge was categorised according to the Place Value Developmental Progression (PVDP). The results highlight the wide range of understanding in each year level and challenge the efficacy of a number magnitude-based progression in place value.

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and explained algebraic relationships in pairs of equivalent equations. Findings indicate that with the introduction of designed instructional material, there was a shift in student noticing of number properties to identify equivalence between pairs of equations. However, identifying the distributive property of multiplication and developing generalisations about the algebraic relationships remained challenging for students.

Calls for high levels of participation and success in Science, Technology, Engineering and Mathematics (STEM) by all Australians exacerbates the need for research into improving participation in mathematics by students from disadvantaged backgrounds. Rogoff's three planes of analysis is used to foreground the institutional plane in a low socioeconomic status school with sustained increased enrolments in Mathematics B (now replaced by Mathematics Methods). Processes in this plane are examined by drawing on semi-structured interviews with a Deputy Principal and Head of Mathematics. Contributing factors appear to be the way that the Mathematics Program and Pastoral Care Program are delivered at the school.

This conceptual paper discusses two frameworks, developed independently by the lead author, that will provide the conceptual foundation for the identification and evaluation of mental computation strategies students demonstrate during an upcoming research project entitled Mental Computation in Year 5. These frameworks will be used by the lead author during an intervention to investigate the application of mental computation strategies in problem solving tasks involving duration of time. It is an intended outcome of the project that the two frameworks will be useful for teachers and students in upper primary school to provide feedback regarding the teaching and learning of mental computation.

It is generally understood that both clear teacher instruction and orderly classroom climates support student achievement in mathematics. However, to what extent does instructional clarity "compensate" for classroom disorder? In this exploratory study, we analyse data from 8,864 Year 8 students sampled by TIMSS 2019 to investigate the relationship between instructional clarity, classroom disorder, and mathematics achievement. The findings demonstrate the deleterious effects of classroom disorder for mathematics achievement, even in the presence of high instructional clarity. The findings contribute to an emerging international body of work and reinforce the importance of teachers having an optimal combination of classroom skills to support student learning.

This paper describes results from a case study about how a primary pre-service teacher (PST) used curriculum materials (CMs) when planning for a mathematics lesson during her final practicum. The data is drawn from a doctoral study (in progress) and results show how the PST initiated an active process of searching for and sifting through CMs on a familiar website to make selections for a lesson. Selections were based on several aspects, including the mathematics focus of her lesson, curriculum connections, her chosen teaching approach and mathematical representation for teaching multiplication. Implications for mathematics Initial Teacher Educators (ITEs) are discussed.

This paper reports on students' experiences of describing and representing variation in hypothetical data. Fifty-six students (8-9 years-old) experienced collecting and working with quantitative data for two years as part of a STEM education project. The task described here was an end-of-year survey question, with three parts about a hypothetical context for surveying students in two different Australian cities: recording the data, describing the potential variation in the data, and creating a representation of what the data might look like when only a descriptive account of the context and variables were provided. The data analysis framework utilised provides a means of determining students' readiness for further development of statistical ideas.

There has been a lot of media attention about 'out-of-field' teaching, but much of it does not capture the complexities of the term or account for the range of knowledge, experience, and attitudes relevant for teaching mathematics in the middle years. In this paper we report on a survey conducted to better understand the diversity of the profession teaching middle school mathematics (Years 7-10) in South Australia. A preliminary analysis indicates that self-identification as a teacher of mathematics is a key contributing factor to confidence in teaching mathematics. We suggest that looking through an identity lens will better inform planning and support for out-of-field teaching in mathematics.

Worldwide interest in Lesson Study (LS) and the opportunities offered for student learning through the use of a Structured Problem-Solving Approach (SPSA), as typically adopted in Japanese LS research lessons in mathematics, have left largely unanswered questions about the extent to which these can be replicated elsewhere. This paper presents a case study of one primary school teacher's learning experiences, and his views about LS and SPSA, as a result of participating in a project introducing SPSA through LS in three Fijian primary schools. The results reveal that engaging in the LS process was instrumental in supporting this teacher's implementation of SPSA in his mathematics classroom. The findings are important for teacher professional learning (PL) in Pacific cultural contexts.

