Conference Proceedings 2006


Identities, Cultures and Learning Spaces
Editors: Peter Grootenboer, Robyn Zevenbergen and Mohan Chinnappan
Table of Contents
MERGA 2006 Conference Proceedings
Stephen Thornton, Peter Grootenboer, Mohan Chinnappan and Robyn Zevenbergen
Implications of Preservice Teachers Intentions to Use Particular Learning Tasks
Anne Scott
List of Reviewers
Judges and Reviewers for MERGA 29
Keynote Address
Creating Learning Spaces
Merrilyn Goos
New Directions for Research on Mathematical Problem Solving
Richard Lesh
Practical Implication Award
To heal and enthuse: Developmental bibliotherapy and pre-service primary teachers' reflections on learning and teaching mathematics
Sue Wilson and Steve Thornton
Becoming a teacher of mathematics: Wenger's social theory of learning perspective
Tracey Smith
Bringing feminist poststructuralism to bear on [mathematics] teacher education
Will Letts
Researching Identity in Mathematics Education: The Lay of the Land
Peter Grootenboer, Tracey Smith and Tom Lowrie
Teacher Identity from a Bourdieuian Perspective
Robyn Zevenbergen
Research Paper
Game Playing to Develop Mental Computation: A Case Study
Paula Asplin, Sandra Frid and Len Sparrow
Grade12 Mathematics Teachers' Views on Curriculum Reform in New South Wales
Paul Ayres and John McCormick
Pedagogical Content Knowledge for Teaching Primary Mathematics: A Case Study of Two Teachers
Monica Baker and Helen Chick
Teachers' Confidence and Beliefs and their Students' Attitudes to Mathematics
Kim Beswick, Jane Watson and Natalie Brown
Investigating Parental Roles of Mathematically Gifted Students
Brenda Bicknell
Authentic Learning in a Year 8 Classroom
Kathy Blum
Reform-Oriented Teaching Practices and the Influence of School Context
Janette Bobis and Judy Anderson
Transforming Practice: Using Collective Argumentation to bring about Teacher Change in a Year 7 Mathematics Classroom
Raymond Brown and Peter Renshaw
Identifying At-Risk Students: Is it Possible in a Tertiary Preparation Course for Adults'
Colin Carmichael, Peter Dunn and Janet Taylor
Mathematics Teachers and Working Mathematically: Responses to Curriculum Change
Michael Cavanagh
Examining the Changed Role of Numeracy Coordinators
Jill Cheeseman and Doug Clarke
The Numeracy Journey: How Long Does it Take to get on Board?
Linda Cheeseman
Probing Teachers' Pedagogical Content Knowledge: Lessons from the Case of the Subtraction Algorithm
Helen L. Chick, Thuy Pham and Monica K. Baker
Historical Perspectives on Mathematical Elegance: To What Extent is Mathematical Beauty in the Eye of the Beholder?
Ken Clements and Nerida Ellerton
Many Dimensions: the Complex Picture of Student Encounters with a Computer Algebra System
Mary Coupland and Kate Crawford
The Leaving Certificate in New South Wales from 1939 to 1962
Stephen Curtis
Primary Students' Knowledge of and Errors on Number Lines
Carmel M. Diezmann and Tom Lowrie
The Singaporean Mathematics Curriculum: Connections to TIMSS
Jaguthsing Dindyal
Organisational Structure for Mathematical Modelling
Katherine Doyle
Introducing Young Children to Complex Systems through Modelling
Lyn English
A Model of Students' Statistical Thinking and Reasoning about Graphs in an ICT Environment
Noleine Fitzallen
Australian year 12 "Intermediate" level mathematics enrolments 2000-2004: Trends and patterns
Helen J. Forgasz
A Justification for Mathematical Modelling Experiences in the Preparatory Classroom
Jillian Fox
Real World Problems: Developing Principles of Design
Peter Galbraith
Identifying Key Transition Activities for Enhanced Engagement in Mathematical Modelling
Peter Galbraith, Gloria Stillman and Jill Brown
Standing on the Outside: A Tale of How Technology Can Engage Those Working on the Margins of a Community of Inquiry
Vince Geiger
Living in the Gap: A Tale of Two Different Types of Researchers
Vince Geiger and Merrilyn Goos
One-Third is Three-Quarters of One-Half
Peter Gould, Lynne Outhred and Michael Mitchelmore
Mathematics Educators: Identity, Beliefs, Roles and Ethical Dilemmas
Peter Grootenboer
The Role of Abstraction in Learning about Rates of Change
Ibrahim Hassan and Michael Mitchelmore
Maori Preservice Primary Teachers' Responses to Mathematics Investigations
