**Proceedings of the Day Conference held at Loughborough University in November 2009**

## Contents

### Research reports

#### 01 Interpretations of, and orientations to, “understanding mathematics in depth”: students in MEC programmes across institutions

Jill Adler^{1}, Sarmin Hossain^{1}, Mary Stevenson^{2}, Barry Grantham^{2}, John Clarke^{3}, Rosa Archer^{4}

^{1}King’s College London, ^{2}Liverpool Hope University, ^{3}University of East London, ^{4}St Marys College Twickenham

In this paper we present initial findings from our study of interpretations and orientations to ‘understanding mathematics in depth’ among students in selected Mathematics Enhancement Courses (MEC) in the UK. The MEC is a 26-week pre-Initial Teacher Education (ITE) ‘mathematics subject knowledge for teaching’ course designed for, and undertaken by, graduates wishing to teach mathematics at secondary level, but do not have a Mathematics degree. It is completed before commencing with a PGCE. A common theme running through the MEC documentation is the importance of ‘understanding mathematics in depth’. We are interested in what and how MEC students interpret and orient themselves towards ‘understanding mathematics in depth’. In designing and conducting our empirical work we have drawn upon a related project in South Africa, which is exploring ‘mathematics for teaching’, specifically what and how mathematics and teaching are co-constituted in mathematics teacher education programmes. The MEC is an interesting empirical context for such study, as it is a mathematics course, or set of courses, specifically designed for future teachers. We have collected data through guided, semi-structured interviews with 18 students and 4 lecturing staff at three different institutions. The interpretations and orientations of MEC students towards mathematics and the notion of ‘understanding mathematics in depth’, we contend, provide additional insight into the developing notion of mathematical knowledge in and for teaching.

#### 02 Symbolic addition tasks, the approximate number system and dyscalculia

Nina Attridge^{a}, Camilla Gilmore^{a} and Matthew Inglis^{b}

^{a}Learning Sciences Research Institute, University of Nottingham, UK ^{b}Mathematics Education Centre, Loughborough University, UK

Several recent theorists have proposed that dyscalculia is the consequence of a disconnect between the so-called ‘approximate number system’ and formal symbolic mathematics. Such theories propose that symbolic exact mathematics is built out of approximate representations of quantity. Here we investigate this proposal by testing whether non-dyscalculic adults appear to use their approximate number systems when tackling symbolic tasks. We find a strikingly similar pattern of responses on two approximate addition tasks, one where participants saw numerosities represented as dots and one where the numerosities were represented with Arabic symbols. These findings are consistent with the view that non-dyscalculic adults do indeed use the approximate number system when dealing with symbolic mathematics.

#### 03 The T-shirt task: Using a mathematical task as a means to get insights into the nature of the collaboration between in-service teachers and researchers

Claire Vaugelade Berg

University of Agder, Kristiansand, Norway

By following a mathematical task, from its design by researchers to its implementation by a teacher, it is possible to get some insights into the collaboration between researchers and teachers. Activity Theory is used as a theoretical approach in this research.

#### 04 Motivating Years 12 and 13 study of Mathematics: researching pathways in Year 11

Rod Bond, David Green and Barbara Jaworski

Mathematics Education Centre – Loughborough University

We report on a collaboration, between 4 teachers in 4 schools and a university team of 3, over a period of 21 months, to enthuse Year 11 students (taken from the top 25% ability range) about mathematics and encourage their further study of mathematics in Years 12 and 13. Each school used a different pathway to achieve these goals: this involved acceleration, enrichment, the Free Standing Mathematics Qualification or an early start to A level. The research was developmental in both studying the practices and processes involved while contributing to teachers’ continuing professional development in mathematics.

#### 05 Computer Based Revision

Edmund Furse

Swansea Metropolitan University

A computer system on the web known as xplus12 has been developed which supports KS3 revision in Number and Algebra. This has been evaluated with a year 8 top-set class on two trials in Number with a little Algebra. Since all the data is stored in a database including timings and all student attempts at answers, it is possible to identify a number of behaviour patterns among the pupils. These include the identification of groups such as “rubbishers” and “rushers” who essentially abuse the system. Although an effect size of 0.9 was found in the first trial it was not statistically significant due to the low numbers of pupils completing the test, and no improvement was found in the second trial. Misconceptions were also identified by the system. A number of suggestions are discussed for improvements of the system including improved examples with animation and explaining answers. Also included are techniques to handle rubbishers and rushers.

#### 06 Children’s Difficulties with Mathematical Word Problems.

Sara Gooding

University of Cambridge, UK

This article reports a study of the difficulties that primary school children experience whilst tackling school mathematical word-problems. A case study of four Year 5 children was conducted; this involved interviews which probed the children’s views of their own difficulties and discussions with the children as they tackled word problems. The data were qualitatively analysed using a thematic analysis approach based on categories of difficulty identified from existing literature. Examples of transcripts and responses which show the children experiencing difficulties are included, as well as the children’s opinions on their difficulties. My interpretation of these findings, including proposed subcategories of difficulty, is also given. The report concludes with suggestions of methods – subject to further research – that teachers may use to help children overcome their difficulties with school mathematical word problems.

