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This paper discusses the way "right conduct" in education - praxis - is embedded in social practices - practice architectures - which enable and constrain conduct in three dimensions: cultural-discursive, material-economic, and social-political, or "sayings", "doings", and "relatings". It is argued that changing professional practices like mathematics education is not just a matter of changing the understandings (cf. sayings), skills, and capabilities (cf. doings) or values and norms (cf. relatings) of practitioners, but also changing the practice architectures that enable and constrain practitioners' actions and interactions. The practice architectures of mathematics education are constructed not only by the knowledge, capabilities, and values internal to traditions in mathematics education but also by meta-practices external to those traditions - particularly the meta-practices of educational administration and policy making, initial and continuing teacher education, and educational research and evaluation. Today, the elaborateness, the rigidity and the compulsions associated with these meta-practices threaten the vitality of education in general, including mathematics education. Educators are thus confronted by an invidious choice. Should they conduct their practice as praxis, oriented by tradition and by considerations of the good for each person and the good for humankind as these are expressed in the conduct of education as a practice, as agents of education? Or should they conduct themselves as the operatives of the education systems in which they find themselves, following the rules and procedures that constitute the functional rationality of those systems?

In the early 1970s, gender differences in mathematics learning outcomes favouring males were identified. Research efforts revealed that learner-related cognitive and affective variables, as well as school-related and societal factors were implicated. Policy changes and funded intervention programs followed and had mixed effects. Both government and research attention have since turned elsewhere. In this paper, I present recent findings on gendered patterns in mathematics achievement and participation rates, and on the effects of technology on mathematics learning outcomes. The data indicate that any narrowing of the gender gap in the past decade now appears to be reversing. While there is a growing tendency to focus on smaller scale, qualitative studies, I argue that there is also the need to continue examining large scale data sources to monitor trends over time. I use three navigational metaphors to challenge thinking on the direction of future Australasian research on gender issues in mathematics education.

In the current shifts in mathematics classrooms teachers are challenged to use effective pedagogy to develop inquiry communities in which all participants are offered opportunities to engage in the reasoning discourse of proficient mathematical practices. The challenge for teachers is to know what pedagogical actions support the development and use of effective mathematical practices. This paper examines how a group of teachers used a purposely designed communication and participation framework as a tool to scaffold development of inquiry communities and the use of progressively more proficient mathematical practices within their classrooms.

This investigation examined the performance of students (9–12 year olds) over a three-year period as they solved graphics-based mathematics items which are commonly found in standardised tests. There were statistically significant improvements in the performance of students (across each of the six language types) in each year of the study. Mean scores for boys were higher than girls on most measures with statistically significant differences in Axis (number line), Map and Retinal-list categories.

National numeracy tests herald a new era in Australian school assessment. The sets of sample test items suggest that understanding information graphics (e.g., maps) will be an important component of these tests. However, an analysis of test items reveals limited types of graphics in sample sets, poor quality graphics, atypical use of graphics, and a lack of consistency in the graphics depicting a common shape. These findings indicate the importance of repeating the analysis with the actual tests.

This paper describes the sense making of 11- 12- year-olds as they interpret two mathematics items which include graphics. In particular, it outlines the changed behaviour (and performance) of students when solving items when slight modifications were made to the graphic or the mathematical language. The results show that performance increased when the graphic was modified but diminished when the language was modified. Implications include the need for test designers to carefully consider the graphic embedded within assessment items.

The construction of knowledge by students continues to be a central concern of mathematics education, and research has shown the importance of theoretical approaches for our understanding of construction of knowledge. Tommy Dreyfus has not only made substantial contributions to this field but has forged links between researchers in this area, and encouraged new researchers. This symposium provides an introduction to Tommy’s research for some MERGA members, a chance to reflect further on it for others, and opportunities for everyone to develop new insights by connecting ideas in these papers.

There is scant research on the role of graphics in students’ mathematical performance. This paper distinguishes between the contextual and informational roles of graphics and provides an overview of the types of information graphics. It also presents findings from a new mathematics instrument that has been used to quantitatively and qualitatively assess students’ performance on information graphics. Key findings using this instrument have provided insights into age, gender and item effects on performance, and difficulties that students experience interpreting graphics.

The diagnostic test results from a second cohort of preservice teachers in a longitudinal project are presented. Data analysis using the Rasch Model showed consecutive intakes’ performances improved. However, diploma-education students consistently performed lower than foundation-education students. Item analyses indicate problem solving, fractions, and interpreting complex diagrams are most difficult. Findings suggest the university needs to address the mathematical needs of at-risk preservice teachers and to develop policies to regulate entry requirements and enhance course content and delivery.

In November 2007 around 14 000 Mathematical Methods students and around 1 500 Mathematical Methods (CAS) students sat a common one hour technology-free examination. The examination covered the same function, algebra, calculus and probability content with corresponding expectations for key knowledge and key skills for both studies. This paper provides some analysis of the examination and items and student performance with respect to both cohorts.

Teachers choose to attend professional development courses and workshops for a range of reasons. New mathematics syllabuses in NSW with increased expectations for student learning outcomes created opportunities for providers to support the implementation of the new syllabuses. The participants at four six-week professional development courses completed surveys indicating motivation for attendance. This paper considers the reasons 109 teachers chose to attend these events and the type of knowledge they value as reported in evaluations of the courses. Responses were categorised using Shulman’s knowledge categories with pedagogical content knowledge, curriculum knowledge and knowledge of learners as most valued.

This paper focuses on the disempowering nature of Years 3, 5, and 7 Aspects of Numeracy testing undertaken by the Queensland Studies Authority, in relation to teachers of Indigenous students. It describes a Deadly Maths project with four far North Queensland regional schools to take back control of outcomes by empowering teachers to undertake their own analyses of the tests and use them for immediate intervention programs. It describes students and teachers’ perceptions of the tests and teachers’ reactions to the project, indicating difficulties in overcoming teacher resistance and reporting success for teachers who did take back some control.

As part of an evaluation of the mathematics education strand of a bachelors degree program a sample of a cohort were interviewed on completion of their mathematics education studies, and another sample from the same cohort were interviewed 5 years after graduation. Interviewees were asked about their perceptions of the value of the course and its impacts on their beliefs about teaching mathematics. Comparison of the responses revealed changes in priorities and evidence of the shifting status of memories. There are implications both for mathematics educators and for understandings of the development and structure of teachers’ knowledge and belief systems.

