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These collected papers are a record of the proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia entitled, "Shaping the Future of Mathematics Education." The conference is held in Fremantle, as was the previous conference a decade earlier. Keynote speakers are discussing issues that are emerging more sharply at the present time as the mathematics education community focuses on the education of Indigenous children, the scope of the mathematics curriculum and ways in which the content might be taught effectively, and the significant role of technologies in teaching and learning in the twenty-first century. Research papers are featured from mathematics educators from all states and territories of Australia, from colleagues in New Zealand, and from overseas--United Kingdom, Singapore, United States of America, India, Thailand, South Africa, and Indonesia. This set of proceedings includes abstracts and full papers for refereed research presentations, short communications of developing research, roundtable discussions, and symposia. Individual papers contain tables, footnotes, figures, references and appendices.

This document presents the proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia (MERGA). The theme of this conference is "Navigating Currents and Charting Directions." The theme reminds us that, although we are constantly pushed to account for the quality and impact of our research, we need to assert some control over our work by making our own research futures. The conference was sponsored by the University of Queensland, Teaching and Educational Development Institute. Volume 1 includes the following: (1) Stars, Compass, and GPS: Navigating Currents and Charting Directions for Mathematics Education Research on Gender Issues (Helen Forgasz); (2) Praxis and Practice Architectures in Mathematics Education (Stephen Kemmis); (3) Facilitating Communities of Mathematical Inquiry (Roberta Hunter); (4) Assessing Primary Preservice Teachers' Mathematical Competence (Karoline Afamasaga-Fuata'i, Paul Meyer, and Naomi Falo); (5) Teachers' Motivation to Attend Voluntary Professional Development in K-10 Mathematics (Judy Anderson); (6) Using National Numeracy Testing to Benefit Indigenous Students: Case Studies of Teachers Taking Back Control of Outcomes (Annette R. Baturo, Tom J. Cooper, Matthew T. Michaelson, and Jessica Stevenson); (7) Recollections of Mathematics Education: Approaching Graduation and 5 Years Later (Kim Beswick and Shelley Dole); (8) Using Paper-Folding in the Primary Years to Promote Student Engagement in Mathematical Learning (Kathy Brady); (9) The Case of Mathematical Proof in Lower Secondary School: Knowledge and Competencies of Pre-service Teachers (Jill Brown, Gloria Stillman, Bjorn Schwarz, and Gabriele Kaiser); (10) Employing Mathematical Modelling to Respond to Indigenous Students' Needs for Contextualised Mathematics Experiences (Kelli Brown); (11) Reconceptualising Agency through Teachers Talking About a Sociocultural Approach to Teaching Mathematics in the Classroom (Raymond Brown and Trevor Redmond); (12) Middle School Students' Interest in Statistical Literacy (Colin Carmichael and Ian Hay); (13) One Secondary Teacher's Use of Problem-Solving Teaching Approaches (Michael Cavanagh); (13) Does Student Success Motivate Teachers to Sustain Reform-Oriented Pedagogy? (Linda Cheeseman); (15) Year Five Students Solving Mental and Written Problems: What Are They Thinking? (Julie Clark); (14) Mathematics for Engineering Education: What Students Say (Mary Coupland, Anne Gardner, and Georgina Carmody); (15) Advancing Research Into Affective Factors in Mathematics Learning: Clarifying Key Factors, Terminology and Measurement (Patricia C. Cretchley); (16) Explorations of Early Childhood: New Entrant Transition in Mathematics (Ngaire M. Davies and Karen Walker); (17) Eliciting Growth in Teachers' Proportional Reasoning: Measuring the Impact of a Professional Development Program (Shelley Dole, Doug Clark, Tony Wright, Geoff Hilton, and Anne Roche); (18) Links between Children's Understanding of Multiplication and Solution Strategies For Division (Ann Downton); (19) Intervention Instruction in Structuring Numbers 1 to 20: The Case of Nate (David Ellemor-Collins and Robert Wright); (20) Interdisciplinary Problem Solving: A Focus on Engineering Experiences (Lyn D. English); (21) Addressing Verbal Memory Weaknesses to Assist Students with Mathematical Learning Difficulties (Maureen Finnane); (22) Validation of an Assessment Instrument Developed for Eliciting Student Prior Learning in Graphing and Data Analysis (Noleine Fitzallen); (24) Using Valsiner (Linda Galligan); (25) CAS Enabled Devices as Provocative Agents in the Process of Mathematical Modelling (Vince Geiger, Rhonda Faragher, Trevor Redmond, and Jim Lowe); (26) Researcher-Teacher Relationships in Mathematics Education (Merrilyn Goos); (27) Towards a Sociocultural Framework for Understanding the Work of Mathematics Teacher-Educator-Researchers (Merrilyn Goos); (28) Identity as a Lens to Understand Learning Mathematics: Developing a Model (Peter Grootenboer and Robyn Zevenbergen); (29) Capturing Students' Thinking about Strategies used to Solve Mental Computations by Giving Students Access to a Pedagogical Framework (Judy Hartnett); (30) A Review of Recent Research in Early Mathematics Learning and Technology (Kate Highfield and Kristy Goodwin); (31) The Development of Students' Use of Justification Strategies (Jodie Hunter and Glenda Anthony); (32) Using Task-Based Interviews to Assess Mathematical Thinking of Primary School Students (Chris Hurst); (33) Who a Student Sits Near to in Maths: Tension between Social and Mathematical Identities (Naomi Ingram); (34) Social Constructivism in the Classroom: From a Community of Learners to a Community of Teachers (Jane Irvin); (35) Primary Teachers' Beliefs about the Use of Mathematics Textbooks (Romina Jamieson-Proctor and Carmen Byrne); (36) Abstraction in Context, Combining Constructions, Justification and Enlightenment (Ivy Kidron and Tommy Dreyfus); (37) How Humanism Can Foster Mediocrity in Early Years Mathematics Education: A Poststructuralist Comparison (Mary Klein); (38) Preservice Teachers and Numeracy Education: Can Poststructuralism Contribute? (Mary Klein); and (39) High Achievers in Mathematics: What Can We Learn from and about Them? (Gilah Leder). Volume 2 contains: (1) The 2007 Common Technology Free Examination for Victorian Certificate of Education (VCE) Mathematical Methods and Mathematical Methods Computer Algebra System (CAS) (David Leigh-Lancaster, Pam Norton, Peter Jones, Magdalena Les, Michael Evans, and Margaret Wu); (2) Focusing Year 8 Students on Self-Regulating their Learning of Mathematics (Andrea McDonough and Peter Sullivan); (3) Feedback about Professional Growth for Teachers of Mathematics: A Developmental