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How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk I consider the proposal that geometry instruction could be improved by infusing it with activities where students use representations of figures to model their experiences with shape and space and I show how our basic research on high school geometry instruction informs the implementing and monitoring of such modeling perspective. I argue that for mathematics education research on instruction to contribute to improvements that teachers can use in their daily work our theories of teaching need to be mathematics-specific.

While there is widespread agreement that all learners of the 21st century need to be numerate and literate, reforming pedagogical practices to achieve such an outcome is challenging for many teachers. This is a report of one aspect of a project which aims to integrate a culturally responsive pedagogical mathematics practice within communities of mathematical inquiry. The focus of this report is on the exploration of dynamic mentoring from the perspective of teachers. The results demonstrate how ‘in the moment’ actions from mentors during lessons cause reflective transformations in pedagogical practices.

This year marks the 25th anniversary of the publication of a National Statement on Mathematics for Australian Schools, which was the first curriculum statement this country had including “Chance and Data” as a significant component. It is hence an opportune time to survey the history of the related statistics education research, consider where we are at the moment, and speculate about where we should be going. Some of the issues to be considered along the way include the relationship of research to the curriculum and its implementation (or not), the rise of a research culture, the juxtaposition of statistical literacy and statistical reasoning, and the importance of context for learning about statistics. The continued interest in affective variables is considered, as are the recent trends related to the pedagogical content knowledge needs of teachers and the influence of advances in technology. Finally some views are canvassed on the new initiatives in STEM education and Mathematics by Inquiry, as well as the possible impact of statistics education researchers in the field.

Let’s Count is a preschool mathematics intervention implemented by The Smith Family from 2012 to the present in ‘disdvantaged’ communities across Australia. It is based on current mathematics and early childhood education research and aligns with the Early Years Learning Framework. Let’s Count has been shown to be effective in enhancing mathematics learning and dispositions of young children, early childhood educators and families through a longitudinal evaluation undertaken from 2012-2015. This paper reports on the development, implementation and evaluation of Let’s Count and its likely future trajectory.

Highly capable mathematics students are not usually considered strugglers. This paper reports on a case study of a Year 6 student, Debbie, her response to a lesson, and her learning involving a challenging mathematical task. Debbie, usually a highly capable student, struggled to complete a challenging mathematical task by herself, but as the lesson unfolded she was able to solve the problem. Rather than relying on teacher explanations during the lesson introduction, Debbie benefitted from the support of her teacher during the lesson and sharing of other students’ thinking when applying her knowledge to an unfamiliar task. [*Debbie is a pseudonym.]

This paper examines the learning by students who were participating in a project designed to promote persistence while working on mathematical tasks. We examined their learning of mathematics concepts and learning about the processes of engaging in mathematical tasks. There were substantial increases in students’ knowledge of angles and also evidence that the students built on their prior knowledge, made connections between concepts, found the tasks rewarding, valued concrete materials and engaged in mathematical communication.

As part of a project exploring the use of challenging mathematical tasks, data from New Zealand teachers and their students were analysed to explore teachers’ actions that encouraged students to persist. Rather than rescuing the students when they needed help, the teachers’ actions included arranging for and encouraging students to work independently and cooperatively, asking questions, providing materials, and getting students to reflect on the process. When the teachers performed these actions, they reported students persisted and were better able to navigate through the zone of confusion.

The following outlines the rationale and structure of a professional learning initiative that seeks to explore teachers’ ways of engaging students more actively in building mathematical connections for themselves. An example of one of the suggested experiences is presented.

This paper explores young children’s drawings (6 years old) in early number and addition activities in Malaysia. Observation, informal interviews and analysis of drawings revealed two types of drawing, and gave insight into the transitional process required for children to utilise drawings in problem solving. We argue the importance of valuing and supporting the early development of children’s drawn mathematical representations to facilitate their successful use in problem solving processes.

Intellectual risk is valued among 21st century skills. Three primary teachers who promoted positive learning within mathematical inquiry collaborated with researchers to design and apply a rubric to assess children’s progress in taking intellectual risks twice during the year. Results suggest that handling setbacks and giving feedback to peers were the most challenging skills initially, but showed significant gains by the end of the year. Teacher interviews discussed challenges that students faced and how positive classroom culture encouraged intellectual risk

This paper reports on a study that examines the effects of problem context on students’ performance. The performance of 151 Year 10 students on six mathematical problems was compared with the performance on fifteen variants with more and less context familiarity (CF) and engagement (CE) across levels of context use (LCU). The latter explanatory variables (CF, CE, and LCU) are used to estimate the strength of the relationship among them and the students’ performance. Results show that neither CF nor CE affect students’ performance but LCU demanded in solving a problem does.

This paper aims to revisit and clarify the term problem context and to develop a theoretical classification of the construct of levels of context use (LCU) to analyse how the context of a problem is used to formulate a problem in mathematical terms and to interpret the answer in relation to the context of a given problem. Two criteria and six indicators form the basis of the construct. While this construct connects to a previous classification of uses of contexts, several theoretical considerations are considered and clarified. Quantitative analysis suggests that the construct is effective in distinguishing between LCU.

Grouping children by achievement levels is a thriving practice in New Zealand primary school mathematics classrooms. In this paper we look at the impact of a formative intervention project—Developing Communities of Mathematics Inquiry—that required a whole-school shift to mixed achievement grouping. Engeström’s Cultural Historical Activity Theory framework is used to explore changes in the teachers’ object/motives over the first year of the project. Teachers’ learning about mixed-ability groupwork focused first on organisational and social structures, then participatory practices, and finally to students and their own mathematical sense-making. Such a shift is characteristic of expansive learning and transformative agency.

Collaboration between mathematicians and mathematics educators may provide a means of improving the quality of pre-service teacher education for prospective teachers of mathematics. Some preliminary findings of a project that investigates this type of interdisciplinary collaboration, both within and across institutions, are reported on in this paper. Interviews were conducted with selected participants to identify the nature of the boundary encounters and brokering involved between disciplinary communities in order to create new practices and transfer these practices to a new institutional context.

This study examined the shared beliefs among mathematics teachers in one secondary school in the United Kingdom across the first term of a school year and almost 4 years subsequently. Leximancer software was used to analyse the language used as teachers responded to questions concerning their beliefs about mathematics, mathematics teaching, mathematics learning, and their department on at least two occasions. The analysis revealed changes of language and evidence of shared beliefs at each point in the study.

Textbooks are a ubiquitous part of classrooms in all levels of education. Whilst textbooks used in tertiary content subjects have been examined in several studies, research focused on textbooks used in mathematics pedagogy subjects is scarce. Using a discourse analytic framework, this paper presents data about the implicit and explicit messages sent to preservice primary teachers by a mathematics pedagogy textbook. Results suggest the majority of messages sent are inclusionary. However, there are contradictions and ambiguities about nature and field of mathematics education within these messages that are also discussed.

