**Proceedings of the New Researcher’s Day and Day Conference held at University of Oxford, Saturday 10th June 2017**

## Contents

**01 Examining prospective mathematics teachers’ pedagogical content knowledge of limit using vignettes**

Melike Göksu Nur, Hatice Akkoç, Hande Gülbağcı-Dede and Betül Yazıcı

*Marmara University*

The aim of this study is to examine upper secondary prospective mathematics teachers’ pedagogical content knowledge (PCK) with regard to students’ difficulties with and misconceptions of limit. The PCK of 27 prospective teachers enrolled in a Turkish teacher preparation program was explored using vignettes which illustrate students’ difficulties with and misconceptions of limit. Data were analysed using content analysis. The findings focus on how participants notice, interpret, and make decisions on how to overcome students’ difficulties and misconceptions.

**02 Trainee teachers’ perceptions of solving word problems and the bar-model as a strategy to support children in solving word problems**

Sarah Ankers

*University of Chester*

Problem solving is one of the three main aims of the National Curriculum for Mathematics in England, but the literature suggests that solving word problems can be difficult for many children. The bar model is a strategy used in Singapore to bridge the gap between the word problem and the symbolic representation of the solution. As a new teacher-educator researcher, I was interested in how my group of PGCE trainee teachers perceived the bar model as a strategy for supporting children in solving word problems. Findings suggest that some trainee teachers experience the same issues in solving word problems as children, and that the introduction of the bar model should be structured alongside the teaching of key mathematical concepts rather than as a discrete topic.

**03 Replication of a British study in Brazil: How do teachers describe mathematical tasks?**

Leonardo Barichello and Rita Santos Guimarães

*University of Nottingham*

British researchers recently used factor analysis to identify seven factors underlying a comprehensive set of adjectives and expressions commonly used to describe mathematical tasks. This part of their study was replicated by the authors of this paper with 415 Brazilian mathematics teachers and the results presented many similarities, but in our analysis only four were similar to factors identified in the original study. In this paper, we will discuss the main results of both studies, compare them and present some possible explanations for the differences.

**04 Retaining heads of mathematics in a performative culture: A case study**

Bryony Black

*The University of Sheffield*

This study explores the experiences and perspectives of heads of mathematics departments (HoMs) in secondary schools in northern England. Taking as a theoretical background Foucault’s work on power, governmentality and discourse the study considers the way in which school leaders’ interpretations of a performative culture affects the work of HoMs. Employing a case study approach and using data collection techniques including focus groups, one-to- one interviews, audio diaries and observations, the aim of the study is to identify the conditions under which retention of those in the role of HoM is likely.

**05 How to look and what to see: Noticing in a mathematics community**

Julian Brown

*School of Education, University of Bristol*

As a new researcher in mathematics education, I am seeking to work through the ontological and epistemological challenges associated with setting aside the modes of observing that I have assumed, consciously or otherwise, and develop seeing in other ways. In this account of a workshop, I will discuss the presentation of different accounts of the same classroom episode constructed with different protocols and reflect on the mediating effect of the protocol and observer. Comparison with the experience of a direct viewing of the same episode provides a context for discussion of what has and has not been seen and what might be distilled as of wider interest, following Jaworski’s guidelines for use of video excerpts – giving an ‘account of’ before ‘accounting for’.

**06 Partial knowledge of understandings needed for proportional reasoning**

Joan Burfitt

*The University of Western Australia*

Improving the function of multiple-choice items could be achieved by awarding credit when a person selects an option which, while incorrect, indicates that they have some knowledge of the tested concept. To support the identification of what might constitute partial knowledge, Rasch Measurement Theory was applied to students’ responses to items on ratios and fractions. This study suggests that the identification of items for which partial credit is warranted requires both qualitative and quantitative analysis.

**07 Evolution and comparison of mathematical abilities in China’s mathematical curriculum standard for elementary schools**

Hongyan Cai and Jian Zhang

*Teachers’ College of Beijing Union University, Beijing, China*

The teaching of mathematics in China’s elementary schools is based on the Mathematical Curriculum Standard, which is a programmatic document issued by the national education authorities, including such abilities as operational ability, spatial imagination, logical thinking and problem-solving ability. In this paper we have roughly compared and summarized the different requirements for mathematical abilities in the Primary Mathematics Syllabus and the Mathematical Curriculum Standard in the past years, and suggest how to train students’ mathematical abilities through specific mathematical examples.

