### Conference Proceedings 2015

Title |

Mathematics education in the margins |

Content |

MERGA 38 Table of contents |

Preface |

MERGA38 preface |

List of Reviewers |

MERGA38 list of Reviewers |

Keynote Address |

Exploring a Structure for Mathematics Lessons that Foster Problem Solving and Reasoning |

Mathematics Education as a Field of Research: Have We Become Too Comfortable? |

Researching and Doing Professional Development Using a Shared Discursive Resource and an Analytic Tool |

Practical Implication Award |

Teacher Actions to Facilitate Early Algebraic Reasoning |

Symposium |

Research Paper |

The challenge of supporting a beginning teacher to plan in primary mathematics |

Contemplating symbolic literacy of first year mathematics students |

Problematising Mathematics Education |

Identity as an Embedder-of-Numeracy: Identifying ways to support teachers to embed numeracy across the curriculum |

Young Children’s Number Line Placements and Place-Value Understanding |

The Role of Cultural Capital in Creating Equity for Pasifika Learners in Mathematics |

The importance of praxis in financial literacy education: An Indigenous perspective |

Coming to do Mathematics in the Margins |

“You play on them. They’re active.” Enhancing the mathematics learning of reluctant teenage students. |

CAS or Pen-and-paper: Factors that Influence Students’ Choices |

The Language Used to Articulate Content as an Aspect of Pedagogical Content Knowledge |

Specialised Content Knowledge: Evidence of Pre-service teachers’ Appraisal of Student Errors in Proportional Reasoning |

Learning from Lessons: Studying the Construction of Teacher Knowledge Catalysed by Purposefully-designed Experimental Mathematics Lessons |

Inclusive Practices in the Teaching of Mathematics: Supporting the Work of effective primary teachers |

Supporting Students to Reason About the Relative Size of Proper and Improper Fractions |

Proportional Reasoning as Essential Numeracy |

A Case Study of the Pedagogical Tensions in Teacher’s Questioning Practices When Implementing Reform-Based Mathematics Curriculum in China |

Improving the Effectiveness of Mathematics Teaching through Active Reflection |

Promoting Teacher Growth through Lesson Study: A Culturally Embedded Approach |

The Self-Efficacy of students with Borderline, Mild and Moderate Intellectual Disabilities and their Achievements in Mathematics |

Identifying Core Elements of Argument-Based Inquiry in Primary Mathematics Learning |

STEM Education: What Does Mathematics Have To Offer? |

The Challenge for Non-first-language-English Academic Publishing in English Language Research Outlets |

The Impact of Let’s Count on Children’s Mathematics Learning |

Comparing the Development of Australian and German 7-Year-Old and 8-Year-Old’s Counting and Whole Number Learning |

Learning at the Boundaries |

The Practice of ‘Middle Leading’ in Mathematics Education |

Teaching Computation in Primary School without Traditional Written Algorithms |

Calculating fo r probability: “He koretake te rima” (Five is useless) |

Students’ Relationships with Mathematics: Affect and Identity |

Using Alternative Multiplication Algorithms to ‘Offload’ Cognition |

Successful Mathematics Lessons in Remote Communities: A Case Study of Balargo |

Differentiated Success: Combining Theories to Explain Learning |

The Mathematics Instructional Leader: What a Difference Crucial Conversations Make |

The Search for Fidelity in Geometry Apps: An Exercise in Futility? |

Pre-service teachers and numeracy in and beyond the classroom |

Gender Differences in Mathematics Attitudes in Coeducational and Single Sex Secondary Education |

Developing a Theoretical Framework to Assess Taiwanese Primary Students’ Geometric Argumentation |

Starting a Conversation about Open Data in Mathematics Education Research |

A snapshot of young children’s mathematical competencies: Results from the Longitudinal Study of Australian Children |

Examining PCK in a Senior Secondary Mathematics Lesson |

Teacher’s Scaffolding over the Year to Develop Norms of Mathematical Inquiry in a Primary Classroom |

Middle Years Students Influencing Local Policy |

Early Years Teachers’ Perspectives on Teaching through Multiple Metaphors and Multimodality |

Young Indigenous Students’ Engagement with Growing Pattern Tasks: A Semiotic Perspective |