Numeracy Skills testing was brought to the forefront following the Action Now: Classroom Ready Teachers report in 2015. The announcement was met with much concern by many pre-service teachers (PSTs). Given the body of research associated with mathematics anxiety and assessment, which suggests that students’ self-efficacy and confidence in their mathematical abilities can be counterproductive, this paper explores the impact of positive reinforcement and affirmation on the knowledge and mathematical ability of a cohort of PSTs facing the Numeracy Test for Initial Teacher Education Students.

This paper explores the construction of classroom contexts facilitative of student engagement in Mathematics. Employing a form of discourse analysis framed within a participation approach to learning, the paper provides insights into the construction of such contexts. The affordances and constraints of constructing such a context are discussed in the light of the writings of one Year 7 teacher as she employed Collective Argumentation to re-construct her classroom context to better engage students in the learning of Mathematics.

This paper describes the pedagogical framework used by YuMi Deadly Maths, a school change process used to improve mathematics teaching and thus enhance employment and life chances for socially disadvantaged students. The framework, called the RAMR cycle, is capable of being used by mathematics teachers for planning and delivering lessons and units of work with minimal training and external support, as demonstrated by three case studies. These, and other cases, suggest that the YuMi Deadly Maths approach is an effective model for scaling up professional development programs where school participation is voluntary and costs have to be minimised.

This paper describes the genesis of YuMi Deadly Maths, a school change process that has been used in over 200 schools to develop mathematics teaching and learning to improve students’ employment and life chances. The paper discusses the YuMi Deadly Maths approach to mathematics content and pedagogy, implemented through a process of PD and school change, and looks at the strengths and weaknesses of the process and the challenges it faces.

There is growing consensus that the process of planning mathematics lessons is as complex as teaching them, yet there is limited research on this. This paper reports on one aspect of a project examining issues in primary teachers’ mathematics planning. The results, taken from a questionnaire completed by 62 primary teachers, indicate that when planning their lessons, teachers give priority to a diverse range of aspects related to their mathematical knowledge for teaching, yet there are similarities in the challenges which they experience. Findings also suggest that team planning can support teachers overcome such challenges. Issues requiring further attention are discussed.

This study explored the potential of a rich assessment task to reveal students’ multiplicative thinking in respect to a hypothetical learning trajectory. Thirty pairs of students in grades 5 and 6 attempted the task. Twenty-two pairs applied multiplicative structure to find the number of items in arrays. However counting and computational errors resulted in a success rate of less than 50%. The rich task provided valuable data about students’ strategic choices and their need to develop computational fluency.

The aim of the larger study, of which this paper is a part, is to investigate the decline in Year 10 male students’ participation in senior calculus mathematics courses at an independent boys’ school located in metropolitan Queensland. This paper draws on Sealey and Noyes’s (2010) relevance framework to conduct document analysis and interviews with leaders and teachers concerning the relevance of mathematics. In both cases results indicated that there was a limited understanding of relevance of mathematics.

Lesson Study (LS) is considered a powerful tool for effecting teacher growth through understanding student thinking and has spread to many countries over the past decade. However, these LS implementations seem to have varied results. Some efforts have been successful and still on-going, while others were somewhat successful but was not sustained, ending up as wasted efforts. The success of introducing an innovation is dependent on the extent the stakeholders are convinced by the innovation. However, getting people to favour change is largely influenced by the extent to which the innovation is aligned with what they value in their practice. This study aims to find out what values are embedded within the construct of Lesson Study and to what extent Japanese mathematics teachers endorse these values. It is also hoped that this study of Japanese mathematics teachers can alert us to potential areas of tension that may arise when LS is implemented in a different national context.

One potential means to develop students’ contextual and conceptual understanding of mathematics is through Inquiry Learning. However, introducing a problem context can distract from mathematical content. Incorporating argumentation practices into Inquiry may address this through providing a stronger reliance on mathematical evidence and reasoning. This paper presents a framework derived from the implementation of multiple, successive Argument-Based Inquiry units to 8-10 year olds. Three key knowledge domains are identified: mathematical, contextual and argumentation knowledge. Key components and roles of each domain are addressed and offered as an initial framework for further research.

Research that has explored students’ interpretations of graphical representations has not extended to include how students apply understanding of particular statistical concepts related to one graphical representation to interpret different representations. This paper reports on the way in which students’ understanding of covariation, evidenced through their interpretation of scatterplots, was applied to the interpretation of split stacked dot plots. The outcomes of the study suggest that incomplete understanding of the characteristics of a graph and the data displayed can lead to students applying knowledge of statistical concepts relevant to one graph type to misinterpret a different graph type.

In 2015, a new MTeach coursework unit, Numeracy for Learners and Teachers, was introduced at Monash University. The impetuses for this unit were the Australian Institute for Teaching and School Leadership numeracy standards for graduate teachers, and the inclusion of numeracy as a general capability in the Australian Curriculum. In this paper, we describe the content and organisation of the unit, and its delivery modes. An evaluation was conducted with students using pre- and post-unit questionnaires and interviews. A major finding was that students’ confidence to incorporate numeracy into their teaching across the curriculum increased after studying the unit.

Australian teachers, recruited via Facebook, completed an online survey about aspects of numeracy. The survey was designed to explore views on numeracy and capacity to respond to numeracy tasks. In this paper, we focus primarily on responses to two numeracy tasks – one numerical, the other requiring critical evaluation. On the first item, 40% answered correctly; on the second, 60% performed at a level expected of people aged 17 or older. The provocative findings warrant further research with a larger sample.

The aim of this paper is to report on barriers to ICT integration in teaching practices from the perspective of early childhood teachers. Six early childhood teachers from a combined private school in Queensland participated in this study. Individual interviews explored the ICT tools used in early childhood programs and the barriers to integration in teaching programs. Results indicate that a range of extrinsic barriers which included lack of certain digital tools, time, access (technical and tools) and professional development opportunities were perceived as the major barriers. Intrinsic barriers were also identified but less frequently. Strongly held philosophies about early childhood teaching practices influenced some teachers’ decisions about incorporating ICT into their programs.

The purpose of this paper is to describe the processes utilised to develop an online learning module within the Opening Real Science (ORS) project – Modelling the present: Predicting the future. The module was realised through an interdisciplinary collaboration, among mathematicians, scientists and mathematics and science educators that drew on the enquirybased approach underpinning ORS as well as structuring devices and working practices that emerged during the course of the module development. The paper is a precursor to further research that will investigate the effectiveness of the module in terms of students’ learning and attitudes as well as the module development team members’ perspectives on the interdisciplinary collaboration that took place.

