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Mathematics: Essential Research, Essential Practice
Editors: Jane Watson and Kim Beswick
Table of Contents
MERGA 2007 Conference Proceedings
Kim Beswick and Jane Watson
List of Reviewers
Keynote Address
Introducing Students to Data Representation and Statistics 63 downloads
Richard Lehrer
Studies in the Zone of Proximal Awareness 60 downloads
John Mason, Helen Drury and Liz Bills
Teaching and Learning by Example 58 downloads
Helen L. Chick
The Beginnings of MERGA 56 downloads
Ken Clements
Practical Implication Award
Empowered to Teach: A Practice-based Model of Teacher Education 63 downloads
Janette Bobis
Children?s Number Knowledge in the Early Years of Schooling 71 downloads
Ann Gervasoni
Early Childhood Mathematics Education Research: What is Needed Now? 57 downloads
Bob Perry and Sue Dockett
International Perspectives on Early Years Mathematics 55 downloads
Jillian Fox
Listening to Student Opinions about Group Assessment 62 downloads
Judith Mousley and Coral Campbell
Listening to Students? Voices in Mathematics Education 55 downloads
Brian Doig, Susie Groves, Coral Campbell, Judith Mousley and Gaye Williams
Research Enriched by the Student Voice 54 downloads
Gaye Williams
Students? Pedagogical Knowledge: A Source of Pedagogical Content Knowledge 54 downloads
Brian Doig and Susie Groves
Trimangles and Kittens: Mathematics Within Socio-dramatic Play in a New Zealand Early Childhood Setting 63 downloads
Shiree Lee
Research Paper
Communicating Students? Understanding of Undergraduate Mathematics using Concept Map 62 downloads
Karoline Afamasaga-Fuata?i
Primary Student Teachers? Diagnosed Mathematical Competence in Semester One of their Studies 57 downloads
Karoline Afamasaga-Fuata?i, Paul Meyer & Naomi Falo
An Online Survey to Assess Student Anxiety and Attitude Response to Six Different Mathematical Problems 62 downloads
Vincent Anderson
Mathematical Investigations: A Primary Teacher Educator?s Narrative Journey of Professional Awareness 59 downloads
Judy Bailey
Describing Mathematics Departments: The Strengths and Limitations of Complexity Theory and Activity Theory 60 downloads
Kim Beswick, Anne Watson & Els De Geest
Three Student Tasks in a Study of Distribution in a ?Best Practice? Statistics Classroom 57 downloads
Anthony Bill & Jane Watson
Teacher Researchers Questioning their Practice 59 downloads
Linda Bonne & Ruth Pritchard
Imagined Classrooms: Prospective Primary Teachers Visualise their Ideal Mathematics Classroom 56 downloads
Kathy Brady
Early Notions of Functions in a Technology-Rich Teaching and Learning Environment (TRTLE) 62 downloads
Jill Brown
Collective Argumentation and Modelling Mathematics Practices Outside the Classroom 56 downloads
Raymond Brown and Trevor Redmond
Visual Perturbances in Digital Pedagogical Media 56 downloads
Nigel Calder
Professional Experience in Learning to Teach Secondary Mathematics: Incorporating Pre-service Teachers into a Community of Practice 56 downloads
Michael Cavanagh and Anne Prescott
Young Children?s Accounts of their Mathematical Thinking 54 downloads
Jill Cheeseman and Barbara Clarke
Mathematical Reform: What Does the Journey Entail for Teachers? 57 downloads
Linda Cheeseman
Year Six Fraction Understanding: A Part of the Whole Story 63 downloads
Doug M. Clarke, Anne Roche and Annie Mitchell
Teaching as Listening: Another Aspect of Teachers? Content Knowledge in the Numeracy Classroom 65 downloads
Ngaire Davies and Karen Walker
Essential Differences between High and Low Performers? Thinking about Graphically-Oriented Numeracy Items 58 downloads
Carmel M. Diezmann, Tom J. Lowrie and Nahum Kozak
High School Students? Use of Patterns and Generalizations 57 downloads
Jaguthsing Dindyal
The Teacher, The Tasks: Their Role in Students? Mathematical Literacy 55 downloads
Katherine Doyle
Informal Knowledge and Prior Learning: Student Strategies for Identifying and Locating Numbers on Scales 55 downloads