Traditionally, craft activities have been incorporated into early childhood mathematical learning experiences as an effective means of fostering curiosity and interest, and introducing abstract mathematical concepts through the use of concrete materials. However, in the primary schooling years creative and active mathematical learning experiences often give way to approaches such as computational drills and rote learning which are less than inspiring for students and often do not result in meaningful understanding. This paper shows that craft activities, in particular paper-folding, can also be valuable in the primary years as a means of promoting affective, behavioural and cognitive engagement in the mathematical learning.

A case study of the preparedness of pre-service mathematics teachers to teach proof at lower secondary is reported. Data collected by questionnaire and problem-centred interviews were subjected to in-depth qualitative content analysis. Knowledge and competencies demonstrated by two future teachers are examined. Both displayed high affinity with proving and a formalist image of mathematics but their inferred abilities to convey this affinity to students at this point in their careers differed. Implications for teacher education and university mathematics are outlined.

This discussion paper highlights some of the issues related to mathematics underachievement experienced by Indigenous students. Mathematical modelling can be implemented as a rich way to provide contextualised mathematics learning experiences able to be couched in significant contexts that Indigenous students can relate to and enjoy. This paper proposes that mathematical modelling will enhance Indigenous students’ motivation to engage in Western concepts of mathematics, hence improving cognition and achievements in this Key Learning Area.

This paper explores teachers’ “agency” as they talk about using a Sociocultural approach to teaching and learning (Collective Argumentation) to mediate activity in the mathematics classroom. The paper examines a re-conceptualisation of teacher agency as evidence in a report by one middle school teacher of a classroom mathematics activity. Employing discourse analysis to examine aspects of teacher activity in the report, the paper relates the development of teacher agency to the appropriation of pedagogical practices and to teacher talk about those practices.

This paper presents the results of a research review into the development of middle school students’ interest in statistical literacy. In particular it reviews the concept of interest and its motivational influence on learning. Findings reveal that very little research has examined the influence of positive affect such as interest on learning in the middle-school statistics context. Further, these findings suggest that interest development will be the result of a complex interplay of classroom influences and individual factors such as: students’ knowledge of statistics, their enjoyment of statistics and their perceptions of competency in relation to the learning of statistics.

This paper reports part of a larger study and examines one teacher’s use of problem-solving teaching approaches in Years 7 and 9. Thirteen problem-solving lessons were observed over an 18 month period during which the teacher devised and used 7 different problem-solving tasks. Three tasks are described in detail and analysed in terms of task structure and implementation, and how the teacher managed wholeclass discussions. Analysis highlights changes in the design and use of the tasks. They became more openended and the teacher improved the quality of whole-class discussions to promote student learning and reflection.

This paper investigates one outcome of teacher professional development, student success, which appears to be an effective mechanism to motivate and sustain teacher change for teachers implementing reformoriented teaching approaches. The reform-oriented pedagogy is inquiry-based and aims to encourage student mathematical thinking in order to increase student achievement in numeracy. It is this outcome of increased student achievement and student enthusiasm for the new teaching approaches that motivated teachers in this reported study to sustain changes to their teaching.

This study investigated strategies used by Year five students to solve multi-digit whole number problems mentally and in written formats. The students participated in semi-structured interviews and the think-aloud strategy was used to determine the students’ metacognition. The results indicated that students experience more difficulty answering oral questions than written questions. In particular subtraction and division questions proved more troublesome for students. The range and type of strategies used was indicative of the level of cognitive understanding of the students.

In the context of curriculum renewal, 111 engineering students were surveyed about their perceptions of the relevance of topics in their core mathematics and science subjects to their later engineering studies. The mathematics topics rated most highly were trigonometric functions and linear algebra. Differences by student major and by stage through the degree are described. The implications of low ratings and ways to address these are discussed.

The literature on affective factors in the learning of mathematics is difficult to interpret because of differences and inconsistencies in terminology and measurement. To advance research in this field of affect, I compare and clarify terminology, and reconcile scales for measurement by examining the factors and research instruments targeted by four research teams. The findings reveal two distinct broad primary areas of interest, namely self-concepts about mathematics, and intrinsic motivations for learning mathematics. The instruments used to measure a range of underlying factors within these two areas are analysed and reconciled, terminology is clarified, and further recommendations are made.

There is current interest in how the mathematics content, understanding, and practices of the new entrant classroom connects with the learning the child experienced within early childhood settings. Positive transitions directly impact on children. They are important to the child, to the parent, to the teachers, and to the centre/school. This paper reports on research investigating the existing transition practices between early childhood settings and primary schools with regard to mathematics learning and teaching. We report on interim findings from four early childhood settings.

Proportional reasoning is required to operate in many mathematical domains in the middle years’ curriculum. It is also a major connecting theme across both mathematics and science. Working together, middle years teachers have the potential to promote students’ proportional reasoning through integrated learning experiences. However, building awareness of the connections between these two curriculum domains is an important first step. This paper reports on one aspect of a large project exploring the connections between mathematics and science curriculum. In this paper, first steps to eliciting teachers’ pedagogical content knowledge in relation to proportional reasoning are discussed.

This paper reports on strategies 26 Year 3 students used to solve a range of division word problems in a oneto-one interview, following the participation of half of the group in a teaching experiment. The focus here is on the strategies used by case study students to solve equivalent groups and times as many division tasks. Results suggest that young children are capable of solving complex division problems given experience with a range of semantic structures for multiplication and division.

Nate was one of 200 participants in a research project aimed at developing pedagogical tools for use with low-attaining 3rd- and 4th-graders. This involved an intervention program of approximately thirty 25-minute lessons over 10 weeks. Focusing on the topic of structuring numbers to 20, the paper describes Nate’s pre- and post-assessments, including major gains on tests of computational fluency. Relevant instructional procedures are described in detail and it is concluded that the procedures are viable for use in intervention.

