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Mathematics education seen both as a field of research and as a field of practice is a multicultural field. For a while, this essential characteristic of the field has been underestimated but this is no longer the case. Both theoretical frameworks and empirical research help us today better approach this phenomenon, understand its impact on the field, reflect on research outcomes and the value of knowledge progressively built as well as on the challenges that we globally face as a community. In this lecture I would like to share with the audience my vision of this evolution and of the potential it offers for the field of mathematics education, relying on my personal experience as a researcher raised in a specific culture but also involved in many international collaborations, and on my engagement in ICMI, the International Commission on Mathematical Instruction, an institution which, for more than one century, has tried to contribute to the development of mathematics education through international exchanges and collaboration.

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

International studies of mathematics achievement have profound influence on mathematics education worldwide in the past 15 years. Results of studies such as TIMSS and PISA have dominated the agenda of discussion in the mathematics education community as well as among policy makers. Much attention however has been paid on the ranking of countries in the league tables generated from such studies, often without due consideration of the nature of these studies, as well as the contextual factors that affect the performance of students from different countries. In this paper, the nature of these international studies of mathematics achievement will be examined. Based on an understanding of the nature of these studies, how the results should be interpreted will be discussed. This includes a proper understanding of the meaning and significance of ranking of countries; the suitability of drawing causal relationships between various variables and student achievement; and the appropriateness or otherwise of learning from the educational practices of high ranking countries, etc. It will be argued that cultural value may be an important factor in explaining differences in educational practices and student achievement. Without due consideration of the cultural and other contextual differences , passing judgement on the performance of students in different countries based on results of international studies is very misleading, and may even be damaging. In learning from other countries, one must first evaluate the cultural values and educational context in one’s own country before deciding on how much can be learned from other countries. Examples of lessons that can be and should be learned from international studies of mathematics achievement will be discussed, and implications for policy makers in education as well as for school mathematics teachers in their classroom practices will be explored.

This paper reports the impact of a collaborative professional experience model on pre-service secondary mathematics teachers’ perspectives and practices within a learning community. Nine pre-service teachers made 12 school visits over one year to observe and co-teach problem-solving lessons in two Year 8 classes. They discussed the lesson with the teacher and the university supervisor, and posted reflective comments to an online forum. Data from questionnaires, interviews, and reflections indicate participation in the learning community helped pre-service teachers link theory and practice, learn from each other, and become more reflective.

On-line quizzes are used to help first year University Mathematics students identify weaknesses in their basic skills and improve them. Quizzes developed as a formative tool have been utilised at JCU for eight years. However, before this research no-one has questioned the effectiveness of quizzes for this task. We present a description of the quizzes used in a core first year Mathematics subject at JCU and provide a statistical overview of their usage and efficacy for the intended task.

Peer group learning is the name we have given to a particular type of collaborative learning that has been implemented as part of an action research project designed to improve teaching and learning of first year university mathematics at James Cook University. Using an innovation-decision process model we analysed the response of academics to the implementation of the peer group learning initiative. The analysis reveals that the action research methodology allows generally positive attitudes regarding the initiative to surface.

As part of an innovative tutorial structure introduced to a first year university mathematics subject, an attendance monitoring system was implemented. The system collected data that was used to investigate the relationship between student attendance and assessment performance which is reported here. The implementation of this system also assisted in the increase of student participation and engagement.

This paper describes how a group of university lecturers are adopting an action research approach to improve the learning experience of students in first year mathematics. Using the three categories of saying/thinking, doing, and relating (Kemmis, 2009) to explore practice, it describes the new practices of the action research team, the established practices of mathematics teaching at university, and the team’s trials at changing elements of that teaching practice.

One of the most significant influences on student engagement is the teacher’s pedagogical practices, including the incorporation of technology into the teaching and learning of mathematics. This paper reports on a qualitative study investigating how the incorporation of iPads into a Year 3 primary classroom during a six month trial influenced teaching and learning practices and student engagement with mathematics. All of the students appear to have had a positive experience during the trial and the classroom teacher believed their engagement with mathematics had improved as a result. Although there were challenges involved in integrating the iPads into mathematics lessons, some teaching practices were adapted to accommodate the technology. The integration of the iPads highlighted the need for teacher professional development and the importance of developing strong Technological Pedagogical Content Knowledge.

Using a case study, the complex challenge of making mathematics education research accessible to secondary mathematics teachers was addressed with two questions. How can we design a method that will meet the challenge of making research usable for mathematics teachers? And what would this method be? To address this challenge we describe a process and product that emerged from using the stages of design-thinking. We invite fellow researchers to join us in the collective mission of bridging the gap between mathematics research and practice, furthermore, we seek to stimulate conversations about what counts as a research output.

This study, drawing on data from the Trends in International Mathematics and Science Study (TIMSS) 2007, examined the influences of self-perceived competence in mathematics and positive affect toward mathematics on mathematics achievement of adolescents in Singapore. Ordinary least squares (OLS) regression analyses revealed the positive influences of self- perceived competence in mathematics and positive affect toward mathematics on mathematics achievement of adolescents in Singapore. Implications of the findings for policy and practice are discussed.

This paper examines the experiences of three teachers who were engaged in a project designed to enhance student engagement in mathematics through the development of real- world context activities which enabled students to use mathematics as a tool for responsible citizenship and the pursuit of social justice.

Four university lecturers at an Australian university have been undertaking an action research project for almost two years to improve first year mathematics teaching. The project is analysed here through the single idea of “connection”. Making new connections or changing the nature of existing connections with colleagues and especially with students is leading to different ways of teaching. Beliefs and practices that had remained, until now, unexamined, are being abandoned, modified, or at the very least, questioned.

This paper reports on initial data from teachers at an Australian Year 9-12 secondary school which is attempting to implement a project-based learning model across the entire curriculum. The eight teachers had diverse prior teaching experiences, including work in primary schools, special education, Year 7-10 secondary schools, and technical and further education. None had studied mathematics at tertiary level and just one listed maths as a main teaching area. Results of an initial survey indicate that most had reasonable levels of personal mathematics competence, and could identify relevant mathematical concepts among the affordances of a particular scenario, but that they struggled to articulate how they would work with students to pursue any of the relevant mathematics in depth.

This study examines the developing beliefs and practices of six beginning primary teachers. Their accounts reveal practices indicative of contemporary approaches to teaching and learning in mathematics. Additionally, a consistency appears to exist between the beliefs and practices of the beginning teachers, and the ideals for mathematics teaching formed during pre-service education. This is noteworthy as it is contrary to the literature that has described how the challenges of beginning teaching can have a significant negative impact on pre- existing beliefs and classroom practices.

Pedagogical content knowledge has been widely acknowledged by researchers and practitioners as a significant factor for improving student knowledge, understanding and achievement. Recently, the knowledge teachers need for teaching has expanded to include teacher horizon content knowledge, “an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (Ball, Thames, & Phelps, 2008, p. 403). This study uses a collective case study design, in which three Kindergarten teachers from Greig Heights Primary School participated in a professional learning and development program designed to enhance aspects of their teacher knowledge. This paper will provide an emerging description of the nature of teacher knowledge, and discuss the potential implications this has for catering for the needs of students at-risk of experiencing difficulties in acquiring early numeracy skills (i.e., number sense knowledge).

This paper examines how mathematical understandings might emerge through student- centred inquiry. Data is drawn from a research project on student-centred curriculum integration (CI) that situated mathematics within authentic problem-solving contexts and involved students in collaboratively constructed curriculum. Participatory action research (PAR) was the methodology employed and mixed methods were used to collect data. The project took place in three primary school classrooms in New Zealand. The findings indicated that mathematics centred on real-life learning was highly engaging and that the measurement and geometric thinking explored went beyond New Zealand curriculum requirements.

For teaching statistics investigations at primary school level, teacher knowledge has been identified using a framework developed from a classroom based study. Through development of the framework, three types of teacher listening problems were identified, each of which had potential impact on the students’ learning. The three types of problems are described, with examples from the classroom along with links to the teacher knowledge framework. It is concluded that teacher knowledge is a necessary condition for avoiding such listening problems.

