**Proceedings of the Day Conference held at Oxford University on 19 Nov 2011**

## Contents

#### 01 Assessing young children’s understanding of multiplication

Patrick Barmby, David Bolden, Stephanie Raine and Lynn Thompson

Durham University

This study is part of a Nuffield Foundation-funded project aimed at developing primary children’s understanding of mathematics via the use of visual representations. As part of this project, a test of primary children’s understanding of multiplication was developed. Although there has been research on developing tests of understanding in other mathematical topics (e.g. fractions) and in particular for older (e.g. late primary or secondary) children, there has been little reported work on tests for multiplication for younger primary children. In this study, therefore, a test of multiplication was constructed based on the range of contexts and representations associated with multiplication. The 19-item test was administered to a sample of mainly Year 3 pupils (n=272) with a small sample of Year 4 pupils (n=18) in 10 primary schools. The age of the pupils meant that test questions were read out by the teacher, and most were multiple choice items. The data obtained from the test were analysed using Rasch analysis in order to examine the reliability of the overall measure, and the validity of individual items. Two items were shown to have poor fit statistics and were excluded from the analysis. Overall, the resulting measure was shown to have a good reliability indicated by a Cronbach α value of 0.79. The analysis of the data was also used to examine the progression in difficulty of the different items, and related to the progression in children’s thinking in multiplication. Recommendations for further improvements in the test are put forward.

#### 02 Reflection on Practice, in Practice: The Discipline of Noticing

Sinead Breen, Aisling McCluskey, Maria Meehan, Julie O’Donovan and Ann O’Shea

St Patrick’s College, Drumcondra; National University of Ireland, Galway; University College Dublin; Cork Institute of Technology; National University of Ireland, Maynooth

This paper outlines the use of John Mason’s Discipline of Noticing by a group of university level mathematics lecturers. We describe the aims that motivated the study, the challenges we faced in using the Discipline of Noticing to reflect on our teaching, and the progress that we have made.

#### 03 Relationships between the influences on primary teachers’ mathematics knowledge

Ian Campton and Julie-Ann Edwards

University of Southampton

This paper reports on one aspect of a small scale research project that aimed to identify areas for improvement in the teaching of mathematics through continuing professional development for primary school teachers in two schools. The findings suggest an emergent conceptual framework of the influences on primary school teachers’ mathematical content knowledge. The relationship between these influences reveals a multi-layered belief system that is well-intended and well-informed at the top level but underpinned by less firmly established levels of subject knowledge and consequent pedagogical approaches.

#### 04 The ‘algebra as object’ analogy: a view from school

Kate Colloff and Geoff Tennant

University of Reading

Treating algebraic symbols as objects (e.g.,. “‘a’ means ‘apple’”) is a means of introducing elementary simplification of algebra, but causes problems further on. This current school-based research included an examination of texts still in use in the mathematics department, and interviews with mathematics teachers, year 7 pupils and then year 10 pupils asking them how they would explain, “3a + 2a = 5a” to year 7 pupils. Results included the notion that the ‘algebra as object’ analogy can be found in textbooks in current usage, including those recently published. Teachers knew that they were not ‘supposed’ to use the analogy but not always clear why nevertheless stating methods of teaching consistent with an ‘algebra as object’ approach. Year 7 pupils did not explicitly refer to ‘algebra as object’, although some of their responses could be so interpreted. In the main, year 10 pupils used ‘algebra as object’ to explain simplification of algebra, with some complicated attempts to get round the limitations. Further research would look to establish whether the appearance of ‘algebra as object’ in pupils’ thinking between year 7 and 10 is consistent and, if so, where it arises. Implications also are for on-going teacher training with alternatives to introducing such simplification.

#### 05 Fractions in context: The use of ratio tables to develop understanding of fractions in two different school systems

Dolores Corcoran and Pamela Moffett

St Patrick’s College, Drumcondra, Dublin and Stranmillis University College, Belfast

The project seeks to investigate the implementation of a number of Realistic Mathematics Education lessons on fractions in two different educational systems. Four teacher participants engaged with trialling an agreed sequence of lessons from an RME textbook in their own classrooms — one Primary 6 classroom in Northern Ireland and three-fifth classes in Southern Ireland. The teaching of lessons was observed by each researcher in her own school system. Nine of the lessons were video recorded, and short video clips were made of children at work during other lessons. Children’s mathematical workings from the lessons were collected and analysed. Similarities and differences in teaching approaches across contexts were examined with a view to identifying some of the supports and constraints experienced by teachers in the implementation of these lessons. In this session, we propose to report on the manner in which three RME lesson contexts provided teachers and children with novel ways of thinking about and working with fractions.