We are facing a serious skills shortage in mathematics, science, and engineering—our efforts to remain globally competitive will be severely hampered if this shortage continues. Numerous recent calls for improving students’ learning in these disciplines and for raising our nation’s levels of innovation and creativity have been made. In response, this discussion paper argues for a future-oriented interdisciplinary approach to mathematical problem solving, one that draws upon engineering. Consideration is given to engineering as a problem-solving domain, the interdisciplinary knowledge and processes that are fostered, and the role of mathematical modelling in solving engineering-based problems. An example of such a problem for the primary/middle school is analysed.

International research demonstrates that between 6% - 8% of students face very significant and persisting mathematical learning difficulties (MD). It is still little understood why some students experience so much difficulty in mastering basic mathematical concepts and skills. This researcher will argue that verbal memory span weaknesses have been reliably associated with MD and present research with Year 2 Queensland students which confirms this finding. The paper explores how verbal memory weaknesses can constrain early maths learning, and reports an intervention designed to circumvent these restrictions with a low-attaining Year 4 student.

This paper reports on the validation of an assessment instrument used to elicit student prior learning in relation to reasoning about data in an ICT environment. A paper-based assessment instrument developed using a theoretical framework about statistical thinking and reasoning in relation to data analysis, graphing, and graph-sense making was completed by year 4, 5, and 6 students. Student responses were analysed using the framework to determine if the assessment instrument elicited responses for all aspects of the framework. The results identified that one aspect of the framework had not been addressed; informing further development of the theoretical framework and changes to the assessment instrument.

Students develop within rich, complex cultural community settings involving teachers and the environmental surrounds. To investigate the multiple perspectives in such a teaching and learning setting a suitable framework incorporating sociocultural practices is needed. The developmental works of Valsiner and associated double stimulation methods of Vygotsky, are proposed here which, it will be argued, assists in the identification and analysis of developmental issues. The application of the theoretical framework presented in this paper is exemplified in the development of numeracy in a 1st year nursing.

This paper considers the potential of Computer Algebra Systems (CAS) to enhance the processes associated with mathematical modelling and application tasks. In doing so, the role of technology in the cyclical development of mathematical models is considered in relation to current literature in this area. The analysis of data drawn from a one year study of three different secondary school classrooms indicates that CAS enabled technologies have a role to play as provocateurs of productive student-student-teacher interaction in both small group and whole class settings.

University-based mathematics educators typically rely on gaining access to teachers and students in schools or teacher education settings to conduct their research. In these circumstances it is more common for teachers to be co-opted into the research agenda than for genuine researcher-teacher collaboration to be realised. This paper proposes a framework for examining research spaces created by such relationships, illustrates its use by comparing three of my own research projects, and discusses implications for researchers working with teachers on mathematics education projects.

Within the mathematics education research community there is growing interest in theories that view teachers’ learning as a form of participation in social and cultural practices. This paper explores what we can learn from research that takes a sociocultural perspective on learning to teach, and how such research might provide a framework for understanding the work of mathematics teacher-educator-researchers. These questions are addressed with particular reference to Valsiner’s zone theory, drawing on studies that take contrasting approaches to its application.

In this discussion article we consider mathematics learning as a process of developing a mathematical identity. This process is constituted by relationships between three key components – the teacher, the students and the discipline of mathematics. It is posited that the teacher’s role is to facilitate the development of students’ mathematical identities by relationally bridging student and the subject. Fundamentally, this requires that mathematics teachers have well-developed personal mathematical identities.

Capturing evidence of the strategies that students use when completing mental computation is difficult to do in everyday classroom practice. Teachers do not have access to or time to analyse verbal protocols as researchers do. This study shows that meaningful written recordings of strategy choice and student thinking while completing computations are possible when a scaffolded framework for teaching the strategies is used and the students are given access to this pedagogical framework in a way they can understand and apply.

The proliferation of technological tools in Australian mathematics classrooms has not been well supported by evidence-based research, particularly in early mathematics learning. This paper reports two stages of document analysis; a review of recent meta-analyses in early mathematics education and technology, and a quantitative analysis of research published in selected mathematics education research journals over the last five years. The initial review highlights potential affordances of technology for mathematics learning predominantly with older students. The quantitative analysis supports this finding but highlights the limited quantity and scope of publications focused on mathematics learning and technology with young children.

This paper examines young students’ development and use of justification strategies when engaged in numeric patterning activities. Drawing on findings from a three-month teaching experiment aimed to improve students’ early algebraic understanding we show how student use of justification strategies can be extended through the use of specifically designed tasks and pedagogical actions. In particular, we examine how students’ ability to participate in mathematical argumentation can be supported through use of justification which triangulates numeric, verbal and visual strategies.

This paper describes part of a larger study of an intervention program involving strategies designed to develop the mathematical thinking of upper primary students. It considers the effectiveness of task-based interviews in identifying the extent to which different modes or levels of thinking were used by eight students in a multiple case study following the implementation of an intervention program. The modes of thinking are based on mathematical, contextual and strategic knowledge.

This paper reports on an investigation into the seating arrangements of a mathematics classroom, and the effect of these arrangements on students’ affect and learning. A seating arrangement is successful depending on whom a student is sitting near. Students need to be surrounded by others whose behaviour does not disrupt or distract them, and who they like and feel comfortable with. The study suggests that adolescent students do not have the power or control to stop other people’s behaviour affecting them, nor do they have the power to sit where they want to ensure their academic identities are being fulfilled. By instituting seating plans, teachers can ensure students’ academic and social needs are met therefore improving student learning through positive discussion and help-seeking.

This paper reports on a Head of a Mathematics Department (the author) in a Brisbane suburban High School and her attempts, over a two year period, to implement a new teaching program involving innovative curriculum resources and a new pedagogy into the first year of high school. The goal of the Head of Department was to support staff to create learning environments that followed a social constructivist philosophy. What was found was that there was a parallel process that occurred for the teachers. The teachers were learners and they needed a community of learners for support within which to reflect, learn and grow.

This paper describes a small-scale investigation of the beliefs teachers hold about student textbooks and the factors that influence their decision to use them in mathematics lessons. Findings suggest that teachers’ decisions to use textbooks in mathematics are influenced by external factors, the teachers’ perceived educational value of textbooks and, the teachers’ personal confidence and competence to teach mathematics. However, there is some evidence to suggest that contrary to previous studies, the 34 Queensland teachers surveyed appear to make less frequent use of textbooks and are more discerning about the manner in which they use textbooks in their classrooms.