Informed by teaching statistics across curriculum areas strategies, this study investigated the skills of 36 pre-service teachers (PSTs) who designed numeracy activities that focused on the statistics strand of the Australian curriculum. The data were analysed using descriptive statistics. The results showed that the PSTs designed numeracy activities focused on the science curriculum area. The results further showed that PSTs emphasised a few year levels and focused on collecting and recording data. The first stage of teaching statistics (designing effective questions) is often ignored in their activities. The approach adopted in this study can be used to identify PSTs' knowledge gaps and their professional learning requirements.

This paper reports on 73 Grades 3 to 6 students' written responses to equal groups, arrays, multiplicative comparison and Cartesian product word problems. It is part of a larger study relating to students' development of multiplicative thinking (MT). The potential of using multiplicative word problems as a diagnostic assessment to reveal students' transition barriers from additive thinking to MT was explored. The development of the assessment and some of the research findings are described here. Findings show that students experience different types of barriers during their transition from additive to MT. Some recommendations are given on how these barriers might be overcome during primary school years.

In this paper we compare the early years mathematics curricula of Germany, South Africa, and Australia in relation to the place value concept. Place value is an important topic as it underpins much of the number work completed by learners in the early years of schooling. We found that there were differences between the three curricula that could be summarised using five themes: namely, number range, place value structure, role of the zero, influence of language, and use of materials. We argue that how the different curricula deal with these five themes influences the quality of learning provided and we highlight key areas of concern. In concluding we identify three important implications for our future research project.

Providing opportunities for students to demonstrate their mathematical thinking can be challenging. This paper reports on a case study conducted with a teacher and her class of Grade 3/4 students which investigated this phenomenon. Following collection of baseline data that showed her students were not demonstrating evidence of mathematical thinking, the teacher implemented teaching approaches designed to address this. The findings indicated the teaching approaches were effective and led to changes in the teacher's practice and professional growth. The study has implications for teachers who are looking to make changes to their practice based on evidence-informed teaching approaches.

With the aim of strengthening mathematics teaching | learning, classroom video is increasingly used as a tool for teachers to reflect on their practice. In the first phase of a two-phase design-based study, a group of primary teachers selected, viewed and discussed video excerpts from mathematics lessons. Engaging with and working to reconcile differences that emerged in the teachers' ideas about practice catalysed productive conversations for their learning. What was noticed and picked up for discussion mediated what conceptual resources were made available for teachers' reflections. Sharing classroom video provided teachers with both a window into the otherwise private practice of others, and a mirror in which to reflect on their own practice.

This study examined the types of mathematical connections established by four secondary school students while constructing concept maps on transformations of the parabola. The use of concepts maps revealed students' understandings--including some misconceptions--of transformations of the parabola and confirmed the usefulness of a model for categorising types of connections.

This paper reports on early years teachers and how often they should devote planning time to anticipating student responses in advance of the lesson. Sixty-five Foundation to Year 2 teachers (students 5-8 years of age) completed questionnaires at the beginning and end of a year-long research-based professional development program. Participants were learning to teach with sequences of challenging tasks. Post-program data showed a shift in the frequency of time participants believed teachers should devote to anticipating student responses prior to teaching. Supporting teachers' mathematical knowledge for teaching with an emphasis on how they plan and anticipate student responses has implications for improving practice and student outcomes.

Behaviourism proposes successful learning to be dependent on the performance of conditioned behaviours that are distinctly observable and objectively measurable. Over the past 100 years, various behaviourist concepts have been superseded by sociocultural and cognitive learning theories, but the entwined areas of assessment in mathematics education have not received the same focus. Outdated beliefs and unquestioned norms surrounding behaviourist assessment ideologies continue to dominate schools' decision-making around mathematics education. This paper includes the theoretical foundations for a PhD that aims to surface the tension of what is said to be valued in mathematics as "success" and what is systematically labelled as "success".