**08 How primary trainee teachers’ beliefs change about mathematics education during their first term of a PGCE course
**Kathryn Clarke

*University of Chester*

Within the UK it is widely acknowledged that there are not enough good teachers of mathematics and that this is having an impact on different facets of society. This paper will explore the experiences of three Primary Trainee Teachers within the world of mathematics education. Their perceptions, confidence and attitude towards mathematics education are explored at different points during their PGCE Primary teaching course to discover whether university led subject knowledge/pedagogy sessions and teaching children on placement can change these views in a positive way, using the framework of the ‘Knowledge Quartet’.

**09 Singapore bar models appear to be the answer, but what then was the question?
**Laura Clarke

*University of Winchester*

The use of Singapore bar models has been a topic of great interest in recent years in response to the success of Singaporean students in international mathematics tests. However, results for English students have improved without the use of these diagrams. This research sought to identify which strategies were used by four Year 6 pupils who had not been taught how to use bar models to solve word problems. Findings suggested that pupils had a useful repertoire of problem solving strategies and could successfully solve a range of worded problems and create a range of diagrams for a variety of purposes to suit their needs for each problem.

**10 Constructivist principles in maths lessons: Pi in the sky?
**Fiona Curtis, Yota Dimitriadi, Marina Della Giusta and Giovanni Razzu

*University of Reading*

Constructivist ideas about learning mathematics have existed at least since the second half of the 20th century, yet teaching in many cases remains transmissive and exam-focused and results in many students resisting further compulsory study after 16. This presentation reports on the results of a survey of over 500 sixth formers regarding their experiences of school mathematics teaching across ten constructivist principles, and compares their experiences in maths with experiences in science, ICT and English. The findings indicate that student experiences are diverse, but that many maths lessons do not embrace constructivist principles.

**11 Mathematics resilience: What is known in the pre-tertiary mathematics education research and what we have found researching non-mathematics-specialists
**Francis Duah

*University of York*

This paper describes a two-phase pilot study that explored mathematical resilience amongst non-mathematics-specialist students in a tertiary institution in the UK. Two cohorts of first-year undergraduate students completed a modified version of an existing mathematical resilience instrument. The association between respondents’ level of mathematical resilience scores and the type of pre-tertiary mathematics qualification they had achieved and their programme of study was explored. Semi-structured interviews were conducted to explore experiences of learning mathematics and the strategies they used to persist with mathematics.

**12 Formal methods for division: Evaluating the benefits of pre-teaching mathematics using a ‘flipped classroom’ approach
**Lorna Earle and Caroline Rickard

*University of Chichester*

This paper briefly outlines our research into supporting trainee teachers in their quest to understand formal methods for division, often considered a tricky topic to understand deeply and to teach effectively. Literature suggested that providing pre-teaching, using a ‘flipped classroom’ approach, might be beneficial and initial findings from our research suggest this to be the case.

**13 Difficulties in teaching calculus concepts in undergraduate courses: The notion of limit of a sequence
**Marius Ghergu

*School of Mathematics and Statistics, University College Dublin, Ireland*

It is commonly agreed that undergraduate students encounter difficulties in understanding the notion of the limit in calculus courses. Both the formal and informal approach do not convey a setting that bridges intuition and mathematical rigour. In this work we explore the formal and informal definition of the limit of a sequence. A case study was conducted on a large group of first year undergraduate students. The findings reveal students’ difficulties in formalising their mathematical observations along with a reluctance to employ formal or informal definitions of a limit of a sequence in their arguments.

**14 Teaching undergraduate mathematics in the United States and the United Kingdom: Four comparative observations
**Barry J. Griffiths

*University of Central Florida*

Increased global mobility has given students and faculty unprecedented opportunities to study and work in different countries, which leads to new challenges for faculty in adapting to classrooms that are more international than ever before. This paper looks at the author’s experience in teaching mathematics in the United States and the United Kingdom, and to students with very different cultural and academic traditions. Four broad themes are used as the basis for comparison, chosen on the basis of their relevance to contemporary issues, with previous literature blended with personal observations, along with qualitative data gained through individual interviews.