The nested epistemic actions model of abstraction in context has been used to analyse a solitary learner’s process of justification. In previous work, we have shown that this process gave rise to the phenomenon of parallel interacting constructing actions. In this paper, we analyse the interaction pattern of combining constructions, and show that combining constructions indicate an enlightenment of the learner. This adds an analytic dimension to the nested epistemic actions model of abstraction in context.

In this paper I argue that humanist understandings of learners can underscore mediocrity in mathematics learning in the early years. Although many children come to school ready and eager to learn mathematics, it can happen that their classroom experiences alienate and disenfranchise them. This sometimes occurs when teachers, deferring to humanist understandings of learners as naturally capable and competent and learning as experiential, teach little mathematics but concentrate on fashioning the learning environment to supposedly make it non-threatening, ‘enjoyable’ and ‘relevant’. In contrast I use the poststructuralist notions of positioning and subjectification to suggest that learners can not be positively positioned in the discourse of mathematics education if they are not given the opportunity to construct robust mathematics and generative and idiosyncratic ways of thinking and reasoning in mathematics.

Numeracy is the confident and critical application of mathematical ideas in a variety of contexts. Numeracy has an ontological dimension, it is a state of being numerate, that has its genesis and growth in mathematics education and life. Preservice teachers are expected to teach their students for numerate participation in a global world, although they themselves oftentimes lack the necessary mathematical foundations and strategic and critical skills. In this paper I describe how an awareness of the poststructuralist processes of subjectification and positioning informed my teaching in numeracy education, and the possible effects.

Success in mathematics is widely regarded as an important gate keeper for many courses and occupations. But does success in mathematics at school influence educational and career paths? Do talented mathematics students have distinctive working habits, are they attracted to a mathematics intensive field or more likely to turn to other areas? These and related issues are explored through information gained from students recognized at secondary school as high achievers in mathematics.

This paper tells of a Year 8 intervention in a school where students are reported to typically have low engagement in learning mathematics. Aiming to help students self-regulate their learning of mathematics, a tool was developed for students to set short-term goals and select, and reflect upon, their developing command of related strategies for organisation and persistence. The results from the intervention were mixed, suggesting the tool has most potential to benefit middle and low achieving students.

This paper provides some insights into teacher professional growth as part of the ten-day Australian Government Summer School for Teachers of Mathematics Programme (the Program). These insights are revealed in three surveys completed during the Program. Preliminary analysis has provided information about participants’ expectations related to classroom practice and the professional growth that took place during the program. This growth represents a shift in focus from the acquisition of new knowledge to thinking more deeply about its applications.

The performance of 29 Grade 6 children on eight number line tasks is descriptively analysed with regard to three types of misconceptions. The first is conceptual; over-generalised part-whole unit-forming. The other two are semiotic misreadings of inscriptions, namely counting lines and decimalising. Success on these number line tasks, possible examples of the measure sub-construct of fractions, is descriptively compared to the children’s performance on tasks assessing conservation and the additivity concept of length measurement, and on multiplication and division items.

This paper reports on the use of “teachable moments” and the role they play in effective teaching of numeracy. Transcripts from three case study teachers’ numeracy lessons were examined to identify teachable moments; qualitative descriptions illustrate the nature of these teachable moments and their potential to enhance students’ understanding. It was found that teachers “missed” teachable moments, incorporated them into the discourse or actively ignored them. The findings indicate that teachers need to identify when to act upon teachable moments to avoid the likelihood of students forming misconceptions about important mathematical concepts.

This paper examines the affective responses of students to an algebra intervention in the last year of primary school. The intervention was based upon extensive use of materials, discussion activities and specifically designed algebra games. Data on student attitudes and beliefs were collected with an eight dimension Likert scale, student interviews, analysis of student sketches and classroom observations. While the overall results indicated student appreciation of the use of materials and educational games, some students perceived their value to be transitory.

In many schools throughout the western world teachers are observing increasing student disengagement with the study of mathematics and technology. The Queensland Education Department implemented a curriculum project designed to trial the use of design practice as a tool for integrating the teaching of mathematics and technology. This paper reports on the outcomes of the trail. It argues that differences in regulatory and instructional discourse adopted by the teachers resulted in different and pedagogically important outcomes.

This study engages mathematics teachers in reflecting on their practice by collecting data electronically from their students during the course of a mathematics lesson. This method of data collection eliminates the burden of manual collation and produces immediate feedback for analysis while the lessons are still fresh in the teachers’ minds. It is possible to pose a number of standard questions across all lessons as well as questions formulated by the teachers as being of interest.

This study focused on primary teachers’ perceptions of their knowledge and understandings of length, area and volume. It also explored their understanding of how children’s growth of measurement concepts and processes develops. Data gained from in-depth interviews revealed that teachers’ knowledge was often implicit and that they struggled to articulate their knowledge of measurement concepts and children’s trajectories of learning.

This paper reports on a part of my ethnographic study about uses of the Internet for teacher professional development and teaching mathematics. Drawing on two case studies of teachers in Queensland, I analyse what factors (personal and contextual) support or inhibit mathematics teachers in making use of the Internet for teacher professional development and for teaching mathematics. The findings support the notion that teachers’ knowledge and beliefs are key determinants in embracing technology as a tool for teaching and learning.

This paper applies a situated perspective on learning to investigate the process of becoming a secondary mathematics teacher. We followed a group of beginning teachers into their early teaching careers. Each participant was interviewed on three separate occasions during their university studies in order to examine how they interpreted their practicum experiences. In the following year, we again interviewed some of the participants to investigate their sense of self as mathematics teachers and to document the issues which they identified as significant in shaping their classroom practices. The beginning teachers saw the culture of the school and pedagogies of more experienced colleagues as the most important factors affecting their professional growth and ability to implement Working Mathematically in the classroom.

This paper considers the intuitive solutions of 26 first year tertiary students to a binomial probability problem on entry to a statistics unit. For this problem a successful solution requires consideration of the sample size. On the basis of a Rasch analysis, students were classified into three groups according to their ability, and the reasoning they used compared. The problem was again posed at the end of the unit and the answers and reasoning used by the students were compared with their earlier responses.