Conceptual body movement in education is an external representation mode that research suggests can support children's learning about mathematical phenomena. Children's learning and understanding of mathematical concepts and processes, such as number structure and relationships, number sequencing, position, or geometric properties, may be supported by experiences using their own body movements. The aim of this paper is to share classroom activities trialled within the Embodied Learning in Early Mathematics and Science project in 7 classrooms focusing on conceptual body movements. The results share what was trialled, and what was observed in relation to children's learning of mathematical processes. Findings revealed that body movement is a helpful mode through which young children can learn and communicate mathematical understanding.

The phenomenon of teachers designing their own instructional materials is gaining more attention in research. Different aspects of this enterprise have been examined--its potential to reveal the complexity of teachers' instructional planning considerations, the design principles employed by teachers to realise instructional goals, among others. In the study reported in this paper, the focus was on its utility as a form of teacher professional development. In particular, evidence was sought for this claim: not only is teacher-designed instructional materials a useful tool for professional development, it can capture usable knowledge teachers appropriate from professional development.

This paper presents an initial analysis of 10 Preschool children's responses to a look-draw-trace-draw task. The findings suggest that figure-tracing helped half the children to produce a more accurate representation of the geometric figure presented to them, in their second drawing.

There is an educational expectation that children's natural drawing will develop into proficient mathematical representations and formal diagrams, yet there is little research available to guide the assessment and development of children's mathematical drawing skills. The aim of this paper is to explore how Year 2 children (approx. 7 years) chose to represent their interpretations of a simple story that is suggestive of the take-away subtraction process, and what changes occurred when the drawing task was repeated 6 months later. Analysis of 13 pairs of drawings revealed changes in what the children drew (categories of number representations) and how they drew it (style). The findings suggest that substantial change in children's representational ability can occur in within 6 months.

The study reported in this paper is on the professional development (PD) of primary school mathematics teachers. Teachers from two primary schools participated in the PD for two years. High-stakes mathematics examination tasks were used to kick start awareness and thinking about teaching for big ideas. Teachers did the tasks and discussed their solutions focusing on how their instruction could facilitate the acquisition of mathematical ideas as a body of connected knowledge. Data presented in this paper show that the tasks teachers worked with at the start of the PD did impact their understanding and instructional practice specific to big ideas in mathematics. Some challenges the teachers faced during the PD are also noted.

Despite curriculum expectations, many students, including a disproportionate number of girls, do not 'just know' (retrieve) single-digit addition facts by Year 3. The current study employed structured interviews to explore which strategies Year 3/4 students (n = 166) used when solving more difficult addition combinations. Results revealed that students preference the near-doubles strategy when the difference between the addends was one, the bridging-through-10 strategy when one of the addends was a nine, and the count-on-from-larger strategy when a derived strategy was more effortful. Moreover, whereas boys were more inclined to use derived strategies, girls were almost three times more likely to use the count-on-from-larger strategy.

In this study, the authors investigate the ways in which young students demonstrate their critical mathematical thinking (CMT). Students aged 5-6 who are beginning their first formal year of education participated in the study. Data is presented from individual clinical interviews undertaken with 16 students. These interviews were analysed using the Critical Mathematical Thinking for Young Students (CMTFYS) framework to identify common patterns in the responses. The findings suggest that these young students beginning school, most often rely on providing explanations and more specifically, justifying, to demonstrate their CMT.

The primary school mathematics syllabuses in Singapore as of the year 2020 reinforces that Big ideas are central to the learning of mathematics. In support of the push to teach for big ideas, a research study is presently underway. A part of it is on the professional development (PD) of primary school mathematics teachers. As part of the PD teachers attempted a mathematical task as measure of the big idea, Equivalence, in an online environment at the start of their PD. Data from the task show that teachers, were generally not cognisant of the big idea of equivalence when solving the task. They were also unable to distinguish between a heuristic (diagrams) and a mathematical idea about relationships, specifically equivalence as in the mathematical task.