Professional Knowledge Required when Teaching Mathematics for Numeracy in the Multiplicative Domain |

Determining a Student’s Optimal Learning Zone in Light of the Swiss Cheese Model |

Student and Parent Perspectives on Fipping the Mathematics Classroom |

Authority and Agency in Young Children’s Early Number Work: A Functional Linguistic Perspective |

Examples in the Teaching of Mathematics: Teachers’ Perceptions |

How Inquiry Pedagogy Enables Teachers to Facilitate Growth Mindsets in Mathematics Classrooms |

Challenging the Mindset of Sammy: A Case Study of a Grade 3 Mathematically Highly Capable Student |

Facebook as a Learning Space: An Analysis from a Community of Practice Perspective |

Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking |

Mentoring to Alleviate Anxiety in Pre-Service primary mathematics Teachers: an orientation towards improvement rather than evaluation. |

Spatial Visualisation and Cognitive Style: How Do Gender Differences Play Out? |

The Practice of Teacher Aides in Tasmanian Primary Mathematics Classrooms |

Qualitative Facets of Prospective Elementary Teachers’ Diagnostic Proceeding: Collecting and Interpreting in One-on-one Interviews |

Describing the nature and effect of teacher interactions with students during seat work on challenging tasks |

Teachers’ talk about Robotics: Where is the Mathematics? |

Teaching Statistics in Middle School Mathematics classrooms: Making Links with Mathematics but Avoiding Statistical Reasoning |

Context counts: The potential of realistic problems to expose and extend social and mathematical understandings |

Theorising about Mathematics Teachers’ Professional Knowledge: The Content, Form, Nature, and Course of Teachers’ Knowledge |

Understanding Geometric Ideas: Pre-service Primary Teachers’ Knowledge as a Basis for Teaching |

Mathematical Language Development and Talk Types in Computer Supported Collaborative Learning Environments |

The Individual Basic Facts Assessment Tool |

Affording and Constraining Local Moral Orders in Teacher-Led Ability-Based Mathematics Groups |

Exploring relationship between scientific reasoning skills and mathematics problem solving |

Developing Adaptive Expertise with Pasifika Learners in an Inquiry Classroom |

Getting out of Bed: Students’ Beliefs |

Improving Student Motivation and Engagement in Mathematics Through One-to-one Interactions |

A Cross-cultural Comparison of Parental Expectations for the Mathematics Achievement of their Secondary School Students |

“I was in year 5 and I failed maths”: Identifying the Range and Causes of Maths Anxiety in first year Pre-service Teachers. |

Enhancing Mathematics (STEM) Teacher Education in Regional Australia: Pedagogical Interactions and Affect |

Mathematics, Programming, and STEM |

Laying the Foundation for Proportional Reasoning |

The Development and Evaluation of an Individualised Learning Tool for Mathematics students with Intellectual Disability: IMPELS |

Capturing Mathematical Learning in an Inquiry Context: There are Some Things Not Easily Measured |

Teacher Professional Growth through using a Critical Mass Mentoring System: Effective Whole School Teacher Professional Development |

Anatomy of a Mathscast |

An Exploration of Strategies That Teachers Use When Teaching Beginning Algebra |

Factors Influencing Social Process of Statistics Learning within an IT Environment |

Identifying categories of Pre-service Teachers’ Mathematical Content Knowledge |

Using Drawings and Discussion to Prompt Young Learners to Reflect Upon and Describe Their Mathematical Understandings |

Language and Mathematics: Exploring a New Model to Teach in Bi/Multilingual Mathematics Classroom |

Exploring the Influence of Early Numeracy Understanding Prior to School on Mathematics Achievement at the End of Grade 2 |

An Irish Response to an International Concern:Challenges to Mathematics Teaching |

An Analysis of Modelling Process based on McLuhan’s Media Theory: Focus on Constructions by Media in Cases of Using Geoboard |

The Knowledge Dimension of Revised Bloom’s Taxonomy for Integration |

Developing an analysing tool for dynamic mathematics-related student interaction regarding affect, cognition and participation |

Thinking Strategies Used by 7th-Grade Students in Solving Number Sense Problems |

