### Conference Proceedings 2012

Title |

Mathematics Education: Expanding Horizons |

Content |

Preface |

PREFACE |

List of Reviewers |

MERGA 35: Reviewers and Judges |

Keynote Address |

Mathematics Education as a Multicultural Field of Research and Practice: Outcomes and Challenges |

MERGA 2012: Where We've Been, Where We Are, and Where We're Going |

What Can and Should We Learn from International Studies of Mathematics Achievement? |

Practical Implication Award |

A Learning Community for Pre-service Secondary Mathematics: Learning With and From Each Other |

Symposium |

Are Online Quizzes an Effective Tool for Mastering Basic Algebra? |

Doing it Differently: The Ups and Downs of Peer Group Learning |

Monitoring and Analysing Attendance in First Year University Mathematics Tutorials |

Working Through the Practice Architectures of First Year University Mathematics Teaching |

Research Paper |

Exploring the Use of iPads to Engage Young Students with Mathematics |

Teaching Algebra Conceptually: The Process of Bringing Research to Practitioners |

Influences of Self-Perceived Competence in Mathematics and Positive Affect toward Mathematics on Mathematics Achievement of Adolescents in Singapore |

Socially Response-able Mathematics Education: Lessons from Three Teachers |

Improving First Year Mathematics Teaching Through Making Connections: An Action Research Approach |

Teaching Mathematics in a Project-Based Learning Context: Initial Teacher Knowledge and Perceived Needs |

Stories From the Classroom: The Developing Beliefs and Practices of Beginning Primary Mathematics Teachers |

Teacher Subject Matter Knowledge of Number Sense |

Mathematics as it Happens: Student-Centred Inquiry Learning |

How Does Teacher Knowledge in Statistics Impact on Teacher Listening? |

An Experienced Teacher's Conceptual Trajectory for Problem Solving |

Mathematical Knowledge for Teaching of MERGA Members |

Emotions and the Development of Statistical Literacy |

The Effects of Creating Rich Learning Environments for Children to Measure Mass |

Developments in Pre-service Teachers' Mathematics for Teaching of Fractions |

Mathematics Knowledge for Teaching: Evidence from Lesson Study |

Characteristics of Problem Posing of Grade 9 Students on Geometric Tasks |

Alternative Starting Point for Teaching Fractions |

Concept Cartoons as a Way to Elicit Understandings and Encourage Reasoning about Decimals in Year 7 |

Block Model Approach in Problem Solving: Effects on Problem Solving Performance of the Grade V Pupils in Mathematics |

Constructing and Consolidating Mathematical Entities in the Context of Whole-Class Discussion |

Male Students' Perspectives Concerning the Relevance of Mathematics - Pilot Study Findings |

Spatial Metaphors of the Number Line |

Reinventing the Wheel: Historical Perspectives on Theories for Interpreting Discourse Patterns in Mathematics Classrooms |

Young Children's Metarepresentational Competence in Data Modelling |

Challenges in Responding to Scaffolding Opportunities in the Mathematics Classroom |

Using Classroom Episodes to Foster Prospective Teachers' Didactical Knowledge: Issues for Teacher Education |

Interpreting Graphs: Students Developing an Understanding of Covariation |

Young Pedestrians' Gendering of Mathematics: Australia and Spain |

The Progress of Grade 1 Students Who Participated in an Extending Mathematical Understanding Intervention Program |

Auditing the Numeracy Demands of the Australian Curriculum |

Gesture Types for Functions |

Evaluating Middle Years Students' Proportional Reasoning |

Singapore Students' Performance on Australian and Singapore Assessment Items |

Developing Teacher Understanding of Early Algebraic Concepts Using Lesson Study |

Designing Opportunities for Prospective Teachers to Facilitate Mathematics Discussions in Classrooms |

Professional Learning for Teaching Assistants and its Effect on Classroom Roles |

Curriculum Leadership: Reforming and Reshaping Successful Practice in Remote and Regional Indigenous Education |

Digital Games for Learning Mathematics: Possibilities and Limitations |

Influences of Metacognitive and Self-Regulated Learning Strategies for Reading on Mathematical Literacy of Adolescents in Australia and Singapore |

Identifying Stages in a Learning Hierarchy for Use in Formative Assessment - the Example of Line Graphs. |

Get Into Vocational Education (GIVE): Motivating Underperforming Students |

Mathematical Proficiency and the Sustainability of Participation: A New Ball Game through a Poststructuralist Lens |

Developing a Culture of Collaboration |

K-2 Make it Count Students' Views of Mathematics |

Variation and Mathematics Pedagogy |

Supporting Secondary Novices' Efforts to Implement Student- and Discourse-Centred Pedagogical Practices |

The Hammer-and-Nail Phenomenon in Mathematics Education |

Impulsive-Analytic Disposition in Mathematical Problem Solving: A Survey and a Mathematics Test |

Teaching Algebra Conceptually: Student Achievement |

Does Knowing More Advanced Mathematics Ensure Effectiveness of Working Towards Demonstrating Specialised Mathematical Content Knowledge of Second-Year Pre-Service Teachers? |

The Development of an Assessment Tool: Student Knowledge of the Concept of Place Value |

Projects, Puzzles and other Pedagogies: Working with Kids to Solve Local Problems |

The Impact of a Professional Learning Intervention Designed to Enhance Year Six Students' Computational Estimation Performance |

An Exploration into Growing Patterns with Young Australian Indigenous Students |

The Four-Three-Four Model: Drawing on Partitioning, Equivalence, and Unit-Forming in a Quotient Sub-Construct Fraction Task |

Virtual Mathematics Education: Using Second Life to Model and Reflect upon the Teaching of Mathematics |