In this paper, preliminary research into an effective approach to numeracy task design is described and analysed using a conceptual framework based on two dimensions – one related to the nature of numeracy and the other concerned with the characteristics of effective mathematical tasks. A case that documents one teacher’s attempt to create a numeracy task for her class of preparatory students is used to illustrate the relevance of the components of the framework to task design from the perspectives of participant teachers. Teachers’ comments provided insight into aspects of the design process that require greater emphasis and also helped identify complexities that require resolution before a coherent framework for effective numeracy task design can be realised.

Learning the sequence of number words in English up to 30 is not a simple process. In NSW government schools taking part in Early Action for Success, over 800 students in each of the first 3 years of school were assessed every 5 weeks over the school year to determine the highest correct oral count they could produce. Rather than displaying a steady increase in the accurate sequence of the number words produced, the Kindergarten data reported here identified clear, substantial hurdles in the acquisition of the counting sequence, as well as a number of acquired potholes.

Reasoning is an important aspect in the understanding and learning of mathematics. This paper reports on a case study presenting one Australian primary teacher’s reflections regarding the role played by a professional learning program in her developing understanding of mathematical reasoning. Examination of the transcripts of two interviews identified changes in her perceptions of mathematical reasoning by mapping interview responses against the Mathematical Reasoning Framework (Herbert et al., 2015). This change indicates that a well planned program of professional learning based on a demsonstration is efficacious in developing teachers’ understanding of mathematical reasoning.

This research paper examines the perspective of the Heads of Learning Area: Mathematics (HOLAMs) within all Western Australian secondary schools as to why they felt capable students were not enrolling in the two higher-level mathematics courses of study. All HOLAMs were invited to participate in a single, anonymous online survey comprising predominantly qualitative items. Key findings indicate perceptions of student awareness that two mathematics courses are not needed for university entrance, there are other viable and less rigorous courses of study available, and students can maximise their Australian Tertiary Admissions Ranking (ATAR) score without completing these mathematics courses.

This study investigated a group of six junior primary school teachers’ learning as they collaboratively inquired into teaching practice they observed together. The focus of the study was on understanding how teachers collaborated around observed teaching practice to improve their pedagogy. The design involved four iterative stages of co-planning, observation, analysis and reflection. Results indicated a shift in participation of group members from seeing themselves as passive observers to active designers proposing improvements in teaching practice to their colleagues. An implication is that collaborative observation and reflection on teaching situated within the enactment of challenging tasks can be effective in supporting teachers to make sense of teaching in new ways.

Despite the importance placed on how children come to solve single-digit addition problems, many children count on to solve these problems when they are expected to use accurate retrieval-based strategies. In this study, we assessed if a subitising intervention improved the rate at which problem-solving practice promoted retrieval, using a multiple baseline across participants design. For two of three participants, problem solving practice was initially ineffective for promoting retrieval before the intervention but after the intervention, retrieval increased significantly as a function of practice. We also examined possible reasons for why this occurred.

A considerable number of children rely on counting to solve single-digit addition problems when they are expected to use accurate retrieval-based strategies. There are different reasons why this may be so. Children may use inefficient counting strategies, produce errors when applying backup strategies or lack sufficient confidence to just state the answer. In this study, children in Years 2-6 (n=94) were assessed on how the solved singledigit problems. Data were analysed to identify five performance groups that represented different patterns of difficulty. The findings highlight how interventions need to be better tailored to suit individual learning needs and indicate how this may be achieved.

This study explores the implications for mathematical knowledge and pedagogical practices of primary teachers in Papua New Guinea making a shift from a transmission approach to a connectionist approach to teaching mathematics. The research participants were engaged in a professional development program designed to support the teaching and learning of mathematics using a connectionist approach. The data suggest that teacher capacity to adopt a connectionist approach was influenced by their subject knowledge and that their subject knowledge influenced classroom pedagogy.

Multiplicative thinking is a critical stage in mathematical learning and underpins much of the mathematics learned beyond middle primary years. Its components are complex and an inability to understand them conceptually is likely to undermine students’ capacity to develop beyond additive thinking. Of particular importance are the ten times relationship between places in the number system and what happens when numbers are multiplied or divided by powers of ten. Evidence from the research project discussed here suggests that many students have a procedural view of these ideas, and that a conceptual understanding needs to be developed. It is suggested that this may be possible through the use of a device called ‘The Marvellous Multiplier’.

Multiplicative thinking is a ‘big idea’ of mathematics that underpins much of the mathematics learned beyond the early primary school years. This paper reports on a current study that utilises an interview tool and a written quiz to gather data about children’s multiplicative thinking. The development of the tools and some of the research findings are described here. Findings suggest that middle and upper primary aged children often have a procedural level of understanding of aspects of multiplicative thinking and that various aspects of multiplicative thinking are partially known, and known in different ways by different children.

Proving that a given set is indeed a subgroup, one needs to show that it is non-empty, and closed under operation and inverses. This study focuses on the first condition, analysing students’ responses to this task. Results suggest that there are three distinct problematic responses: the total absence of proving this condition, the problematic understanding of subgroup’s definition, and the inaccurate application of the relevant metarules. For the purposes of this study there has been used the Commognitive Theoretical Framework.

Undergraduate mathematics students consider Group Theory as a challenging topic. This study aims to investigate the interrelation of cognitive, affective and pedagogical issues of students’ first encounter with this module. The results suggest that there is interdependence between cognitive difficulties, affective reactions, involving a wide spectrum of emotions, and pedagogical activities, particularly the teaching and learning processes, in relation to coping strategies. This “triangular” interdependence of the three aspects is described as the “trilateral interlock of learning”.

Most studies which investigate mathematics teacher noticing cast perception into a passive role. This study develops an ecological analysis of mathematics teachers’ noticing in order to investigate how teachers actively look for information in classroom environments. This method of analysis is applied to data collected as an experienced primary teacher of mathematics moved between desks and selected students to stop and talk to. An ecological lens draws attention to an active process of looking, employed by the teacher, which locates particular properties in the mathematical representations created by students.

The aim of this paper is propositional and is based on research findings which suggest that success in mathematics teaching and reform is contingent upon having key personnel in schools to lead curriculum reform. Based on the outcomes of a large national study on successful practice in the teaching of numeracy for some of Australia’s most disadvantaged learners, it was found that, among other practices, the appointment of a numeracy leader alongside the use of effective and appropriate digital tools (in particular apps) supported teachers to implement strong and effective numeracy practices. This paper presents a rationale for a renewed focus on practices that will enable success for all Australian students, but most particularly those who are most at risk of mathematical mortality.