Short Communication (abstract only) |

A Focus Question Approach to the Teaching of Mathematics This is a presentation on a focus question approach to teaching mathematics (FFQA) which is the title of my thesis. It is proposed to investigate the impact of the FFQA at the commencement of each mathematics lesson on the learning and motivation of students. The style of five questions that I propose to investigate has the first four questions as instrumental style questions that focus on procedural knowledge, with the final question using a relational understanding approach with some of the questions being open ended investigational style questions focusing on conceptual knowledge. The research is ongoing. |

A Problem Solving Lesson: Pre-service Teachers Initiation to Lesson Study Three pre-service teachers (PST) who had no prior experience with lesson study had to use a lesson study approach to plan and teach a problem solving lesson. This paper documents how the three PSTs were initiated into the Japanese style of lesson study and then how as a team they went about planning their research lesson on problem solving for a primary three class and then teaching it. The focus is on some of the issues that surfaced when preparing this problem solving lesson on magic squares and how they addressed them. |

Breaking down Barriers Engaging cohorts including less quantitatively-adept students and educating them about the value of Statistics has its challenges. This talk will outline two successes: the first resulted in a first-year Statistics for Business course increasing student satisfaction scores from under 3.5 out of 5 to 4.72 whilst maintaining ‘challenge’ scores and reducing Failure rates previously exceeding 25% to 7-12%; the second is a national project-based learning activity (piloted in the Hunter Region in 2014) which facilitates boundary encounters (between secondary, tertiary, and industry sectors and students having varied backgrounds and areas of interest) and develops key communication, research and quantitative skills. |

Building upon the Language Model of Mathematics The language model of mathematics is a useful framework to conceptualise the teaching and learning of mathematics from a constructivist perspective. The model currently proposes that students move along two dimensions (visual and verbal) towards increasing levels of mathematical abstraction. We present the case for theorising the existence of a third dimension, the gestural, by drawing upon established theories of learning within mathematics and also from brain based learning. Examples will be provided on how the addition of the gestural dimension can enhance mathematics education at all levels. |

Changing Practices in Indigenous Communities Educating teachers in Indigenous communities to use collaborative mathematics is a challenge. This is particularly the case in developing countries such as Papua New Guinea. First there are the issues around the perception of authority and questioning in school. Second, there is the issue of class atmosphere and teacher pedagogical knowledge. Third is the issue of meeting different needs in large under-resourced classes. Finally there is the issue of teacher professional learning. This paper discusses one attempt that has led to an increased awareness and use of the role of questioning in the classroom. It points out some of the aspects of teacher development and what seemed to be contributing to change. |

Conceptual Connectivity in Mathematics Human environmental interactions involve general conceptual connectivity processes such as categorisation, abstraction and generalisation. These are linked to the development of mathematics concepts, but research in this area is relatively new in mathematics education. A conceptual connectivity lens, however, has been used in cases where there are difficulties in mathematics learning, such as developmental dyscalculia, as well as in studies of mathematical pattern and structure with young gifted children. This presentation suggests that such studies support the determination that individual differences in processing of environmental information are an important way forward in understanding what underpins mathematics conceptual development. |

Examining a Students’ Resource for Reconstructing the Limit Concept at Need: A Structural Abstraction Perspective This presentation examines a student’s learning of the limit concept of a sequence compatible with his strategy of making sense, through which the structural abstraction framework evolves and is further refined. The attention is focused on a student’s generic representation of the limit concept that allows him to generate meaningful components specific to particular contexts. Further, a sketch of the basic ideas of structural abstraction is given, and the use of the generic representation as a resource to reconstruct the meaning of the concept at need is discussed. Additionally, the importance of structural abstraction for learning mathematics is elaborated. |

Exploring Students’ Views on using iPads in Mathematics The use of iPads in education is increasing, with increasing numbers of studies focussing on teacher use of this tool in mathematics teaching and learning. As a stakeholder group, the views of students must also be investigated. As part of a larger case study, the views of Year 5 to Year 12 students from one Victorian school were sought about the use of iPads in mathematics. A number of concerns related to the perceived negative impact of iPad use in mathematics learning arose and will be further explored in the presentation. |