Michael Drake
Documenting the Knowledge of Low-Attaining Third- and Fourth-Graders: Robyn?s and Bel?s Sequential Structure and Multidigit Addition and Subtraction 60 downloads
David Ellemor-Collins, Robert Wright and Gerard Lewis
Interdisciplinary Modelling in the Primary Mathematics Curriculum 55 downloads
Lyn English
Students? Tendency to Conjoin Terms: An Inhibition to their Development of Algebra 55 downloads
Judith Falle
Towards ?Breaking the Cycle of Tradition? in Primary Mathematics 54 downloads
Sandra Frid and Len Sparrow
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 60 downloads
Ann Gervasoni, Teresa Hadden and Kathie Turkenburg
Technology-Enriched Teaching of Secondary Mathematics: Factors Influencing Innovative Practice 62 downloads
Merrilyn Goos and Anne Bennison
Supporting an Investigative Approach to Teaching Secondary School Mathematics: A Professional Development Model 64 downloads
Merrilyn Goos, Shelley Dole and Katie Makar
Identity and Mathematics: Towards a Theory of Agency in Coming to Learn Mathematics 55 downloads
Peter Grootenboer and Robyn Zevenbergen
Categorisation of Mental Computation Strategies to Support Teaching and to Encourage Classroom Dialogue 62 downloads
Judy Hartnett
Student Experiences of VCE Further Mathematics 55 downloads
Sue Helme and Stephen Lamb
Video Evidence: What Gestures Tell us About Students? Understanding of Rate of Change 53 downloads
Sandra Herbert and Robyn Pierce
The Role of Dynamic Interactive Technological Tools in Preschoolers? Mathematical Patterning 56 downloads
Kate Highfield and Joanne Mulligan
Students Representing Mathematical Knowledge through Digital Filmmaking 56 downloads
Geoff Hilton
What Does it Mean for an Instructional Task to be Effective? 56 downloads
Lynn Hodge, Jana Visnovska, Qing Zhao and Paul Cobb
A School-Community Model for Enhancing Aboriginal Students? Mathematical Learning 55 downloads
Peter Howard and Bob Perry
Benchmarking Preservice Teachers? Perceptions of their Mentoring for Developing Mathematics Teaching Practices 59 downloads
Peter Hudson
Relational or Calculational Thinking: Students Solving Open Number Equivalence Problems 61 downloads
Jodie Hunter
Scaffolding Small Group Interactions 57 downloads
Roberta Hunter
Numeracy in Action: Students Connecting Mathematical Knowledge to a Range of Contexts 63 downloads
Chris Hurst
A Story of a Student Fulfilling a Role in the Mathematics Classroom 57 downloads
Naomi Ingram
Secondary-Tertiary Transition: What Mathematics Skills Can and Should We Expect This Decade? 55 downloads
Nicolas Jourdan, Patricia Cretchley and Tim Passmore
The Power of Writing for all Pre-service Mathematics Teachers 58 downloads
Keith McNaught
?Connection Levers?: Developing Teachers? Expertise with Mathematical Inquiry 63 downloads
Katie Makar
Acquiring the Mathematics Register in te reo M?ori 56 downloads
Tamsin Meaney, Uenuku Fairhall and Tony Trinick
Teaching Ratio and Rates for Abstraction 62 downloads
Mike Mitchelmore, Paul White and Heather McMaster
Setting a Good Example: Teachers? Choice of Examples and their Contribution to Effective Teaching of Numeracy 61 downloads
Tracey Muir
Developing the Concept of Place Value 70 downloads
Mala Saraswathy Nataraj and Michael O. J. Thomas
Interdisciplinary Learning: Development of Mathematical Confidence, Value, and the Interconnectedness of Mathematics Scales 67 downloads
Dawn Kit Ee Ng and Gloria Stillman
Mathematical Methods and Mathematical Methods Computer Algebra System (CAS) 2006 - Concurrent Implementation with a Common Technology Free Examination 55 downloads
Pam Norton, David Leigh-Lancaster, Peter Jones and Michael Evans
A Concrete Approach to Teaching Symbolic Algebra 56 downloads
Stephen Norton and Jane Irvin
Developing Positive Attitudes Towards Algebra 58 downloads
Stephen Norton and Jane Irvin
Changing Our Perspective on Measurement: A Cultural Case Study 57 downloads
Kay Owens and Wilfred Kaleva