"Balai Besar Guru Penggerak Daerah Istimewa Yogyakarta (BBGP DIY)" conducted a professional development course aimed at enhancing primary school teachers' capacity to develop and implement STEM lesson plans in their classrooms. As part of the course, teachers were asked to record their lessons, which were later analysed to identify the mathematical content involved and how teachers delivered this content during the STEM lessons. The findings indicated that, although mathematics appeared to be less pivotal than other STEM disciplines in the lesson plans, the STEM activities provided rich opportunities for developing students' mathematical content knowledge. Most of the teachers realised this and were able to deliver the mathematical content within the context of STEM education. This paper presents an insight into how primary school teachers in Indonesia deliver mathematical content in their STEM lessons and highlights the role of mathematics in STEM education.

In this paper we report on an aspect of the findings of a larger three-phase study exploring the factors that influence teachers implementing pedagogies that cultivate students' STEM capabilities and 21st century skills. Data were collected through an online questionnaire, semi-structured interviews, focus groups and case studies. This paper will focus on the findings from the first phase of this study and initial analysis of focus groups and semi-structured interviews data. Preliminary findings show that participants hold mixed beliefs concerning student proficiency in mathematics and there are common factors that influence decisions concerning the use of pedagogical practices that support students' mathematical proficiency and the development of students' 21st century skills. These factors include teachers' personal beliefs and attitudes, perceived time and curriculum constraints, student behaviour and students' academic ability.

This paper reports on the development of six student teachers' knowledge of instructional strategies (KOIS) for teaching proportions during a 2-month practicum in China. Development of four subcomponents was explored through Content Representation (CoRe) questionnaires and follow-up interviews. Data was analysed deductively and levels of each subcomponent determined based on a scoring rubric. Implications include that practicum is capable of developing KOIS effectively as final scores were at the maximum level with some participants showing greater initiative within their KOIS than others. Implications include that a further level in the rubric could be considered to reflect when creativity is shown in PCK.

The current study explored the reasons for students' preferences for the teach-first and task-first lesson structures, and whether students' preferences were influenced by their perceptions of the teacher's preference. Students (n=18) from two composite Year 3 and 4 classes (aged 8-10 years) completed a post-lesson drawing task and participated in a semi-structured interview following a series of lessons. Findings indicated students had a variety of reasons for their preference of lesson structure. Most focus students reported noticing aspects of the teacher's enjoyment during instruction. The results have implications for the way teachers inadvertently influence their students' own enjoyment of and preferences for instructional approaches.

We explore how relational trust was evidenced within the leadership activity of one mathematics leader using cultural-historical activity theory as a theoretical perspective. We use data from interview and observation sources to suggest that relational trust was a motive object of that mathematics leader's activity. Findings contribute knowledge about how relational trust dimensions, originally situated within middle leadership theory, were evidenced within that mathematics leader's activity whilst leading a professional learning experience in their school.

Teaching towards big ideas provide opportunities for teachers to think deeply about content and pedagogy in order to support their students to see connections in mathematics. However, teachers may not always activate or mobilise their knowledge in classroom situations. This paper looks into how a teacher, Peter, think about the tasks in his instructional materials he crafted to uncover what he may notice about the affordances of the tasks for teaching proportionality.

Two case studies of Australian primary school students tracked changes in their data interpretation and representation over three years. Students were engaged in predictive reasoning tasks based on their interpretation of a data table showing temperature change over time. Students' explanations and graphical representations were collected at the beginning of Years 3 and 4 and the end of Years 4 and 5. The first case study was a student mathematically weaker than her peers while case study two was within the average range for her year. Despite differences in starting points, both case studies followed a similar developmental sequence of predicting, interpreting and representing, with the first case generally lagging one stage behind the second case. Similarities and contrasts between the two students are discussed.