Mapping school students’ aspirations for STEM careers Declining enrolments in STEM disciplines and a lack of interest in STEM careers is concerning at a time when society is becoming more reliant on complex technologies. We examine student aspirations for STEM careers by drawing on survey data from 8235 school students in Years 3 to 11 who were asked to indicate their occupational choices and give reasons for those choices. These data are also examined in relation to student SES, gender, prior achievement and educational aspirations. The analysis provides a strong empirical basis for understanding current student interest in STEM and exploring implications for educational policy and practice. |

Mathematical Thinking in a Context of ‘General Thinking’: Implications for Mathematics Education This new project explores the similarities and differences of mathematical thinking and ‘general thinking’, as well as related motivational and emotional aspects, focusing on how these differ in educational contexts. It will examine assumptions of the underlying feature of mathematics curriculum design and pedagogy, for example, that linear structure is the most efficient means of building mathematical knowledge or that number-based knowledge is a reliable indicator of mathematical skill. Insights gained will be used to improve the current paradigms in course structure and pedagogy for classroom mathematics in order to develop a structure better aligned to student capabilities and potentials. |

Paternal influence on school students’ aspirations for STEM careers There is a growing awareness of the important and differential influence fathers have on child lifestyle behaviours compared to mothers. This ‘paternal’ influence could potentially carry across to children’s early career aspirations. A sample of n = 8235 school students in Years 3 to 11 were asked to indicate their occupational choices, give reasons for those choices and also provide information about their parents education and occupation. Using regression analysis, associations between paternal and maternal education levels and occupations with children’s STEM career aspirations were modelled. The findings provide further evidence of the potential differential influence parents have on their child’s aspirations. |

Pre-service Teachers’ Views on Mathematics Homework Practices Literature suggests that homework plays an important role in mathematics learning yet, in the Australian context, there is limited related research on this issue. This exploratory study sets out to better understand pre-service teachers’ intentions and practices concerning mathematics homework. Using a survey design, we analysed data collected from a questionnaire administered to 98 (71% response rate) pre-service teachers (PSTs), all in the third year of their BEd program and completing a third course in mathematical methods as well as professional experience. Contrary to our expectation, the difference in perceptions among PSTs teaching upper and lower primary grades were not statistically significant. |

Primary-Middle Pre-Service Teachers reported use of the Mathematics Textbook The 1999 TIMSS video study highlighted a heavy reliance on the mathematics textbook in Australian classrooms (Hiebert et al., 2003). This promoted further investigation by Vincent & Stacey (2008) who have documented the differences between mathematical textbooks and concerns with regard to problem solving. However, there is much anecdotal evidence to suggest that the role of the textbook may be changing and that the emergence of digital technologies may in fact replace the mathematics textbook (Hu, 2011). Hence, this exploratory study intends to a brief insight into the current status of the mathematics textbook and its use within Australian classrooms. |

Promoting Financial Literacy in Pre-service Teacher Education through On-line Modules Opening Real Science (ORS) is 3-year Australian Government project led by Macquarie University supported by the Office for Learning and Teaching under the Enhancing the Training of Mathematics and Science Teachers Scheme (ETMST). ORS is developing a series of modules for implementation in teacher education programs, some of which focus on financial literacy: budgeting, investing and protecting, and modelling. The modules will be designed for active learning incorporating digital literacy themes to showcase implementation of technology integration into curriculum. Currently there are several trials in progress at three partner Australian universities. Evaluation data will inform the design-based approach to program re-development aimed at building the mathematical competence, and confidence of teachers. |

Promoting the Development of Foundation Content Knowledge in all Primary Pre-service Teachers A feature of Linsell and Anakin’s (2013) concept of foundation content knowledge is that all pre-service teachers should have a growth oriented disposition and extend their knowledge, whether or not it is initially strong. This study reports on the use in mathematics pedagogy classes of introductory problems designed to encourage all first year primary pre-service teachers to become aware of the features of foundation content knowledge and to extend their own knowledge. Eighty-one percent of those pre-service teachers whose foundation content knowledge was not initially strong considered the introductory problems helpful, compared to 61% of those whose knowledge was strong. |