This study explores how teachers understand and translate curriculum statements into concepts that are better suited to supporting student progress with rich mathematical problem-solving tasks. We report the actions of one experienced teacher of primary mathematics in charting the sequence of concepts and processes relevant to the above issue. We contend that the study embellishes Pedagogical Content Knowledge dimensions of the framework advanced by Ball, Hill and Bass (2005). Implications for field experiences of prospective teachers of mathematics are discussed.

Although there has been considerable interest in teachers’ knowledge for teaching mathematics for nearly a quarter of a century, little attention has been paid to the knowledge of mathematics teacher educators. The responses of 57 MERGA members to an online survey addressing beliefs about mathematics and its teaching, mathematics content knowledge and mathematics pedagogical content knowledge are reported. Teacher educators found the items addressing pedagogical content knowledge more difficult than mathematics content questions or endorsing beliefs, and the type of employment appeared to be a more important influence on outcomes than the level of mathematics studied.

Emotions play an important part in learning and are known to influence human development. The positive emotions of enjoyment and interest are thought to contribute to learning in distinctive ways, yet this distinction tends to be blurred in some learning research. This paper explores the role these two emotions play in the development of children’s statistical literacy. It focuses on the responses of 220 middle-school children to just six self- descriptions, three assessing interest and three assessing enjoyment. Analysis of these responses suggests that the two emotions are difficult to differentiate empirically, but that differences in reported levels of enjoyment and interest may depend on students’ perceptions of competence in the specific tasks with which they are engaged. Implications of this for teachers and researchers are discussed.

This paper reports on a design experiment regarding young children’s concepts of mass measurement. 119 year one and two children were interviewed using a clinical interview both before and after the teaching period comprising five lessons that offered rich learning experiences regarding concepts of mass. The results of the interviews were that the majority of these Year 1 and 2 children moved from using non-standard units to using standard units and instruments for measuring mass.

The mathematics required for teaching is increasingly becoming an important issue for research (Ball, Thames & Phelps, 2008). This study examines the development and quality of the mathematical content knowledge of a cohort of pre-service primary teachers. We commenced the study of the impact of a Model-Based Teaching and Learning (MBTL) approach on the development of pre-service teachers’ conceptual knowledge in the domain of fractions (Forrester & Chinnappan, 2011). In this present study we found further evidence for the robustness of MBTL as an effective instructional strategy in promoting conceptual knowledge.

In this study we take up the issue of teacher knowledge by characterising the quality of one teacher’s Mathematics Knowledge for Teaching (Ball, Thames and Phelps, 2008) by examining her Pedagogical Content Knowledge (PCK) that was accessed during a Lesson Study observation in a Chinese primary school. Results show that the participant’s PCK was robust in the way she weaved knowledge of content and her students. We explore implications for the development and sharing of this knowledge in the context of her further involvement in our Lesson Study.

This is an exploratory study into the individual problem-posing characteristics of 480 Grade 9 Singapore students who were novice problem posers working on two geometric tasks. The students were asked to pose a problem for their friends to solve. Analyses of solvable posed problems were based on the problem type, problem information, solution type and domain knowledge. With the open-ended task, the students tend to over-condition their problems and to produce more problems with implicit assumption. How the findings can contribute to research in problem posing in schools is discussed.

We investigate the viability of a new approach to initial fraction instruction. We establish the need to empirically investigate whether the proposed approach shares the strengths of currently used approaches, specifically, whether students will (a) construe problems based on the proposed approach as experientially real, and (b) bring up ideas that could be build upon in subsequent fraction instruction. We then present an analysis of sixteen student interviews from a school in southern Mexico (ages 8 and 9). The analysis supports the conjecture that the proposed approach to initial fraction instruction can be viable, and thus warrants further research attention.

This paper is located within the research into encouraging learners to reason mathematically and become engaged with concepts rather than just procedures. It reports on a research in progress examining two techniques to be used at the beginning of a sequence of lessons in order to elicit students’ prior knowledge and understanding of topics about to be studied. The research will be conducted with two Year 7 classes of girls as they are introduced to units of work on decimals and fractions. Two techniques will be used to elicit the students’ prior understandings: Concept Maps and Concept Cartoons. This paper reports on the ideas behind Concept Cartoons and the development of Concept Cartoons for eliciting understandings of decimals. The paper briefly outlines the research into difficulties with decimals before discussing the rationale and design of the cartoons. The paper is concluded with a discussion of how concept cartoons may or may not elicit different understandings to using concept maps.

The teaching of mathematics involves problem solving skills which prove to be difficult on the part of the pupils due to misrepresentation of the word problems. Oftentimes, pupils tend to represent the phrase “more than” as addition and the word difference as “- “. This paper aims to address the problem solving skills of grade five pupils employing the block model approach which is based on concrete - representation – abstract principle of teaching mathematics.

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

A pilot study was conducted with Year 10 males (N = 154) preparing to make their senior mathematics subject choice. Survey data revealed that students did not understand the different dimensions of relevance of mathematics. Additionally, a statistically significant difference in the level of agreement concerning relevance was identified between students choosing Mathematics A and those choosing both Mathematics B and C. Students choosing both Mathematics B and C perceived mathematics as relevant for facilitating their career pathway, while Mathematics A aspirants acknowledged the relevance of mathematics was less influential, reporting their choice was guided by their mathematical ability.

This paper examines spatial metaphors in the English language associated with the number line, in particular metaphors of direction and motion, and how these are manifested in actual spatial practices associated with number. It considers how these metaphors are culturally influenced, and how the influences of other cultures, such as Arabic, produce inconsistencies that can contribute to confusion in the classroom. It also considers how the metaphors of number vary in some other languages, and how this leads to both different spatial representations of numbers and more challenges in multilingual and multicultural classrooms. In particular, it pays attention to the implications of variety in these metaphors for the mathematics education of Indigenous students in Australia being taught by English speaking teachers.

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

This paper reports findings from an activity implemented in the final year of a 3-year longitudinal study of data modelling across grades 1-3. The activity engaged children in designing, implementing, and analysing a survey about their new playground. Data modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. The core components of data modelling addressed here are children’s structuring and representing of data, with a focus on their display of metarepresentational competence (diSessa, 2004). Such competence includes students’ abilities to invent or design a variety of new representations, explain their creations, understand the role they play, and critique and compare the adequacy of representations. Reported here are the ways in which the children structured and represented their data, the metarepresentational competence displayed, and links between their metarepresentational competence and conceptual competence.

This paper reports on a study that explored the use of cognitively challenging tasks with low- attaining mathematics students and in particular, their teachers’ attempts at scaffolding. A major finding was that responding appropriately to scaffolding opportunities was challenging for the teachers. In this paper two main factors are discussed which impacted on the teachers’ responses to scaffolding opportunities: teacher knowledge of appropriate tools and representations for particular mathematical concepts, and teacher response to student prior knowledge and understanding.

After briefly analysing a classroom episode, we discuss aspects of the teacher’s didactical knowledge, namely in its mathematical and instructional dimensions, as reflected in the answers of three prospective teachers to a written assignment based on the episode. We then raise some issues regarding initial teacher education, anchored in the notion of didactical knowledge.

Students’ development of an understanding of covariation was the focus of a research project that investigated the way in which 12 Year 5/6 students engaged with the learning environment afforded by the graphing software, TinkerPlots. Using data generated from individual interviews, the results demonstrate that upper primary students use their understanding of covariation to draw conclusions about trends in data evidenced in the graphs they created as well as their knowledge of the context. Implications for the Australian Curriculum are considered.

People aged 20-39 were stopped in the streets of Victoria (Australia) and Madrid (Spain) to gauge their views on the gendering of mathematics. The findings suggested that for respondents from both countries, if stereotyped beliefs are held they were more strongly associated with the traditional male stereotype, that is, that males are considered more suited to pursuits in mathematics. However, in general, the Spanish respondents held stronger views than the Australians that mathematics was gender neutral, that is, that it, and related fields, are equally suited to males and females.

The Extending Mathematical Understanding (EMU) Program is a specialised mathematics program that aims to accelerate the learning of Grade 1 students who struggle with learning school mathematics. Forty-two students participated in an EMU Program in 2010 as part of the Bridging the Numeracy Gap (BTNG) project. Analysis of students’ mathematics knowledge at the beginning of the EMU Program highlighted how diverse was this group of students. The students’ mathematics knowledge was assessed again at the beginning of the following year in order to evaluate the effectiveness of the program for accelerating learning. Overall the students made very good progress and their learning was maintained.