Ngarewa Hawera
Issues in Teaching Mathematics to Aboriginal Students
Peter Howard and Bob Perry
Students in a Mathematical Community of Inquiry: What do They Think?
Jodie Hunter
Structuring the Talk Towards Mathematical Inquiry
Roberta Hunter
The Development of a Community of Practice and its Connection with Mentoring in Low Socio-Economic Secondary Schools in New Zealand
Barbara Kensington-Miller
What to Leave Out When Preservice Mathematics Education goes from Four Years to One: A Poststructural Account
Mary Klein
Catering for Individual Differences: Lessons Learnt from the Australian Mathematics Competition
Gilah C. Leder
Departing from the Traditional Long Division Algorithm: An Experimental Study
Issic Leung, Regina Wong and Wai-sum Pang
In the Hands of the Learner: The Impact of Self-Assessment on Teacher Education
Sandi Tait-McCutcheon and Brenda Sherley
Emerging Issues in the Investigation of the Construct of Partitive Quotient
Lucas McGee, Lisa Kervin and Mohan Chinnappan
What does Effective Teaching for Numeracy Look Like? The Design of an Observation Schedule
Tracey Muir
Improving Early Numeracy Through a Pattern and Structure Mathematics Awareness Program (PASMAP)
Joanne Mulligan, Marina Papic, Anne Prescott and Mike Mitchelmore
Building Powerful Understanding by Connecting Informal and Formal Knowledge
Trish O'Toole
'Is that right?': Asking questions and appealing for help in mathematics
Angela Page
Using the Internet in Teaching Mathematics in Primary School
Sitti Maesuri Patahuddin and Shelley Dole
Linking Powerful Mathematical Ideas and Developmental Learning Outcomes in Early Childhood Mathematics
Bob Perry, Sue Dockett, Elspeth Harley and Nicole Hentschke
The Notion of Carried-Number, between the History of Calculating Instruments and Arithmetic
Caroline Poisard
An Investigation of Pre-service Secondary Mathematics Teachers? Beliefs as They Begin Their Teacher Training
Anne Prescott and Michael Cavanagh
Promoting Change in Teacher Practices: Investigating Factors which Contribute to Sustainability
Ruth Pritchard and Fiona McDiarmid
The VideoPaper: Issues in Implementation of a Multimedia Tool for Professional Self-Dialogue and Communication in Mathematics Education
Robin Rider and Robert Hunting
Mathematical teacher professional development incorporating an external critical friend
Pauline Rogers
Unpacking the Rules of Class Discussion: Young Children Learning Mathematics within a Community of Inquiry
Abigail Sawyer
Self-Study Through Narrative Inquiry: Fostering Identity in Mathematics Teacher Education
Tracey Smith
Describing and Exploring the Power of Relational Thinking
Max Stephens
Student Thinking about Eigenvalues and Eigenvectors: Formal, Symbolic and Embodied Notions
Sepideh Stewart and Michael O. J. Thomas
Developing Guidelines for Teachers Helping Students Experiencing Difficulty in Learning Mathematics
Peter Sullivan, Judy Mousley and Robyn Zevenbergen
Moving into Third Space: High School Students' Funds of Knowledge in the Mathematics Classroom
Steve Thornton
Equity and Technology: A Case Study
Colleen Vale
Scaffolding Numeracy: Pre-service Teachers' Perspective
Irina Verenikina and Mohan Chinnappan
Classroom Arrangements That Benefit Students
Margaret Walshaw and Glenda Anthony
Supporting Learning in Early Algebra: A Model of Professional Learning
Elizabeth Warren
Teacher Professional Development in Patterns and Algebra: Being Sensitive to a Teacher's Zone of Proximal Development
Elizabeth Warren, Tom Cooper and Janeen Lamb
Teachers' Knowledge of their Students as Learners and How to Intervene
Jane Watson, Kim Beswick and Natalie Brown
Spreadsheets in Year 5 Chance and Data: A Professional Development Experience
Anne Williams
Student-Engineered 'Space to Think'
Gaye Williams
Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model
Asmamaw Yimer and Nerida F. Ellerton
Students' Perspectives on the Nature of Mathematics
Jenny Young-Loveridge, Merilyn Taylor, Sashi Sharma and Ngarewa Hawera
Using ICTs to Support Numeracy Learning Across Diverse Settings
Robyn Zevenbergen and Steve Lerman
Short Communication (abstract only)
Breaking the Cycle: Maths Anxiety and Pre-service Primary Student Teachers
Gillian Frankcom