The Teaching for Metacognition project, a hybrid model of PD, integrates the “training model of PD” with sustained support for teachers to integrate knowledge gained from the PD into their classroom practice. This paper is based on the work of the teachers in the project in twotier communities of practice. It examines the perceptions of the project teachers about working collaboratively and reflecting on their practice when integrating the knowledge they acquired during the knowledge-building workshops of the project. From the reflective journals of the teachers it is evident that working collaboratively provided them with the encouragement and support that is needed to begin to experiment with new approaches to teaching and reflecting on their teaching in school-based groups provided them with much needed insights as to how they may improve on what they did.

As teaching is a cultural activity embedded in a unique context, professional development opportunities for teachers may be best placed amid the relevance of everyday classroom practice, and supported by an innovative curriculum. This progress paper reports on data collected as part of doctoral research studying the changes in mathematical knowledge and beliefs of three year 5/6 teachers as they implemented a four-week, innovative curriculum unit. Early analysis of the case of Year 5 teacher Mark pointed to reflection on the pivotal influence of teachers’ stated and inferred beliefs about mathematics and mathematics teaching on classroom practice, and the vagaries of didactic contracts in a change environment.

This paper reports on a mathematics intervention initiative, Prepare 2 Learn, designed taking into account research literature and elements of other successful programs. The program was intended to prepare students for their upcoming mainstream mathematics lessons as well as make them aware of the impact they can have on their own learning through their actions and attitudes. The intervention resulted in the students reaching the expected standard or beyond for their year level as well as positively changing the way they saw themselves as mathematics learners. The following paper focuses on sustainable changes to one students’ mathematics learning and disposition.

Although the psychological literature has demonstrated that spatial reasoning and mathematics performance are correlated, there is scant research on these relationships in the middle years. The current study examined the commonalities and differences in students’ performance on instruments that measured three spatial reasoning constructs and two mathematics content areas. There were no gender differences in terms of performance on the three constructs that measured students’ spatial visualisation, mental rotation and spatial orientation. There were strong positive relationships between the students’ spatial reasoning and mathematics performance (r=0.66), with over 44% of shared variance between the two dimensions. Our study highlights the importance of spatial reasoning in the mathematics curriculum and the necessary promotion of this dimension as a general numeracy capability.

International research suggests that early mathematical competences predicts later mathematical outcomes. In this paper, we build on our previous study of young children’s mathematical competencies (MacDonald & Carmichael, 2015) to explore the relationship between mathematical competencies at 4-5 years, as measured by teacher ratings, and later results on Years 3, 5, 7 and 9 NAPLAN numeracy tests. Data from a nationally-representative sample of 2343 children participating in the Longitudinal Study of Australian Children (LSAC) are examined. In line with overseas studies, we report moderate correlations between pre-entry mathematics and later NAPLAN results. However, analysis of individual growth trajectories suggests that in fact early mathematics predicts the initial (Year 3) level, but not subsequent growth. This suggests that early mathematical competences are important for enhancing outcomes in early schooling, but that the quality of mathematics education provided in the schooling years is critical for future development.

Pedagogical content knowledge is widely considered an essential and complex facet of mathematics teacher knowledge, but little research has focused on PCK at the senior secondary level. This study explores some of the complexities of PCK in a teacher’s lesson for senior secondary students by analysing data from lesson observation, the teacher’s own commentary on the lesson, and students’ perceptions of the teacher’s knowledge and actions. Findings suggest that classroom norms relating to the kinds of problems students are typically required to solve can affect priorities about what is taught, so that what is attended to and valued by students may impact on teachers’ PCK demands.

An empirical study was conducted with the aim to develop teachers’ confidence and proficiency with teaching mathematics through inquiry. The study followed 41 primary teachers and compared a regular mathematics lesson to a lesson taught using an inquiry approach; 19 of these teachers were also followed over three years. Lessons were coded on the extent of intellectual quality in the lesson across six dimensions. Higher order thinking showed the most gains over time. Implications for research and practice are given.

Much is known about preschool children’s mathematics learning and the role of play in that learning. Many early childhood educators are quite adept at observing and documenting the mathematics learning of the children in their settings. These teachers ‘notice’ the children’s mathematics but they are not the only ones to ‘notice’. In this paper, children’s noticing of their own and others’ mathematics is investigated through a pilot study of a small number of preschool and first-year-of-school children. Content analysis of observations and children’s discussions with researchers provides the basis for the early development of an analytical framework for children’s ‘noticing’ mathematics.

The mathematical proficiencies in the Australian Curriculum: Mathematics of understanding, problem solving, reasoning, and fluency are intended to be entwined actions that work together to build generalised understandings of mathematical concepts. A content analysis identifying the incidence of key proficiency terms (KPTs) embedded in the content descriptions from Foundation to Year 9 revealed a much lower representation of “actions” relating to the proficiency reasoning than to the other three proficiencies. A generalised model of patterning is proposed to provide an interrelated view of the proficiencies and to further support the development of generalised understandings in mathematics education.

There is growing evidence that students learn best when they are presented with academically challenging tasks that focus on problem solving and reasoning (Kilpatrick, Swafford, & Findell, 2001). Guided by this evidence, the Australian Curriculum: Mathematics (Australian Curriculum, Assessment and Reporting Authority, 2010) provides a new framework for teaching mathematics in an attempt to enhance student learning outcomes and address concerns regarding the “syndrome of shallow teaching” (Stacey, 2003, p. 119). Not only does the curriculum provide a new definition for problem solving as one of four proficiency strands (Understanding, Problem Solving, Reasoning, and Fluency) but it also provides a clear rationale for the teaching of problem solving. However despite the amount of policy advice and resources to support problem-solving practices, there are growing concerns that many Australian students are given limited opportunities to engage in problems other than those of low procedural complexity. Stacey (2003) contends that students are often asked to follow procedures without reason, solve low complexity problems with excessive repetition and given limited opportunities for reasoning or classroom discourse. The following report investigates this phenomenon through research undertaken as part of the Encouraging Persistence, Maintaining Challenge (EPMC) project. This study aims to explore how primary teachers describe their efforts to engage students in complex problem solving through their choice of task. In particular the features of problem solving tasks and the likely level of cognitive demand are examined.

Many primary pre-service teachers (PSTs) who are enthused by tertiary courses that espouse and model a socio-constructivist approach to teaching mathematics, revert to a traditional approach when they encounter mathematics teaching during professional experience. An intervention was designed to translate the initial pedagogical intent of four mathematically competent primary PSTs into classroom practice. Soon after completion of their first unit in mathematics teaching, they took part in a professional experience learning community focussed on teaching mathematical problem solving. We report their reflections and the impact of the program on their future professional experiences of mathematics teaching. Results suggest that the program could serve as a model for the provision of professional experience to primary teacher education students specialising in mathematics.