Numeracy is a general capability to be developed in all learning areas of the Australian Curriculum. We evaluated the numeracy demands of the F-10 curriculum, using a model of numeracy that incorporates mathematical knowledge, dispositions, tools, contexts, and a critical orientation to the use of mathematics. Findings of the history curriculum audit, presented in this paper, highlight the distinction between the numeracy demands and opportunities of the curriculum, and uncover mismatches between claims made about numeracy in the curriculum materials.

This paper reports on the different gesture types employed by twenty-three Year 10 students as they endeavoured to explain their understanding of rate of change associated with the functions resulting from two different computer simulations. These gestures also have application to revealing students’ understanding of functions. However, interpretation of gesture is problematic but classification of gestures assisted in the analysis of the video- recorded interviews probing participants’ conceptions of rate of change. This paper builds on the classifications reported in previous research. Five additional gesture types are presented, which provide insights into students’ thinking about rate of change, and hence functions.

Proportional reasoning is a key aspect of numeracy that is not always developed naturally by students. Understanding the types of proportional reasoning that students apply to different problem types is a useful first step to identifying ways to support teachers and students to develop proportional reasoning in the classroom. This paper describes the development of a diagnostic instrument that aims to identify situations in which students can apply proportional reasoning and the types of reasoning they use.

This study describes Singapore students’ (N=607) performance on a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia’s NAPLAN and Singapore’s PSLE. In addition, the MPI had a corresponding question which encouraged students to describe how they solved the respective tasks. In particular, the investigation considers two tasks the cohort found difficult to solve—a relatively complex task from the Singaporean item bank and a novel task from the Australian tasks.

This paper reports on the use of lesson study as a professional development tool. In particular the paper focuses on the way in which the teachers increased their understanding of how tasks, classroom activity and teacher actions scaffolded student learning of early algebraic reasoning of equivalence and the commutative principle. Teacher voice is used to illustrate how lesson study cycles caused the teachers to reflect and review their own understandings of early algebraic concepts and how their students considered the concepts.

How prospective teachers can best be prepared to teach effectively in mathematics classrooms is a topic of current concern. In this paper, we describe our exploration of ways in which prospective teachers were supported to translate what they learnt in mathematics methods classes into pedagogical practice. We illustrate how the use of discourse routines, enacted in iterative cycles of guided rehearsals, disrupted previous beliefs about teaching and learning mathematics and led to them more confidently respond in pedagogically appropriate ways.

The Swan Valley Cluster of schools for the Make It Count project identified the professional learning of teachers and teaching assistants as a key factor in improving numeracy outcomes for urban Indigenous children. Initial training for assistants began in late 2010 and took the form of workshops based on a modified First Steps in Mathematics Number program. It was continued in 2011 and lead to a pilot program in training assistants to plan for targeted mathematics learning for individuals and small groups of children. This paper reports on the success of the pilot with regard to the improved confidence and ability of the assistants to assume greater responsibility for teaching, as well their development as integral members of professional learning communities.

The importance of leadership in changing schools and building quality programs is the focus of this paper. While leadership is often seen as a management aspect of school life, the role of leadership in curriculum change may be quite different from that of school leadership vis- a-vis the principal. In small remote schools where there are many factors impacting on school reform, this paper explores curriculum leadership where the schools successfully performed against numeracy benchmarks. Features of curriculum leadership are drawn from these cases to develop a framework for considering effective leadership in remote Indigenous contexts.

Drawing from Gee’s learning principles developed from the digital games environment, we provide a critical analysis of the difference between using these principles in a literacy environment as opposed to a mathematical environment. Using stimulated recall, primary school-aged students played with a number of contemporary digital games. Feedback was sought. This was compared with the descriptions provided by experienced adult gamers. Both players provided insights into the cognitive process used by gamers when engaging with games. Collectively, these sources allow us to propose that the learning principles may restrict deep learning processes for mathematical learning.

This study, drawing on data from the Programme for International Student Assessment (PISA) 2009, explored the influences of metacognitive and self-regulated learning strategies for reading on mathematical literacy of adolescents in Australia and Singapore. Ordinary least squares (OLS) regression analyses revealed the positive influences of metacognitive learning strategies and control strategies for reading on mathematical literacy of adolescents in Australia and Singapore. In contrast, the two components of self-regulated learning strategies for reading—memorization and elaboration—had negative influences on mathematical literacy of adolescents in Australia and Singapore.

This paper discusses issues arising in the design of questions to use in an on-line computer-based formative assessment system, focussing on how best to identify the stages of a learning hierarchy for reporting to teachers. Data from several hundred students is used to illustrate how design decisions have been made for a test on interpreting line graphs.

This study explores the effects of a vocational education-based program on academic motivation and engagement of primary school aged children. The Get Into Vocational Education (GIVE) program integrated ‘construction’ and the mathematics, English and science lessons of a Year 4 primary classroom. This paper focuses on investigating the components of the GIVE program that led to student changes in mathematical academic motivation and engagement resulting in outstanding gains in NAPLAN Numeracy results. The components proposed to have contributed to effectiveness of the GIVE program are: teacher and trainer expectations, task mastery and classroom relationships. These findings may be useful to researchers and educators who are interested in enhancing students’ mathematical academic motivation.

In Australia we face a serious problem in that over the last twenty years the quality of students’ mathematical knowledge and abilities has “deteriorated to a dangerous level” (Brown, 2009, p. 3). Too few students want to study further mathematics and are “unlikely to continue its study voluntarily” (Commonwealth of Australia, 2008) or pursue careers where high levels of mathematical proficiency are needed. In this paper I make use of the poststructuralist notion that ‘proficiency’ is a state of being daily constituted in classroom practice to (a) at a theoretical level, rethink how it might be ignited and sustained, (b) analyse contemporary interactional strategies that commonly though unknowingly obviate expressions of proficiency and (c) through a combined psychological/poststructuralist lens, nominate three (3) tentative indicators of instructional practice necessary for students to achieve and maintain a state of being ‘proficient’ as defined in the Australian curriculum: mathematics (Australian Curriculum Assessment and Reporting Auathority [ACARA], 2010). It is hoped that an additional, poststructuralist reading of the complexity and tenuous productivity of the learning process might interrupt commonly held assumptions that currently inform research on proficiency in mathematics.

This paper explores the role of professional learning community and collegial discussion as important supports for developing teacher expertise in the teaching and learning of mathematics in rural and remote regions of Queensland, Australia. The research reported in this paper is from the first year of a longitudinal research and development project. Findings suggest that teachers in rural and remote schools with a small number of mathematics teachers may benefit from access to the mathematics professional learning community of a larger rural school.

Data from an attitude survey administered to students in grades K–2 from four schools participating in the Make it Count project are reported in this paper. Few differences were found in the attitudes and beliefs of the Indigenous and non-Indigenous participants. The relevance of these findings for students’ longer term mathematics performance is also considered.

This discussion paper put forwards variation as a theme to structure mathematical experience and mathematics pedagogy. Patterns of variation from Marton’s Theory of Variation are understood and developed as types of variation interaction that enhance mathematical understanding. An idea of a discernment unit comprising mutually supporting variation interactions is proposed and used as a building block for a mathematics pedagogy that resembles a developmental sequence from coarse idea to precise definition. Classification of plane figures is used as a pedagogical example for illustration.

Novice secondary mathematics teachers attempting teaching consonant with NCTM (1991) Professional Standards for Teaching Mathematics experience stresses related to those attempts. Foremost among those stresses are challenges while orchestrating student-centred, whole-class discussions. Such discussions can create uncertainty and stress as novices consider students’ mathematical ideas, ask questions to push the class’ thinking forward, and choose between multiple competing directions that such discussions may take. Novices’ self- reports suggest social resources, particularly teacher learning communities, offer promising support for attempting such teaching.

"For a person with a hammer, everything looks like a nail" is a proverb that can be used to highlight the phenomenon that students tend to rely on familiar ideas as opposed to taking time to think about and analyse a problem. Presented in this theoretical paper is the usefulness of the hammer-and-nail metaphor, other related theoretical constructs, pedagogical causes of student impulsive behaviours, and pedagogical suggestions for addressing them.