Enhancing Student Achievement in Mathematics: Identifying the Needs of Rural and Regional Teachers in Australia 57 downloads
Debra Panizzon and John Pegg
The Growth of Early Mathematical Patterning: An Intervention Study 62 downloads
Marina Papic and Joanne Mulligan
Whole Number Knowledge and Number Lines Help to Develop Fraction Concepts 58 downloads
Catherine Pearn and Max Stephens
Identifying and Analysing Processes in NSW Public Schooling Producing Outstanding Educational Outcomes in Mathematics 61 downloads
John Pegg, Debra Panizzon and Trevor Lynch
Teachers Research their Practice: Developing Methodologies that Reflect Teachers? Perspectives 58 downloads
Ruth Pritchard and Linda Bonne
Teacher Professional Learning in Mathematics: An Example of a Change Process 55 downloads
Pauline Rogers
Seeking Evidence of Thinking and Mathematical Understandings in Students? Writing 56 downloads
Anne Scott
Utilising the Rasch Model to Gain Insight into Students? Understandings of Class Inclusion Concepts in Geometry 56 downloads
Penelope Serow
Exploring Teachers? Numeracy Pedagogies and Subsequent Student Learning across Five Dimensions of Numeracy 57 downloads
Jane Skalicky
The Complexities for New Graduates Planning Mathematics Based on Student Need 57 downloads
Carole Steketee and Keith McNaught
Students? Emerging Algebraic Thinking in the Middle School Years 58 downloads
Max Stephens
A Framework for Success in Implementing Mathematical Modelling in the Secondary Classroom 57 downloads
Gloria Stillman, Peter Galbraith, Jill Brown and Ian Edwards
Eliciting Positive Student Motivation for Learning Mathematics 57 downloads
Peter Sullivan and Andrea McDonough
Learning from Children about their Learning with and without ICT using Video-Stimulated Reflective Dialogue 62 downloads
Howard Tanner and Sonia Jones
Dependency and Objectification in a Year 7 Mathematics Classroom: Insights from Sociolinguistics 59 downloads
Steve Thornton
Pedagogical Practices with Digital Technologies: Pre-service and Practicing Teachers 64 downloads
Colleen Vale
Procedural Complexity and Mathematical Solving Processes in Year 8 Mathematics Textbook Questions 54 downloads
Jill Vincent and Kaye Stacey
Designing Effective Professional Development: How do we Understand Teachers? Current Instructional Practices? 56 downloads
Jana Visnovsk
?Doing Maths?: Children Talk About Their Classroom Experiences 66 downloads
Fiona Walls
The Role of Pedagogy in Classroom Discourse 64 downloads
Margaret Walshaw and Glenda Anthony
Australian Indigenous Students: The Role of Oral Language and Representations in the Negotiation of Mathematical Understanding 62 downloads
Elizabeth Warren, Janelle Young and Eva deVries
Student Change Associated with Teachers? Professional Learning 57 downloads
Jane Watson, Kim Beswick, Natalie Brown and Rosemary Callingham
Choosing to Teach in the ?STEM? Disciplines: Characteristics and Motivations of Science, ICT, and Mathematics Teachers 56 downloads
Helen M. G. Watt, Paul W. Richardson and James Pietsch
Percentages as Part Whole Relationships 61 downloads
Paul White, Sue Wilson, Rhonda Faragher and Mike Mitchelmore
My Struggle with Maths May Not Have Been a Lonely One: Bibliotherapy in a Teacher Education Number Theory Unit 56 downloads
Sue Wilson
Students? Conceptual Understanding of Equivalent Fractions 71 downloads
Monica Wong and David Evans
Statistics Teachers as Scientific Lawyers 61 downloads
Joanne Woodward and Maxine Pfannkuch
Developing Pedagogical Tools for Intervention: Approach, Methodology, and an Experimental Framework 57 downloads
Robert Wright, David Ellemor-Collins and Gerard Lewis
Pedagogy and Interactive Whiteboards: Using an Activity Theory Approach to Understand Tensions in Practice 63 downloads
Robyn Zevenbergen and Steve Lerman
Short Communication (abstract only)
?I Have a Fear of Maths and it Does Worry Me a Bit as a Future Teacher?: The Cycle of Maths Anxiety
Gillian Frankcom