#### 06 Teachers of mathematics to mathematics teachers: a TDA Mathematics Development Programme for Teachers

Cosette Crisan and Melissa Rodd

Institute of Education, University of London

To address the shortage of mathematics teachers in England, serving teachers, qualified in subjects other than mathematics yet teaching secondary mathematics, were eligible to participate in a Mathematics Development Programme for Teachers (MDPT) commissioned and funded by the Teacher Development Agency (TDA). A research project was set up to investigate how teachers in our 2010-11 cohort developed into mathematics teachers within this Programme. This report indicates how (1) learning new mathematics, (2) developing a view on the nature of mathematics and (3) teaching mathematics in different ways, contribute to a mathematics teacher identity, yet there was a discrepancy between the teachers’ espoused confidence in being a mathematics teacher and their technical mathematical competence.

#### 07 Individual Differences in Generalisation Strategies

Rebecca Crisp*, Matthew Inglis*, John Mason** and Anne Watson**

*Mathematics Education Centre, Loughborough University, **Department of Education, University of Oxford

We report a study which investigated how mathematics graduates and engineering undergraduates studied sequential tables of values with the aim of deriving a function. By recording the eye-movements of participants as they tackled this task, we found that there were substantial individual differences in the strategies adopted by participants. However, these strategy choices appeared to be unrelated to both the mathematical background of participants, and their success rates.

#### 08 Approaches to Learning of Undergraduate Mathematicians

Ellie Darlington

University of Oxford

The approaches to learning (ATLs) adopted by undergraduate students have been heavily researched, particularly since Marton and Säljö (1984) first wrote of a deep/surface approach dichotomy. Using this terminology, a study was conducted which aimed to research the ATLs of first-year undergraduate mathematicians, specifically relating to what these are and the ways in which the students themselves perceived them to have evolved over time. The results revealed that an overwhelming majority of undergraduate mathematicians at Oxford University adopt strategic ATLs, which they claimed were due to the nature of university study. It was established that it was, in fact, the nature of their specific course that resulted in this approach, as the nature of the new mathematics being studied, assessment demands and question formats in their department were contributing factors. In this article, I shall detail the findings of the quantitative research conducted using the ASSIST (Tait, Entwistle and McCune 1998) questionnaire.

#### 09 Evaluating the impact of a Realistic Mathematics Education project in secondary schools

Paul Dickinson*, Susan Hough*, Jeff Searle** and Patrick Barmby**

*Manchester Metropolitan University, **Durham University

Over the past 30 years, researchers at the Freudenthal Institute in the Netherlands have developed a mathematics curriculum and a theory of pedagogy known as Realistic Mathematics Education (RME). This curriculum uses realisable contexts to help pupils to develop mathematically. In 1991, the University of Wisconsin, in collaboration with the Freudenthal Institute, started to develop a middle school curriculum based on RME called ‘Maths in Context’. A related Mathematics in Context (MiC) project was carried out in England in 2004 to 2007 at Manchester Metropolitan University (MMU) with Key Stage 3 pupils. This initial pilot project was evaluated by Anghileri (2006). In 2007, the ideas behind the project were extended to include Key Stage 4 pupils, particularly those studying towards Foundation GCSE Mathematics, and given the project title Making Sense of Mathematics (MSM). MSM has been running as a pilot project in some Manchester schools since 2007. Both these projects were recently evaluated by Durham University, with the revaluation of test data from the original MiC project using Rasch analysis, interviews with teachers from both projects, and observations of the RME approach in lessons. This paper presents the findings from the Durham University evaluation and discusses the impact of RME on both pupils and teachers.

#### 10 What are the factors that influence the frequency of mathematics register in one linguistic code than in another?

Danyal Farsani

School of Education, University of Birmingham

This study is to draw attention to some of the significant factors that have increased the frequency of English mathematics register by both classroom teacher and learners in a bilingual Farsi/English mathematics lesson. The extent to which code-switching occurs does depend on the speaker’s linguistic proficiency. The students who were more fluent in English were more likely to respond in English, and the ones who were more fluent in Farsi responded in Farsi most of the time. This study reveals not only that being communicative competence in a particular language can dominate the conversation in that specific language, but also how the written mode can influence the verbal counterpart.

#### 11 Portuguese pre-service elementary teachers’ knowledge of geometric transformations: an exploratory study.

Alexandra Gomes

CIEC/IE – U. Minho

No one questions the fact that teachers’ knowledge plays a crucial role in teaching. Research on teachers’ knowledge indicates that content knowledge is influential on instruction. Even though there is plenty of research on teachers’ knowledge of number and operations, the same doesn’t happen with geometry. In Portugal, a new mathematics programme for elementary school introduces geometric transformations from 1st grade. Since this is a rather new topic in the elementary curriculum, it seems important to understand what knowledge (future) teachers have on the topic. In this paper, we present findings from an exploratory study, conducted with future elementary teachers designed to evaluate their knowledge on geometric transformations.