The Likelihood-to-Act (LtA) survey and a mathematics test were used in this study to assess students’ impulsive-analytic disposition in the context of mathematical problem solving. The results obtained from these two instruments were compared to those obtained using two widely-used scales: Need for Cognition (NFC) and Barratt Impulsivity Scale (BIS). The exhibited correlations of the LtA scores with the NFC, BIS, and a math test provide evidence of the criterion validity of the analytic LtA items, and suggests further revision of the impulsive LtA items to improve the overall measurement validity of the LtA scale. Students LtA scores were found to be marginally correlated to their math scores and correlated to their confidence levels in the math items.

This paper reports findings from the second year of a two-year study designed to develop approaches to teaching algebra in years 9 and 10. The aim of the research was to explore and develop teaching approaches to assist students to acquire a conceptual understanding of algebra, and to document the impact of these approaches on student outcomes. Previous work (Linsell, 2009) has detailed the strategies that students use to solve equations but there has been no research to date on teaching approaches that make use of these findings. The two main components of this study were a qualitative description of teaching approaches based on videos, teachers’ diaries and meeting notes, and a quantitative analysis of student outcomes based on results from a previously developed assessment tool (Linsell, Savell, Johnston, Bell, McAuslan, & Bell, 2006). Approaches to teaching were not uniform between teachers, but all made extensive use of the diagnostic assessment information and recognised that algebraic thinking pervades the entire mathematics curriculum. Assessment of the students at the end of either one or two year’s participation in the study showed significant improvements in algebraic strategies and knowledge compared to baseline, though there was little improvement during the second year. Furthermore, the measures of student outcomes displayed significantly higher values for students in this study compared to year 9 and year 10 students in Secondary Numeracy Project (SNP) schools, which are fairly representative of the population.

The theoretical understanding that underpins a teacher’s foundation knowledge draws on their common content knowledge (CCK) and influences their mathematics’ teaching (Rowland, Turner, Thwaites, & Huckstep, 2009). Teachers who have specialised content knowledge (SCK) demonstrate a unique kind of content knowledge which is more than knowing the content (Ball, Thames, & Phelps, 2008). This study reports on a comparison of two second-year pre-service teachers who had varied mathematical content knowledge (MCK) at the beginning of their Bachelor of Education course. It investigated whether knowing more advanced mathematics or foundation knowledge (Rowland et al., 2009) facilitates working towards demonstrating SCK. The results draw on a qualitative analysis, categorising lesson observation and interview responses using foundation knowledge and connections constructs of the ‘Knowledge Quartet’ framework (Rowland et al., 2009). Both pre-service teachers experienced course opportunities that consolidated foundation knowledge and demonstrated connections. During their lesson observation they relied on procedural explanations and neither especially demonstrated working towards SCK.

The importance of student understanding of the concept of place value cannot be underestimated. Place value is a ‘gate keeper’ in developing mathematical understanding. The purpose of this study was to examine and develop a teacher-made test of place value knowledge. The questions were developed using the progressions from the Number Framework (Bobis, Clarke, Clarke, Thomas, Wright et al., 2005). An exploratory factor analysis was used to evaluate the assessment tool. The analysis of student responses to the test questions revealed a three-factor structure that supported the existing literature on the progression in learning of place value ideas, by identifying the critical key ideas that underlie the concept of place value. The results validated the tool as a test of place value knowledge that could be used to assess the performance of Yrs. 3-9 students.

Engaging and extending middle years students in mathematics is a continual challenge. One of the aims of the Australian Curriculum: Mathematics is to ensure that students are “confident, creative users and communicators of mathematics” (ACARA, 2011). Use of mathematical models and/or problems has been suggested as methods of achieving this aim, and mathematical investigations have been shown to improve student engagement. This paper looks to build on these ideas and combine them with the framework of Knowledge Producing Schools (KPS) (Bigum & Rowan, 2009) to determine whether, when students are working on a community based project of their choice, students become “confident, creative users and communicators of mathematics” (ACARA, 2011).

This paper reports on the analysis of a study of a professional learning intervention focussing on computational estimation. Using a multiple case study design it was possible to describe the impact of the intervention of students’ beliefs and computational estimation performance. The study revealed some noteworthy impacts on computational estimation performance and that beliefs were altered to a lesser extent.

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

This paper demonstrates the explanatory power of Kieren’s framework for rational number knowing (1988, 1992, 1993, 1995), renamed here the four-three-four model, by describing the different approaches of Grade 6 students to a quotient context task (sharing three or seven custard tarts between five people) using Kieren’s terminology of partitioning, equivalence, and unit-forming.

Two of the challenges faced by mathematics teacher educators involve the issue of pre- service teachers’ mathematical content knowledge and the impact this has on their PCK, and the perceived gap between the theoretical knowledge received in their teaching course and the practical knowledge gained in the classroom while on professional experience. This paper examines the use of what is essentially a teacher educator tool or strategy, designed to give pre-service teachers a realistic environment in which to address these two challenges. The study uses data collected from pre-service teachers who participated in virtual lessons using Second Life, post-lesson interviews and a teacher educator survey to document the advantages and disadvantages of using such an approach. The results indicate that the use of Second Life has practical implications for teacher educators as an additional tool for modelling and reflecting upon the teaching of mathematics.

The Pattern and Structure Mathematical Awareness Program (PASMAP) is an early mathematics program designed to promote structural thinking. PASMAP pedagogy removes the structure commonly provided for students in order to challenge them to construct their own, focusing student attention on spatial and numerical patterns and leading them to formulate their own generalisations. In this paper, we give some examples of Kindergarten PASMAP tasks that develop and link the themes of grid patterns, number patterns, multiplication and base ten numeration. Work samples drawn from a recent evaluation study are used to illustrate the range of students’ structural development.

Data from a larger study were used to identify variables that best predict children’s perceived usefulness of mathematics [PUM]. Gender differences in PUM scores were also explored using a sample of 300 grade 7 children and 225 parents from February to May 2011 in Mozambique. Surveys and interviews were used to collect data. Consistent with traditional beliefs it appeared that mathematics is viewed as more useful for boys than for girls. Education of parents, school geolocation, and number of siblings were statistically significant predictors of children’s perceived usefulness of mathematics.

This paper reports further findings from a study on categorisation of typical problems in the learning of ratio originally presented at the 2011 conference. The paper focuses specifically on the seven categories identified; examines strategies used by two classes of Primary 6 students to solve problems, and presents data in the form of student journal writing; interviews; voice recordings and a pupil perception survey. Pre- and post-tests were administered. Outcomes reveal the positive effects of learning ratio through categorisation.

The value of a consideration of large numbers and exponentiation in primary and early secondary school should not be underrated. In Indian history of mathematics, consistent naming of, and working with large numbers, including powers of ten, appears to have provided the impetus for the development of a place value system. While today’s students do not have to create a number system, they do need to understand the structure of numeration in order to develop number sense, quantity sense and operations. We believe that this could be done more beneficially through reflection on large numbers and numbers in the exponential form. What is reported here is part of a research study that involves students’ understanding of large numbers and powers before and after a teaching intervention incorporating historical ideas. The results indicate that a carefully constructed framework based on an integration of historical and educational perspectives can assist students to construct a richer understanding of the place value structure.

The importance of understanding the various uses of the literal symbol in algebra, and in particular the idea of generalised number, is well documented in the literature. Many research findings have also reported student difficulties with this vital and central concept. This research study examines the use of a combination of historical and educational research ideas as a way of enhancing students’ understanding of generalised number. The results suggest that this approach helped some students to make some generalisations and to understand the difference between specific unknown and generalised number.

The detrimental consequences of mathematics anxiety have repeatedly been evidenced empirically, yet little is known of its influence on secondary school students in Asia. This study thus examined its origins and impact on 294 secondary students in Singapore through interviews and surveys. Results revealed an average anxiety level of 44% and a negative correlation with achievement. Of the top 5 situations that worried students, 4 were test- related. Nonetheless, highly anxious students continued to persevere and enjoy the subject.

This study focuses on a distinction between process- and object-oriented discourses when characterising the discourse of university students’ summaries of lectures and examining connections between students’ discourse and the discourse of lectures. Results show that students’ discourse in general tends to be process-oriented, by their use of active verbs and little use of nominalisations. Students’ summaries of process-oriented lectures also tend to be more process-oriented, but the differences between individual students are larger than differences caused by variations of the discourse in the lectures.