The maths anxiety of 29 pre-service primary teachers, which was measured using a questionnaire, was exhibited in writing a response to two different stimulus statements. These students used the word fear on many occasions and other metaphors to describe their anxiety about the sort of teacher they might become and whether they might break the cycle of maths anxious teachers producing maths anxious children. Their words are used to illustrate this desire to break the cycle.

Activity Theory as a Framework to Analyse the Positive Influence of Formative Assessment on Student Learning
Trish O?Toole

In this short communication, I provide an example of a student?s work to illustrate the power of assessment for learning. This approach has been adopted to support both teachers and students to come to aspects of mathematical learning. Drawing on the literature on assessment and how it is integral to learning rather than as the end product to show what learning has occurred, I provide examples of one case to illustrate the power of assessment for learning. By drawing on aspects of second generation activity, I provide examples of aspects of the learning milieu to frame the analysis of the student?s work. Through the use of activity theory, a coherent approach to understanding the complex milieu of classroom learning environment can be developed.

An Insight into Norwegian Students? Thoughts about Mathematics
Kirsti Kislenko

Students? beliefs about mathematics were the focus of a pilot study based on fieldwork carried out in Norway in early 2005. A web-based Likert-scale questionnaire about beliefs in mathematics was administered to students in 6 schools from one urban area. Two hundred and seventy-six students from grades 7 (12-13 years), 9 (14-15 years), and 11 (16-17 years) completed the questionnaire. Despite lacking interest in mathematics, students acknowledged the usefulness, importance, and need to work hard in mathematics.

Autobiographical Research and Mathematics Curriculum
Andy Begg

Research methods such as narrative inquiry and autobiographical research are increasingly accepted in education. In this paper I discuss why and how I used autobiography in my research, the value of this to my work in mathematics education, and my emerging view of school mathematics, curriculum, and the related development processes.

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

This paper is a report of a sustained professional development project in South Australia in which a small group of preschool teachers worked with the authors to develop their own skills in facilitating young children?s mathematical learning through investigative approaches and their own assessment of this learning through the use of learning stories. After providing some background information about the project, this paper considers the impact of the project on the early childhood educators themselves and their growth in knowledge, skills, and confidence in early childhood mathematics.

CAS in the Middle Secondary Years: Strengths, Weaknesses, Opportunities and Threats
Robyn Pierce

Computer Algebra Systems (CAS) are used in middle secondary classrooms as a tool to support learning and sometimes in preparation for senior secondary mathematics. This paper presents an analysis of strengths, weaknesses, opportunities, and threats identified in the literature and perceived by twelve secondary teachers working with year 9 and 10 students. CAS is valued for calculation and manipulation capabilities, the option of alternative representations, the opportunity for systematic exploration, and for prompting rich discussion. However the technical overhead, initial workload for the teacher, and unresolved questions about the contribution of machine and by-hand work to learning must also be considered.

Defining Teacher Knowledge Needed in the Teaching of Statistics at Primary School Level
Tim Burgess

A study of teacher knowledge, necessary for and used in the teaching of statistics investigations, was conducted in four New Zealand primary classrooms in 2006. This presentation reports on the framework that was developed to describe the components of teacher knowledge with regard to statistics. The framework integrates six dimensions of statistical thinking with four types of teacher knowledge: knowledge of content, both common and specialised; and pedagogical content knowledge, related to both students and teaching. Video and stimulated-recall interview data were analysed in relation to the framework to develop descriptions of knowledge used in the real-time tasks of teaching.

Exploring Data Representation and Statistical Reasoning through Integrated Investigations in a Grade 2 Classroom
Karen Ahearn

Grade 2 students, experienced in the use of technological tools in learning, were engaged in measurement and data investigations related to an integrated unit ?How I have changed?. Data analysis skills were developed as students were encouraged by the mathematics specialist teacher to pose and interpret questions, experiment with data handling methods, draw inferences, and make connections with prior representations. This pilot study focused on the impact of students? use of Excel software on their understanding of the relationship between the data and different graphical representations. The study also identified both student misconceptions and advanced development of proportional reasoning.

Improving Procedures for Effective Teaching
Murray Black, Farida Kachapova and Ilias Kachapov

The paper gives a brief overview of the relations between procedural and conceptual knowledge and emphasizes the importance of developing effective mathematical techniques. The paper describes and analyses some teaching strategies used by the authors in a classroom in order to improve students? learning of some mathematical techniques and their understanding of the relevant concepts. We show how the substitution method can be applied to some classes of mathematical problems where traditionally other methods are used. The problems are completing the square, solving quadratic equations, evaluating the limits of indeterminate form 0/0 and integrating rational functions with a quadratic in the denominator. We also analyse how the method of probability trees is used in problems about conditional probability and the ways to improve this teaching strategy.