We show how the RBC model for abstraction in context can be used to follow the emergence of a learner’s knowledge constructs and to identify in detail the learner’s partially correct constructs (PaCCs). These PaCCs are used to explain the learner’s inconsistent answers and provide added insight into processes of knowledge construction. The research process is illustrated by means of an example from elementary probability. We thus demonstrate the analytic power of the RBC model for abstraction in context.

Prominent policy guidelines suggest that effective mathematics pedagogy assists students to “make connections” between various types of mathematical knowledge and between mathematical knowledge and real-life phenomena. This paper describes the perspectives and practices of two early years teachers involved in a reform that encourages teachers to help students make connections between forms of disciplinary knowledge and between disciplinary knowledge and real-life experience. The purpose of this description is to reveal strategies that assist students learn to make mathematical connections.

This paper describes how an early career teacher used technology in a lesson about capacity utilising calculators, a bank of PCs and an Interactive White Board. Data analyses of the digital recordings of the lesson using a framework for classroom practice indicated mostly student-centred learning and opportunities to experiment and talk about their activities. However, there was limited evidence of deepened understandings of mathematics. Findings from this study warrant further investigation into the usefulness of a tool to refine classroom practice.

Technology, such as dynamic geometry software (DGS), is available in many secondary mathematics classrooms. Whilst studies have highlighted the motivational aspects of DGS to explore geometric concepts, there exists a need to explore specific strategies for using technology in the mathematics classroom as a teaching tool, as opposed to a display tool. The project presented explored a phase approach to incorporate technology into the teaching/learning cycle to facilitate developmental progression in a secondary mathematics classroom. This paper presents the findings of a project which linked theory and practice within a technological environment.

Interactive whiteboard (IWB) technology is present in a large number of Australian primary schools. Despite the rapid increase in availability of this technology over the last five years, previous research suggests that the technology is being used for sophisticated transmission style teaching as opposed to constructivist approaches. This paper presents findings of a project which considered the implementation of IWB technology in three Victorian primary mathematics classrooms. The case studies explored the teaching strategies adopted by three teachers as they embarked on the use of IWB technology as an integral component of mathematical activities with the support of professional development.

This paper explores the perceptions of teachers of five-year-old children and the expectations these teachers have of the children when they begin school in terms of their level of mathematics knowledge. Teachers were asked about their beliefs in relation to mathematics teaching and learning and how they thought children best learned mathematics. This was matched with classroom observations to compare teacher practice with teacher beliefs. A significant finding was the general lack of attention that teachers paid to the knowledge and skills that children had when they started school. There was a lack of consistency between what teachers said they believed in compared with what they actually did in the classroom.

Numeracy is a complex construct and has become an essential capability for the twenty-first century. This paper reports the case of one teacher, Alice, with respect to her positioning of numeracy in a reform curriculum. By analysing her conversations, Alice’s numeracy pedagogy was described and visually represented according to five dimensions of practice. The case of Alice, as presented in this paper, demonstrates the potential for gaining insight into the numeracy beliefs and practices of individual teachers and for further understanding of the complex nature of numeracy teaching.

Understanding that mathematics is not just an arbitrary collection of rules to follow is basic to good mathematics learning, but studies show that many classrooms exhibit little mathematical reasoning. In order to better understand the nature of reasoning in schools, this study examined the modes of explicit reasoning in the explanations, justification and proofs of several topics in four textbooks. Eight distinct modes of reasoning are identified, illustrations of these are given, and their characteristics are discussed.

Forty-one students from Years 3 to 10 completed a clinical interview on the topic of fractions. Four of the questions from the 20-30 minute interview involved the fraction 3/4; students’ responses to these questions are analysed in this paper. When asked to illustrate 3/4, the most popular model chosen by students was a circle, yet fewer than 25% of the students knew that a circle divided into 4 parts with unequal areas (using 3 vertical lines) did not represent 3/4. Such students are unable to identify the appropriate attribute which is relevant to the given model.

This study uses number sentences involving one and two unknown numbers to identify some key junctures between relational thinking on number sentences and an ability to deal with sentences involving literal symbols. Number sentences involving two unknown numbers distinguish between students who are restricted to computational approaches and those who can genuinely engage in relational thinking. Furthermore, such sentences allow the identification of different stages of relational thinking.

Kindergartens in China offer structured full-day programs for children aged 3-6. Although formal schooling does not commence until age 7, the mathematics program in kindergartens is specifically focused on developing young children’s facility with simple addition and subtraction. This study explored young Chinese children’s strategies for solving basic addition facts as well as their intuitive understanding of addition via interview methods. Results indicate a strong impact that teacher-directed teaching methods have on young children’s cognitions in relation to addition.

Self-efficacy has the potential to facilitate or hinder our mathematics learner’s motivation, use of knowledge, and disposition to learn. This paper examines the use of a questionnaire and classroom discussions to access the self efficacy of 64, year four to six students. The questionnaire and classroom discussions gathered data on the students affective, cognitive, and conative psychological domains of functioning. The findings from the questionnaire and classroom discussions are presented and discussed in regard to their relationships with the students’ self-efficacy.

There are difficulties accessing students’ thinking about mathematical concepts, although methods such as task observation and interviews provide some useful information. In recent years it has become possible to use functional magnetic resonance imaging (fMRI) techniques to access brain activity while students are thinking about mathematics. In this study we have used this technique to examine brain activity while students were processing graphical and algebraic representations of function. Results show some evidence for increased difficulty of translation between these formats for linear compared with quadratic functions. We also describe regions of the brain that are involved in the translations.

This paper uses a textual analysis of two documents prepared by the mathematics community and the mathematics education research community to the National Numeracy Review in 2007 to uncover and compare knowledge legitimation within these two fields. The paper shows that knowledge within these disciplines is based on different epistemic devices, and hence that debates surrounding mathematics education arise, at least in part, from differing ways of viewing knowledge.