#### 07 Some initial findings from a study of children’s understanding of the Order of Operations

Carrie Headlam and Ted Graham

University of Plymouth

This paper presents some of the initial findings of a study into the strategies used by children to solve arithmetic and algebraic problems requiring the appropriate use of the order of arithmetic operations. The research has utilised graphics calculators which have been programmed with Key Recorder Software as a data collection tool. This has enabled the researchers to analyse the children’s approaches to some of the questions posed by observing their calculator keystrokes. Interviews with both teachers and pupils will be used to link the pupils’ strategies with the teaching methods used, and an initial analysis of observed misconceptions has been carried out. Initially, this study has involved children in the UK and in Japan, where teaching methods differ substantially.

#### 08 The role of attention in the learning of formal algebraic notation: the case of a mixed ability Year 5 using the software Grid Algebra

Dave Hewitt

School of Education, University of Birmingham

The learning of formal algebraic notation is seen as a challenge for many students (Van Amerom, 2003). The act of symbolising is not so much a problem. Hughes (1990) showed that very young children are able to symbolise in order to record how many items were placed into a tin. The problem is more concerned with interpreting and using someone else’s notation, in this case, the socially agreed convention of formal algebraic notation. In activities where students are asked to find rules for pictorial patterns, they can often find rules but expressing those rules in formal notation is seen as difficult…

#### 09 Lower secondary school students’ attitudes to mathematics: Evidence from a large-scale survey in England

Jeremy Hodgen^{a}*, Dietmar Küchemann^{a}, Margaret Brown^{a} and Robert Coe^{b}

^{a}King’s College London, ^{b}University of Durham

In this paper we present some preliminary data from the ESRC funded ICCAMS project about current student attitudes to mathematics at Key Stage 3 in England. We compare attitudes by sex and by attainment. Whilst the data largely confirms existing findings, an unexpected result was that a very high proportion of students responded that, for mathematical success, the effort was more important than ability. We also present some interview data concerning student attitudes.

#### 10 Simon Says: Direction in Dialogue

Jenni Ingram, Mary Briggs and Peter Johnston-Wilder

University of Warwick

There has been a steady increase in the quantity of mathematics education research focusing on language, discourse and interaction. A wide variety of theoretical frameworks and methodological approaches have been taken including discursive psychology, commognition, and discourse analysis. This paper explores the use of a conversation analysis approach to analysing interactions in mathematics classrooms. In particular what this approach can tell us about the structure of interactions and the use of repair in the negotiation of mathematical meanings.

#### 11 The relationship between number knowledge and strategy use: what we can learn from the priming paradigm

Tim Jay

Graduate School of Education, University of Bristol

Priming methods involve showing a stimulus for a short amount of time (the prime), followed by a second stimulus (the target), which children are asked to perform some operation on. If there is a strong association between the prime and target for a particular child, then the operation on the target will be facilitated by the presence of the prime. This paper describes a project in which priming methods are used to add to our understanding of strategy development for simple addition problems. Children were asked to complete two activities; a priming trial designed to demonstrate priming effects for doubling, and a set of addition problems where participants were asked to explain how they arrived at their answers. Approximately half of the participants used counting strategies (count on from first, count on from smallest), while half used non-counting strategies (decomposition, tie or retrieval). Results indicate that a priming effect for doubling relationships but only for the group of children using non-counting strategies. This result could help to explain the relationship between the development of number knowledge and the development of new strategies.

#### 12 Aspects of a teacher’s mathematical knowledge in a lesson on fractions

Bodil Kleve

Oslo University College

This paper is about a mathematics teacher, and how aspects of his mathematical knowledge surfaced in a 5th grade (11 years old) fractions lesson in Norway. The teacher’s responses to pupils’ (unexpected) comments and questions, ‘contingent moments’, are discussed. Difficulties in dealing with improper fractions, which were mirrored in the pupils’ inputs in the lesson, are discussed. Considerations are made whether the problems the pupils expressed can be traced back to aspects of the teacher’s mathematical knowledge.

#### 13 Post-16 maths and university courses: numbers and subject interpretation

Peter Osmon

Department of Education and Professional Studies, King’s College London

The low take-up of mathematics post-16 and consequences for the traditional STEM (science, technology, engineering, and maths) subjects in higher education are well known. The effect on the newer IT-based subjects, like computing and communications engineering, and the commerce-based subjects, like business and management, economics, and finance is less widely recognised but is at least an equal cause for concern. Most university courses in these subjects are populated with students with no maths beyond GCSE, despite the evident need for better mathematical foundations- perhaps a year of post-16 maths. The scale of this effect and the consequences for these subjects in many university courses are described along with potential implications for the AS-level curriculum.