Mathematical Modelling in CAS Clothing
Vince Geiger, Rhonda Faragher and Trevor Redmond

This paper considers the potential of Computer Algebra Systems (CAS) to enhance the processes associated with mathematical modelling and application tasks. In doing so, the role of technology in the cyclical development of mathematical models will be theorised. Finally, a theoretical framework will be outlined for a classroom-based investigation into the implementation of CAS technologies into classroom contexts where mathematical modelling and applications are a focus.

Mathematically Gifted Students Managing School Transfer
Brenda Bicknell

This paper reports on the school transfer of 15 mathematically gifted Year 6 and Year 8 students. The data are extracted from a longitudinal qualitative study that examines student and parent perspectives, and programme provision for mathematically gifted and talented students before and after a change of schools. Two groups of primary school students made the transfer to intermediate school (Years 7 and 8) or to a Years 7-13 high school. Another group of students from intermediate school made the transfer to high school. The students? and their teachers? and parents? perceptions of the transfer are described.

Measuring the Effectiveness and Efficiency of Language-In-Use for Algebra Learning: A Multi-Level Nested Modelling and DEA Approach
Robert de la Serna

This study investigates the effectiveness and efficiency of mathematics instruction in two languages (English vs. Cebuano/English Code-Switching) on the performances of Filipino algebra students in 11 classes of a Philippine private high school. Conducted in 2005-2006, a quasi-experiment addressed the question: Between the two languages of instruction, which promotes better algebra learning among Filipino bilingual students? The analyses are limited to evaluating the performances of high-ability students (comprising 3 classes) by using Multi-level Nested Modelling to compare the impact of languages-in-use on student achievement, and Data Envelopment Analysis (DEA) ? a novel economic modelling approach ? to measure the relative efficiencies of student learning outcomes.

Misconceptions in Locating Negative Decimals on the Number Line
Wanty Widjaja, Kaye Stacey and Vicki Steinle

This paper explores misconceptions revealed when pre-service teachers locate negative decimals on a number line. Written responses from 96 pre-service teachers to tests and group worksheets and video-recorded observation of their classroom discussions and interviews provide extensive data. Three misconceptions are identified. Two relate to incorrect analogies between the positive and negative parts of the number line. The other is a ?repair? to overcome an inconsistency, made more likely by intuition that negative decimals are very small. Implications for teaching are drawn.

Myths and Positioning: Insights from Hermeneutics
Steve Thornton

School mathematics values abstract reasoning over practical knowledge, propagating the myth of reference. School mathematics also makes claims that mathematics is essential for effective functioning in society, thus propagating the myth of participation. This paper uses hermeneutics to examine a worksheet used in a year 7 mathematics classroom to illustrate the myths of reference and participation. The continuation of these myths, together with discourse that is localising and limiting, devalues students? informal knowledge and positions them as subservient to mathematics rather than as subjects having mathematical agency.

Pre-service Primary Teachers Developing Positive Attitudes Towards Teaching Mathematics
Julie Clark

There is ongoing concern about the negative attitude large numbers of pre-service primary teachers have towards mathematics. The participants in this study were students in year-long primary mathematics curriculum studies courses that focussed on beliefs and attitudes, alongside content and pedagogy. Throughout the year, pre-service teachers personally experienced supportive and effective mathematics communities. As a result, many pre-service teachers developed a deeper conceptual understanding of mathematics. The majority of the pre-service teachers expressed increased confidence in their ability to teach mathematics and a willingness to continue gaining skills and knowledge in mathematics pedagogy.

Proportional Reasoning: A Global or Localised Development?
Vince Wright

The study of 29 year 8 students investigated whether the development of proportional reasoning was localised or generalised. It found support for localised early development and a generalised later development. Multiplicative thinking with whole numbers was found to be a necessary, but not sufficient, condition for proportional reasoning.

Reform and Assessment Practice: The Need for an Investigation
Julie Anderson

Public education in Victoria is currently undertaking a major reform process involving a number of initiatives including reporting and assessment. This paper will describe the identification of a gap between assumptions made by Department of Education and Training about teachers? assessment practice and about what teachers actually do when they assess student progress. A description and findings of a pilot study is included justifying the need for a wider investigation.

Revisions and Extensions of a Pirie-Kieren-Based Teaching Model
Peter Hughes

A teaching model based on Pirie-Kieren Theory was devised for the teaching of strategic thinking for the New Zealand Numeracy Project. This was used in the design of teacher material that was progressively available to most New Zealand primary teachers in the period 2001 to 2006. The ways in which teachers mis-implemented the model in their classrooms led to some significant alterations. In particular, the meaning of imaging is better defined and support for real-time formative assessment of students to inform the teaching is revised and extended.