#### 12 Movement, language and mathematics: an interplay on the journey towards confidence with formal notation

Dave Hewitt

School of Education, University of Birmingham

A mixed ability group of 21 9-10-year-old students were taught over a three lesson period using the software Grid Algebra. They gained considerable confidence with reading formal algebraic notation over this time, and a key feature was the blended space created whereby the notation could be read in terms of physical movements on a grid as well as mathematical operations. Three episodes from the lessons are discussed which exemplified the changing dynamic between movement, language and mathematics.

#### 13 A Talk Framework for Primary Problem Solving

Mike Hickman

Faculty of Education and Theology, York St John University

Part-time postgraduate primary student teachers at York St John University are currently taking part in a pilot project exploring how digital audio recordings may provide opportunities to engage in closer consideration of and reflection on their mathematical problem-solving performance. The project considers how thinking aloud, supported by digital audio recording may support student teachers’ learning and levels of confidence in teaching primary problem solving. Having produced digital recordings of problem-solving activities within university-based taught sessions, participants are given their digital recordings to listen to and analyse using a ‘talk framework’ in a stimulated recall situation. This framework includes the proposed categories of ‘exploratory transformative’ and ‘exploratory encoding’. While initial findings suggest the very process of audio recording and the associated verbalisation may adversely impact upon problem-solving performance, stimulated recall provides potentially valuable opportunities to reflect upon learning.

#### 14 Lower attaining primary trainee teachers’ choice of examples: the cases of Naomi and Victor.

Ray Huntley

University of Gloucestershire

This paper reports on selected findings of a doctoral study exploring primary trainee teachers’ choices of mathematical examples and the relationship between these and their mathematical subject knowledge. Through a combination of interviews and lesson plans gathered from the final school placement of one cohort of B.Ed trainees, and measures of mathematics attainment before and during their curse, the choice of examples by two lower attaining trainees, known as Naomi and Victor, are considered. This paper presents aspects of the data relating to Naomi and Victor and raises issues of concern about their approaches which will impact on pupil learning.

#### 15 Pre-service teachers’ understandings of learning to use digital technologies in secondary mathematics teaching

Rosalyn Hyde and Julie-Ann Edwards

University of Southampton

One of the biggest challenges facing pre-service mathematics teachers is that of learning how to make effective use of digital technologies in the classroom in order to enhance the learning of their students. For initial teacher educators, the challenge is to enable the development of teachers who have the capability to respond flexibly to new technologies and who are able to evaluate and reflect on the impact of such technologies on learning. We report on data collected as part of a research project investigating ways in which pre-service mathematics teachers can develop more effective skills in using digital technologies to enhance teaching and learning in the classroom. We examine this evidence using Mishra and Koehler’s (2006) model of Technological Pedagogical Content Knowledge (TPCK). The emerging understandings of pre-service teachers’ learning are considered in the context of their learning experiences during their initial teacher training course and in terms of charting the learning journeys they undertake on the course. The project outcomes point towards ways forward in enabling more effective learning by pre-service teachers in the use of technologies for mathematics teaching.

#### 16 Exploring algebraic thinking in post-16 mathematics: the interpretation of letters

Martin Jones

Havant Sixth Form College, Hampshire, UK

Students in their first year of post-16 mathematics were given a test consisting of items requiring algebraic reasoning. This was based on work by Kuchemann with secondary school students. The responses were analysed to assess students’ level of algebraic thinking and their results compared with their public examination results. This paper includes a summary of the analysis and a discussion of the implications.

#### 17 Professional development of Turkish mathematics teachers within a computer-supported learning environment: changes in beliefs

Umit Kul

University of Leicester, UK

The purpose of this study is to investigate the degree to which a professional development (PD) course based on the use of dynamic geometry affects the beliefs of school teachers with regard to three aspects: the nature of mathematics, teaching mathematics and learning mathematics. The PD course was designed to provide six teachers with a better theoretical and practical understanding of mathematics teaching and learning through interacting with computer-based mathematical activities that were consistent with the constructivist paradigm. The primary intention was to find out how participants in such a learning environment form their beliefs. The potential shifts in beliefs of the participants were identified using a pre-and post-course mathematical beliefs questionnaire. Overall, the results indicate that such efforts transformed teachers’ beliefs to some extent in favour of the constructivist view.