Teachers’ Beliefs about Knowledge of Content and Students and its Effect on their Practice This study investigated mathematics teachers’ beliefs about teachers’ knowledge of content and students (Ball, Thames, & Phelps, 2008) about particular mathematical content and its effect on teaching practice. Two teachers participated in the study. Data were collected through classroom observations and an interview. The interview was based on An, Kulm, Wu, Ma, and Wang (2002) and focused mainly on teachers’ beliefs about knowledge of students’ thinking, approach to planning the mathematics instruction, students’ homework, and importance and approach to grading homework. The study indicated both teachers believed the importance of teachers’ understanding the way students think about a certain mathematics subject or the difficulties they experience with it. Nevertheless, it is seemed the teachers’ beliefs had no effect on their teaching practice. Moreover, they had limited awareness of how to identify students’ difficulties. |

Teaching out-of-field: Meanings, representations and silences Teaching out-of-field is a concern internationally, and in Australia, and is linked to social, economic and educational costs for students and teachers along with an ethical and social justice issue for the community. At the national level, out-of-field teaching is most often represented as a problem of teacher quality involving less qualified teachers. Using a critical lens, meanings and representations of government policy and stakeholder perspectives and practices are analysed. The findings show how teaching out-of-field occurs and is legitimated and reveal the opportunities for contesting these positions to improve the outcomes for students and out-of-field teachers. |

The Australian Mathematics Competition: What’s the Score? The Australian Mathematics Competition (AMC) is a problem-solving competition for Primary and Secondary students. Each paper has 30 problems graded from routine to baffling, challenging and rewarding students of all abilities. The competition’s quality depends on the collective effort of dozens of Mathematics Educators (Primary to University) who write and scrutinise the papers in several stages. Our current work is to ensure the AMC provides a reliable challenge for students. Tools for calibrating the performance of questions and papers across a range of question types are improving the competition, measured by the relative performance of each question, and by each paper’s aggregate score. |

The Pattern and Structure of the Australian Curriculum-Mathematics The mathematical proficiencies in the Australian Curriculum—Mathematics describe the processes students are engaged in while developing mathematical concepts (ACARA, 2014). This presentation focuses on how the proficiencies: understanding, problem solving, reasoning and fluency, may work together to build patterns of thinking which can lead to generalised understandings of mathematical concepts. The authors connect the combined role of these proficiencies with a proposed Generalised Model of Patterning (McCluskey, Mitchelmore, & Mulligan, 2013), highlighting the role of patterning in the development of conceptual understandings within and beyond mathematics. |

Understanding Mathematics: Teacher Knowledge, Task Design and Evaluating Students’ Mathematical Reasoning This presentation describes a research project designed to understand the relationship between teachers’ conceptual understandings of mathematics, the tasks they design for their students and their evaluation of students’ responses to tasks. Using Timperley’s (2008) Teacher Knowledge Building and Inquiry Cycle, Year 5 and 6 primary teachers and leaders at a range of career stages engaged in tasks to highlight the connection between what students need to know, what teachers need to know and what teachers need to learn. The implications for developing teachers’ understandings of mathematics will be discussed in terms of system-level professional learning. |

Poster (abstract only) |

Round Table (abstract only) |

Investigating Mathematical Inquiry The aim of this Round Table is to bring together a community of researchers who focus on the teaching, learning, assessment, and research of a mathematical inquiry approach. We invite those interested in the study of mathematical inquiry to discuss their work or aspects of inquiry that are in need of research. A few questions are listed below to provoke conversation. Bring your own! 1. What shared and unshared perspectives do we have of mathematical inquiry? 2. What are purposes of mathematical inquiry? 3. How can mathematical inquiry be used to assess learning? 4. What signature practices characterise inquiry pedagogy in mathematics education? 5. How is mathematical inquiry similar to or different from inquiry in other content areas, such as science? 6. How does the teaching of mathematical inquiry fit into the broader repertoire of pedagogies used by teachers in the course of a year? 7. What challenges do teachers and students face in adopting mathematical inquiry? 8. Does an inquiry approach benefit children with different backgrounds differently? 9. What are key benefits and drawbacks of learning mathematics through inquiry? 10. Do particular strands of mathematics fit better with inquiry? 11. Does mathematical inquiry improve learning in mathematics? 12. Is mathematical inquiry scalable? 13. How can different paradigms contribute to a diversity of insights into mathematical inquiry? 14. What key research areas are strongly tied to mathematical inquiry (e.g., argumentation, socio-mathematical norms, collaboration)? 15. What are possible programs of research for mathematical inquiry? |