This paper presents a hypothesised learning trajectory for a Year 3 Indigenous student en route to generalising growing patterns. The trajectory emerged from data collected across a teaching experiment (students n=18; including a pre-test and three 45-minute mathematics lessons) and clinical interviews (n=3). A case study of one student is presented as a representative of high achieving students’ progression and shifts in learning. Results suggest that students are capable of functional thinking, which contradicts the notion of young students can only engage with recursive pattern sequences. In addition, particular teaching actions assisted in promoting shifts in developing students’ capability to generalise growing patterns.

This paper presents findings from classroom observations of one teacher (Beth). It focusses on the development of conceptual understanding of fractions with her students, articulated in Kieren’s sub-constructs (Kieren, 1980,1988), and Hansen’s progressions (Hansen, 2005). The study covers three lessons within a six week unit. Findings from this study suggest there is a need for teachers to understand and teach a wider range of fraction constructs than appears to occur currently; how assessment data available might be used more effectively to plan quality lessons; and highlights the importance of a varied range of models to connect conceptual and procedural understanding of fractions.

Teachers of senior secondary mathematics are required to contend with a number of challenges including covering the prescribed curriculum, differentiating the content for a range of learners, and preparing students for externally imposed assessment tasks. The flipped classroom is gaining in popularity as an approach that can be used to address these challenges. This paper provides a framework that can be used to interpret the affordances of a flipped classroom within the context of teaching secondary mathematics and the motivational factors that influence the uptake of the approach. Data analysed through the framework showed that students believed that the approach enabled them to have autonomy over their learning and achieve their goals. The study has implications for senior secondary teachers and students, particularly in terms of meeting the challenges of curriculum coverage and preparation for externally imposed assessment tasks.

This paper reports on the use of the RAMR framework within a curriculum project. Description of the RAMR framework’s theoretical bases is followed by two descriptions of students’ learning in the classroom. Implications include the need for the teacher to connect student activities in a structured sequence, although this may be predicated on the teacher’s own structural understanding of mathematics.

Language is frequently discussed as barrier to mathematics word problems. Hence this paper presents the initial findings of a linguistic analysis of numeracy skills test sample items. The theoretical perspective of multi-modal text analysis underpinned this study, in which data was extracted from the ten sample numeracy test items released by the Australian Council for Educational Research (ACER) in 2015. The initial data presented here maps the core content of these items to the Australian Curriculum and identifies the lexical density of each sample item. The findings indicate that the sample test items typically correlate to Year 6 of the national curriculum but each have high lexical density.

This case-study explores the impact of a 12 week in-class intervention designed to encourage creativity, inquiry and exploration as a normal and expected part of mathematics lessons, with a particular focus on supporting the learning of highly capable and gifted students. Fred is a mathematically highly capable grade five student whose personal learning focus in mathematics changed from simply ‘getting the answers right’ and striving for ‘A pluses’ to being more willing to think beyond the set mathematics task to include imagination and creativity.

Researchers have argued that there are strong links between primary school students' competence with fraction concepts and operations and their algebraic readiness. This study involving 162 Years 5/6 students in three primary schools examined the strength of that relationship using a test based on familiar fraction tasks and a test of algebraic thinking utilising number relations and equivalence. A strong relationship was found between the two, and with some fraction tasks embodying high potential to anticipate algebraic thinking.

This paper presents initial data from a study being undertaken into the potential effects of a mentoring program for teacher education students who have self-identified as suffering from mathematics anxiety. The first phase of the study saw 8 primary teacher education students opt into a program matching them in pairs with 4 mentor teachers who had been selected by their principals after meeting 6 criteria that identified them as highly capable mentors in mathematics education. The mentees worked with the mentors in classrooms for 8 weeks. After the program the students were interviewed. Data from these students is explored here with particular emphasis on an artefact presented by one of the students about her journey through the process.

A history-infused lesson package developed by a team of teachers in a professional learning community was used to teach introductory calculus in a secondary school. First, we report a quasi-experimental design that showed that students in the experimental group performed significantly better than students in the control group. Second, we report on the qualitative data collected from a larger group of students who were generally positive about the use of the history-infused lessons.

A crucial step towards improving the conceptual use of digital technology (DT) in the mathematics classroom is to increase teacher involvement in the development of tasks. Hence, this research considers some teacher factors that might influence DT algebra task development and implementation in secondary schools. We observed and assisted one group of three teachers as they designed and implemented DT tasks. Our preliminary analysis examines the richness of the two tasks produced by one group and seeks to explain the difference between them. The results suggest the intervention provided with respect to task design led to improved Pedagogical Technology Knowledge for the teachers, and hence a richer task. The delivery of the intervention could be of assistance in focusing professional development programs so they may better facilitate the training of teachers in the use of digital technology in teaching mathematics.

This paper examines the factors that hinder students’ success in working with and understanding the mathematics of quadratic equations using a case study analysis of student error patterns. Twenty-five Year 11 students were administered a written test to examine their understanding of concepts and procedures associated with this topic. The findings indicate that students’ success was inhibited by errors that arose from a lack of procedural understanding regarding fractions, algebraic processes, and conceptual understanding regarding algebraic conventions. Without this prerequisite knowledge, working with and understanding the nature of quadratics was hindered.

A central premise of this project is that teachers learn from the act of teaching a lesson and that this learning is evident in the planning and teaching of a subsequent lesson. We are studying the knowledge construction of mathematics teachers utilising multi-camera research techniques during lesson planning, classroom interactions and reflection. This paper reports on the learning of two Year 7 teachers, one in Melbourne and one in Chicago, teaching the same initial lesson focusing on division, remainders and context. Both teachers claimed to have learned about their students’ mathematical thinking after teaching the initial lesson, but found planning a second lesson to accommodate this learning challenging.

This paper introduces a revised model for the development of basic computation skills. The model draws on four key phases, which have proven to be important for the development of calculation strategies and stresses the use of gestures and the verbalisation of concrete and mental images. This seems to be of crucial importance for children with special needs as the case of Mia illustrates. Context is a university based intervention program that seeks to support children who struggle with the learning of basic arithmetic concepts and skills.

As part of ongoing design-based research exploring financial literacy teaching and learning, ten tasks termed “financial dilemmas” were trialled by 14 teachers and more than 300 Year 5 and 6 students in 4 government primary schools in urban Darwin. Drawing on data related to two tasks - Catching the bus and Buying bread - this article explores insights into problem context and task design principles. The findings highlight that unfamiliar, novel, and imaginable problem contexts, while pedagogically demanding for teachers, are valued by students and have the potential to broaden their horizons.

This study surveyed and analysed four secondary school students’ writing about a square. Sfard’s discursive approach to understanding mathematical discourse was used to analyse the responses collected from 214 Australian secondary school students. The results showed that geometric knowledge was developed experientially and not developmentally. This in turn helps refining the development of a geometric learning progression, with the accompaniment of a set of validated assessment tools and learning tasks that seeks to deepen teachers’ understanding of geometric reasoning and support student learning.