Te Poutama Tau (TPT): An Indigenous Response to the Numeracy Development Project 2002-2006
Wini Emery and Leeana Herewini

Maori medium education in New Zealand has been in existence for more than 20 years. Te Poutama Tau (TPT), similar to the mainstream Numeracy Project, is responsive to the Ministry of Education?s strategy for improving levels of literacy and numeracy in New Zealand schools. The major difference between the Numeracy Project and TPT is that TPT is situated within the context of M?ori development, including the maintenance and revitalisation of the M?ori language. Since 2002, there have been four evaluation reports tracking the progress in Te Poutama Tau. The reports have been used to show trends in student achievement, and the teaching of mathematics via Te Reo M?ori.

Teaching Geometry with CAS in the Junior Secondary Classroom: A Case Study
Warren Palmer

Computer Algebraic System (CAS) calculators are currently being trialled in a number of junior secondary New Zealand mathematics classrooms. In this case study, two geometry lessons are described where the teacher has successfully incorporated the technology into his teaching so that a problem that might otherwise have proved to be a stumbling block is surmounted, and insights into the nature of mathematical proof are explored. The observed lessons reflect a problem solving approach advocated by the teacher himself and they appear to be engaging yet challenging for the students. The lessons illustrate, in a positive manner, what can be accomplished once a teacher has attained a solid grasp of the relatively new technology.

The Cognitive and Pedagogical Affordances of Digital Learning Tools on Early Mathematical Development
Kristy Goodwin, Joanne Mulligan and John Hedberg

Significant technological change has impacted on the representational modalities employed in mathematics learning. Yet, studies evaluating their impact and efficacy are not entirely unequivocal. This study investigates the unique contribution of interactive, digital technologies in the learning of early mathematical concepts in the first years of schooling. The study describes the pedagogical and technological affordances upon differences in observed learning outcomes. The impact of digital, pedagogical interventions on children?s internalised representations of concepts is explored. Case studies of the development of children?s mathematical representations are presented through the use of digital agents, such as learning objects, interactive whiteboards, and wireless mobile learning devices.

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

Using a design approach, this study monitors the influence of patterning tasks on the mathematics learning of 10 Kindergarten children. The children were engaged in a Pattern and Structure Mathematics Awareness Program over 15 weekly teaching episodes. Children were pre- and post-tested using an interview assessment of pattern and structure (PASA) and a standardised mathematics test. Nine of the 10 children showed impressive growth in their ability to represent and symbolise simple and complex patterns, arrays, grids, partitions, borders, and growing patterns. They also showed substantial improvement in mathematical skills such as counting in multiples, sequencing, similarity and congruence, and co-linear structure.

The Impact of Didactical Contract on Students? Perceptions of their Intentional Learning Acts
Troels Lange and Tamsin Meaney

This paper considers how the didactical contract between students and their teacher is influenced by students? foregrounds and backgrounds and the impact that this has on their learning of mathematics. It uses examples of two students in two primary classrooms, one in Denmark and the other in New Zealand. It is possible to see in both situations that students develop non-conventional mathematical learning. We speculate on how students? perceptions of the didactical contract are affected by considerations such as those to do with social interactions and the impact of these perceptions and consideration on possibilities for learning.

Using Cabri Geometry to Explore the Geometric Properties of Parallelograms in Year 7 Mathematics Classrooms
Sahar Bokosmaty

A teaching experiment was conducted to explore the impact of Cabri Geometry on 6 Year 7 students? understanding of parallelograms. Students worked in pairs, and were guided through activities designed to introduce the software and to encourage discovery and exploration of specific geometric properties. Students were also required to use this knowledge to solve geometric problems. Over several lessons, data were collected via taperecorded observations of student interactions and discussions, work samples, and test results. The dynamic investigation promoted deeper understanding of the underlying geometric properties of parallelograms and students? abilities to solve geometric problems.