Perspective (Greg McPhan, John Pegg, and Stefan Horarik); (4) Fraction Number Line Tasks and the Additivity Concept of Length Measurement (Annie Mitchell and Marj Horne); (5) "Zero Is Not a Number": Teachable Moments and Their Role in Effective Teaching of Numeracy (Tracey Muir); (6) Students' Attitude Towards Using Materials to Learn Algebra: A Year 7 Case Study (Stephen Norton and Will Windsor); (7) Teaching Mathematics and Technology through Design Practice (Stephen Norton and Tom J. Cooper); (8) Engaging Mathematics Teachers in Professional Learning by Reflecting on Their Pedagogical Practice (Richard O'Donovan); (9) Primary Teachers' Perceptions of Their Knowledge and Understanding of Measurement (Michelle O'Keefe and Janette Bobis); (10) Use of the Internet for Teacher Professional Development and for Teaching Mathematics: Supports and Inhibitors (Sitti Maesuri Patahuddin); (11) A Situated Perspective on Learning to Teach Secondary Mathematics (Anne Prescott and Michael Cavanagh); (12) The Hospital Problem Revisited. Tertiary Students' Perceptions of a Problem Involving the Binomial Distribution (Robyn Reaburn); (13) The Identification of Partially Correct Constructs (Gila Ron, Rina Hershkowitz, and Tommy Dreyfus); (14) Making Connections: Promoting Connectedness in Early Mathematics Education (Abigail Sawyer); (15) Engagement versus Deep Mathematical Understanding: An Early Career Teacher's Use of ICT in a Lesson (Anne Scott, Ann Downton, Donna Gronn, and Adam Staples); (16) Investigating a Phase Approach to Using Technology as a Teaching Tool (Penelope Serow); (17) The Introduction of Interactive Whiteboard Technology in the Primary Mathematics Classroom: Three Case Studies (Penelope Serow and Rosemary Callingham); (18) School Readiness: What Do Teachers Expect of Children in Mathematics on School Entry? (Brenda Sherley, Megan Clark, and Joanna Higgins); (19) Gaining Insight into Alice's Pedagogy with Respect to Five Dimensions of Numeracy (Jane Skalicky); (20) Modes of Reasoning in Explanations in Year 8 Textbooks (Kaye Stacey and Jill Vincent); (21) What Does Three-Quarters Look Like? Students' Representations of Three-Quarters (Vicki Steinle and Beth Price); (22) Some Key Junctures in Relational Thinking (Max Stephens); (23) Chinese Young Children's Strategies on Basic Addition Facts (Huayu Sun); (24) Self-Efficacy in Mathematics: Affective, Cognitive, and Conative Domains of Functioning (S. L. Tait-McCutcheon); (25) Neuropsychological Evidence for the Role of Graphical and Algebraic Representations in Understanding Function (Michael O. J. Thomas, Anna J. Wilson, Michael C. Corballis, and Vanessa K. Lim); (26) Speaking with Different Voices: Knowledge Legitimation Codes of Mathematicians and Mathematics Educators (Steve Thornton); (27) Recognising Different Starting Points in Aboriginal Students' Learning of Number (Kaye Treacy and Sandra Frid); (28) Deepening the Mathematical Knowledge of Secondary Mathematics Teachers Who Lack Tertiary Mathematics Qualifications (Colleen Vale and Alasdair McAndrew); (29) Indigenous Students' Early Engagement with Numeracy: The Case of Widgy and Caddy (Elizabeth Warren, Janelle Young, and Eva De Vries); (30) Building Informal Inference in Grade 7 (Jane Watson and Julie Donne); (31) Proportional Reasoning: Student Knowledge and Teachers' Pedagogical Content Knowledge (Jane Watson, Rosemary Callingham, and Julie Donne); (32) Counting on 2007: A Program for Middle Years Students Who Have Experienced Difficulty with Mathematics (Allan White); (33) How Group Composition Can Influence Opportunities for Spontaneous Learning (Gaye Williams); (34) Success and Consistency in the Use of Heuristics to Solve Mathematics Problems (Khoon Yoong Wong); (35) Fractions as a Measure (Monica Wong and David Evans); (36) Mixing Colours: An ICT Tool Based on a Semiotic Framework for Mathematical Meaning-Making about Ratio and Fractions (Andy Yeh and Rod Nason); (37) Secondary School Students Investigating Mathematics (Joseph Yeo); (38) Teaching Area and Perimeter: Mathematics-Pedagogical-Content Knowledge-in-Action (Kai Kow and Joseph Yeo); (39) Problem Solving Activities in a Constructivist Framework: Exploring How Students Approach Difficult Problems (Oleksiy Yevdokimov and Tim Passmore); (40) Creating Equitable Practice in Diverse Classrooms: Developing a Tool to Evaluate Pedagogy (Robyn Zevenbergen, Richard Niesche, Peter Grootenboer, and Jo Boaler); (41) The Role of Information Graphics in Mathematical Proficiency (Carmel Diezmann and Tom Lowrie); (42) A Longitudinal Study of Student Performance on Items Rich in Graphics (Tom Lowrie); (43) Standardised Assessment in Mathematics: The Tale of Two Items (Tracy Logan and Jane Greenless); (44) Graphics and the National Numeracy Tests (Carmel Diezmann); (45) The Construction of Knowledge: Theoretical Approaches (Tommy Dreyfus, Michael O. J. Thomas, Jill P. Brown, and Gaye Williams); (46) Motivation and Engagement in School Mathematics During the Middle Years (Catherine Attard); (47) Moving beyond the Script: Addressing Numeracy Needs of Low Achieving Students through Quality Professional Development (Moira Blair and Anuja Singh); (48) MERGA: Including the X and Y in Mathematics Education Research (Peter Grootenboer and Naomi Ingram); (49) Issues in Bridging between Senior Secondary and First Year University Mathematics (Michael Jennings); (50) Student Achievement in Mathematics: Learning through Home School Partnership (Honor Ronowicz and Gaynor Terrill); (51) The Role of Mathematics Competitions (Brenda Bicknell); (52) Scaffolding Students' Understanding of Geometric Properties Using Dynamic Geometry Software (Sahar Bokosmaty); (53) The Next Big Teaching Resource: Interactive White Boards But Where is the Research? (Philip C. Clarkson); (54) Errors Made by Student Teachers When Writing Test Items (Jaguthsing Dindyal); (55) Insights from Pre-Service Secondary Mathematics Teachers on their Practicum Experience (Jaguthsing Dindyal); (56) Improving Language for Problem Solving (Benedicte Esterman); (57) Reflections on Exponential Functions (Sandra Herbert and Farhang Afshar); (58) Coaching and Mentoring Numeracy Lead Teachers to Improve Student Learning: The Journey of Two Year Seven and Eight Teachers (Judith Mills); (59) Connecting the Points: Students Learning Decimal Place Value (Bruce Moody and Jenny Young-Loveridge); (60) The Role of Pattern and Structure in Early Mathematics Learning: An Evaluation Study in the First Year of Formal Schooling (Joanne Mulligan, Lyn English, and Mike Mitchelmore); (61) Motivation and Engagement in Mathematics: The Transition from Primary to Secondary School (Karen Skilling); (62) Achieving Computational Fluency in Multi-Digit Multiplication and Division (Kristen Tripet); and (63) Financial Modelling with Matlab (Peter Watson and Jiling Cao). (Individual papers contain references.) [Cover title varies.]