Time is crucial in our society. However, it would appear that there is limited research on the learning and teaching of time. Curricula appear to place an undue emphasis on the reading of time measuring tools. We argue that key ideas of succession, duration, and measurement should be central to learning about time. Drawing upon available research, this theoretical study developed a framework of core ideas that underpin a full understanding of time, can inform curricula, and drive future research.

Although an understanding of time is crucial in our society, curriculum documents have an undue emphasis on reading time and little emphasis on core underlying ideas. Given this context, a one-to-one assessment interview, based on a new framework, was developed and administered to investigate students’ understanding of core ideas undergirding the notion of time: succession, duration and measurement. This paper reports on the development and implementation of the interview and initial results for Year 3/4 students.

The research reported here uses a pre/post-test model and stimulated recall interviews to assess teachers’ statistical reasoning about comparing distributions, when enrolled in a graduate-level statistics education course. We discuss key aspects of the course design aimed at improving teachers’ learning and teaching of statistics, and the resulting different ways of reasoning about comparing distributions that teachers exhibited before and after the course.

Mathematics anxiety in primary pre-service teachers’ affects their future teaching of mathematics and achievement of students. Data collected via Critical Incident Technique were used to investigate this anxiety as perceived and identified by first year pre-service teachers. This paper proposes the application of the Quality of Life conceptual framework of being, belonging and becoming, as a lens for analysis of these reflections to elucidate the concepts of identity and projective identity. This paper makes a contribution to the frameworks through which primary pre-service teachers’ maths anxiety, and its implications for their identity development, might be understood.

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

Visual and analytic strategies are features of students’ schemes for spatial tasks. The strategies used by six students to anticipate the folding of nets were investigated. Evidence suggested that visual and analytic strategies were strongly connected in competent performance.

This paper outlines a framework to explain the early development of place-value understanding based on an analysis of data from 84 five- to seven-year-old children from diverse cultural and linguistic backgrounds. The children were assessed individually on number knowledge tasks (recalled facts, subitizing, counting, place-value understanding) and strategies for solving word problems (addition, subtraction, multiplication, division). Children were categorised as working at one of four levels, each reflecting an increasing awareness of the structure of place value.

Concern is often raised about the performance of Australia’s ‘best’ mathematics students on international studies, relative to past students and other countries. An important consideration in developing strategies to ensure our most proficient mathematics students experience learning growth is an understanding of the distribution and concentration of high achieving students across schools, with subsequent implications for policy and teacher training. In this paper, student performance on the Numeracy test from the NAPLAN assessment is explored by grouping and comparing students at different levels of performance.

The teaching and learning of Geometry has been identified in much of the literature as being problematic and the mathematics strand where many teachers feel least knowledgeable and least confident to teach. This paper describes a school-based project which sought to develop teacher knowledge and confidence in this strand via the use of Professional Learning Communities (DuFour & Reeves, 2016) and Instructional Coaching.

According to Clements (2003), Dinham (2013) and Sullivan (1992, 2011, 2012) there is an urgent need for change to the way in which mathematics is taught in Australian Schools. The five question approach (FQA) to teaching mathematics, developed during my thirty years of secondary teaching, occurs at the commencement of each mathematics lesson. It is the subject of my doctoral research, currently at the early data analysis stage. The research investigates if the FQA results in an increase in student academic achievement, perceived and / or actual, and engagement.

Arguably, the NAPLAN Numeracy test is regarded as an effective instrument in gauging students’ ability to integrate multiple pieces of information during the course of solving real-life problems (Lawson & Chinnappan, 2016). In this study, students were administered the conceptions of mathematics questionnaire (Crawford, Gordon, Nicholas & Prosser, 1998) alongside the grade 9 non-calculator NAPLAN numeracy test. We report on the results of two correlational analysis (n=61 and n=68) for students taking two different introductory level undergraduate mathematics courses. The results raise questions about the claim that NAPLAN Numeracy tests are effective in assessing knowledge connectedness in mathematical thinking.

Prior research studies on senior secondary students’ use of advanced calculators (graphics and CAS calculators) have found that some students tended to either underutilise calculators by preferring to solve even calculator-required questions by hand, while others over-rely on calculators for computations and methods that can be replicated easily by hand. Calculator use is also known to be influenced by assessment requirements. Thus this study aims to investigate the questions in the senior secondary mathematics written examination papers in Singapore and Victoria for the kinds of expected calculator use. Preliminary findings will be presented and the implications discussed.

Numeracy skills testing is being introduced for pre-service teachers (PSTs) across Australia in the coming years. ACER and the Australian Government (Dept. of Education and Training, 2015) state that these skills tests are intended to demonstrate that PSTs are in the top 30% for numeracy. Given that the South Australian Literacy and Numeracy Strategy (Department for Education and Child Development, 2013, p.8) partially defines numeracy as an “ability to use mathematical information to solve problems” it is logical to assume that categories of mathematical thinking should be evident in the numeracy test items. As part of their initial analysis, the authors are exploring an application of the National Council of Teachers of Mathematics’ (NCTM) five processes of mathematical thinking (Representation, Reasoning and Proof, Communication, Problem Solving, and Connections) as a potential for mathematical characterisation of such numeracy items.

This study aims to investigate the influence of mathematical modelling on students’ real-world problem solving skills. In mathematical modelling, students have to convert real-world problem scenarios into mathematical problems. They then solve the mathematical problems using known mathematical techniques before interpreting and 689 translating the solutions for the scenarios. Using pre and post surveys, scoring of students’ real-world problem solving skills, classroom observations as well as focus group discussions, this study investigated how mathematical modelling influences Grade 9 students’ perception of mathematics in practical situations, as well as how mathematical modelling influences teachers’ practices to provide students with the opportunities to form links between theoretical and real-world situations.

Previous researchers have explored the relationships between a teacher’s beliefs about the mathematics discipline and school subject, their beliefs about the teaching of mathematics and their teaching practice. In Australia, there are many teachers who cross subject boundaries to teach mathematics as an out-of-field teacher. Exploring out-of-field teachers’ beliefs about mathematics and the teaching of mathematics will enable us to understand more about the way in which their in-field beliefs, school context and practices influence their beliefs and practice when teaching mathematics. Some preliminary findings from Out-of-Field Teaching: Sustaining Quality Practices Across Subjects project will be presented.

Use of digital environments has the potential to allow us to break down barriers between learning and assessment by allowing us to gather data about students' proficiency as they interact in the learning environment. This work describes a pilot of a learning system including an online game, online digital activities, and in-person classroom activities all aligned to a common learning progression describing how students learn the concept of area. This work demonstrates how activities can be designed to align to the progression and presents preliminary evidence of both learning and assessment from a tryout with more than 300 students.