Using Counter-Examples and Paradoxes in Teaching Probability: Students? Attitudes
Murray Black, Farida Kachapova, Sergiy Klymchuk and Ilias Kachapov

The paper presents and analyses students? attitudes towards using counter-examples and paradoxes as a pedagogical strategy in teaching/learning of a first-year university course in probability theory and applications. Our intentions of using this strategy were: to achieve deeper conceptual understanding; to reduce or eliminate common misconceptions; to advance one?s statistical thinking, which is neither algorithmic nor procedural; to enhance generic critical thinking skills ? analysing, justifying, verifying, checking, proving; to increase motivation and interest in the subject; and make learning more active and creative. The majority of the students reported that the strategy was effective and made learning more challenging, interesting, and creative.

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

Communicating mathematics to distance students is often difficult. This presentation reports on preliminary research in three areas. (1) The trial of an electronic handwriting tool in MSN Messenger in a large mathematics service course in Semester 1 2006 for online synchronous group work and individual consultations. (2) The use of a tablet PC and a computer software program ?Camtasia? to record live lectures involving electronic handwriting for transmittion to distance students. (3) The use of pre-recorded online ?Breeze? presentations to highlight particular ?tricky questions?. The presentation will demonstrate the use of these technologies, discuss some of their advantages and disadvantages, and report on initial results of the research.

Wanted: One Great Maths Teacher!
Pamela Perger

The New Zealand Teachers Council identifies factors important for teacher registration. These factors must also be considered the foundation criteria for effective teachers of mathematics. What is it that makes an effective mathematics teacher? This article presents a picture of the ideal mathematics teacher as described by a group of Year 7 and 8 Pasifika students (11-12 year olds). The criteria identified by these students fell into two categories. One category addressing factors that were associated with personal qualities of the teacher and the second category that related to the learning environment the students saw as most supportive to their mathematical learning.

Year 12 Students? Participation in Higher Mathematics Courses
Mohan Chinnappan, Stephen Dinham, Tony Herrington and Dale Scott

Australia?s future prosperity and our ability to compete in the global arena demand that our education systems develop human capital that is highly skilled and knowledgeable. Participation in mathematics, in particular higher mathematics, is an important prerequisite for young Australians if we are to develop the range of skills that underpin this capital. This is a report on a study in progress focusing on the concern that participation in Mathematics at Year 12 and at universities in Australia is declining. In the context of this larger issue, we identify a number of questions we will be investigating with students, teachers, universities, and other stakeholders.

Poster (abstract only)
Round Table (abstract only)
An Investigation of Mathematics Strategies in Traditional School Contexts and Real-World Contexts
Julie Clark and Kathy Brady

The purpose of this roundtable is to describe a proposed research project and to seek advice, ideas, and suggestions from participants. The project will address two main questions. Firstly, how do mathematical strategies used in traditional school contexts compare with strategies used in real-world contexts? Research continues to highlight the lack of connections that students make between mathematics used in school and in everyday life. The National Council for Teachers of Mathematics emphasise the importance of students being able to use mathematics in varying situations (2000). In this study students will be presented with basic mathematics tasks in different contexts. For example, they will be asked how much change they would get from $5 for something that cost 65 cents. Next, they will be asked to answer a similar question presented traditionally: 500 ? 65. The second research question associated with this project is: What perceptions do teachers have of students? mathematical strategies? Fennema and Franke (1992) indicate that it is important for teachers to understand students? mathematical thinking and to have good knowledge of various instructional approaches. Teachers will be interviewed concerning their perceptions of students? mathematics learning. During the interviews they will watch DVD excerpts of students solving problems and be asked to make comments on the strategies used.

M?ori Student?s Perspective on Their Mathematical Journey Through M?ori Medium
Leeana Herewini

The focus of this round table is to invite colleagues to inform, contribute to, and explore some of the key themes that arose in this study. These being the transition from primary school to secondary school, the role of students? perspectives in mathematics research, and mathematics education in an indigenous language. This study set out to explore the perspectives of 10 year 8 students on their mathematics learning and in particular the transition from Kura Kaupapa M?ori (M?ori medium primary schools) to secondary school (M?ori medium education in the Waikato region has extended from K?hanga Reo to wharekura). The Kura Kaupapa M?ori in this region are at least 10 years old. The 10 students interviewed in this study were identified as successful learners of mathematics in their year 8 programme, with outcomes at Stage 7 or above on the number framework. The students? teachers had participated in professional development in Te Poutama Tau/Numeracy project. Kaupapa M?ori methodology and interviews were key approaches undertaken in this study. Interviews took place with students and a questionnaire was given to the students? year 9 teachers. The interview process was a powerful vehicle in accessing student voice. Results indicated there were differences between the students? perceptions of their year 8 and year 9 mathematics programmes. The students viewed mathematics positively and had formed opinions regarding mental strategies and where and how to use them. Some students were able to discern what constituted effective teaching and had formed ideas on the value of the mathematics taught they had learned. An interesting area to emerge in the research was the self report by the girls in the study on their ability.