This paper describes part of a study which involved 29 third year pre-service student teachers answering Maths Anxiety and Maths Self-efficacy questionnaires and questions regarding previous and current mathematical experiences. Results indicate that previous depth of mathematics learning is not a factor in the level of Maths Anxiety and neither is the level of success in current mathematics teacher education courses, and that Maths Anxiety is highly correlated with Maths Self-efficacy.

Developing Identity as a Mathematical Thinker
Kay Owens

The argument in this paper is that identity as a mathematical thinker develops through self-directed learning within a supportive community of practice. This paper discusses how identity as a mathematical problem solver and investigator develops through selfregulation. This development is illustrated by considering students undertaking a mathematics and technology subject in a primary teacher education degree. It shows how students set goals, plan, organise, self-evaluate, record keep and structure their learning environment to achieve self-regulation. The role of the tutorial group and technology is also important in establishing their identity as a mathematical problem solver and investigator.

Engagement of Boys in Middle School Mathematics
Beth Southwell and Catherine Attard

Following a concern for the engagement of boys in a NSW Year 6 classroom, action research was undertaken to explore the effect of the use of computer technology and the change in role of the teacher from a giver of knowledge to a facilitator, on the motivation and engagement of the boys in the class. Data were obtained from a number of sources, including a Motivation Scale, focus group discussions, observations, student reflective logs and a teacher diary. Initial results indicated that while boys responded enthusiastically to the challenge of manipulating data using the Tinkerplots software and developing their own questions and research topics, there was no indication that they preferred to use technology.

Frameworks: Climbing Aids or Entrapments?
Vince Wright and Jenny Young-Loveridge

This paper reported the influence of the Numeracy Development Projects (NDP) in revision of the number strand of the New Zealand mathematics curriculum. It documented how four types of teaching and learning research frameworks were synthesised to provide evidence for validity of the number framework and associated pedagogies. Examination of students' responses across number domains showed consistency and suggested a general growth path. Strategy stage norms to set expected levels of achievement were described and challenges posed for mathematics teaching and learning.

If You Don't Listen to the Teacher, You Won?t Know What to Do
Pamela Perger

Everyone has beliefs about how learning should take place and what the best practices are to enable this to happen. Although it is believed that students' beliefs about 'best practice' will mirror those of their teachers, and change as they change teachers (Kershner & Pointon, 2000; Kloosterman, Raymond, & Emenaker, 1996) the importance of listening to the 'students voice' is becoming recognised (McCullum, Hargreaves & Gipp, 2000). This paper reports on one aspect of a larger study that explored Pasifika student achievement in mathematics at Year 7. What is it that these students consider 'best practice' in learning mathematics? Do their beliefs truly mirror those of their teacher?

Inaccurate Mental Computation: Identifying Flawed Thinking
Ann M. Heirdsfield and Janeen T. Lamb

This paper reports on a study of five, Year 2 students' strategy choice, flexibility and accuracy when answering 20 addition and subtraction mental computation questions. All five students were identified as being inaccurate. However, two students employed a range of calculation strategies while the other three students remained inflexible in their strategy choice, choosing low order strategies. Individual interviews were conducted to identify these aspects of calculation. Two conceptual flowcharts developed by Heirdsfield (2001b) were utilised to identify factors and relationships between factors that impact on mental computation. Use of these flowcharts provides an avenue for identification of the breakdown in the structures thereby providing an understanding of where the child is operating and how they may be moved forward.