This short communication will report some initial findings from a multiple case study that investigated the use of consumer and financial literacy as a tool to improve student engagement with mathematics in low socio-economic schools. Data from one of the participating schools will be discussed and the impact of the study in relation to improving student engagement, building community awareness of consumer and financial literacy and improving teacher capacity will be shared.

Australian Teachers are saying “a rewarding experience”, “a great competition”, “21st Century learning at its best”, “a resounding success…motivates and engages students”. Mentors are saying “I was inspired by their keenness”, “provides students a unique opportunity”. What are they talking about? The successful annual National Schools Statistics Poster Competition! This talk will: outline the national project-based learning activity which facilitates boundary encounters (between secondary, tertiary, and industry sectors and students having varied backgrounds and areas of interest) and develops key communication, research and quantitative skills; and describe coming additions as part of a national initiative to assist students with statistics.

Modelling ideas are now emphasised in the teaching and learning of statistics (e.g., Kawakami, 2015). This presentation reports on a case study examining how Year 9 students (14-15 year olds) created and utilized data models in estimating and predicting Japanese population by 5-year age group. The analysis illustrated that the students perceived the role of data and the necessity of shuttling between data and context in developing data models to understand, estimate and decide on the trend of population. Findings will inform educators on how to strengthen the link between statistical ideas and modelling ideas in the teaching and learning of statistics.

Epstein and Sheldon (2006) posit that students’ learning and development are enhanced when: (a) families-parents/carers, (b) the school and (c) the local community work together to guide and support them. Similarly, Goos (2004) found that to build successful, sustainable long-term collaborative mathematics programs, parental and community involvement is pivotal. When these principles were implemented in a mathematics homework program which enabled parents to become involved and assist with their children’s learning both students and parents developed more positive feelings and attitudes towards mathematics (Van Voorhis, 2011). This presentation explores the introduction of a mathematics club in two primary schools in low SES areas of Sydney and the activities undertaken to engage the school community.

This report discusses the preliminary stages of a study into the impact of reconceptualising curriculum design on the teaching of fractions in Kindergarten to Year 2. A mixed methods approach is engaged using a questionnaire to explore current teacher practices and a diagnostic assessment instrument to determine Year 3 students’ current understanding of fractions. Teachers will have training on using the new curriculum documentation and will be interviewed and observed in the classroom implementing both the old and new approaches.

Students' perceptions of what teachers do and what students themselves do that helps them learn gives an insight into what might be effective teaching and learning strategies in a mathematics classroom. This paper looks at student perceptions of teaching in general but also specifically in relation to the teaching and learning of mathematics. Data was collected from students at two South Australian schools via an online survey on a range of topics two of the topics examined the students perceptions of what teachers did to help them learn and what they as students did that helped them learn. The initial analysis of the results of the survey will be presented and will highlight areas that students think are most important in learning mathematics.

The term 21st Century Teaching & Learning highlights the 4Cs of collaboration, critical thinking, creativity and communication and is often connected with the use of technologies, in particular with the opportunities that being online offers. This presentation reports on a study examining students perceptions of the mathematics teaching and learning at a large suburban high school in South Australia that has made a significant effort to develop 21st Century Teaching and Learning across the school which incorporated a significant focus on the use of technologies. This presentation will report on students impressions of the effectiveness of the pedagogies used in mathematics and more generally on aspects of blended learning which was a focus for the changed pedagogy. The study indicated that over 90% of the students indicated that the students for the new approaches were effective or very effective for learning mathematics. Also, the student text responses indicated that they were using it to access materials out of hours, with senior year’s students commenting how they found it very useful for revision when the mathematics teachers uploaded the Interactive Whiteboard recordings.

Financial literacy is often taught from a skills perspective focusing on budgeting. In this presentation, we discuss teaching financial literacy for social justice in mathematics classrooms. Conventional financial literacy curricula are taught from a deficit position focusing on the individual and their ability to make ‘effective’ financial decisions. We explore the role of identity and socio-economic status (SES) in financial literacy education and find that individuals are ‘blamed’ for their SES, or ‘affirmed’ for their high SES under the conventional FLE curricula. A more compassionate approach to FLE for mathematics classrooms is discussed in this presentation.

reSolve: Mathematics by Inquiry is an Australian Government funded project to develop and disseminate a suite of high quality, innovative mathematics resources for students and teachers from F to Year 10 incorporating contemporary mathematics pedagogy exemplifying an inquiry approach. The guiding principles behind the project are elaborated in the reSolve: Mathematics by Inquiry Protocol, which addresses the elements of mathematical purpose, challenge and access, and development of a supportive knowledge-building culture. The poster will give examples of resources that exemplify the Protocol and offer participants an opportunity to discuss critical questions related to mathematics by inquiry.

Recent research findings indicate that using multimodal learning experiences to teach students about different models can be an effective teaching approach. Using a social semiotic lens within a participationist framework, this paper reports on a professional learning collaboration with a primary school teacher designed to explore the use of different metaphors, models and modalities. This case study was conducted in a teacher’s Year 2 classroom over two terms, with the focus on one specific child’s journey towards understanding the part-part-whole relationship model. Video was the predominant research tool. The initial findings have shown that the teacher was able to use specific gestures and language to support the concrete model being used. This paper explores how the metaphors, models and modalities were intertwined in the classroom discourse.

In the Encyclopedia of Mathematics Education, the term “Inquiry-based Mathematics Education” (IBME) is described as a student-centered paradigm of teaching mathematics and science, in which students are invited to work in ways similar to how mathematicians and scientists work. This means they have to observe phenomena, ask questions, look for mathematical and scientific ways of how to answer these questions … interpret and evaluate their solutions, and communicate and discuss their solutions effectively. (Dorier & Maass, 2014, p. 300) The aim of this Round Table on IBME is to extend conversations within a community interested in teaching, learning, assessment and research on mathematical inquiry. We invite those interested in mathematical inquiry to participate and help to create a richer understanding of work being done in this area. In 2015, a Round Table to initially gauge interest in mathematical inquiry was held. Because of the large number of participants, there was little opportunity for discussion. At this Round Table, the aim is to separate into smaller areas of interest around particular themes. Possible strands around IMBE include: • Early Years • Primary • Secondary • Tertiary Mathematics • Initial Teacher Education • Professional Development • Assessment and Research Measures • Engaging with Stakeholders, Community and Policymakers • Classroom Norms and Argumentation • Theoretical and Methodological Perspectives • Social Justice and Inclusion • Affect • 21st Century Skills • Curriculum and Resources • Areas Needing More Research A possible outcome could be a symposium, publication and/or online community. New and previous participants equally welcome!