This is a record of the proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (MERGA). The theme of the conference is "Mathematics: Essential research, essential practice." The theme draws attention to the importance of developing and maintaining links between research and practice and ties in with the joint day of presentations with the 21st biennial conference of the Australian Association of Mathematics Teachers (AAMT). This special feature highlights the benefits of collaboration between researchers, practicing classroom teachers, and curriculum developers. Volume 1 contains the following papers: (1) The Beginnings of MERGA (Ken Clements); (2) Teaching and Learning by Example: The Annual Clements/Foyster Lecture (Helen L. Chick); (3) Introducing Students to Data Representation and Statistics (Richard Lehrer); (4) Studies in the Zone of Proximal Awareness (John Mason, Helen Drury and Liz Bills); (5) Empowered to Teach: A Practice-based Model of Teacher Education (Janette Bobi); (6) Communicating Students' Understanding of Undergraduate Mathematics using Concept Maps (Karoline Afamasaga-Fuata'i); (7) Primary Student Teachers' Diagnosed Mathematical Competence in Semester One of their Studies (Karoline Afamasaga-Fuata'i, Paul Meyer and Naomi Falo); (8) An Online Survey to Assess Student Anxiety and Attitude Response to Six Different Mathematical Problems (Vincent Anderson); (9) Mathematical Investigations: A Primary Teacher Educator's Narrative Journey of Professional Awareness (Judy Bailey); (10) Describing Mathematics Departments: The Strengths and Limitations of Complexity Theory and Activity Theory (Kim Beswick, Anne Watson and Els De Geest); (11) Three Student Tasks in a Study of Distribution in a "Best Practice" Statistics Classroom (Anthony Bill and Jane Watson); (12) Teacher Researchers Questioning their Practice (Linda Bonne & Ruth Pritchard); (13) Imagined Classrooms: Prospective Primary Teachers Visualise their Ideal Mathematics Classroom (Kathy Brady); (14) Early Notions of Functions in a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill Brown); (15) Collective Argumentation and Modelling Mathematics Practices Outside the Classroom (Raymond Brown and Trevor Redmond); (16) Visual Perturbances in Digital Pedagogical Media (Nigel Calder); (17) Professional Experience in Learning to Teach Secondary Mathematics: Incorporating Pre-service Teachers into a Community of Practice (Michael Cavanagh and Anne Prescott); (18) Young Children's Accounts of their Mathematical Thinking (Jill Cheeseman and Barbara Clarke); (19) Mathematical Reform: What Does the Journey Entail for Teachers? (Linda Cheeseman); (20) Year Six Fraction Understanding: A Part of the Whole Story (Doug M. Clarke, Anne Roche and Annie Mitchell); (21) Teaching as Listening: Another Aspect of Teachers' Content Knowledge in the Numeracy Classroom (Ngaire Davies and Karen Walker); (22) Essential Differences between High and Low Performers' Thinking about Graphically-Oriented Numeracy Items (Carmel M. Diezmann, Tom J. Lowrie and Nahum Kozak); (23) High School Students' Use of Patterns and Generalizations (Jaguthsing Dindyal); (24) The Teacher, The Tasks: Their Role in Students' Mathematical Literacy (Katherine Doyle); (25) Informal Knowledge and Prior Learning: Student Strategies for Identifying and Locating Numbers on Scales (Michael Drake); (26) Documenting the Knowledge of Low-Attaining Third- and Fourth-Graders: Robyn's and Bel's Sequential Structure and Multidigit Addition and Subtraction (David Ellemor-Collins, Robert Wright and Gerard Lewis); (27) Interdisciplinary Modelling in the Primary Mathematics Curriculum (Lyn English); (28) Students' Tendency to Conjoin Terms: An Inhibition to Their Development of Algebra (Judith Falle); (29) Towards "Breaking the Cycle of Tradition" in Primary Mathematics (Sandra Frid and Len Sparrow); (30) Exploring the Number Knowledge of Children to Inform the Development of a Professional Learning Plan for Teachers in the Ballarat Diocese as a Means of Building Community Capacity (Ann Gervasoni, Teresa Hadden and Kathie Turkenburg); (31) Technology-Enriched Teaching of Secondary Mathematics: Factors Influencing Innovative Practice (Merrilyn Goos and Anne Bennison); (32) Supporting an Investigative Approach to Teaching Secondary School Mathematics: A Professional Development Model (Merrilyn Goos, Shelley Dole, and Katie Makar); (33) Identity and Mathematics: Towards a Theory of Agency in Coming to Learn Mathematics (Peter Grootenboer and Robyn Zevenbergen); (34) Categorisation of Mental Computation Strategies to Support Teaching and to Encourage Classroom Dialogue (Judy Hartnett); (35) Student Experiences of VCE Further Mathematics (Sue Helme and Stephen Lamb); (36) Video Evidence: What Gestures Tell Us About Students' Understanding of Rate of Change (Sandra Herbert and Robyn Pierce); (37) The Role of Dynamic Interactive Technological Tools in Preschoolers' Mathematical Patterning (Kate Highfield and Joanne Mulligan); (38) Students Representing Mathematical Knowledge through Digital Filmmaking (Geoff Hilton); (39) What Does it Mean for an Instructional Task to be Effective? (Lynn Hodge, Jana Visnovska, Qing Zhao and Paul Cobb); (40) A School-Community Model for Enhancing Aboriginal Students' Mathematical Learning (Peter Howard and Bob Perry); (41) Benchmarking Preservice Teachers' Perceptions of their Mentoring for Developing Mathematics Teaching Practices (Peter Hudson); (42) Relational or Calculational Thinking: Students Solving Open Number Equivalence Problems (Jodie Hunter); (43) Scaffolding Small Group Interactions (Roberta Hunter); (44) Numeracy in Action: Students Connecting Mathematical Knowledge to a Range of Contexts (Chris Hurst); and (45) A Story of a Student Fulfilling a Role in the Mathematics Classroom (Naomi Ingram). Volume 2 contains the following papers: (1) Secondary-Tertiary Transition: What Mathematics Skills Can and Should We Expect This Decade? (Nicolas Jourdan, Patricia Cretchley and Tim Passmore); (2) The Power of Writing for all Pre-service Mathematics Teachers (Keith McNaught); (3) "Connection Levers:" Developing Teachers' Expertise with Mathematical Inquiry (Katie Makar); (4) Acquiring the Mathematics Register in te reo Maori (Tamsin Meaney, Uenuku Fairhall and Tony Trinick); (5) Teaching Ratio and Rates for Abstraction (Mike Mitchelmore, Paul White and Heather McMaster); (6) Setting a Good Example: Teachers' Choice of Examples and their Contribution to Effective Teaching of Numeracy (Tracey Muir); (7) Developing the Concept of Place Value (Mala Saraswathy Nataraj and Michael O. J. Thomas); (8) Interdisciplinary Learning: Development of Mathematical Confidence, Value, and the Interconnectedness of Mathematics Scales (Dawn Kit Ee Ng and Gloria Stillman); (9) Mathematical Methods and Mathematical Methods Computer Algebra System (CAS) 2006--Concurrent Implementation with a Common Technology Free Examination (Pam Norton, David Leigh-Lancaster, Peter Jones and Michael Evans); (10) A Concrete Approach to Teaching Symbolic Algebra (Stephen Norton and Jane Irvin); (11) Developing Positive Attitudes towards Algebra (Stephen Norton and Jane Irvin); (12) Changing Our Perspective on Measurement: A Cultural Case Study (Kay Owens and Wilfred Kaleva); (13) Enhancing Student Achievement in Mathematics: Identifying the Needs of Rural and Regional Teachers in Australia (Debra Panizzon and John Pegg); (14) The Growth of Early Mathematical Patterning: An Intervention Study (Marina Papic and Joanne Mulligan); (15) Whole Number Knowledge and Number Lines Help to Develop Fraction Concepts (Catherine Pearn and Max Stephens); (16) Identifying and Analysing Processes in NSW Public Schooling Producing Outstanding Educational Outcomes in Mathematics (John Pegg, Debra Panizzon and Trevor Lynch); (17) Teachers Research their Practice: Developing