#### 18 Mixed methods in studying the voice of disaffection with school mathematics

Gareth Lewis

University of Leicester

Disaffection with school mathematics is a complex phenomenon as well as a serious problem. It is clearly related to affect, but the study of affect in mathematics education is also problematic. A case is made that it is necessary to study the phenomenon beyond the quantitative study of attitude in order to understand better the complex and multi-dimensional nature of disaffection and to understand the subjective experiences of students who are disaffected. In order to do this, new methods and approaches are needed. This paper reports on a study of disaffected students of mathematics in a Further Education College. It describes the novel methods used to understand disaffection as a motivational and emotional phenomenon. The paper outlines a range of quantitative and qualitative methods used to elicit the subjective reality of disaffected students in relation to mathematics. It provides an opportunity to evaluate these methods and their efficacy in capturing the dynamic nature of the motivational and emotional reality behind the phenomenon of disaffection.

#### 19 Explicit and Implicit Pedagogy: variation theory as a case study

John Mason

University of Oxford and Open University

Variation theory, promoted by Ference Marton and colleagues (Marton and Booth 1997, Marton and Trigwell 2000, Marton and Tsui 2004, Marton and Pang 2006) and augmented by Watson and Mason (2002, 2005) has roots going back at least as far as Isocrates (Papillion 1995). It proposes that learners must experience variation in the critical aspects of a concept, within limited space and time, in order for the concept to be learnable. But the presence of variation does not in itself guarantee that that variation will be experienced. As Kant implied, a sequence of experiences does not guarantee an experience of that sequence. Implicit variation theory assumes that the presentation of variation is sufficient in order for learners to learn what is intended, whereas explicit variation theory incorporates some degree of explicitness in the interaction between teacher and student. The conjecture is proposed that tension between explicitness and implicitness is present in all attempts both to implement theories in practice and to justify or analyse pedagogical choices using theories, of whatever kind.

#### 20 Grammatical structure and mathematical activity: comparing examination questions

Candia Morgan*, Sarah Tang*, Anna Sfard**

*Institute of Education, University of London; **University of Haifa

The project “The Evolution of the Discourse of School Mathematics through the Lens of GCSE examinations” is studying the ways in which the mathematical activity expected of students has changed over the last few decades by analysing the discourse of examination papers, using linguistic tools. In this paper, we present one aspect of this analysis, comparing the grammatical complexity of sentences in questions from 1987 and 2011. We discuss the implications of differences in the grammatical structure for the nature of the mathematical activity demanded of students.

#### 21 Teacher, do you know the answer? Initial attempts at the facilitation of a discourse community

Siún NicMhuirí

CASTeL, St Patrick’s College, Dublin City University

My research involves a teaching experiment I undertook in my own primary classroom. The aim of the research was to facilitate a mathematical discourse community where students would explain and justify their mathematical thinking and question the reasoning of others. It was envisaged that students would regularly engage in cognitively demanding tasks and take responsibility for determining what was mathematically correct by discussing different possible solutions. The lesson presented here was the first recording of the experiment and focused on initial attempts at exploring equivalent fractions in the context of sharing pizzas between people. The contributions of students show different levels of mathematical understanding and engagement with the task. The whole class discourse is analysed with reference to the four components of the Math-Talk Learning Community (MTLC) framework (Hufferd-Ackles, Fuson and Sherin 2004). These components are questioning, explaining mathematical thinking, the source of mathematical ideas and responsibility for learning. Both teacher and student actions in these key areas are explored. Analysis of teacher questions was carried out using question categories developed by Boaler and Brodie (2004).

#### 22 Applied Mathematics = Modeling > Problem solving?

Peter Osmon

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

The latest stage of an ongoing investigation of the case for reforming Level-3 mathematics, with modeling replacing traditional applied mathematics, is reported. The case rests on the potential for improvement in learners’ ability to use mathematics knowledge and improvement in take-up. Issues around teaching model making are identified and discussed and also the importance of mathematical modeling in understanding knowledge and progress in science.

#### 23 The epiSTEMe pedagogical approach: essentials, rationales and challenges

Kenneth Ruthven, Riikka Hofmann, Christine Howe, Stefanie Luthman, Neil Mercer and Keith Taber

University of Cambridge

The goal of the epiSTEMe project is to develop a research-informed pedagogical intervention in early-secondary physical science and mathematics, suited to implementation at scale in the English educational system. This paper provides an overview of the pedagogical essentials of the epiSTEMe approach and their supporting rationales and identifies some of the main practical challenges encountered.