**15 Mathematics teacher educator noticing: A methodology for researching my own learning
**Tracy Helliwell

*University of Bristol, Graduate School of Education*

I present here my interpretation of Mason’s four interconnected actions within the Discipline of Noticing as a methodological framework for my research, that is, what I notice as a Mathematics Teacher Educator working with a group of teachers. My research concerns how and what I am learning in becoming a mathematics teacher educator. As a method of research, I am using the Discipline of Noticing as a way of developing my expertise. I explore what constitutes data in such a study and how working with this data, in a way guided by the methodological framework, supports my learning as a mathematics teacher educator.

**16 On learning number: The ‘-ty’ and ‘-teen’ confusion
**Dave Hewitt

^{1}and Alf Coles

^{2}

^{1}*Loughborough University,*

^{2}*University of Bristol*

The aim of this paper is to draw out implications for the early learning of number from a detailed analysis of the work of one student, over an eight month period. The background to this work was our research interest in exploring the potential of approaching number without a focus on objects and cardinality, particularly for currently low-attaining students. We report on the difficulties that arose from a confusion of ‘-teen’ and ‘-ty’ numbers, for the student. We conclude that there is an argument for: (a) delaying work on ‘-teen’ numbers until students have worked on number structure more generally; (b) adopting a dual naming to regularise our naming system*.*

**17 The early take-up of Core Maths: Emerging findings
**Matt Homer, Rachel Mathieson, Indira Banner and Innocent Tasara

*School of Education, University of Leeds*

Core Maths, a new Level 3 qualification equivalent in ‘size’ to an AS-level but intended to be studied over two years, is a response to England’s poor post-16 mathematics participation. Core Maths has been taught in Early Adopter institutions since 2014, with first examination in summer 2016. This paper reports on the historic development of Core Maths, and the early findings of a large-scale three-year mixed-methods project funded by the Nuffield Foundation intended to provide clear practical and policy guidance to secure improved post-16 mathematics participation in England.

**18 Words and contexts: Teaching mathematics vocabulary
**Jenni Ingram, Nick Andrews and Andrea Pitt

*University of Oxford*

Research into vocabulary learning emphasises the need to teach vocabulary in authentic contexts and through making connections to students’ prior knowledge or experiences. In mathematics, we have vocabulary that students are only likely to meet in their mathematics lessons, words that are widely used with a similar meaning, and words widely used with very different meanings. In this paper, we explore the discussions from the workshop about what might be meant by an authentic context in mathematics for these different types of words, and how these contexts may, or may not, support students in making connections.

**19 Sharing chocolate bars: Year 8 students’ use of narrative, visual and symbolic representations of fractions
**Dietmar Küchemann

*University of Nottingham*

In this paper we discuss incidents from video recordings of a lesson and of a follow-up interview arising from a story about ‘sharing chocolate bars’. The students, from a relatively ‘low attaining’ Year 8 (Grade 7) class, made use of various representations of fractions. They sometimes made good sense of a representation by linking it to the story, but it often proved challenging to make fruitful connections between representations. And representations were sometimes used procedurally in ways that fitted what students remembered or (re-constructed) about fractions, rather than with how the procedures and outcomes might have related to the story.

**20 Teachers’ perception of mathematical tasks: What teachers ‘saw’ in a published task, and how this changed following enactment of the same task
**Matt Lewis

*University of Oxford*

What do teachers perceive when presented with a mathematical task? I analysed three teachers’ lesson planning notes and interview responses to investigate their intended use of a published task, before and after their observation of teaching using the same task. I relate pedagogical decision-making to teachers’ perception of the task, described in terms of teachers’ beliefs, knowledge and goals. From a mathematics teacher educator perspective I suggest the Japanese practice of kyozai kenkyuu as a means to inform teacher decision making before teaching the lesson.

**21 Studying the link between classroom dialogue and the implementation of rich tasks in post-16 mathematics with Underground Mathematics
**C.J. Rauch, Marc De Asis, Alvin Leung, Tatiana Rostovtseva and Sharon Walker

*University of Cambridge*

Underground Mathematics (UM) develops online resources to support teaching and learning in post-compulsory mathematics and holds a strong belief that dialogue is inherent to mathematics. To fully understand the implementation and use of its resources, the project evaluation has employed a mixed-methods approach. Classroom observations were conducted in A level classrooms across England using UM materials. Dialogue was analysed using the Cam-UNAM Scheme for Educational Dialogue Analysis. This paper presents insights from the data analysis and argues in favour of a methodology in which dialogue is central to research in mathematics education.