Methodologies that Reflect Teachers' Perspectives (Ruth Pritchard and Linda Bonne); (18) Teacher Professional Learning in Mathematics: An Example of a Change Process (Pauline Rogers); (19) Seeking Evidence of Thinking and Mathematical Understandings in Students' Writing (Anne Scott); (20) Utilising the Rasch Model to Gain Insight into Students' Understandings of Class Inclusion Concepts in Geometry (Penelope Serow); (21) Exploring Teachers' Numeracy Pedagogies and Subsequent Student Learning across Five Dimensions of Numeracy (Jane Skalicky); (22) The Complexities for New Graduates Planning Mathematics Based on Student Need (Carole Steketee and Keith McNaught); (23) Students' Emerging Algebraic Thinking in the Middle School Years (Max Stephens); (24) A Framework for Success in Implementing Mathematical Modelling in the Secondary Classroom (Gloria Stillman, Peter Galbraith, Jill Brown and Ian Edwards); (25) Eliciting Positive Student Motivation for Learning Mathematics (Peter Sullivan and Andrea McDonough); (26) Learning from Children about their Learning with and without ICT using Video-Stimulated Reflective Dialogue (Howard Tanner and Sonia Jones); (27) Dependency and Objectification in a Year 7 Mathematics Classroom: Insights from Sociolinguistics (Steve Thornton); (28) Pedagogical Practices with Digital Technologies: Pre-service and Practicing Teachers (Colleen Vale); (29) Procedural Complexity and Mathematical Solving Processes in Year 8 Mathematics Textbook Questions (Jill Vincent and Kaye Stacey); (30) Designing Effective Professional Development: How do we Understand Teachers' Current Instructional Practices? (Jana Visnovska); (31) "Doing Maths:" Children Talk About Their Classroom Experiences (Fiona Walls); (32) The Role of Pedagogy in Classroom Discourse (Margaret Walshaw and Glenda Anthony); (33) Australian Indigenous Students: The Role of Oral Language and Representations in the Negotiation of Mathematical Understanding (Elizabeth Warren, Janelle Young and Eva deVries); (34) Student Change Associated with Teachers' Professional Learning (Jane Watson, Kim Beswick, Natalie Brown and Rosemary Callingham); (35) Choosing to Teach in the "STEM" Disciplines: Characteristics and Motivations of Science, ICT, and Mathematics Teachers (Helen M. G. Watt, Paul W. Richardson and James Pietsch); (36) Percentages as Part Whole Relationships (Paul White, Sue Wilson, Rhonda Faragher and Mike Mitchelmore); (37) My Struggle with Maths May Not Have Been a Lonely One: Bibliotherapy in a Teacher Education Number Theory Unit (Sue Wilson); (38) Students' Conceptual Understanding of Equivalent Fractions (Monica Wong and David Evans); (39) Statistics Teachers as Scientific Lawyers (Joanne Woodward and Maxine Pfannkuch); (40) Developing Pedagogical Tools for Intervention: Approach, Methodology, and an Experimental Framework (Robert Wright, David Ellemor-Collins and Gerard Lewis); (41) Pedagogy and Interactive Whiteboards: Using an Activity Theory Approach to Understand Tensions in Practice (Robyn Zevenbergen and Steve Lerman); (42) International Perspectives on Early Years Mathematics (Jillian Fox); (43) Early Childhood Mathematics Education Research: What is Needed Now? (Bob Perry and Sue Dockett); (44) Trimangles and Kittens: Mathematics Within Socio-dramatic Play in a New Zealand Early Childhood Setting (Shiree Lee); (45) Children's Number Knowledge in the Early Years of Schooling (Ann Gervasoni); (46) Listening to Students' Voices in Mathematics Education (Brian Doig, Susie Groves, Coral Campbell, Judith Mousley, and Gaye Williams); (47) Students' Pedagogical Knowledge: A Source of Pedagogical Content Knowledge (Brian Doig and Susie Groves); (48) Research Enriched by the Student Voice (Gaye Williams); (49) Listening to Student Opinions about Group Assessment (Judith Mousley and Coral Campbell); (50) Profiles of Thinking Skills and Levels of Motivation in a Problem-Solving Task (Sarah Buckley, Mary Ainley and Pip Pattison); (51) An Investigation of Mathematics Strategies in Traditional School Contexts and Real-World Contexts (Julie Clark and Kathy Brady); (52) Maori Student's Perspective on Their Mathematical Journey Through Maori Medium (Leeana Herewini); (53) Some Methodological Considerations in the Estonian Study about Students' Beliefs in Mathematics: Is Triangulation Necessary? (Kirsti Kislenko); (54) Progress in Mathematics--Learning through Home School Partnership (Denise Smith and Gaynor Terrill); (55) Exploring Data Representation and Statistical Reasoning through Integrated Investigations in a Grade 2 Classroom (Karen Ahearn); (56) Reform and Assessment Practice: The Need for an Investigation (Julie Anderson); (57) Autobiographical Research and Mathematics Curriculum (Andy Begg); (58) Mathematically Gifted Students Managing School Transfer (Brenda Bicknell); (59) Improving Procedures for Effective Teaching (Murray Black, Farida Kachapova and Ilias Kachapov); (60) Using Counter-Examples and Paradoxes in Teaching Probability: Students' Attitudes (Murray Black, Farida Kachapova, Sergiy Klymchuk and Ilias Kachapov); (61) Using Cabri Geometry to Explore the Geometric Properties of Parallelograms in Year 7 Mathematics Classrooms (Sahar Bokosmaty); (62) Defining Teacher Knowledge Needed in the Teaching of Statistics at Primary School Level (Tim Burgess); (63) Year 12 Students' Participation in Higher Mathematics Courses (Mohan Chinnappan, Stephen Dinham, Tony Herrington and Dale Scott); (64) Pre-service Primary Teachers Developing Positive Attitudes towards Teaching Mathematics (Julie Clark); (65) Measuring the Effectiveness and Efficiency of Language-In-Use for Algebra Learning: A Multi-Level Nested Modelling and DEA Approach (Robert de la Serna); (66) Te Poutama Tau (TPT): An Indigenous Response to the Numeracy Development Project 2002-2006 (Wini Emery and Leeana Herewini); (67) "I Have a Fear of Maths and it Does Worry Me a Bit as a Future Teacher:" The Cycle of Maths Anxiety (Gillian Frankcom); (68) Using Electronic Handwriting and Tablet PCs to Enhance Distance Students' Understanding of First Year Mathematics at University (Linda Galligan, Birgit Loch, Janet Taylor and Christine McDonald); (69) Mathematical Modelling in CAS Clothing (Vince Geiger, Rhonda Faragher and Trevor Redmond); (70) The Cognitive and Pedagogical Affordances of Digital Learning Tools on Early Mathematical Development (Kristy Goodwin, Joanne Mulligan and John Hedberg); (71) Revisions and Extensions of a Pirie-Kieren-Based Teaching Model (Peter Hughes); (72) An Insight into Norwegian Students' Thoughts about Mathematics (Kirsti Kislenko); (73) The Impact of Didactical Contract on Students' Perceptions of their Intentional Learning Acts (Troels Lange and Tamsin Meaney); (74) The Impact of an Intervention on the Development of Mathematical Pattern and Structure in the First Year of Schooling (Joanne Mulligan, Mike Mitchelmore, Coral Kemp, Jennie Marston and Kate Highfield); (75) Activity Theory as a Framework to Analyse the Positive Influence of Formative Assessment on Student Learning (Trish O'Toole); (76) Teaching Geometry with CAS in the Junior Secondary Classroom: A Case Study (Warren Palmer); (77) Wanted: One Great Maths Teacher! (Pamela Perger); (78) Building Early Childhood Educators' Knowledge, Skills and Confidence in the Facilitation and Assessment of Young Children's Mathematical Learning (Bob Perry, Elspeth Harley and Sue Dockett); (79) CAS in the Middle Secondary Years: Strengths, Weaknesses, Opportunities and Threats (Robyn Pierce); (80) Myths and Positioning: Insights from Hermeneutics (Steve Thornton); (81) Misconceptions in Locating Negative Decimals on the Number Line (Wanty Widjaja, Kaye Stacey and Vicki Steinle); and (82) Proportional Reasoning: A Global or Localised Development? (Vince Wright). [Individual papers contain references, figures, tables, and appendices.