#### 24 Investigation of the relationship between calculus students’ cognitive process types and representation preferences in definite integral problems

Eyup Sevimli and Ali Delice

Marmara University

This study focuses on the students’ cognitive process and preference of representation. We try to find an answer for that problem “How do students’ preferences of the multiple representations change in definite integral problems according to the type of cognitive process”. The participants of the study are 26 undergraduate students who enrolled Calculus II course. The preferences of the student representation determined by the Representation Preferences Test and their type of cognitive process evaluated with Mathematical Process Instrument. Results show that the participants generally prefer algebraic representation. The visual type of participants’ preference tendencies are influenced by input representations.

#### 25 A data collection process for an embedded case study focusing on the teacher-teaching assistant partnership in the mathematics classroom

Paul Spencer and Julie-Ann Edwards

University of Southampton

This paper discusses the progress to date of an NCETM/ESRC case studentship project which focuses on the partnership between teachers and teaching assistants in secondary mathematics classrooms. The research background and rationale for the study are explained, and the development of an innovative system of classroom observation to track the movement of the teacher and teaching assistant during mathematics lessons is discussed. Examples from the pilot study are employed to illustrate how this data tracking system is being used to triangulate the teachers’ and teaching assistants’ interview responses and identify how the teacher and teaching assistant work collaboratively in the classroom environment.

#### 26 An analysis of pre-service mathematics teachers’ performance in modelling tasks in terms of spatial visualisation ability

Halil İbrahim Taşova and Ali Delice

Marmara University

This study aims to identify pre-service teachers’ spatial abilities and to explore the effects of these abilities on performance in mathematical modelling tasks. Following a case study research approach, mixed methods were used for data collection. Participants were 75 pre-service teachers studying an MA degree without dissertation at a state university in Turkey. In order to identify pre-service teachers’ spatial abilities, data was collected using a Mental Rotation and Spatial Visualisation Test. In order to investigate the effects of spatial abilities on performance during the solution process and on the visualisation process, pre-service teachers participated in modelling activities. Descriptive statistics were used to analyse the qualitative data. Results indicated that almost half of the pre-service teachers had high-level spatial abilities. It was also found that pre-service teachers’ mathematical modelling abilities were not sufficiently developed and that their spatial visualisation abilities were weaker than their mental rotation abilities. Moreover, the result that pre-service teachers who had the higher spatial ability also had better performance in modelling tasks than the other pre-service teachers implied a direct relationship.

#### 27 Impact of the Mathematics Lesson Structure reform in Seychelles on pupils’ achievement

Justin Valentin

King’s College London

This paper draws on secondary achievement data and describes the pupils’ achievement during the first years of a mathematics teaching reform. Cross-sectional analyses of the data showed no improvement in performance during the first years of the reform. However, during the same period, variations in the pupils’ scores reduced. The fact that it is difficult to make claims about the impact of reform in the absence of experimental data, findings reported in this paper have become a rational to extend these analyses beyond the descriptive statistics to include data from other sources.

#### 28 To what extent might role play be a useful tool for learning mathematics?

Helen Williams

Roehampton University, London and Marlborough Primary School, Falmouth

The work discussed here forms the beginnings of my PhD research, investigating the mathematical potential of role play in a primary school where a role play area is chosen by the children from Reception to Year 6, and is assigned for mathematics. Currently, I am collecting and analysing video- and audio-taped data collected as a non-participant observer in a Y4 classroom with two groups of childen of eight years of age. Susequent to each observation, selected video clips are reviewed with the children. The focus of my research is currently 1) whether there is there any identifiable mathematics happening; 2) the level of involvement of the participants; and 3) how what is happening relates to what else is going on in the classroom. Some broader educational themes are arising from the data and are outlined here, including the benefits and disadvantages of using video, reproposal, exploratory talk and the role of metacognition in learning mathematics.

#### 29 Mathematics teachers make statistical inference based on the distribution of sampled values

Kai-Lin Yang

National Taiwan Normal University, Taiwan

The study presented in this paper illustrated part-characteristics underlying mathematics teachers’ alternative understanding or misunderstanding of the confidence interval (CI) – related concepts. Firstly we developed assessment instruments to explore teachers’ understanding of CI-related concepts. We found that mathematics teachers knew mean, variance and some properties of a random variable, what a CI for a proportion measures, and the relationships between sample sizes, confidence level and the width of a CI for a proportion. In addition, they were able to calculate a CI for a proportion. However, they did not integrate their understanding of a random variable appropriately when estimating a parameter for a random variable and did not transfer their understanding of CIs for a proportion to CIs for a mean. One critical characteristic underlying their thoughts was that they made statistical inference mainly based on the distribution of sampled values.