Eighteen Aboriginal students, in Years 1 to 11 at a remote community school, were interviewed using standard counting tasks, and a ‘counting’ task that involved fetching ‘maku’ (witchetty grubs) to have enough to give a maku to each person in a picture. The tasks were developed with, and the interviews conducted by, an Aboriginal Research Assistant, to ensure appropriate cultural and language contexts. A main finding was that most of the students did not see the need to use counting to make equivalent sets, instead using an ‘estimation’ strategy, even though they were able to count.

A professional learning program for unqualified practising secondary mathematics teachers regarding senior secondary mathematics teaching is described in this article. Professional learning episodes, artefacts and reflections of three teachers who participated in the program are analysed to identify the development of these teachers’ mathematical and pedagogical content knowledge. The findings indicate that a program designed for senior secondary mathematics can enable practising teachers to deepen and broaden their understanding of junior secondary mathematical pedagogy.

This paper reports on a component of a research project, Young Australian Indigenous students Literacy and Numeracy (YAILN), a longitudinal study investigating learning and teaching activities that support Young Indigenous Australian students as they enter formal schooling. In Queensland students are allowed to attend a non compulsory year of schooling, Preparatory (Prep), if they reach the age of 5 years by June in the year that they enrol. In YAILN one of the participating Indigenous schools’ preparatory intake included Indigenous students who had not reached the required age for Prep. Numeracy understandings for two of these students, Widgy and Caddy were tracked during their pre prep year of schooling. Both Widgy and Caddy are from low socio-economic families. Their pre and post test numeracy results and the interview conducted at the beginning of their ‘official’ preparatory year suggest that this extra year of schooling enhanced their knowledge of mathematics and has (a) put them on an even footing with students from more advantaged backgrounds as they enter Prep, and (b) given them a distinctive advantage over other Indigenous students who have not had equivalent experiences.

This study reports on the second phase of a design experiment involving classroom implementation of a sequence of four lessons introducing informal inference supported by TinkerPlots software to a grade 7 class. A Beginning Inference Framework was used as an implicit foundation for the teachers and as an explicit rubric for assessing students’ observed outcomes. Outcomes were judged in relation to saved TinkerPlots files annotated with student-completed text boxes and to individual interviews with 12 of the students.

This report considers the responses of 1205 students to two chance and data problems involving proportional reasoning and the interventions suggested by 44 teachers for four typical incomplete or inappropriate student responses. Students’ responses reflect the relative difficulty of the two items, whereas the teachers’ suggested interventions display a wide range of pedagogical content knowledge, including aspects of content knowledge and knowledge of students as learners. Rubrics are provided for both students’ and teachers’ responses and analysis supports the view that teachers find it difficult to assist students without directly telling them the answer. Suggestions are made for assisting teachers to improve feedback for students.

The Counting On 2007 project was designed to support the professional development of teachers in identifying and addressing the learning needs of middle years students in mathematics. It was based on earlier models and included changes designed to simplify and encourage further and ongoing involvement of schools. One change was a simplified assessment process that provoked initial concerns that teachers would not develop an appreciation of their students’ specific difficulties nor a deep understanding of the learning framework in number and thus the students’ learning would suffer as a consequence. Implemented in 122 schools across NSW, the findings for the program indicated it was successful in assisting students who had struggled with mathematics with 66% of the students increasing their place value understanding by one or more levels and similarly with 65% for multiplication/division, while providing a vehicle for teacher professional learning.

Classroom video, and video-stimulated interviews of small group work, in a Grade 5/6 classroom are used to show ways group composition can influence learning opportunities. Vygotsky’s (1933/1966; 1978) learning theory on the spontaneous creation of knowledge as compared to the guidance of an expert other frames this group analysis. Illustrations from two groups show how opportunities to spontaneously create new knowledge can be limited or enhanced by psychological factors associated with the inclination to explore that have been linked to resilience in the form of optimism (Seligman, 1995, Williams, 2003). This study contributes to our knowledge on forming groups to promote deep learning. It raises questions about other ways in which learning may be influenced by optimistic orientation and about building this personal characteristic to enable deep learning.

The ability to solve mathematics problems is the main goal of mathematics education in many countries. This ability depends on coordinating several types of knowledge and mathematical processes, especially heuristics. Commonly used heuristics include guess and check, draw a diagram, logical argument, and simplifying the problem. This paper describes the heuristics used by a sample of Primary 5 (n = 221) and Secondary 1 (n = 64) Singapore pupils to solve problems like this one: “There are 100 buns to be shared by 100 monks. The senior monks get 3 buns each and 3 junior monks share 1 bun. How many senior monks are there?” The pupils solved two sets of problems; the second set consisted of parallel problems to the first set but was administered a few months later. The pupils’ written solutions were analysed according to the heuristics used. A comparison of the heuristics used between the two parallel tests shows that some pupils did not use similar heuristics to solve parallel problems. This issue of consistency in heuristic use should be further researched to unravel its implications for the teaching of heuristics.

Simultaneous co-ordination of the referent unit, symbolic notation and pictorial representations are one aspect of understanding fractions as a measure. The notion that students must identify the referent unit, fractional parts of the referent unit, and apply it as an accurate method of measurement was investigated. Using different types of pictorial representations, students were required to represent the quantity being measured and conversely, identify the quantity represented. Four questions from a larger study that used fractions as a measure were examined. Nine students were interviewed to gain further insight into their thinking on these questions and their misconceptions identified.

This paper reports on the research and development of an ICT tool to facilitate the learning of ratio and fractions by adult prisoners. The design of the ICT tool was informed by a semiotic framework for mathematical meaning-making. The ICT tool thus employed multiple semiotic resources including topological, typological, and social-actional resources. The results showed that individual semiotic resource could only represent part of the mathematical concept, while at the same time it might signify something else to create a misconception. When multiple semiotic resources were utilised the mathematical ideas could be better learnt.

This paper describes a research study to find out the ability of Singapore secondary school students in attempting open investigative tasks. The results show that most high-ability students had no experience in open mathematical investigation and they did not even know how to start. Providing sample problems in the tasks for students to investigate did not seem to help them understand the requirements of the tasks. The implication of these findings on research methodology using paper-and-pencil tests will be discussed.