Primary Pre-service Teachers? Understanding of Place Value
Romina Jamieson-Proctor

The development of place value understanding is an essential foundation concept that enables students to have a strong number sense. However, place value is a common problem area experienced by students and pre-service teachers alike. This paper reports the results of an initial investigation of pre-service teachers? understanding of the symmetry of the place value system. The results suggest that pre-service teachers' mathematical content knowledge with respect to the symmetry of the place value system is weakest for fractions and when they are asked to generalise their understanding to a base a system. Overall more of the pre-service teachers had a pre- or uni-structural understanding of the symmetry of the place value system.

Teacher Misconceptions about Projectile Motion
Anne Prescott and Michael Mitchelmore

Student misconceptions of projectile motion in the physics classroom are well documented, but their effect on the teaching and learning of the mathematics of motion under gravity has not been investigated in the mathematics classroom. An experimental unit was designed that was intended to confront and eliminate misconceptions in senior mathematics secondary school students studying projectile motion as an application of calculus to the physical world. The approach was found to be effective, but limited by the teacher's own misconceptions. It is also shown that teachers can reinforce student misconceptions of motion because they cannot understand why students have difficulty understanding it.

The Initial Professional Development of Teachers Learning to Use a Framework for Determining Students' Strategic Thinking
Peter Hughes

Derived initially from the observation of children's methods of counting, Mathematics Recovery and Count Me in Too, the New Zealand Numeracy Projects have, as a starting point for the training of teachers, the understanding and use of a strategy framework that traces children's development in number reasoning. Research indicated the usefulness of teachers interviewing students in their own class, and viewing video clips of strategic reasoning across of a wide range of ages of student. This paper outlines how interviews and the video clips are incorporated into the initial stages of teacher Professional Development in learning to use the strategy number framework.

Understanding mathematics anxiety in a New Zealand secondary classroom
Naomi Ingram

Since finishing a Masters in Mathematics Education, I have taught mathematics concepts in industry and in a secondary classroom, and have frequently come across able students who have difficulty learning and achieving to their academic potential in mathematics. These difficulties seem to stem from anxiety that the students experience when doing mathematics. This is case study of Year 10 students (14-15 years old) who are in the top achievement class of an Otago secondary school. Students were chosen for this class by the school at the beginning of Year 9 because they demonstrated excellence in one or more fields, not necessarily mathematics. Through mainly qualitative methods (classroom observations, questionnaires, and individual interviews), the following is being investigated:  the mathematical identities of students in the class;  the identification of maths anxiety and potential maths anxiety in students;  interventions to improve the mathematics experience of anxious and potentially anxious students.

Poster (abstract only)
Round Table (abstract only)
Approaches for Teaching the Division of Fractions
Leyton Walker

When contemplating the division of two common fractions, the 'invert and multiply' algorithm, does not develop naturally from using manipulatives. (Borko; Eisenhart; Brown; Underhill; Jones & Agard. 1992) suggest it is for this reason, that it is unlikely that children will invent their own 'invert and multiply' algorithm. Before a student can be expected to 'invent' this algorithm, knowledge of whole number division and basic fraction concepts, including the notion of equivalent fractions is essential. (Sharp, 1998). The purpose of this round table discussion is to take cognisance of the suggestions attested to by Borko et al. and Sharp and examine the teaching approaches adopted by classroom teachers, as they relate to the process of division with fractional numbers. To highlight this, six Year 7 and Year 8 teachers were asked to solve and then describe their mathematical approaches and processes used to calculate 2/3 รท 1/2; illustrate the meaning of the operation and describe the means by which they would explain their respective methods for solving problems involving the division of fractions with their students. The opportunity to examine the use of mathematical equipment that may be used to support and illustrate the mathematical process of division with fractions, will also be examined with the view to generating a conceptual representation of the meaning of the division of fractions. References Borko., H; Eisenhart., M; Brown, C.A., Underhill, R., Jones, D., & Agard, P. (1992). Learning to teach hard mathematics: do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education 23 194-222. Sharp, J. (1998). A constructed algorithm for the division of fractions. In: The Teaching and Learning of Algorithms in School Mathematics. Morrow, L.J. & Kenny, M.J. (editors) Reston, VA: National Council of Teachers of Mathematics.