**22 Before ***Pythagoras***: A brief cultural history of the **** Pythagorean relation**Leo Rogers

^{1}and Sue Pope

^{2}

^{1}

*Independent researcher,*

^{2}

*Manchester Metropolitan University.*

*Pythagoras’ theorem* is standard fare for schoolchildren around the world, often presented as a result to be memorised, rather than appreciated (it is a remarkable result), understood and proved. Rarely do learners have the opportunity to find out about the extensive history of this important mathematical result, or to question whether *Pythagoras* actually existed. In this paper, we highlight some sources that might allow teachers to offer a different experience.

**23 Completing the square: The cultural arbitrary of Oxbridge entrance?
**Rachel Sharkey

*University of Manchester*

I report on part of a research project using a case study approach drawing on qualitative methods to examine a preparation programme for students applying to study mathematics at elite universities, in an independent school. I draw on a Bourdieusian conceptual framework to analyse teachers’ social and cultural capital and show how this can provide experiences that are thought to confer distinction. Here I look at some of the skills which I think could be argued to become arbitrary, rather than functional, in practice and to what extent a student who acquires these skills gets ‘use’ and/or ‘exchange’ value from these skills.

**24 Developing a concrete-pictorial-abstract model for negative number arithmetic
**Jai Sharma and Doreen Connor

*Nottingham Trent University*

Research findings and assessment results persistently identify negative number arithmetic as a topic which poses challenges to learners. This study aims to build on existing research which identifies four conceptualisations involved in negative number arithmetic: unary, binary, and symmetric operations, and magnitude, to develop a concrete-pictorial-abstract (CPA) model which uses the vertical number line and ‘number bar’ manipulatives as the key representations. A trial group taught using the CPA model are found to have made significantly greater increases in post-assessment scores compared to a control group who were taught using a non-CPA approach.

**25 Mathematics and examination anxiety in adult learners: Findings of surveys of GCSE Mathematics students in a UK Further Education college
**Jenny Stacey

*Chesterfield College*

Surveys of my adult learners of GCSE maths indicate around a third describe themselves as anxious or very anxious about mathematics, and about half have the same responses for exams. Furthermore, females of any age educated in Great Britain have a proportionally higher rate of anxiety about mathematics and exams than their non-GB educated or male peers. Maths and exam anxiety were rarely linked, and there was little difference between the comments on exams of the maths anxious and non-maths anxious groups, most citing exam pressure. As a result of their experiences in an adult classroom, learners’ views became more positive.

**26 A case study to explore approaches that help teachers engage with students’ development of mathematical connections
**Nicola Trubridge and Ted Graham

*Plymouth University*

This research study considers the Collaborative Connected Classroom (CCC) model and how it might be implemented within a school via a programme of sustained CPD that incorporates: research sharing; engagement with activities that bridge theory and practice; active collaboration and exploration of ideas. This paper reports the findings when looking at which aspects of the CCC model engaged teachers themselves and then which of these tasks were used with their learners to develop mathematical connections.

**27 Different levels for basic knowledge and skills assessment and mathematical rigorousness
**Pei Wang

^{1}and Jian Zhang

^{2}

^{1}*Basic Courses Department of Beijing Union University, Beijing, China,*

^{2}*Teachers’ College of Beijing Union University, Beijing, China*

China’s *Mathematical Curriculum Standard* is a programmatic document issued by the national education authorities in which comprehensive recommendations have been proposed for teachers on how to carry out assessment. Four different levels have been proposed, such as ‘understanding, comprehension, mastering and application’. In this paper we use some mathematical examples to explain such four different levels and discuss the mathematical rigorousness embodied in such examples.

**28 Report from the Critical Mathematics Education Working Group meeting
**Pete Wright

*UCL Institute of Education*

This BSRLM working group met for the second time with the aim of discussing ways of promoting research that brings about positive social change through mathematics education. The meeting began with a presentation and discussion around student teachers’ apparent reluctance to share their pedagogical rationale with learners and the implications of adopting open-ended approaches to learning mathematics for the achievement of working-class students. This was followed by a discussion of the aims of the working group, its structure and organisation, and the focus for future meetings and activities.