This document represents volumes 1 and 2 of the proceedings of the 24th annual conference of the Mathematics Education Research Group of Australasia (MERGA) held at the University of Sydney, June 30-July 4, 2001. In volume 1, papers include: (1) "Connecting Mathematics Education Research to Practice" (Judith Sowder); (2) "Understanding, Assessing, and Developing Young Children's Mathematical Thinking: Research as a Powerful Tool for Professional Growth" (Doug Clarke); (3) "Visions of Practice: Getting the Balance Right" (Tracey Smith and Tom Lowrie); (4) "The Mathematics Enhancement Project: Combining Research and Development" (Wilfredo Alangui, Jessie Autagavaia, Bill Barton, and Albert Poleki); (5) "Multiple Learning Contexts: A Vehicle for Changing Preservice Primary Teachers' Mathematical Beliefs, Knowledge, and Practices" (Sharne Aldridge and Janette Bobis); (6) "CGI Down Under" (Glenda Anthony, Brenda Bicknell, and Jan Savell); (7) "Year 9 Mathematics Schemes in New Zealand" (Robin Averill and Megan Clark); (8) "Assessing Algebraic Expectation" (Lynda Ball, Kaye Stacey, and Robyn Pierce); (9) "The Gender Stereotyping of Mathematics: Cultural Dimensions" (Anastasios Barkatsas, Helen Forgasz, and Gilah Leder); and (10) "Collaborative Learning and Student Change: A Case Study" (Mary Barnes). Volume 2 papers include: (1) "Fastback Practices and the Classroom Culture" (Patricia Hosking and Mal Shield); (2) "Learning Mathematics: The Voices of Aboriginal Children" (Peter Howard and Bob Perry); (3) "Recognizing the Difference between Additive and Multiplicative Thinking in Young Children" (Lorraine Jacob and sue Willis); (4) "Identifying Testable Teacher Beliefs through Teacher-Led Professional Development" (Stephen Keast); (5) "Testing for Additivity in Intuitive Thinking of Area" (Gillian Kidman); (6) "A Poststructuralist Analysis of Mathematical Inquiry in a Year 6 Classroom: The Rhetoric, Realization, and Practical Implications" (Mary Klein); (7) "Correcting Mathematical and Attitudinal Deficiencies in Preservice Teacher Education: The Conservative Effect of Blaming the Victim" (Mary Klein); (8) "Australian Year 7 Benchmarks in Numeracy: Student Performance on Matching TIMSS Items" (John Lindsey and Barry McCrae); (9) "The Influence of Visual Representations on Mathematical Problem Solving and Numeracy Performance" (Tom Lowrie); and (10)"Misperceptions in Mathematics and Music: An Intra- and Inter-Disciplinary Investigation Employing Applied Metacognition via Computer-Aided Learning" (John Malone, Sam Leong, and Martin Lamb). (ASK)

This document contains Volumes 1 and 2 of the proceedings of the 23rd annual conference of the Mathematics Education Research Group of Australasia Incorporated (MERGA) held at Fremantle, Western Australia, July 5-9, 2000. Papers in Volume 1 include: (1) "Bridging Practices: Intertwining Content and Pedagogy in Teaching and Learning To Teach" (Deborah Loewenberg Ball); (2) "Looking Backwards, Looking Forwards, and Looking for Direction" (Andy Begg); (3) "Bridging the Gap: A Challenge for the Dual Community" (John A. Malone); (4) "Collaborative Problem Solving in Senior Secondary Mathematics Classrooms" (Merrilyn Goos); (5) "Changing the Professional Knowledge of Teachers" (Janette Bobis and Peter Gould); (6) "Teachers, Students and Research: One Possibility for Teaching Early Numeracy" (Philip Clarkson); (7) "Mapping Children's Stages of Arithmetical Solution Methods" (Peter Gould and Robert Wright); (8) "Reviewing Literature Relevant to a Systemic Early Numeracy Initiative: Bases of CMIT" (Robert Wright and Peter Gould); (9) "Emergence of Mathematical (In)Competence and Identity" (Pat Forster and Peter Taylor); (10) "Is There More to Numeracy Than Meets the Eye? Stories of Socialization and Subjectification in School Mathematics" (Mary Klein); (11) "Mental Effort and Errors in Bracket Expansion Tasks" (Paul Ayres); (12) "Developing Task Specific Criteria: A Preliminary Report" (Dawn Bartlett); (13) "Construction of a Numeration Model: A Theoretical Analysis" (Annette Baturo); (14) "Modes of Representation in Students' Explanations" (Brenda Bicknell); and (15)"What Is Taught versus What Is Learnt: The Case of Linear Measurement" (Philippa Bragg and Lynne Outhred). Papers in Volume 2 include: (1) "Classroom and Teacher Effects in Mathematics Achievement: Results from TIMSS" (Stephen Lamb and Sue Fullarton); (2) "3-Dimensional Geometry: Assessment of Students' Responses" (Chrisitine Lawrie and John Pegg); (3) "Mathematics and Gender: Beliefs They Are a-Changin'" (Gilah Leder and Helen Forgasz); (4) "Informal Assessment Questions Used by Secondary School Mathematics Teachers" (Sanka Liyanage, Kathryn Irwin, and Mike Thomas); (5) "Grade 1 Children in Problem-Posing Contexts: Problem Solving Prior to Completing the Task" (Tom Lowrie); (6) "Knowledge and Strategies Students Employ To Solve Open-Ended Problem-Solving Activities" (Tom Lowrie, Rod Francis, and Geoff Rogers); (7) "Mental Computation, Number Sense and General Mathematics Ability: Are They Linked?" (Alistair McIntosh and Shelley Dole); (8) "Designing Constructivist Computer Games for Teaching about Decimal Numbers" (Janine McIntosh, Kaye Stacey, Calvin Tromp, and Darren Lightfoot); (9) "Scaffolding: A Suitable Teaching Characteristic in One-to-One Teaching in Maths Recovery" (Bronwyn McMahon); (10) "Low Achieving Mathematics Students' Attitudinal and Achievement Changes as a Result of Using an Integrated Learning System" (Campbell McRobbie, Annette Baturo, and Tom Cooper); (11) "Teaching for Abstraction: Reconstructing Constructivism" (Michael Mitchelmore and Paul White); (12) "Graphical Representations of Statistical Associations by Upper Primary Students" (Jonathan Moritz); (13) "Reasoning and Expressing Probability in Students' Judgements of Coin Tossing" (Jonathan Moritz and Jane Watson); (14) "The Practicum as Context: Two Snapshots" (Judith Mousley and Georgina Herbert); and (15) "A Comparative Analysis of the 1996-1999 Calculus TEE Papers" (Ute Mueller and Patricia Forster). (ASK)

Papers from the 22nd Annual Conference of the Mathematics Education Research Group of Australasia Incorporated include the following: (1) "Making Sense of Primary Mathematics" (Gillian M. Boulton-Lewis); (2) "Seeking a Rationale for Particular Classroom Tasks and Activity" (Peter Sullivan); (3) "Research in Mathematics Education in Australia: What it Was--What it Is--What it Might Be" (J.P. Keeves); (4) "Successful Percent Problem Solving for Year 8 Students Using the Proportional Number Line Method" (Shelley Dole); (5) "Cumulative and Exploratory Talk in a Collaborative Learning Classroom" (Mary Barnes); (6) "A Case Study of Teacher Endorsement of an Integrated Learning System" (Annette R. Baturo, Campbell J. McRobbie, Tom J. Cooper, and Gillian C. Kidman); (7) "Enactivism and Mathematics Education" (Andy Begg); (8) "The Writing of Explanations and Justifications in Mathematics: Differences and Dilemmas" (Brenda Bicknell); (9) "The Mathematical Achievement of Children in the Count Me In Too Program" (Janette Bobis and Peter Gould); (10) "Mathematics, ICT and Effective Teaching" (Rod Bramald and Steve