Developing Pedagogical Strategies to Promote Structural Thinking in Early Mathematics |

The Influence of Gender, Parents, and Background Variables on Perceived Usefulness of Mathematics among Grade 7 Students in Mozambique |

Problem Categorisation in Ratio - A Closer Look |

Student Understanding of Large Numbers and Powers: The Effect of Incorporating Historical Ideas |

The Concept of Generalised Number: Valuable Lessons from the History of Algebra |

Mathematics Anxiety in Secondary School Students |

Students' Summaries of Mathematical Lectures: Comparing the Discourse of Students with the Discourse of Lectures |

Identity and Ethnomathematics Projects in Papua New Guinea |

Let's Count: Evaluation of a Pilot Early Mathematics Program in Low Socioeconomic Locations in Australia |

Trialling a Professional Statistical Literacy Hierarchy for Teachers |

The Classicist and the Frequentist Approach to Probability within a TinkerPlots2 Combinatorial Problem |

Diffusion of the Mathematics Practical Paradigm in the Teaching of Problem Solving: Theory and Praxis |

Coordinating Known and Unknown Quantities in a Multiplicative Context: Problem Conceptualization, Affordances and Constraints |

Strategies Used by Students to Compare Two Data Sets |

Exploring Student Reflective Practice during a Mathematical Modelling Challenge |

Steps in Developing a Quality Whole Number Place Value Assessment for Years 3-6: Unmasking the "Experts" |

On Diagnosis and Intervention: Helping Students with Special Needs Learn Fraction Ideas Involving Decimal Numbers |

Chinese Primary Students' Mathematical Task Types Preferences |

Mathematical Modelling for Singapore Primary Classrooms: From a Teacher's Lens |

Empirical Evidence for Niss' Implemented Anticipation in Mathematising Realistic Situations |

Supporting Teachers in Choosing and Using Challenging Mathematics Tasks |

Insights into Ways that teachers plan their Mathematics Teaching |

Students' Ways of Knowing and Learning Mathematics and Their Ways of Interacting with Advanced Calculators |

Pedagogical Content Knowledge in Mathematical Modelling Instruction |

Developing Mathematical Knowledge through Social Justice Pedagogy with Young Adult Arab Women |

Developing Mathematical Resilience among Aboriginal Students |

Use of Practical Worksheet in Teacher Education at the Undergraduate and Postgraduate Levels |

The Effect of Small-Group Game Play Activities on Number Sense Performance |

Average Revisited in Context |

Teaching for Abstraction: Collaborative Teacher Learning |

Australian Pre-Service Teachers Overseas Tour: Implications for Mathematics Teaching and Learning |

Investigating Pre-service Teachers' Mathematics Anxiety Using the Revised Mathematics Anxiety Scale (RMARS) |

A Revolving Model of Pre-service Teacher Development in Mathematics |

Nature of an Attitudes toward Learning Mathematics Questionnaire |

Profiling Students' Capacities to Link Number and Algebra in Years 5, 6 and 7 in Nanjing, Chin |

Problem Posing in Mathematical Investigation |

Predictive Validity of Numeracy Entry Requirements for University: Pre-service Teachers' Mathematics Knowledge and Attitudes |

Short Communication (abstract only) |

A Longitudinal Study Preparing Preservice Teachers to Learn and Teach Mathematics with Technology 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. |

Alleviating Maths Anxiety through Mentoring in an Emotional Intelligence Framework 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. |

An Analysis of Students' Strategies for Area Measurement and its Curricular Implications 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. |

Classroom Goal Structure, Achievement Goals, and Achievement: A Multilevel Mediational Analysis of Longitudinal Data 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. |

Critical Reflection as an Important Pathway to Pre-service Teachers' Development (A Snapshot) 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. |

Developing an Intervention Program for Students at Risk Drawing on the Strengths of Successful Existing Programs 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. |

Developing Computational Fluency in Multi-Digit Multiplication: A Learning Trajectory Approach 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 Statistical Literacy: Student Learning and Teacher Education 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. |

Do not Call on Me: Mathematics Anxiety among Students with Learning Differences 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. |

Early Childhood Mathematics: The Case of More or Less 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. |

Examining Opportunities and Constraints in the Use of Context Based
Experiences for Engaging Indigenous Australian Students in the Learning
of 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. |

Harnessing the Power of Cloud Computing for Mathematical Modelling in Two Different Schools 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: Programming as a Means to Motivate Exploration of Foundational Algebraic Concepts 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. |

Investigating the Interrelationships Between Teachers' Pedagogical
Content Knowledge and Student Achievement Within Vocational High School
Context 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. |

Problem@Web: A Project around an On-line Problem Solving Mathematical Competition in Portugal 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: studentsâ€™ 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. |

Seeing is Believing: Building Mathematical Knowledge for Teaching Through Demonstration Lessons 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 Student Engagement and Disengagement in Mathematics 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 (HadrÃ¨, 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. |

The Use of Open-Ended Tasks as an Instructional and Assessment Tool 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. |

Transforming Children's Mathematical and Scientific Development: A Longitudinal Study 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. |

Translation of Word Problems by Year 6 Lower Ability Students: An Action Research Project 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. |

Using Students' Algebraic Thinking to Support Teacher Learning 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. |

Poster (abstract only) |

Round Table (abstract only) |

Developing Self-Regulated Teacher Learners to Improve Student Outcomes in Mathematics 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. |

Models and Modelling for the Future 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. |

Professional Conversations among Mathematics Educators 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. |

Students' Conceptions of Equality 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 student's 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. |

Teachers' Mathematical Knowledge and Practice 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 |

Teaching Mathematics Respectfully: Preparing Culturally Responsive Mathematics Teachers 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. |