Indigenous students may find mathematics in schools difficult because there is discontinuity between cultural mathematics and school mathematics. One of the reasons for this is that their teacher’s identity as a mathematical thinker may not link to their cultural ways of thinking. In Papua New Guinea, there is a subject to assist student teachers to develop their own and hence their students’ consonance between cultural and school knowledge. In the subject, student teachers undertake a project to link culture and mathematics. The question for this research is to explore how student-teacher identity as a mathematical thinker is enhanced when they explore the cultural setting of their mathematics. From 239 reports collected over a 10 year period, 60 were analysed to explore the impact of sociocultural contexts on identity as a mathematical thinker. The document analysis informed the argument that such projects encourage teachers to express both culture and school mathematics and to identify with their cultural mathematical ways of thinking and to value these in school education.

The Let’s Count pilot early mathematics program was implemented in five early childhood educational contexts across Australia during 2011. The program used specifically formulated materials and workshops to enlist the assistance of early childhood educators to work with parents and other family members of children in their settings to help develop these children’s awareness, confidence and skills in early mathematics. The pilot program was evaluated by the authors of this paper using a multi-methods approach. The evaluation was focused on the success of the Let’s Count program in bringing early childhood educators, parents and other family members together, to enhance children’s mathematical engagement, learning outcomes and dispositions.

Teachers now require high levels of statistical literacy in order to take advantage of the many statistical reports analysing assessment data that are provided by system authorities. In this report 16 items from the Attitudes and Statistical Literacy Instrument (ASLI) are used with 704 teachers to provide a hierarchical scale of teacher ability to interpret these assessment data. Using Rasch analysis, three levels of ability are identified, related to reading values, comparing values, and analysing a data set. Implications are drawn for professional learning for teachers and for further research.

This article seeks to address a pedagogical theory of introducing the classicist and the frequentist approach to probability, by investigating important elements in 9th grade students’ learning process while working with a TinkerPlots2 combinatorial problem. Results from this research study indicate that, after the students had seen the systematic construction of the event space via combinatorial analysis, they viewed the sample space as an essential property that regulated the results of the distribution of each sum’s theoretical frequency.

In this paper, we discuss the diffusion (of an innovation) and relate it to our attempt to spread our initial design of a mathematics practical paradigm in the teaching of problem solving.

In line with continuing efforts to explain the demanding nature of multiplicative reasoning among middle-school students, this study explores the fine-grained knowledge elements that two pairs of 7th and 8th graders deployed in their attempt to coordinate the known and unknown quantities in the gear-wheel problem. Failure to conceptualize the multiplicative relation in reverse, mainly due to the numeric feature of the problem parameters and inherent inverse proportional relationship, led the students to use more primitive fallback strategies.

One of the common tasks of inferential statistics is to compare two data sets. Long before formal statistical procedures, however, students can be encouraged to make comparisons between data sets and therefore build up intuitive statistical reasoning. Such tasks also give meaning to the data collection students may do. This study describes the answers given by beginning university students to tasks involving comparing data sets in graphical form, originally designed for students between Grades 3 to 9. The results show that whereas all the students had successfully completed either pre-tertiary mathematics or a bridging mathematics course many had similar difficulties to students of a younger age. In particular, they did not use a measure of centre or proportional reasoning when appropriate.

This paper seeks to compare the reflective writings of two cohorts of students (Year 4/5 and Year 8/9) participating in mathematical modelling challenges. Whilst the reflections of the younger cohort were results oriented, the older cohort’s reflections spoke more to the affective domain, group processes, the use of technology and the acquisition of mathematical knowledge. This study supports the idea that with scaffolding middle years students can engage in reflective practice to develop mathematical modelling skills.

Upper primary school students’ difficulties with decimal place value are widely recognised due largely to the availability of assessment tools that accurately identify students’ misconceptions in this domain. Yet, many students’ decimal difficulties seem to stem from a lack of understanding of whole number place value for which there are few, if any, comprehensive assessment tools at the upper primary school level. The inevitable result of this is that teachers tend to assume upper primary students have a deeper understanding of place value than is often the case and limit their teaching to fairly routine tasks that do not expose student misconceptions. This paper describes the derivation of a research-based, multi-dimensional Hypothetical Learning Trajectory for whole number place value. It explains the process involved in identifying the key aspects of place value and includes indicative examples of the assessment items used to evaluate this.

Despite the emphasis that children should have a robust sense of number and a thorough understanding of fraction (National Mathematics Advisory Panel, 2008), many students continue to struggle with these concepts. Booker Diagnostic Assessment Framework (Booker, 2011) can inform decision about teaching that improves students’ learning outcomes. Focusing on decimal numbers, this paper reports how effective use of the framework and instructional materials can help three teachers challenge their students’ misconceptions with mathematics, leading to improved learning outcomes.

The number of ethnic Chinese students in schools across Australian cities is small but increasing. It is important to understand how these students socialise into the Australian (mathematics) education system, so that we can better facilitate their education experiences in ways which optimise their potential to learn. This paper reports on part of a larger study which seeks to deepen our knowledge in this area. The research question addressed by this paper is: what are the preferences amongst three types of mathematical tasks of Grade 5 and 6 students from Chongqing, China? Through the administration of a questionnaire to 1109 students, it was found that across the topics of Number and Geometry, contextualised tasks were the most preferred by Chinese students. ‘Challenging’, ‘easy to do’ and ‘involving a model’ were students’ reasons for preferring particular task types. The significance of providing all students with a diverse range of task types, at the same time providing them with opportunities to be challenged and to experience success, are emphasized.

Limited Singapore research indicated a lack of exposure of modelling tasks at primary levels. Teacher reflection is used as a tool in design research cycles exploring the potentials of modelling tasks in a Singapore primary five classroom. Findings reveal that the teacher identified three potentials of a modelling task on children’s mathematisation process: the task provided a platform for children to (a) identify variables and form relationships between them, (b) relate school-based math learning to real-world experiences, and (c) justify their mathematical models. Implications on the promotion of modelling tasks at primary schools as well as teacher education are drawn.

Mathematisation of realistic situations is an on-going focus of research. Classroom data from a Year 9 class participating in a program of structured modelling of real situations was analysed for evidence of Niss’s theoretical construct, implemented anticipation, during mathematisation. Evidence was found for two of three proposed aspects. In addition, unsuccessful attempts at mathematisations were related in this study to inability to use relevant mathematical knowledge in the modelling context rather than lack of mathematical knowledge, an application oriented view of mathematics or persistence.

The research reported here was motivated by curiosity about ways of suggesting tasks and activities to teachers that assist them in their planning and which allow them to see the purpose of the suggested tasks. A group of junior secondary teachers involved in a larger project worked through a set of tasks that sought to address the content they were about to teach, and then completed a survey about those tasks. A comparison group also completed the survey. There was consensus across the groups of teachers about the potential of the tasks, they felt they could implement them and that such tasks were preferable to comparable textbook exercises. There are implications for both teacher educators, and for resource developers.

Insights into aspects of teachers’ planning reported here were gathered as part of a broader project examining aspects of the implementation of the Australian curriculum in mathematics. In particular, responses of teachers to a survey of various aspects of decisions that inform their planning are discussed. While there is diversity in processes teachers use for planning, a consistent theme was that teachers make active decisions at all stages in the planning process. There were slight differences in the ways that primary and secondary teachers plan. There are important implications from our findings for those who support teachers in the transition to the Australian Curriculum: Mathematics.

In this paper, the relationships between students’ beliefs about knowing and learning mathematics, and how they engage with calculators, are investigated. An online survey was conducted for 964 Singaporean and 176 Victorian senior secondary students. Students’ connected knowing–deep approach conception of mathematics was found to be associated with their use of calculators as collaborator, and their separate knowing–surface approach conception of mathematics was associated with use of calculators as master. Gender differences in students’ beliefs were also found.

This paper posits that teachers’ pedagogical content knowledge in mathematical modelling instruction can be demonstrated in the crafting of action plans and expected teaching and learning moves via their lesson images (Schoenfeld, 1998). It can also be developed when teachers shape appropriate teaching moves in response to students’ learning actions. Such adaptive development of teachers’ pedagogical content knowledge may in turn be supported by their knowledge of the mathematical modelling process and Ang’s (to appear) proposed framework for planning mathematical modelling instruction.