Place value is a foundational competency for primary school mathematics and for this reason we have sought to investigate what the recent and current academic conversations are around this important concept. In this paper we present a survey of literature presented in the Australasian, European and Southern African contexts through a review of purposively selected conference proceedings and journals to establish what the conversations have been about the teaching and learning of place value in these research communities from 2013 to 2022.

It is well accepted and understood amongst the mathematics education community that numeracy is the responsibility of all teachers, across all levels of schooling. However, the way numeracy is understood and actioned across the Early Childhood, Primary and Secondary contexts is different. This paper reports on a case study of secondary school teachers seeking to be more intentional in embedding numeracy across the curriculum. The findings indicate that dialogue and support to see, and make, numeracy connections result in greater cohesion in terms of understanding numeracy and making numeracy more visible for students.

Climate change is a complex issue that requires collective actions from various disciplines, including educating future generations about sustainability. Some countries such as Indonesia and Australia have included sustainability into their school curricula, which require teachers to design and implement tasks related to a sustainability issue. However, teachers may not have the capability and understanding to do so. The challenges are difficult for mathematics teachers because designing a mathematics task pertaining to sustainability is rarely explored. Therefore, this paper draws from mathematics education literature to provide and discuss a new framework for designing sustainability-related, mathematics tasks (green mathematic tasks).

The interactional role of language use in the mathematics classroom is explored in the last few decades. This paper adds to the knowledge base by exploring different ways of talking that children and their teacher use in a New Zealand primary geometry English-medium classroom. Bakhtin's concept of speech genres is used for an analysing transcript of one audiovisually recorded whole-class interaction episode from a geometry lesson in a Year 5/6 classroom. The analysis suggests that the teacher and children use various genres to participate in classroom interactions. The use of several genres provides insights into what children and the teacher construct as mathematical in the real sense. The paper finished with a few implications for teaching and research.

Teachers often perceive barriers to integrating technology with Mathematics. In the study, teachers participated in professional development about considerations for and examples of integrating Digital Technologies into Mathematics learning in primary schools meeting the "Australian Curriculum: Mathematics and Digital Technologies" requirements. This exploratory multiple case study included preand post-surveys, mentoring conversations and interviews to explore their beliefs and perceived barriers to integrating mathematics and digital technologies. This paper provides insights into their perceived barriers to integrating digital technologies.

Big Ideas in school mathematics can be seen as overarching concepts that occur in various mathematical topics in a syllabus. Although there has been much interest recently in the understanding of Big Ideas, there is little research done in the assessment of Big Ideas thinking. In this paper, we discuss our research on Big Ideas in School Mathematics. The study consists of three sub-studies: the first sub-study on developing an instrument to measure Big Ideas; two sub-studies on measuring students' and teachers' Big Ideas at test-points before and after a professional development on Big Ideas for primary and secondary school teachers and students.

In this paper we report our development of instruments to measure Big Ideas in school mathematics. In tackling the issue of assessment fatigue among students, we present an aggregated approach to measure students' knowledge of Big Ideas.

Fostering students' bidirectional conception of the equal sign (viewing the equal sign as indicating an equivalence of two sides rather than a 'show result' symbol) is challenging, and students' misconception of the equal sign is persistent. Some studies mention that in China, the pedagogical approach to introducing the equal sign supports students' development of bidirectional sense toward the equal sign. Built up on this body of literature, this paper further investigates the Chinese pedagogy from the researcher, students and the teacher's perspectives.

South Australia's PISA performance has been in constant decline since 2003 with the proportion of PISA participants meeting the Australian national proficient standard dropping from 73% (in 2003) to 50% (in 2018). In contrast, Singapore is a consistently strong performer. To better understand student readiness in answering PISA questions, this paper reports a curriculum comparison between Australia and Singapore for Years 9 and 10 in Measurement and Geometry. The findings highlight the similarities and differences in the topics covered in both countries' curricula and raise questions about potential implications for student outcomes in PISA.