Promoting Positive Emotional Engagement in Mathematics of Prospective Primary Teachers Good teaching is described as that which is “charged with positive emotion” (Hargreaves, 1998, p.835). Yet, primary pre-service teacher education programs predominantly focus on the development of knowledge and pedagogy while affective aspects, including emotions, are only implicitly treated (Gootenboer, 2008). To date, research exploring the role emotions play in the process of learning to teach mathematics has received little attention (Hogden & Askew, 2007). The round table will begin by outlining the rationale and theoretical underpinnings of a trans-Tasman research project that aims to deepen primary pre-service teachers’ [PST] emotional and intellectual engagement in learning to teach mathematics. The Mathematics Emotional Engagement [MEE] project aims to develop and assess the effectiveness of an innovative teaching approach designed to promote positive emotional engagement in learning and teaching mathematics. The study explores the impact of a three-step interventional framework, referred to as ‘AIR’, that utilises a series of research-based instructional activities involving preservice primary teachers in: (1) Attending to their existing emotional responses towards the learning and teaching of mathematics; (2) Interpreting the causes and potential impact of existing emotional responses; and (3) Responding to their emotions with strategies to ameliorate negative affects on their learning and teaching of mathematics. Data from the first stage of the project—developing and refining AIR instructional strategies—will provide the stimulus for discussion amongst participants. |

Senior Secondary Students’ Pre-calculus and Calculus Understanding There are substantial and ongoing concerns in the Australian and international secondary and tertiary education sectors about students’ transition from secondary to tertiary mathematics. Declining enrolments in advanced mathematics in secondary schools and less stringent university entry requirements are seen as a major concern for the future of STEM education in Australia. In this round table, I will present data collected from secondary school students on precalculus and calculus topics. These data were collected from two groups of students: those studying intermediate mathematics in the last two years of secondary school; and those studying both intermediate and advanced mathematics. The results suggest that there are distinct differences in students’ procedural and conceptual understanding depending on which mathematics they studied in the last two years of secondary school. Students who studied both intermediate and advanced mathematics performed considerably better in all questions, not only on the calculus questions but also on junior mathematics pre-calculus topics such as gradient of a straight line. The data also showed that both groups of students had difficulty identifying lines parallel to axes, as well as explaining the meaning of the definition of the derivative. This presentation is part of a two-year state-wide longitudinal project that is investigating the transition from secondary to tertiary mathematics. |

Working Across Disciplinary Boundaries in Pre-service Teacher Education In Australia, a suite of national projects has been funded by the Australian government to promote strategic change in mathematics and science pre-service teacher education. This round table session will share some of the interdisciplinary strategies being trialled in one project, Inspiring Mathematics and Science in Teacher Education (IMSITE), and invite feedback from participants on the transferability of strategies to other institutional contexts and the sustainability of these strategies over time. The specific objectives of the IMSITE project are: • to develop and validate a repertoire of strategies for combining knowledge of content and pedagogy in mathematics and science; and • to connect academics from different communities of practice – mathematics, science, education – in order to collaboratively design and implement these new teacher education approaches. Six universities and 23 investigators – mathematicians, scientists, and mathematics and science teacher educators – are the core participants in the project, with more universities to be added in 2015. The first half of the round table session will showcase interdisciplinary strategies such as: • Collaborative development and delivery of new content and pedagogy courses by mathematicians and mathematics educators; • Reciprocal tutoring by mathematicians and mathematics educators into each other’s courses; • Peer observation by mathematicians and mathematics educators of each other’s teaching; • Development of a mathematics specialisation in primary pre-service programs. The remainder of the session will invite discussion of challenges to interdisciplinary collaboration (“siloing” of disciplines, inflexible workload and course funding models, cultural differences between the disciplines) and ways to overcome these. |