The study involved a group of Middle Eastern Muslim women learning mathematics through social justice pedagogy. The findings suggest that the involvement of students in social justice issues has not lead into a decline in their opportunity to learn mathematical content as set in the course. However, this approach has helped them to develop significant mathematical knowledge beyond the basic numeracy skills expected of them. In addition, this study demonstrated how a focus on mathematical content can be maintained while students are involved in social issues in mathematics learning.

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

We have applied the ‘practical paradigm’ in teaching problem solving to secondary school students. The key feature of the practical paradigm is the use of a practical worksheet to guide the students’ processes in problem solving. In this paper, we report the diffusion of the practical paradigm to university level courses for prospective and practising teachers. The higher level of mathematics content would demand higher order thinking skills. Learners without a model of problem solving would often revert to solving by referring to many examples of the same ‘type’ of problem. Polya-type problem solving skills framed by the practical worksheet was used as an attempt to elicit more effective problem solving behaviour from them. Preliminary findings show that they were able to use the practical worksheet to model their solution of problems in the courses.

This study examines the effect of small-group game play before and after a Number Sense Test (NST). Fifty-seven 7-year old children were assigned into two conditions: (a) NST-Game Play, in which students sat for the NST before game play; or (b) Game Play-NST, in which students experienced game play before sitting for the NST. Findings suggested that Game Play-NST students outperformed the NST- Game Play students in number sense performance. Having Game Play-NST experience appeared to be more effective in enhancing students’ performance in number sense.

This paper analyses the responses of 247 middle school students to items requiring the concept of average in three different contexts: a city’s weather reported in maximum daily temperature, the number of children in a family, and the price of houses. The mixed but overall disappointing performance on the six items in the three contexts indicates the need for concerted efforts to link numeracy across the curriculum as required in the new Australian Curriculum.

Working collaboratively with the researchers, a small team of teachers developed and taught a lesson based on the Teaching for Abstraction model (White & Mitchelmore, 2010). This paper reports how one teacher learned about the model and implemented it in her lesson. It was found that she had assimilated several key features of the model, such as starting with several embodiments of the target concept and guiding students to look for similarities between them. However, she had some difficulty focusing students’ attention clearly on the underlying mathematical similarity. It was concluded that teachers need to experience more examples of Teaching for Abstraction before they can to reify and apply the model faithfully.

Australian pre-service teachers have to acquire layers of knowledge as school classrooms are multicultural in composition. To what extent does the experience of an overseas professional experience tour contribute to the development of pre-service teachers in meeting recognised professional teaching standards? This paper describes the perceptions of Australian pre- service mathematics teachers who participated in educational and cultural activities during planned tours to Malaysia. The data set was collected through the use of questionnaires, interviews and focus group discussions.

Pre-service primary (elementary) teachers’ mathematics anxiety affects their engagement with and future teaching of mathematics. The study measured the range of mathematics anxiety in 219 pre-service teachers starting a teacher education course in an Australian university. They responded to the Revised Mathematics Anxiety Scale (RMARS) and a set of demographic questions. Age differences in anxiety were found to be significant, and relationships were found between the RMARS scores and students’ self-perceptions of their current mathematics anxiety levels.

This paper presents a theoretical model for the development of knowledge for teaching, based on a study of primary pre-service teachers (PSTs) in their final year of an Initial Teacher Education Programme. This model arose from findings about PSTs’ perceptions of knowledge for mathematics teaching that were related to knowledge of the curriculum, mathematical content knowledge and knowledge of school contexts. The theoretical model describes how PSTs developed this knowledge for teaching mathematics. There are three phases in this model; recognising crucial aspects of teaching, reconceptualising these aspects, and realizing these aspects in a mathematics classroom. In conclusion I discuss the possible use of this model for my work as a teacher educator.

Students’ attitudes toward mathematics and its learning have been subject to numerous studies in the past six decades. These studies treat such attitudes as both desirable learning outcomes and correlates of mathematics achievement. Many Likert-type attitude scales have been devised to measure significant constructs underlying mathematics-related attitudes, such as confidence, anxiety, and utility of mathematics. The psychometric properties of these attitude scales may be culture and age dependent. As part of a research project called Singapore Mathematics Assessment and Pedagogy Project (SMAPP), an effort was made to devise and validate an attitude toward learning mathematics scale that can be used with lower secondary school students in Singapore. This paper explains the use of exploratory and confirmatory factor analyses to reduce an initial 57-item questionnaire to one with 24 items that cover these six dimensions: Checking solutions, Confidence, Enjoyment, Use of IT in mathematics learning, Multiple solutions, and Usefulness of mathematics. The data comprise responses from about 890 Secondary 1 (Grade 7) students in 2010, who took the 57-item questionnaire, and another 850 students who took the 24-item questionnaire in 2011. The nature of the final questionnaire is discussed. This effort contributes to the continual effort to devise validated attitude scales that are suitable for different cultures and student groups.

This study investigates how 217 students in Years 5, 6 and 7 from three schools in Nanjing, China, link number and algebra (called relational thinking in this study). It categorizes their performances in terms of five levels, and uses these levels to create profiles of algebraic thinking across Years 5, 6, and 7. The study examines the interrelationships between the different types of questions used, and highlights the importance of connecting students’ arithmetic learning and the development of their algebraic understanding.

This paper reports on the types of problems that high-achieving students posed when given investigative tasks that were constructed by opening up some mathematical problems. A teaching experiment was conducted to develop the students’ thinking processes during mathematical investigation, and each student was videotaped thinking aloud during a pretest and a posttest. The findings show that some students were unable to pose the original intended problems and what Krutetskii (1976) called problems that ‘naturally follow’ from the task, including extending the task to generalise. The implications of the difficulty encountered by these students for teaching and research will also be discussed.

This paper presents the findings of a study that assessed the numeracy competency of more than 200 students enrolled in pre-service primary teacher education. The Mathematics Thinking and Reasoning assessment consisted of 10 tasks that included 2-digit computations and proportional reasoning. Students rated their attitudes towards mathematics at primary and secondary school, and currently. Analysis of the data showed that university admission status was not related to students’ scores on the assessment tasks. The correlation between meeting the University Entrance Numeracy Requirements and the total correct was very modest. The implications of these findings for ITE providers are presented.

Current approaches in technology integration narrowly focus on technology alone (Harris, Mishra, & Koehler, 2009). Mishra and Koehler (2006) propose integration of multiple aspects of technology, pedagogy, and content knowledge (TPACK). This poster presents preliminary analysis of data from a four-course sequence (technology, curriculum, student teaching, and research) on the preparation of preservice teachers in integrating technology. Data from journals, curriculum, questionnaires, and sample work (projects, lesson plans, portfolios) were content analysed (Neuendorf, 2002). The goal is to examine the process of becoming technology integrators and the role of a technology course, perceptions, teaching experiences, and TPACK in that process.

Mathematics anxiety in pre-service primary teachers can be described as a debilitating lack of confidence in one’s ability both to use mathematics in a functional way and to teach mathematical content (Uusimaki & Nason, 2004). This problem is a widespread and much researched phenomenon (Haylock, 2001; Trujillo & Hadfield, 1999; Wilson, 2011). The use of mentoring to ameliorate maths anxiety has also been explored, although to a lesser extent (Beswick, Callingham, Ashman & McBain, 2011; Hudson & Hudson, 2007; Hudson & Peard, 2006). The proposed study will incorporate elements of Emotional Intelligence (Goleman, 1995; Salovey & Mayer, 1990) and Communities of Practice (Lave & Wenger, 1991) to create a mentoring system for pre-service teachers suffering from maths anxiety. Excellent mathematics teachers who rate highly on an emotional intelligence scale will mentor pre-service teachers in the hope of developing the pre-service teachers’ confidence to become enthusiastic and effective mathematics educators.

The concept of area-measurement is particularly challenging for students as it encompasses the two critical domains – geometry and numbers. Most students face difficulty in connecting the visual, spatial and numerical aspects related to area- measurement. In line with the objectives of most curriculum, the present study tries to explore the connection between students' formal learning of area-measurement and its applications in other contexts through task based interviews. Analysis of results shows that students' greater reliance on using formal procedures limits their use of own strategies. There is a need for including contextual and visual experiences in the curriculum.