Keeping it Going: Challenges in Sustaining Numeracy Practice
Deborah Gibbs and Marilyn Holmes

New Zealand, along with many other countries, has been investigating ways of raising children's achievements in mathematics by improving teachers? professional knowledge, skills and confidence. The Numeracy Development Project (NDP) in New Zealand grew out of The New South Wales Department of Education and Training initiative 'Count Me In Too' in 2000, and was further developed by 'research evidence about mathematics education, effective teaching, teacher learning, effective facilitation practice, and educational change' (Ministry of Education, 2004). Pivotal to the success of the project are facilitators, principals and lead teachers who work with classroom teachers to effect changes in teaching practice. Schools participating in NDP can expect continuous and focussed support throughout their year long professional development. Sustaining the numeracy momentum within a school is a difficult task and can be further complicated by the employment of untrained numeracy teachers or provisionally registered teachers with varying levels of understanding. During their time in the project, lead teachers have mainly an administrative responsibility. However, in subsequent years, it is magnified to include a complex, multi-layered facilitation role. This small study by Deborah Gibbs and Marilyn Holmes highlights two challenging aspects: (a) The needs of untrained numeracy teachers as they try to come to grips with the numeracy project as well as the school?s mathematics programme; and (b) the complexity of the lead teacher's responsibility in mentoring new untrained staff. Discussion generated around this study will highlight the implications for facilitators in numeracy and pre-service educators. References Ministry of Education. (2004). The Numeracy Story continued. What is the evidence telling us? Wellington: Learning Media.

Non-specialist teachers of mathematics: Pedagogical challenges in the Middle Years of Schooling
Barbara Tadich

This round table discussion will focus on the efficacy, pedagogies and practice of nonspecialist teachers of mathematics in the middle years of schooling in a regional Victorian setting. Case study data, collected through questionnaires and interviews of 26 junior secondary teachers with no mathematics methods in their training, from six, rural, coeducational Victorian Government schools, will be presented. Current middle years' reform and initiatives such as Victorian Essential Learning Standards (VELS) provide guidelines for effective teaching strategies to engage young adolescents generally. However, in order to provide expanded learning opportunities in mathematics, teachers require professional knowledge of their subject. Being able to present an understanding to student of big ideas, the main branches and concepts of mathematics, and providing a sense of their interconnections aids students to engage in mathematics. This research project was undertaken to investigate whether non-specialist mathematics teachers' pedagogical practices in secondary schools were influenced by their limited training and mathematical pedagogical knowledge. The findings indicate that lack of method training and knowledge of pedagogical content impacts on both the teachers and the students' mathematical engagement at junior secondary level. Through this round table I invite other researchers to discuss their experiences of working with middle years mathematics teachers to seek their input in the possible development of further work in investigating the challenges of non-specialist mathematics teachers in junior secondary schools and overall student engagement in mathematics.

The use of hand-held technology in the learning of statistical concepts
Derek Smith

Researchers and educational policymakers have given encouragement to the use of electronic technology in the teaching and learning of school mathematics and in the assessment of senior mathematics and statistics courses. If used appropriately (judiciously) is hand-held technology able to offer secondary mathematics and statistics teachers and their students a significantly richer mathematics learning experience? Little research into hand-held technology and statistics has been done and during this round table discussion, the researcher describes the use of hand-held technology [graphical calculators] in three, N.C.E.A. Level 3, statistics classrooms in a large co-educational urban secondary school and the impact that they had on teacher pedagogy, student learning and understanding of statistical concepts will be presented. The roundtable is to discuss aspects of this study and to invite other researchers to share their experiences in working with secondary school teachers in statistics. The feedback provided in this round table is to assist and inform the researcher about planning, implementing and appropriate methodology for a more in-depth study in hand-held technology use and statistical literacy and thinking.