Higgins); (11) "Mathematics Teachers and the VCE: Broadening the Educational Landscape" (Christine Brew, Gilah Leder, and Glen Rowley); (12) "An Exploration of Gender Differences in Subject Choice and Belief among Year 12 Students" (Peter C. Brinkworth); (13) "Speaking with Authority in Episodes of Mathematical Discourse" (Raymond A.J. Brown and Peter Renshaw); (14) "Do Games Help the Learning of Probability?" (Tim Burgess); (15) "Invented Algorithms: Teachers Face the Challenge" (Anne Buzeika); (16) "Developing Performance Assessment Tasks in Mathematics: A Case Study" (Rosemary A. Callingham); (17) "Children's Informal Composite and Truncated Partitioning Strategies" (Kathy Charles and Rod Nason); (18) "Jumping to Conclusions: Data Interpretation by Young Adults" (Helen L. Chick); (19) "Conceptual Modelling of Functions by an Experienced Teacher" (Mohan Chinnappan and Mike Thomas); (20) "The Use of Concept Mapping Procedure to Characterize Teachers' Mathematical Content Knowledge" (Mohan Chinnappan, Michael Lawson, and Rod Nason); (21) "Equals, Expressions, Equations, and the Meaning of Variable: A Teaching Experiment" (Tom J. Cooper, Anne M. Williams, and Annette R. Baturo); (22) "Assessing Diagram Quality: Making a Difference to Representation" (Carmel M. Diezmann); (23) "Enhanced Mathematics Learning: Does Technology Make a Difference?" (Katherine Dix); (24) "Of Course I Can['t] Do Mathematics: Ethnicity and the Stereotyping of Mathematics" (Helen J. Forgasz, Gilah C. Leder, and Tasos Barkatsas); (25) "Student Participation: Phenomenon and Analysis" (Patricia A. Forster); (26) "A Tale of Two Cities: When Mathematics, Computers, and Students Meet" (Peter Galbraith, Chris Haines, and Mike Pemberton); (27) "Technology, Mathematics, and People: Interactions in a Community of Practice" (Peter Galbraith, Peter Renshaw, Merrilyn Goos, and Vince Geiger); (28) "Teachers Exploring Numeracy Learning and Teaching in the Early Years of Schooling" (Ann Gervasoni); (29) "Understanding Mathematical Text through Peer Explanations" (Merrilyn Goos); (30) "Emerging Themes in Statistics Education" (Sharon Gunn); (31) "Mental Addition and Subtraction Strategies: Two Case Studies" (Ann Heirdsfield); (32) "The Development of a Framework of Growth Points to Monitor Students' Comprehension of Algebra in Grades 7-9" (Marj Horne); (33) "Influences on Secondary Mathematics Curriculum in Victoria" (John Horwood); (34) "Deaf Students Solving of Arithmetic Word Problems" (Merv Hyde, Des Power, and Robyn Zevenbergen); (35) "Mathematical Knowledge in a Mathematics Teaching Episode" (Clive Kanes); (36) "Learning Styles in Secondary Mathematics Classrooms" (Stephen Keast); (37) "Grade 4, 6 and 8 Students' Strategies in Area Measurement" (Gillian C. Kidman); (38) "Agency/Numeracy: A Poststructuralist Analysis of the Relationship between Classroom Mathematics and Numeracy in New Times" (Mary Klein); (39) "Assessment of Students' Understanding in Geometry: The Difficulties in Writing Good Questions" (Christine Lawrie and John Pegg); (40) "Extended Common Assessment Tasks in VCE Mathematics: Validity, Reliability and other Issues" (David Leigh-Lancaster and Ken Rowe); (41) "Free Problem-Posing: Year 3/4 Students Constructing Problems for Friends to Solve" (Tom Lowrie); (42) "Children's Generated Word Problems: A Case Study" (K.J. Maguire); (43) "The Learning Networks of Isolated Teachers: The Use of Audiographics Communication Technology to Build a Virtual Mathematics Department" (H. Elaine Mayo and Kevin Hannah); (44) "An Analysis of the Teacher's Proactive Role in Supporting the Development of Symbolizations" (Kay McClain and Paul Cobb); (45) "Learning Calculus with Supercalculators" (Barry McCrae, Gary Asp, and Margaret Kendal); (46) "Teacher Mediation of Student Constructions of Algebra Knowledge" (Brenda Menzel and David Clarke); (47) "How to Teach Generalisations in Mathematics?" (Michael Mitchelmore); (48) "The Conjunction Fallacy and Longitudinal Development of Chance Expression" (Jonathan B. Moritz and Jane M. Watson); (49) "Perceptions of Mathematical Understanding" (Judith A. Mousley); (50) "Graphics Calculators in the Public Examination of Calculus: Misuses and Misconceptions" (Ute Mueller and Patricia A. Forster); (51) "Secondary Mathematics Teachers' Responses to Computers and Their Beliefs about the Role of Computers in Their Teaching" (Stephen Norton); (52) "Visual Representations in First Year Statistics" (Lynne Outhred and Pamela F. Shaw); (53) "An Analysis of Year 12 Students' Performances on Basic Algebra Questions" (John Pegg and Robyn Hadfield); (54) "Some Language Issues in the Assessment of Secondary Mathematics" (Bob Perry, Peter Howard, and Brian Miller); (55) "Computer Algebra Systems Facilitate Positive Learning Strategies" (Robyn Pierce); (56) "The Conflict between Teachers' Beliefs and Classroom Practices" (Mal Shield); (57) "Understanding Decimals: The Path to Expertise" (Kaye Stacey and Vicki Steinle); (58) "Accessibility of Applications Tasks" (Gloria Stillman); (59) "Choices of Computational Strategies" (Paul Swan and Jack Bana); (60) "Reducing the Incidence of Mathematical Misconceptions in 'Middle Band' Students" (Philip Swedosh); (61) "Children's Understanding of the Number System" (Noel Thomas and Joanne Mulligan); (62) "Differences between the Sexes in Mathematics Achievement in Australia" (Tilahun Mengesha Afrassa and J.P. Keeves); (63) "The Cross-Cultural Perspective of Teachers' Beliefs and Their Influence on Teaching Practices: A Case Study of Two Teachers Teaching Secondary Mathematics in Australia and Indonesia" (Theresia Tirta Gondoseputro); (64) "Graphics Calculators in Victorian Secondary Schools: Teacher Perceptions of Use" (Patrick Tobin, Alla Routitsky, and Peter Jones); (65) "Cabri Geometry: A Catalyst for Growth in Geometric Understanding" (Jill Vincent and Barry McCrae); (66) "The Relationship between the Purported Use of Assessment Techniques and Beliefs about the Uses of Assessment" (Elizabeth Warren and Steven Nisbet); (67) "Longitudinal Understanding of Conditional Probability by School Students" (Jane M. Watson and Jonathan B. Moritz); (68) "Mathematics: More Time, More Ticks!" (Martin Watts, Sue Miller, and Leo Crameri); (69) "Teacher Action Theories and the Use of Group Work in Upper Primary Mathematics Classrooms" (Allan White); (70) "Exploring Diversity: Year 2 Students' Responses to Questions Concerning Simple 2D Shapes" (Jacquelyn Whitland and John Pegg); (71) "Novice Students' Conceptual Knowledge of Statistical Hypothesis Testing" (Anne M. Williams); and (72) "Students' Optimism, Pessimism and Achievement in Mathematics: A Longitudinal Study" (Shirley M. Yates). (ASK)