This longitudinal study examines the mediational role of achievement goals between previous achievement, gender, and classroom goal structure, on the one hand, and subsequent achievement, on the other. Secondary students from 115 math classes in Singapore participated in this study. Multilevel path analyses showed that at the class level, mastery classroom goal structure predicted subsequent performance through mastery approach goals, and classes with higher previous achievement and with more girls had higher subsequent performance through lower performance avoidance goals. At the student level, previous achievement predicted mastery approach goals, which in turn predicted subsequent achievement.

Singapore has placed great emphasis on teacher preparation and pre-service teachers’ development. Such preparation and development can be greatly enhanced through the mediation of both pre-service training at the National Institute of Education and extended classroom learning experiences in schools. Insights gleaned from the pre-service teachers cognitive and affective contents of the weekly reflection logs and perspectives of their experiential learning during practicum and a retrospective analysis of the pre-service teachers’ reflective thoughts on their cooperating teachers’ pedagogical practices, setting and charting their own goals and development will be discussed.

This presentation will outline a planned study of a mathematics intervention program that draws on the successful features of existing and previously reported programs. Currently successful existing programs are being reviewed and the successful characteristics identified. Based on this, a new program will be designed. This communication will outline the proposed research approach that aims not only to evaluate the components of the approach but also to describe the data collection methods including cognitive and affective outcomes of the intervention.

Learning trajectories illustrate a pathway of learning in mathematical domains. Knowing how students develop particular mathematical knowledge informs the construction of learning trajectories, providing a link between conceptual understanding and task selection (Simon & Tzur, 2004). Much of the research in multi-digit multiplication has focused on strategies used by children to solve problems, with limited research outlining successful pathways to developing understanding and how these apply to the classroom setting (Bobis, 2007; Fuson, 2003). This presentation focuses on articulating a learning trajectory for multi- digit multiplication through an examination of relevant research. This trajectory will be the basis for further study and investigation through a teaching experiment with Year 5 students. It is anticipated that the study will demonstrate how tasks can be used to effectively promote the learning process.

Developing students’ statistical literacy is a challenging task for practicing teachers, requiring the development of new perspectives and professional knowledge in statistics. This project aims to study the development of students’ statistical literacy from elementary to secondary education and is centred on two main issues: the characterization of key aspects of students’ statistical literacy, particularly the ability to conduct statistical investigations, and the understanding of the development of statistical and didactical knowledge to teach this subject. The project includes teaching and teacher education experiments, using a mixed-method approach and stressing collaborative work amongst teachers and teacher educators.

Anxiety can impact the teaching and learning of mathematics at all levels of instruction; and anxiety can affect many types of learners. Student with diagnosed learning differences are perceived to be more anxious about mathematics than other students. This survey was an investigation into some of these beliefs and feelings towards mathematics by students with diagnosed learning differences. Among the selected students the overall levels of anxiety were lower than expected. Student anxiety levels were reduced and statistically significantly after taking one mathematics course; however, these students still prefer not to engage in further mathematics applications.

Young children possess informal knowledge of mathematics that is broad, complex and sophisticated. They engage in significant mathematical thinking and reasoning in many contexts. This recognition has catalysed recent reforms on the need to bring a higher level of focus on early childhood learning, in general, and early childhood mathematics, in particular (Malaysian Government, 2012; Australian Government, 2011). Early quantitative reasoning begins to develop as early as the first 2 years of life. These reasoning demonstrate robust sensitivity to numerical information in the environments including counting, numerosity and systems for representing and discriminating small and large sets. Previous studies (Geary, 1994; Clements & Sarama, 2007) have shown that the development of numeracy skills begin before children commence their formal schooling experiences. Two concepts that are foundational to their ability to perform arithmetic operations are children’s ability to discriminate between the concepts of more and less. In this exploratory study we consider pre-schoolers’ (3- to 4-Year-Olds) understanding of these twin concepts. Results show that children tend to have more difficulties with the concept of more than less. Implications of the results are discussed for further investigations of studies of numeracy in early childhood mathematical thinking and numeracy.

This communication will describe proposed research that examines a teaching approach aimed at improving numeracy outcomes for Indigenous Australian students. This approach uses aspects of local Indigenous Australian culture as a context for learning numeracy. The research will explore the effectiveness of teaching culture in conjunction with numeracy while examining the relationship of culture to the dimensions within numeracy. This study will also explore the engagement of students in such context based learning experiences and quantify the performance of students participating in this style of learning.

This paper describes how two schools: one form Singapore and one from Australia, harnessed the power of the emerging Cloud Computing technology to establish a cohesive social network amongst teachers between the two countries as they co-developed Mathematical Modelling lesson units. Both schools adopted the research-based Teaching for Understanding (TfU) framework as a common language and teaching philosophy for the teachers to develop and build the units. This project has demonstrated how the schools have synergistically combined new technology (Cloud Computing), innovative pedagogy (TfU) and content knowledge (Math Modelling) to enrich students’ learning experiences and enhance their understanding of the mathematical ideas and concepts. This model of collaboration exemplifies how schools in different parts of the world can use new technologies to teach the 21st Century skills to students.

iMPaCT-Math is a project involving the development and implementation of a set of learning modules for high-school algebra students to make connections across multiple representations: (a) statements in a program, (b) computational process; (c) graphical output, and (d) underlying mathematical concepts. These programming-related activities provide an experiential-visual context for students to engage in mathematical thinking and reinforce foundational concepts like Cartesian coordinates and slopes. Results of pilot-testing of activities in the first three modules (coordinate system, variables, and linear equations) will be shared and professional development for algebra teachers on classroom implementation will be discussed during our presentation.

Students who attend vocational education are less motivated, have reluctance towards learning (Sahin & Fındık, 2008), do not like academic subjects (Lewis, 2000), encounter problems with mathematics (Green, 1998; Lewis, 2000; Scarpello, 2005), and compared to other secondary students, they are usually less successful in mathematics (Bottom & Korcheck, 1989; Berberoğlu & Kalender, 2005). The knowledge of teachers who teach in vocational high schools, therefore, matters in raising student achievement (Bottoms & Presson, 2000). The present study deals with teachers’ pedagogical content knowledge and its impact on student achievement.

The Problem@Web Project intends to study mathematical problem solving in a context that goes beyond the classroom. The research field involves inclusive web-based mathematical problem solving competitions aimed at middle graders (children aged 10 to 14 years-old). In this poster we address two main strands of the project: students’ creativity in mathematical problem solving, and the impact of digital technologies in students’ approaches when solving problems and communicating results. We provide a brief theoretical support and a short analysis of some empirical data. We also give a summary of the research developments and data gathering for the several questions addressed.

The focus of this presentation is to outline an approach to research that aims to explore the potential of demonstration lessons as a model for teacher learning. The intent of the lessons is to allow teachers to observe the process of planning, teaching, and assessment within the context of real time classrooms. The sessions for teachers involve the preparation of lesson plans, demonstration teaching, and subsequent interrogation. The data collection will include interviews, observations and surveys and will seek to explore and describe the relationship between demonstration lessons and teacher learning.

Teachers’ perceptions of their students’, including whether they perceive them to be engaged or not, influences the teaching strategies they adopt, their responses to students and the efforts they make in the classroom (Hadrè, Davis & Sullivan, 2008). It is therefore important to explore the nature of those perceptions. This study explores teachers’ perceptions of Year 7 students’ engagement and disengagement in mathematics. Thirty-one Year 7 mathematics teachers from ten high schools located in the Sydney metropolitan region were interviewed as part of a larger project investigating student motivation, engagement and achievement in mathematics. Interviews reveal the sources from which teachers’ (‘accurate’ and ‘inaccurate’) perceptions of student engagement are based and the usefulness of conceptualising such perceptions as a ‘spectrum’ of engagement to disengagement.

Open-ended mathematical tasks have the potential to not only engage students in constructive thinking but also hels them make connections to mathematical concepts they have learnt. A high quality open-ended task does not necessarily imply that the students will engage cognitively in higher order thinking to solve the task. Teachers play a crucial role in mediating the learning process in the implementation of such tasks. A guiding framework to characterise the varied degree of “openness” of the mathematical tasks is developed to ease teachers into the facilitator’s role. Rubrics are designed to provide feedback both to both the teachers and students for effective task implementation.