This paper examines the influence of teacher’s mathematics pedagogical content knowledge (MPCK) in the teaching of area and perimeter to Grade 4 pupils. Lessons of a beginning teacher were studied to determine the activities and teaching strategies used to bring out ideas associated with area and perimeter. Observable MPCK-in-action outcomes are also being studied through video-taping the beginning teacher. In addition, this paper also studies how concept of area and perimeter is developed in class. The complex interplay between area and perimeter concepts was sometimes handled well by the beginning teacher, whereas on other occasions, gaps in mathematics pedagogical content knowledge had the potential to cause misconceptions for pupils.

The paper describes results of a teaching experiment with five high school (Year 10 and 11) students. Four qualitative characteristics were established: the first step of solution, main information extracted from the problem, generalisation from a problem and completion of solution. From these characteristics the corresponding quantitative indices were introduced and analysed. The structure of two of them, specific SFS and common SHP,are given in detail. Investigation of quantitative indices and their qualitative characteristics gives an opportunity to find out more about interrelations between different stages of the problem-solving process.

With Australia performing so poorly in terms of equity in mathematical achievement on the PISA scores, there is an increasing recognition for practices that may stem the inequities in education in this country. This paper explores an approach that has been found to be highly successful in the United States and links it to current issues in Australian education. Practical considerations are made regarding the application and implementation of such reform pedagogy when particular nuances of Australian issues are considered. In particular, the development of a tool to evaluate reform pedagogy is the focus of this paper.

Current teaching practice has been greatly enriched through valuable research into students’ acquisition of deep understanding and skills in early multiplication and division. Research in multi-digit multiplication and division is still relatively undeveloped. This proposed study will explore how students achieve computational fluency in multiplication and division. This paper outlines the background literature and research into multiplication and division. It then proposes questions to be explored and the intended methodology to determine effective pedagogical approaches that encase both mental and written computational methods.

The Numeracy Development Project has been systematically introduced into New Zealand schools since its conception in 2001. For many teachers, successful implementation of this mathematics education reform required a transformation of long held beliefs and practices. This shift involved moving from a view of mathematics learning as individualistic and passive, towards one in which students come to do and understand mathematics through participating in collaborative activity. This change in pedagogy and the required content knowledge has brought a range of challenges for many teachers. The purpose of this short presentation is to share the journey of two lead teachers (coaches) working with a mentor to enhance their understanding of number in the mathematics classroom. The mentor was a school adviser and the coaches came from schools that had previously been involved in the Numeracy Development Project. The process took place over six months and was aimed at improving both pedagogy and content knowledge of the in-school coaches and other teachers within their schools. A particular emphasis was placed on improving student learning in multiplicative and proportional thinking.

This presentation documents the learning journey of a group of six Year 5/6 students from a low decile school. These students worked with the first author over several sessions to build their understanding of fractions (i.e., tenths, hundredths), and the links between these fractions and the corresponding decimal places. Structured materials (plastic tubing cut to different lengths) based on a linear measurement model of the number line were used to help students appreciate the relationships between 1 whole, 1/10, and 1/100. Selected excerpts from the learning sessions are used to illustrate the insights that emerged during learning sessions. The findings are interpreted using the theoretical frameworks of Piaget and Vygotsky.

Learning to write good test items is an important aspect of the teacher preparation programmes in Singapore. This paper reports on the types of errors in writing test items in mathematics made by student teachers who were following the pre-service course for teaching at primary level. An analysis of the errors reveals that student teachers demonstrate some key shortcomings in: Use of language, mastery of content knowledge, use of diagrams as support, and use of appropriate context when writing the items.

Financial modelling is an accessible and practical way to expose students to the power of mathematical modelling and the Excel spreadsheet with its extensive collection of financial functions is often incorporated into courses on financial mathematics. At AUT University we have chosen to focus on the Excel spreadsheet for our introductory courses and this has served us well as students have developed modelling skills and Excel skills at the same time as being exposed to financial functions. In our more advanced course we have introduced students to the financial toolbox in Matlab. In this study we report on some experiences of this initial group of students as they grapple with the different mindset needed to relearn from a Matlab perspective functions they had mastered using Excel and explore the potential that Matlab offers to move beyond what Excel can offer. Our report will also identify some of the limitations we have uncovered as we have developed our course and allow us to demonstrate why financial modelling provides a fruitful source of worthwhile examples to help students to develop modelling skills.

This study proposes to look at the nature of language required in a middle school mathematics classroom through a teaching experiment. The teaching intervention proposes to provide explicit teaching of the mathematical and non-mathematical language and orchestrate subsequent opportunities for students to engage in substantive communication through cooperative groupwork. The emphasis will be on conceptual understanding through language acquisition rather than solely through learning procedures. The study proposes to examine the impact of the teaching intervention on student motivation and engagement in mathematics.

In this study, a sample of 73 pre-service secondary mathematics teachers, enrolled in the PGDE programme at the National Institute of Education in Singapore, completed a survey about their views on their practicum experience. Preliminary results show that most of the student teachers acknowledged that the practicum was a good learning experience but they differed in what they reported was the most important aspect of their learning about the teaching of mathematics and about what they perceived as their most and least memorable experiences while teaching mathematics during the practicum.

This research will examine factors affecting engagement in Mathematics as students make the transition from primary and secondary schooling. This presentation will consider the background literature and include discussions on factors affecting motivation, engagement, and student learning in the middle years of schooling, such as teacher-student relationships, teacher content knowledge, and pedagogy. The research questions will be presented and an outline of the proposed methodology will be provided. Understanding the factors that affect students’ learning at this significant stage of education will better inform educators about the issues that may influence their subject choices beyond middle schooling.

A pilot study, analysing twenty-one Year 10 students’ reflections on a unit on exponential functions taught using real-world problems, and supported by the use of CAS calculators. The real-world contexts explored during the unit provided a focus for follow-up interviews. Detailed analysis of the transcripts revealed four themes: Impact of technology; difficulty in expression; prediction; choice of scenario. Findings support results from related research reported in the literature and highlight the importance of the choice of real-world scenarios. Analysis revealed variation in students’ preference relating to scenarios. This variation provides a challenge in the preparation of appropriate material.