This document contains the proceedings of the 25th annual conference of the Mathematics Education Research Group of Australia (MERGA) held at the University of Auckland, New Zealand. The focus of this meeting is mathematics education in the South Pacific. Presentations are centered around the topic of numeracy in primary or elementary school. Papers include: (1) "A Samoan Perspective on Pacific Mathematics Education" (Karoline Afamasaga Fuata'i); (2) "The Process of Introducing New Tasks Using Dynamic Geometry into the Teaching of Mathematics" (Colette Laborde); (3) "Effectiveness and Efficiency of Calculator Use for Computation in Years 8-10" (Jack Bana and Rachel Shipley); (4) "A Problem-Based Schema Analysis in Algebra" (Mohan Chinnappan); (5) "An Investigation of Disjuncture between Graphing in School Mathematics and School Physics" (Patricia A. Forster); (6) "Single or Combination Grades in Mathematics in the Early Years" (Marj Horne); (7) "A Model for Teaching Numeracy Strategies" (Peter Hughes); (8) "Teacher Development for Inquiry Based Instructional Practice in Mathematics: A Poststructuralist Postscript" (Mary Klein); (9) "Patterns of Misperceptions in Linear Transformations: Four Illustrations" (Martin Lamb, Sam Leong, and John Malone); (10) "Common Errors in Mental Computations of Students in Grades 3-10" (Alistair McIntosh); (11) "Rethinking Curriculum: An Ethical Perspective" (Jim Neyland); (12) "Monitoring Effective Use of Computer Algebra Systems" (Robyn Pierce and Kaye Stacey); (13) "Professional Development of Teachers of Mathematics through Extended Collegial Dialogue: The ACT Mathematics Quality Teacher Program" (Steve Thornton and Kerrie Blain); and (14) "Validation of the New Zealand Number Framework" (Jenny Young-Loveridge and Vince Wright). The second volume also includes 22 short communications. (KHR)

The second volume of this proceedings contains the papers presented at the 21st annual meeting of the Mathematics Education Research Group of Australasia. Full papers include: (1) "Teachers Interpreting Algebra: Teachers' Views about the Nature of Algebra" (Brenda Menzel and David Clarke); (2) "Long Odds: Longitudinal Development of Students' Understanding of Odds" (Jonathan Moritz); (3) "An Arithmetic Game with Strategic Components--Playing Procedures of Primary School Children" (Doris Mosel-Gobel); (4) "Ability Grouping: Some Implications for Building Mathematical Understanding" (Judith Mousley); (5) "Ethnographic Research in Mathematics Education: Using Different Types of Visual Data Refined from Videotapes" (Judith Mousley); (6) "A Research-Based Framework for Assessing Early Multiplication and Division Strategies" (Joanne Mulligan); (7) "The Use of Mathematical Function Plotters in Teaching Introductory Calculus" (Bruce Nevill); (8) "Children's Matching of Melodies and Line Graphs" (Steven Nisbet and John Bain); (9) "Identifying Separate and Connected Knowing in Mathematics Education" (Jude Ocean); (10) "Developing the Angle Concept through Investigations" (Kay Owens); (11) "Is There a Need for a Mathematics Intervention Program in Grades 3 and 4?" (Cath Pearn); (12) "Year 3 Students' Place-Value Misconceptions: Another Look At MAB" (Peter Price); (13) "Teaching Mathematically Gifted Students in Primary School" (Ian Putt); (14) "Orchestrating Different Voices in Student Talk about Infinity: Theoretical and Empirical Analyses" (Peter Renshaw and Raymond Brown); (15) "Measuring Attitudes towards Mathematics in Early Childhood and Primary Teacher Education" (Carolyn Roberts, Patricia Cretchley, and Chris Harman); (16) "A Survey of Graphics Calculator Use in Victorian Secondary Schools" (Alla Routitsky and Patrick Tobin); (17) "The Search for Pattern: Student Understanding of the Table of Values Representation for Function" (Julie Ryan and Julian Williams); (18) "Parent Newsletters Supporting Mathematics in the Junior Primary School" (Jan Savell); (19) "Challenging Beliefs about Mathematics Learning and Teaching Using an Electronic Learning Community" (Sandra Schuck and Gerry Foley); (20) "Mathematics Textbooks: Messages to Students and Teachers" (Mal Shield); (21) "Problem Solving through Problem Posing: The Experience of Two Teacher Education Students" (Beth Southwell); (22) "Teachers Building Number Sense Amid the Challenges of Change: Some Case Studies" (Len Sparrow and Alistair Mcintosh); (23) "The Invisible Wall Project on Problem Solving Processes: Concepts and Methods of Interpretive Work with High Resolution Data" (Martin Stein); (24) "The Incidence of Misconceptions of Decimal Notation amongst Students Grades 5 to 10" (Vicki Steinle and Kaye Stacey); (25) "Kindergarten Students' Progress in the Count Me In Too Project" (Rita Stewart, Bob Wright, and Peter Gould); (26) "Task Context and Applications at the Senior Secondary Level" (Gloria Stillman); (27) "Different Forms of Mathematical Questions for Different Purposes: Comparing Student Responses to Similar Closed and Open-Ended Questions" (Peter Sullivan, Elizabeth Warren, Paul White, and Stephanus Suwarsano); (28) "Student Choice of Computation Methods" (Paul Swan and Jack Bana); (29) "Mathematical Misconceptions--We Have An Effective Method for Reducing Their Incidence but Will the Improvement Persist?" (Philip Swedosh and John Clarke); (30) "Dynamic Scaffolding and Reflective Discourse: Successful Teaching Styles Observed within a Project To Teach Mathematical Thinking Skills" (Howard Tanner and Sonia Jones); (31) "A Computer Environment To Encourage Versatile Understanding of Algebraic Equations" (Mike Thomas and Diane Hall); (32) "Teacher Beliefs about the Learning and Teaching of Mathematics: Some Comparisons" (Danielle Tracey, Bob Perry, and Peter Howard); (33) "Exploring the Impact of CAS in Early Algebra" (David Tynan and Gary Asp); (34) "Computers Are Taking Mathematics into the Next Century: Gender Differences in the Attitudes of Secondary Mathematics Students to the Use of Computers" (Colleen Vale); (35) The Mathematics, Technology and Science Interface: Implementation in the Middle School" (Grady Venville, John Malone, John Wallace, and Leonie Rennie); (36) "Progressing through the van Hiele Levels with Cabri-geometre" (Jill Vincent); (37) "What Is 'Meaning' in Gender Research?" (Margaret Walshaw); (38) "Students' Understanding of the Concept of the Variable" (Elizabeth Warren); (39) "Numeracy Benchmarks for Years 3 and 5: What about Chance and Data?" (Jane Watson); (40) "'This Is a Funny Game--You Can't Say Who's Going To Win!': Three Case Studies of Children's Probabilistic Thinking" (Jenni Way); (41)"Beliefs about the Use of Calculators in an Upper Primary Mathematics Classroom: A Partial Application of the Theory of Planned Behaviour" (Allan White); (42) "Metacognition within Mathematics: A New and Practical Multi-Method Approach" (Jeni Wilson); (43) "An Overview of a Research-Based Framework for Assessing and Teaching Early Number" (Bob Wright); (44) "Language, Mathematics and Social Disadvantage: A Bourdieuian Analysis of Cultural Capital in Mathematics Education" (Robyn Zevenbergen); (45) "Mathematical Saturation within Workplace Contexts" (Robyn Zevenbergen). (ASK)