A 3-year longitudinal study integrates a pedagogical approach focused on patterns and structural relationships in mathematics to science learning through novel experiences in data modelling and problem solving. Students are engaged in an innovative program, usually withdrawn in small groups and taught by the research team in collaboration with the teacher on a weekly basis for a 2-year period. The study tracks three cohorts of students, initially involved in a related study1 when in Kindergarten, through to Grades 2, 3 and 4. In addition, two new cohorts of mathematically able students are being tracked from Kindergarten to Grade 2.

Although there is general endorsement among mathematics educators and researchers on the significance of translations in mathematical comprehension, there is substantial evidence that students struggle to accurately translate verbal, tabular, graphical and algebraic representations of mathematical relations (Gagatsis & Shiakalli, 2004; Galbraith & Haines, 2000; Porzio, 1999; Wollman, 1983). This action research set out to investigate the effects of using translation on raising the understanding of word problems involving fractions among Year 6 lower ability students. The “Fast Food Approach” (FFA) was designed by integrating elements from Polya’s four stages of problem solving (1957) and The Problem Wheel (Lee, 2008) with the intent of providing a structure to enhance students’ ability to translate in order to better “understand the (word) problem”. The study employed a single- group pre-test and post-test design involving an intact Year 6 class of lower ability students from a primary school in Singapore. The findings indicated that FFA may be used to enhance the process of problem solving, as well as improve students’ attitude towards and self-confidence in mathematics and problem-solving abilities.

This paper reports a preliminary study aimed to utilise the potential of students' thinking and learning to support teacher learning. The paper begins with a review of research on algebra teaching and learning with the objective of identifying and analysing tasks that relate research on students' algebraic thinking with the practice of teaching. It then discusses a task based on students' algebraic thinking developed and tried with a cohort of middle school teachers from Mumbai. Insights from the study and its implications for continuous teacher professional development are discussed.

Student disengagement in mathematics in the middle years and the related issues of underachievement and lower participation rates in higher levels of mathematics have been linked to teachers’ understanding of mathematical content and their pedagogy (Ryan & Williams, 2007). Research has identified the on-going learning of teachers as key not only to improving their own knowledge, but also valued student outcomes (Timperley, Wilson, Barrar & Fung, 2007). The round table will begin by outlining the theoretical underpinnings of a professional learning program, Empowering Teachers of Mathematics (ETM) and its two complementary foci: its impact on teachers (knowledge, beliefs, practices and ability to self-direct their learning); and the measurement of student outcomes (engagement and achievement). The session will provide an opportunity for international colleagues to discuss issues affecting the (a) sustainable growth of teacher mathematical content and pedagogical content knowledge; (b) the engagement of middle-years students in mathematics; and (c) how self- directed learning in teachers can be supported in a program of professional learning.

The terms, models and modelling, have been used variously in the literature, including in reference to solving word problems, conducting mathematical simulations, creating representations of problem situations (e.g., constructing explanations of natural phenomena), and creating internal, psychological representations while solving a particular problem. This proposed roundtable session will focus on the models and modelling perspective first initiated by Richard Lesh. From this perspective, models may be viewed as conceptual systems or tools comprising operations, rules and relationships that can describe, explain, construct, or modify an experience or a complex series of experiences. Modelling involves the crossing of disciplinary boundaries, with an emphasis on the structure of ideas, connected forms of knowledge, and the adaptation of complex ideas to new contexts (English, in press; Hamilton, Lesh, Lester, & Brilleslyper, 2008). The roundtable will begin with a review of the models and modelling perspective and will then be open to discussion on issues including (but not confined to): 1. Models and modelling in different nations, including sharing and collaboration; 2. The role of task context (including interdisciplinary themes) and task design; 3. The nature of the mathematical ideas and processes embedded in modelling problems; 4. Transfer of learning (e.g., from model-eliciting activities to model exploration and model adaptation activities); 5. Models and modelling with young children; 6. Across-grade sharing of modelling products (e.g., grades 2 and 7 students sharing their solutions to a given modelling problem); 7. Towards the future: advancing models and modelling.

The development of survey questions to address aspects of Pedagogical Content Knowledge (PCK) in mathematics for an Australian Teaching and Learning Council project led to intense dialogue and challenging discussion among the project team. Using the questions developed as a prompt, similar opportunities were provided to mathematics educators within the project team’s participating institutions. These sessions had very positive feedback. MERGA members are invited to participate in a similar experience in this round table, and to reflect upon this approach for useful professional learning.

Don’t we know enough about students’ conceptions of equality already? In this round table discussion, we will present preliminary findings from three lines of inquiry that suggests that students’ conceptions of equality are more complicated than previous theoretical frameworks indicate. This research stems from recent studies in New Zealand that suggest primary and intermediate students have difficulty solving missing number problems that involve the concept of equality. Rather than viewing students’ erroneous responses as problematic, we are probing into how students express their conceptions of equality. We are using diverse methods to analyse students’ written, verbal, and non-verbal responses to arithmetic missing number problems collected from different assessment contexts. Two lines of inquiry involve extant data analysis from samples of over 400 Year 4 (8 and 9 year old) and Year 8 (12 and 13 year old) students that participated in the National Educational Monitoring Project in 2009. In these inquiries, one involves quantitative analyses of students’ written responses and the other involves quantitative and qualitative analyses of students’ oral responses that were video recorded in one-on-one interviews with an adult assessor. The third line of inquiry is a prospective classroom-based study that involves microanalysis of video recorded interactions between pairs of Year 6 (10 and 11 year old) students as they work together to solve a series of missing number problems. Because we are proposing a more complicated theoretical framework of students’ conceptions of equality, we seek feedback, as well as critique, about the strengths and limitations of our diverse methods of inquiry.

The day-to-day work of teachers of mathematics is governed by a range of factors not least their knowledge of mathematics that is relevant to their professional work. An emerging field of research is theorising about understandings of mathematics that teachers need to develop in order to support practices that will optimise mathematical learning (Ball, 2000; Ball et al., 2001; Ma, 1999). The outcome of this stream of research has significant implications for the interpretation of teachers’ classroom practices, teachers’ professional standards, teacher education programs, quality professional development programmes, and, ultimately for arguments about professional strength of the community teachers of mathematics. This Round Table Discussion session will provide a forum for researchers to share their views and findings on the following issues. Colleagues are also invited to suggest new lines of inquiry that has the potential to add to the current debates about teachers’ mathematical knowledge and the teaching of mathematics. _ Teacher’s understanding of subject matter knowledge from a pedagogical perspective; _ Mathematical Knowledge for Teaching (Ball et al, 2008); _ Relations between courses in mathematics and practice; _ Teachers’ subject matter knowledge and representational fluency _ Teachers’ subject matter knowledge and students learning outcomes

Culturally responsive teaching is widely seen to promote equity of access to achievement. The literature indicates that teacher respect is essential for culturally responsive teaching. Teachers showing respect to students has been a recurring theme in recent New Zealand studies; one student stating that the most important thing to teach prospective mathematics teachers was “to respect the kids”. In this round table we present findings from a study into the nature of respect in teachers’ interactions in senior secondary school mathematics classrooms. Our study drew from mathematics students’ and teachers’ perspectives of teacher practice gathered using teacher and student questionnaires and semi-structured interviews, and videos of twelve mathematics lessons. Participants included six Year 12 and 13 mathematics classes and their teachers across three multi-ethnic city schools. Findings included that mathematics teachers show respect to students through their pedagogical practices, professionalism, disposition, and knowing their students well. Using students’ names and providing timely and perceptive individual assistance with mathematical tasks are examples of respectful practices. Some differences between teachers’ and students’ views emerged with students placing greater emphasis than their teachers on professionalism and the constructive treatment of student errors. The perceived challenges to teachers demonstrating respect included factors relating to student behavior, curriculum content, and the diversity of students’ mathematical abilities and ethnicities. This round table forum will begin with a short presentation of findings to initiate discussion regarding how our mathematics teacher education practice can promote respectful teacher behaviours and how teacher respect can contribute to culturally responsive mathematics teaching.