Significant technological tools have impacted on students’ learning of Mathematics. In this study, I wish to seek evidence that dynamic geometry enhances students’ understanding of certain geometric properties and can serve as a vehicle for discovering unknown properties. The aim of the study is to explore the impact of dynamic geometry on 40 Year 9 high school students’ understandings of geometric properties. A mixedmethodology will be utilized in order to confirm the findings from different data sources; such as interview, video and audio recordings, computer savings, students’ workbooks, and pre- and post-tests.

Schools are rapidly adopting Interactive White Boards (IWBs) [or alternatively termed Smart Boards]. The rapidity of uptake has not been in step with any well designed, comprehensive research effort that might inform patterns of adoption and implementation by teachers, nor any sufficient understanding about the impact IWBs might have on students and their learning in mathematics and other areas of the curriculum. This is not the first time that attractive resources have been rapidly taken up into the teaching of mathematics, based more on faith. Whether this matters is an interesting question to ask. In this short communication, I concentrate on insights from some of the relevant IWB research literature, noting gaps and deficiencies that need to be addressed. In particular it is noted that usually only self-report methods are used, there is a lack of studies that describe the impact of IWB use on cognition and that document actual changes in classroom interaction. As well studies often do not clearly focus on IWBs. These shortcomings make it difficult to assess the impact of IWB technology, if not impossible, in terms of any changes in the quality of student learning. Teachers certainly like them, but is this enough?

What role do competitions play in a mathematics programme? This question is examined from the multiple perspectives of fifteen students (11-13 year olds) identified as mathematically gifted and talented, and their teachers and parents. A variety of different types of mathematics competitions were subscribed to including local, national, and international competitions. There was varied access to the competitions and differing perspectives; students and parents shared similar views about the value of competitions but there was a difference of opinion among the teachers.

An evaluation study aims to validate a new conceptual framework for mathematics learning based on the development of pattern and structure. It will evaluate the effectiveness of a school-entry mathematics program built on this framework using classroom observations and an interview-based student assessment. The Pattern and Structure Mathematics Awareness Program (PASMAP) focuses on unit of repeat, multiplicative reasoning, spatial structuring, and congruence and similarity. The program will be evaluated in Kindergarten classes from four large primary schools in Brisbane and Sydney.

MERGA has enjoyed a strong and rich history since its inception over 30 years ago. In that time it has been instrumental in supporting and developing a prominent group of mathematics education researchers who have made a significant contribution to the field. The purpose of this Round-Table Discussion is to consider how MERGA and its members might continue to promote, support and encourage mathematics educators from generation X and generation Y (and beyond), so they can stand on the shoulders of those who have gone before, and continue to build the stellar reputation of mathematics education research in Australasia. This discussion is open to both experienced and novice researchers, and hopefully it will lead to some strategies and ideas for succession planning that will ensure the health of MERGA and mathematics education research into the future.

The New Zealand Numeracy Development Project (NDP) was introduced to combat the disparity between high and low achieving students. It originated as a pilot project in 1999 as an extension and modification of the Australian ‘Count Me In Too’ project, and was implemented by the Ministry of Education in 2001. Whilst some improvements have been noted, there is evidence of continuing disparity between Maori and Pakeha student achievement, and underachievement of these students remains a concern. A vital component of NDP was that teachers were provided opportunities for professional development. A Ministry of Education Quality Teacher Research and Development (QTRD) program was introduced in 2007, supporting teachers in action research to raise student achievement. This presentation analyses the experiences of a group of teachers who undertook the QTRD course to develop more effective teaching practices to enhance the learning of low achieving Maori and Pasifika students. We provide complementary perspectives on our participation in the project, and the dual benefits of increased student achievement and ongoing professional learning for the teachers. We argue that quality, theory-based professional development and reflective practice are necessary to enhance the teaching and learning of disadvantaged students. In the Round Table we hope to stimulate discussion with Australian, New Zealand, and other international colleagues about: • The nature of successful professional development partnerships between teachers and university researchers; • Characteristics of effective numeracy education programs for disadvantaged students; • Challenges in working with teachers and students in low socio-economic or disadvantaged areas.

The Home School Partnership Numeracy facilitators will start the round table discussion by presenting the findings from three small studies: i) Two case studies investigating if schools change the way they communicate student achievement to parents as a result of participating in the Home School Partnership project. ii) A small research project involving all of the schools across the Waikato region that have been involved in Home School Partnership projects in the last three years. The international evidence cited in Alton-Lee (2003) positively supports the enhancement of student learning through home and school partnerships. In New Zealand, two Best Evidence Syntheses research also highlight the importance of establishing effective relationships between home and school (Alton-Lee, 2003; Biddulph, Biddulph, & Biddulph, 2003). The Home School Partnership project reflects the acknowledgement of parents as first teachers and the desire to continue to encourage parents to confidently interact and communicate with their children about mathematics. Effective relationships within the school community encourage parents to take an active role in the shared responsibility of their children’s education. Aspects that might be considered in this discussion include: Background information about home and school partnerships; successful learning communities involving facilitators, lead teacher and lead parents; and communication of student achievement to parents. Alton-Lee, A. (2003). Quality teaching for diverse students in schooling: Best evidence synthesis. Wellington: Ministry of Education. Biddulph, F., Biddulph, J. & Biddulph, C. (2003). The complexity of community and family influences on children’s achievement in New Zealand: Best evidence synthesis. Wellington: Ministry of Education.

The purpose of this roundtable is to describe a research proposal and seek feedback and advice on the proposed study. The research project aims to investigate current concerns of a downturn in motivation and engagement levels as students move through the middle years of schooling (Years 5 to 8 in New South Wales). Current research has shown at this time of transition from primary schooling to secondary many of these students begin to perceive mathematics to be a special domain in which smart students succeed and others merely get by or fail. Although students feel mathematics is important, many are not pursuing mathematics in the later years of school. This choice is seriously influenced by attitudes towards and performance in mathematics and significantly shaped by school mathematical experiences. While there is a wealth of research investigating issues surrounding the motivation and engagement of students in mathematics during the middle years, there is a gap in the research in terms of longitudinal studies. Through this longitudinal study there will be opportunity investigate if and how levels of motivation and engagement change within students as they progress through the transition from primary to secondary schooling. Within the study factors affecting any changes in motivation and engagement will be investigated and factors that play a role in increasing or sustaining levels motivation and engagement in mathematics will be identified.