The first volume of this proceedings contains the papers presented at the 21st annual meeting of the Mathematics Education Research Group of Australasia. Papers include: (1) "Research, Rhetoric and Reality" (Lyn English); (2) "I Keep Six Honest Serving Men" (Peter Galbraith); (3) "The Intension/Intention of Teaching Mathematics" (Stephen Lerman); (4) "One Teacher's Problem Solving Beliefs and Practices: Influences and Coherence" (Judy Anderson); (5) "Towards the Study of the Processes and Effects of Internationalisation in Mathematics Education" (Bill Atweh, Phil Clarkson, and John Malone); (6) "The Use of Action Research To Assist the Transition into Teaching" (Bill Atweh and Ann Heirdsfield); (7) "The Effectiveness of a No Goal Approach in Solving Pythagorean 2-Stage Problems" (Paula Ayres); (8) "Factors Influencing Predictions about Randomly Generated Sequences" (Paula Ayres and Jenni Way); (9) "Towards a Framework for Analyzing Power Relationships in Small Group Discussions" (Mary Barnes); (10) "Year 5 Students' Available and Accessible Knowledge of Decimal-Number Numeration" (Annette Baturo); (11) "Mathematics Curriculum and Development" (Andy Begg); (12) "The Impact of An Early Number Project on the Professional Development of Teachers" (Janette Bobis and Peter Gould); (13) "Arithmetic, Pre-Algebra, and Algebra: A Model of Transition" (Gillian Boulton-Lewis, Tom Cooper, H. Pillay, and L. Wilss); (14) "The Invisible Wall Project: Problem Solving Processes of Pupils Who Work on Problems with a Goal Which Cannot Be Reached" (Birgit Burchardtz and Martin Stein); (15) "'Why Are We Assessed in Mathematics?': The Views of Students" (Ken Carr); (16) "Developing a Framework for Viewing Affective and Knowledge Factors in Teaching Primary Mathematics" (Jean Carroll); (17) "Teachers' Understandings of the Role of Learning Activities" (Min-Pyng Chen); (18) "Showing and Telling: Primary Students' Outcomes in Data Representation and Interpretation" (Helen Chick and Jane Watson); (19) "Representing the Connectedness of Mathematical Knowledge" (Mohan Chinnapan, Michael Lawson, and Rod Nason); (20) "Beginning Teachers' Problems with Fundamental Mathematics" (Phil Clarkson); (21) "Investigating Students' Understanding of the Relationships among Quadrilaterals" (Penelope Currie and John Pegg); (22) "Rethinking What It Means To Understand" (Lyn English); (23) "New Times for Mathematics in Vocational Education and Training" (Gail FitzSimons); (24) "Exploring Avenues of Reflective Practice with Pre-Service Mathematics Teachers" (Sandra Frid, Ted Redden, and Chris Reading); (25) "Students' Perspectives on Using Computers and Graphing Calculators during Mathematical Collaborative Practice" (Vince Geiger); (26) "Identifying the Dilemmas in Early Mathematics Teaching" (Ann Gervasoni); (27) "'I Don't Know If I'm Doing It Right or I'm Doing It Wrong!' Unresolved Uncertainty in the Collaborative Learning of Mathematics" (Merrilyn Goos); (28) "Visual Reasoning and Teaching Styles in Mathematics Classrooms" (Kevin Hannah); (29) "Flexible/Inflexible: Clare and Mandy's Story" (Ann Heirdsfield); (30) "We Really Put Our Minds To It: Cognitive Engagement in the Mathematics Classroom" (Sue Helme and David Clarke); (31) "Maori Mathematics Education: The Challenge of Providing Immersion Programmes For Preservice Maori Primary Teachers" (Leeana Herewini); (32) "Towards an Explanation of Curriculum Control" (John Horwood); (33) "How Sixth Grade Students Explaining Connections between Common and Decimal Fractions" (Robert Hunting, Lauren Oppenheimer, Cath Pearn, and E. Nugent); (34) "Mathematics Curriculum Change in the Northern Territory: What Do Teachers Really Think?" (Rosemary Jacob and Sandra Frid); (35) "The Effective Use of Calculators" (Sonia Jones and Howard Tanner); (36) "New Knowledge/New Teachers/New Times: How Processes of Subjectification Undermine the Implementation of Investigatory Approaches to Teaching Mathematics" (Mary Klein); (37) "Affective Factors and the Improvement of Algebraic Problem Solving" (Saraswathi Kota and Mike Thomas); (38) "Tertiary Mathematics: Perceptions of School Levers and Mature-Age Students" (Gilah Leder and Helen Forgasz); (39) "There's More Than Meets the Eye: The Relationships between the User, the Teacher and the Technology" (Tom Lowrie); (40) "Pre-Service Teachers' Understandings of Word Problems" (Kevin Maguire); (41) "Protasis: A Technique for Fostering Professional Development" (John Mason); (42) "Towards a Psychology of Knowing-To" (John Mason and Mary Spence); (43) "School-Based Assessment in VCE Mathematics: Ten Years On" (Barry Mccrae, John Dowsey, and Max Stephens); and (44) "A New Trilemma" (D.F. Mckenzie). (ASK)

This document contains both volumes of the proceedings of a conference of the Mathematics Education Research Group of Australasia (MERGA). Three keynote papers by Gordon Knight, J. Michael Shaughnessy, and Robyn Zevenbergen, and a practical implications paper by Lyn English are included. Seventy-five conference papers are included in these proceedings that focus on people in mathematics education, interactions among them, and the instructional materials employed. In the Short Communication Papers section there are 28 brief papers and poster sessions pertaining to the same broad topic. Topics treated in this section include graphics calculators, numeracy, use of imagery, professional development, and teaching stochastics. (DDR)

This document contains papers presented at the 19th annual conference of the Mathematics Education Research Group of Australasia. Topics of the presentations include learning research, mathematical representations, problem solving, strategic learning behaviors, algebraic thinking and learning environments, teaching and learning of algebra, assessment, disabilities, calculators, collective argumentation, teachers' beliefs and practice, primary mathematics, differential calculus, teachers' knowledge, trigonometry and geometry, professional development, issues in teaching, standardizing the curriculum, team writing, statistics, Newman error analysis, gender issues, Internet, transition to secondary mathematics, computers and technology, negative numbers, subtraction, aboriginal educators' views, graphics calculators, language, area, probability, word problems, classroom communication, mathematical investigations, ethics and morality, integrating science and mathematics concepts, students' attitudes, instructional computing, expository writing, mathematical autobiographies, problem posing, misconceptions, discussion-based teaching, the Riemann integral, diagrams for solving word problems, fairness and fractions in early childhood, children's probability judgments, phenomenology of writing-to-learn, teachers' beliefs about teaching behaviors, and linear programming. An author index and a subject index are also included. (JRH)

These proceedings contain 4 keynote addresses and 88 papers concerning the theme of the conference: Contexts in Mathematics Education. The four keynote addresses are: (1) "Contemplating Cultural Context" (Bill Barton); (2) "Conceptualising Cultural and Social Contexts in Mathematics Education" (Alan Bishop); (3) "Mathematical Learning in the Social Context" (George Booker); and (4) "Childrens' Construction of Mathematical Ideas" (Carolyn Maher). The 88 contributed papers deal with mathematical content ranging from prenumber to calculus, learners ranging from preschool to adult, teachers, technology, affect, assessment, metacognition, psycholinguistics, and a wide variety of social issues. (MDH)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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