Teacher knowledge is a significant issue for mathematics education (Sullivan, 2008a). Educational research has increasingly tried to identify characteristics of teacher knowledge for effective mathematics teaching and their influence on student learning outcomes (Bobis, Higgins, Cavanagh and Roche, 2012). With teacher performance standards now well iterated, the questions of “what [teacher] knowledge matters more, and why” (Bobis et al, 2012, p.1), remain central to improving mathematics education in Australia. This round table will outline the rationale and theoretical underpinnings of a study into the relationship between primary teachers’ subject matter knowledge and pedagogical content knowledge domains (Ball, Thames and Phelps, 2008). The Mathematics and Task Design Project is a research study involving 65 primary teachers within a system of schools. The study explores: (i) teachers’ understandings of area and volume when solving problems; the level of the tasks teachers design for their students; and, (iii) the nature of the feedback that teachers provide when analysing student work samples, and (iv) the relationships between each of these critical aspects of teacher knowledge. Data from the Task Design aspect of the study will be used to provide the stimulus for a discussion of the challenge of identifying and designing frameworks that can be applied to consistently analyse and describe the levels of opportunity and challenge within mathematical tasks.

Data revolution enables citizens to have access to an enormous amount of complex data sets. These emerging data sets include: 1) large-scale open databases, 2) the growing use of big data, and 3) novel tools and ways of visualising data. The aim of this Round Table is to bring together a community of researchers, who focus on the role of increasingly open data access within mathematics education, the access to “big data”, teaching and learning of mathematics/Statistics, data visualisation in open data contexts, new statistical literacies in a new open data and big data era, the increasing need of research tools that affords opportunities to: 1) study classroom practices in sophisticated ways provided the access to "big data" and the open-access nature of information, and 2) ways of visualising data. I invite those interested in big data" and the increasingly open-access information to discuss their work or aspects of big data" and the increasingly open-access data in Mathematics/Statistics Education that are in need of research. A few questions are listed below to provoke conversation. Bring your own! 1. What is the role of open-access information (open data) and big data in Australian Mathematics classrooms? 2. What is the role of open-access information (open data) and big data in Mathematics research? 3. How can open-access information (open data) and big data be used in Mathematics (or Statistics) teaching and learning? 4. How do we need to change the way we think in terms of the nature of data and its availability, the ways in which data is displayed and used, and the skills that are required for its interpretation in order to take into account the full complexity of data? 5. What views must be included in a framework for teaching mathematical/Statistical literacy or data literacy in a new open data and big data era? 6. What the instructional and research dilemmas in data revolution era?

The Mathematics Support Teacher (MST) intervention was designed for students who have been identified as having severe learning difficulties in mathematics. The MSTs provide intensive mathematics teaching support with the aim to accelerate the students’ progress. The students were provided with four to five additional half hour lessons per week over a 15 to 20 week period. Initial involvement in the intervention has resulted in accelerated gains for the majority of these students. This case study involved interviewing past students who had maintained their learning gains after participating in the intervention in either 2011 or 2012. The students’ past teachers were also interviewed. Interviews were conducted using three schools from different regions across New Zealand. This round table forum presents impact stories of students, teachers and family after involvement in a mathematics intervention. It will provide an opportunity for participants to discuss the findings and consider the conditions and implications for students and their mathematics learning.

Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define knowledge needed by teachers to effectively teach mathematics, and a plethora of terms such 681 as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge have been used to describe aspects of such knowledge. Here, I put forward a new model for teacher knowledge that embraces aspects of earlier models and which focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. Its power lies in the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The new model proposed here considers the various cognitive and affective components and domains that teachers require to enact ‘powerful teaching’. Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. It follows that a deeper level of knowledge might enable a teacher to respond better and indeed to plan and anticipate contingent moments. Taking this further and considering teacher knowledge as ‘dynamic’, I suggest that instead of responding to contingent events, powerful teaching is about provoking contingent events. In order to place genuine problem solving at the heart of learning, the idea is to actually plan for contingent events, to provoke them, and ‘set them up’. The proposed model attempts to represent that process.

Governments and school systems are investing resources into science, technology, engineering and mathematics (STEM) education to address perceived shortages in these fields in university enrolments as well as in the workforce. Many initiatives have been implemented to capitalise on the increased focus on the STEM subjects in schools with universities frequently leading the way by offering engaging and worthwhile programs for students and teachers to promote the STEM subjects. However, many of these programs are ‘one off’ or ‘once a year’ events for small numbers of students in participating schools, potentially having little ongoing impact on the larger cohort of students in each school. A different initiative provides ongoing professional learning for teams of teachers of all STEM subjects from a small number of schools and focuses not just on improving pedagogy in each of the subject areas but encourages teachers to work in teams to design integrated approaches to curriculum. Our experiences with this professional development program has raised many questions about secondary teachers’ engagement and participation in working together in multidisciplinary teams to design integrated units of work. This roundtable will seek to explore the challenges in bringing such teams together, the challenges in implementing integrated approaches in secondary school contexts, and the opportunities afforded by integrating curriculum to enhance connections for students.

Some mathematics learners need additional support to enable them to achieve at their expected level in relation to the New Zealand curriculum and the mathematics standards. Effective mathematics teachers can accelerate the learning of small groups of students in a relatively short period of time when they provide targeted support within their daily classroom programmes. This roundtable will begin by outlining the Accelerated Learning in Mathematics intervention model where teachers are given an opportunity to design an intervention response to accelerate the learning of students struggling in mathematics. We will discuss the early findings of a research project that examined the conditions in twelve schools that enabled a successful and sustainable intervention. Effective collaboration across a small group of teachers was identified as a common feature that promoted changed teacher practice and accelerated learning for targeted students. We will consider the different models of collaboration that lead to a successful intervention for these schools. Results showed that the collaborative nature developed in these schools led to an increased derivatisation of practice, a shared responsibility for accelerated outcomes, a greater willingness to take risks, and teachers became more reflective in their practice which led to deeper conversations around teaching and learning.

A recent national study aimed to provide an evidence base for best practice in mathematics education in Australia. Schools were included in the study on the basis of growth in NAPLAN 680 numeracy from 2011 to 2013, and 2012 to 2014, in Years 3 to 5 and Years 7 to 9. Using Bronfenbrenner’s (1989) Ecological Systems Theory as a structural framework, the study aimed to identify goal orientations and costs associated with achieving these goals at system, school, classroom, and individual level. Mastery goal orientations were defined as having a focus on connections within mathematics, relevance, and cooperation, whereas performance goal orientations were defined as having a focus on skill development, achievement, and competition. Findings suggest that successful schools, regardless of their social context, were consistent in their focus on mathematics across the school, in classrooms and among individuals. Mastery approaches were associated with reduced costs in terms of students’ perceptions of effort, social costs and mathematics difficulty. Teacher enthusiasm for mathematics teaching emerged as a key factor. Issues that arose from the study will provide starting point for discussion.