Profiles of Thinking Skills and Levels of Motivation in a Problem-Solving Task
Sarah Buckley, Mary Ainley and Pip Pattison

In the following project, we investigated students? use of mathematical thinking skills in an interactive, computer problem-solving task. Within the program Between The Lines (BTL) (Ainley & Hidi, 2002), a range of statistical information was presented. Students were required to integrate these different types of mathematical information in line with constructive assessment. This approach promotes problem solutions that move beyond the simple reproduction of answers and formulae and are representative of higher order thinking (Clarke, 1996). Note-taking is a process known to facilitate learning and the development of ideas (Kiewra, 1989), thus the program also asked students to take notes as they worked through the information. These notes were then used to create a problem solution. Students? motivational reactions (i.e., their interest levels) were also monitored throughout the task. Two-hundred and eighty-six Year 8 students (151 males and 135 females) participated in the project. The SOLO taxonomy (Biggs & Collis, 1982) was used to code students? notes whereas their solutions and notes were marked for understanding of the material presented. Associations between students? level of thinking and understanding were investigated in relation to levels of on-task interest and other indicators of motivational engagement. In this roundtable presentation, examples of students? problem solutions and notes will be presented. Profiles of students? thinking, for example, higher order versus low order thinking, and their relationship with on-task motivation will be examined. Discussion will focus on interpretation and implications of the data.

Progress in Mathematics ? Learning through Home School Partnership
Denise Smith and Gaynor Terrill

The Home School Partnership Numeracy facilitators will start the round table discussion by presenting the findings from three small studies: i) New Zealand Council for Education Research (NZCER) National Exploratory Home School Partnership Research ii) A case study on increasing parents? confidence in order to support their children?s learning in numeracy iii) A case study on the clarity of communication between home and school in relation to student achievement in numeracy. The international evidence cited in Alton-Lee (2003) positively supports the enhancement of student learning through home and school partnerships. In New Zealand, two Best Evidence Syntheses research studies also highlight the importance of establishing effective relationships between home and school (Alton-Lee, 2003; Biddulph, Biddulph, & Biddulph, 2003). The Home School Partnership project reflects the acknowledgement of parents as first teachers and the desire to continue to encourage parents to confidently interact and communicate with their children about mathematics. Effective relationships within the school community encourage parents to take an active role in the shared responsibility of their children?s education. Immigrants, refugees and parents who sometimes speak English as a second language are the focus of the community partnerships, especially Pasifika families. Pasifika students are identified through National Numeracy data as achieving well below National benchmarks. Aspects that might be considered in this discussion include: the establishment and sustainability of home and school partnerships; successful learning communities involving facilitators, lead teacher, and lead parents; mathematics as a ?frightening focus? for parents and community sessions in parents? first language.

Some Methodological Considerations in the Estonian Study about Students? Beliefs in Mathematics: Is Triangulation Necessary?
Kirsti Kislenko

A study about students? beliefs and attitudes towards mathematics was carried out in Estonia in early 2006. The study aimed, firstly, to investigate students? views towards mathematics and the underlying rationale, and secondly, to gain an understanding of the regular mathematics classroom activities in Estonia. The study used methodological triangulation, that is to use different methods on the same object of the study (Cohen, Manion, & Morrison, 2000). Firstly, a web-based Likert-scale questionnaire with 98 statements about beliefs in mathematics was administered to seven schools in an urban area in Estonia. Five hundred and eighty students from grades 7 (14 yrs.), 9 (16 yrs.), and 11 (18 yrs.) completed the questionnaire. Secondly, 26 semi-structured interviews were conducted with students and teachers for illuminating their thoughts about mathematics, and the learning and teaching of mathematics. Thirdly, during a three month period I participated in 11 teachers? lessons observing 55 mathematics lessons. Field notes gathered reflected a general picture of the classroom activities ? teachers? activities, methods, behaviour, relationship with students, and so on; and also personal impressions from different situations that I found interesting. Participants of the round table will be asked to consider the following questions: What are the appropriate tools for these kinds of investigations? What are the strengths and weaknesses of the methods used in my study? What are the possible strengths and weaknesses in combining quantitative and qualitative data collection methods?