#### 14 The role of proof validation in students’ mathematical learning

Kirsten Pfeiffer

School of Mathematics, Statistics and Applied Mathematics, NUI Galway

The study of proofs is a major obstacle in the transition from school mathematics to university mathematics. Given the importance of argumentation and proof in the spectrum of mathematical activities, the incoming students’ understanding, appreciation and knowledge of the nature and role of proof must be considered. I describe the results of an exploratory study of first-year mathematics undergraduates’ criteria and learning process when validating mathematical arguments or proofs. The study is based on a series of written tasks and interviews conducted with first-year honours mathematics students at NUI Galway. I presented the whole class with numerous proposed proofs of mathematical statements and asked them to evaluate and criticise those. The first year students’ written comments on different and partly incorrect ‘proofs’ of mathematical statements revealed some information about their criteria when validating mathematical arguments. In recently held interviews with eight randomly chosen students, I focussed on the learning experience during the process of proof validation. Considering the observed learning effect and its large potential extent during the process of proof validation I propose its practice in the teaching of mathematics.

#### 15 An exploration of mathematics students’ distinguishing between function and arbitrary relation

Panagiotis Spyrou, Andonis Zagorianakos

Department of Mathematics, University of Athens, Greece

This paper focuses on students’ awareness of the distinction between the concepts of function and arbitrary relation. This issue is linked to the discrimination between dependent and independent variables. The research is based on data collected from a sample of students in the Department of Mathematics at the University of Athens. A number of factors were anticipated and confirmed, as follows. Firstly, student difficulties involved vague, obscure or even incorrect beliefs in the asymmetric nature of the variables involved and the priority of the dependent variable. Secondly, there were some difficulties in distinguishing a function from an arbitrary relation. It was also thought that additional problems occur in the connotations of the Greek word for function, suggesting the need for additional research into different linguistic environments.

#### 16 Identifying and developing the mathematical apprehensions of beginning primary school teachers

Fay Turner

Faculty of Education, University of Cambridge

In this paper I present a summary of a four-year study into the development of mathematical apprehensions in beginning elementary teachers using the Knowledge Quartet as a framework for reflection on, and discussion about, mathematics teaching. The term mathematical apprehension is used as an inclusive term to cover both mathematical content knowledge and conceptions of mathematics teaching. Evidence from three case studies suggests that focused reflection using the Knowledge Quartet facilitated the development of mathematical content knowledge and promoted positive changes in conceptions about mathematics teaching. Experience and working with others in classrooms and schools were also seen to influence development and change in the teachers’ apprehensions. However, individual reflection was found to have a mediating role on the influence of these two social factors.

#### 17 What Might We Learn From the Prodigals? Exploring the Decisions and Experiences of Adults Returning to Mathematics

Robert Ward-Penny

Institute of Education, University of Warwick

This paper reports on a research project which explored the decision-making process and the experiences of adults who had returned to mathematics after a significant period of time away. Data was gathered using a combination of a questionnaire and follow-up interviews with selected participants. This paper presents some of the key findings together with some examples from the stories of these learners. Finally, it argues that these ‘prodigals’ offer a vivid reminder of the role of mathematics as cultural capital, and an additional perspective on many issues of current interest in mathematics education.

#### 18 Design Decisions: A Microworld for Mathematical Generalisation

Eirini Geraniou, Manolis Mavrikis, Celia Hoyles, Richard Noss

Institute of Education, London Knowledge Lab

This paper provides the preliminary analysis of a study in which year 7 students interacted with eXpresser; a microworld designed to support students’ the transition from the ‘specific’ to the ‘general’ by constructing figural patterns of square tiles and finding rules to describe their model constructions. We present evidence that supports three key design decisions of eXpresser and discuss how these features facilitate students’ expression of generalisation.

#### 19 Do students’ beliefs relating to the teaching of primary mathematics match their practices in school?

Caroline Rickard

Initial Teacher Education, University of Chichester, UK

This paper reports research findings from the second year of a small-scale, longitudinal case study undertaken with undergraduate students at the University of Chichester. This four-year project seeks to explore the impact of the beliefs of students in Initial Teacher Education (ITE) upon their teaching of primary mathematics, noting placement constraints. Data collection involved observations and interviews, and took place in the latter half of a six-week block of school experience.

#### 20 BSRLM Geometry working group: tasks that support the development of geometric reasoning at KS3

Sue Forsythe; Keith Jones

University of Leicester; University of Southampton

Students at Key Stage 3 (i.e., aged 11-14) in English schools are expected to learn the definitions of the properties of triangles, quadrilaterals and other polygons and to be able to use these definitions to solve problems (including being able to explain and justify their solutions). This paper focuses on a pair of Year 8 students (aged 12-13) working on a task using dynamic geometry software. In the research, the children investigated triangles and quadrilaterals by dragging two lines within a shape (i.e., the diagonals of a quadrilateral, or base and height of a triangle) and noting the position and orientation of the lines which gave rise to specific shapes. Following this, the students were asked to use what they had found in order to construct specific triangles and quadrilaterals when starting with a blank screen. While the research is currently ongoing and is using a design research methodology, the evidence to date is that the task has the potential to scaffold students’ thinking around the properties of 2D shapes and hence support the development of geometric reasoning.

#### 21 Working group on trigonometry: meeting 4

Notes by Anne Watson

Department of Education, University of Oxford

These notes record the discussion at the fourth meeting of this working group. The focus was on the history of trigonometry, and discussing three different approaches to teaching it which have appeared in recent readings.