**Proceedings of the Day Conference held at the University of Northampton in November 2007**

## Contents

#### 1 Fallacies in mathematics

Andrew Aberdein

Florida Institute of Technology, USA

Paper presented at the June 2007 Day Conference, Sheffield Hallam University

This paper considers the application to mathematical fallacies of techniques drawn from informal logic, specifically the use oflsquo;argument schemes’. One such scheme, for Appeal to Expert Opinion, is considered in some detail.

#### 2 Prospective mathematics teachers’ pedagogical content knowledge of definite integral: the problem of limit process

Hatice Akkoç^{1}, Sibel Yeşildere^{2} and Fatih Özmantar^{3}

Universities of Marmara^{1}, Dokuz Eylül^{2} and Gaziantep, Turkey^{3}

This study investigates prospective mathematics teachers’ pedagogical content knowledge (PCK) of definite integral. Considering the notion of PCK as described by Shulman (1986, 1987), we will investigate prospective mathematics teachers’ knowledge of student difficulties in relation to the limit process to define definite integral. For that purpose, four prospective mathematics teachers were observed during their micro-teaching and were interviewed afterwards. Micro-teaching videos, interview transcripts, prospective teachers’ lesson plans and teaching notes were analysed. In this presentation, we will discuss how prospective teachers addressed student difficulties for the limit process when constructing the area under a curve from the sum of rectangular areas and consider the implications in terms of PCK.

#### 3 Improving learning in mathematics at Key Stage 2

Jenni Back and Tony Beauchamp

King’s College, London; NCETM independent research consultant

The paper describes a pilot adaptation, for teachers at Key Stages 1 and 2, of the professional development model outlined inlsquo;Improving Learning in Mathematics’ (Swan 2005). Initially, suitable tasks for children were trialled with a Year 5 class of low attaining children over the course of an academic year. The children’s attitudes to maths, and their SATs scores, were monitored over the same period. Work on the professional development course followed and the course has been trialled with a group of 14 primary school teachers. The paper presents some initial analysis of data gathered from the initial study as well as preliminary observations of teachers’ practice prior to the commencement of the CPD course.

#### 4 Mathematics education -lsquo;a field in disarray’?: the story of a search for a methodology

Alf Coles

Kingsfield School, South Gloucestershire and University of Bristol

In this paper I review some recent reviews of mathematics education research, which seem at first sight to confirm a judgement by Steen (1999) that it ‘is a field in disarray’. As a PhD student, looking at competing methodologies there can seem a bewildering complexity from which to choose. It appears that there are relatively few (three to five, depending on authors) theoretical perspectives or paradigms that guide the field, but a plethora of methodologies that can be used with them. One thing I realise would help me, is if authors were explicit about the ‘not’s’ of any approach they use. In this vein I reflect on my own use of enactivist methodology, and its affordances and constraints.

#### 5 Exploratory factor analysis of student-teachers’ perceptions of 3D-descriptive geometry education in Mozambique

Daniel Dinis da Costa

School of Education, Newcastle University

This study explores the factors underlying student-teachers’ perceptions of 3Ddescriptive geometry education in Mozambique. A grounded theory mixed-method approach was used to gather data from six focus groups, ten interviews and a questionnaire with 120 participants. Principal Component Analysis (PCA) for a fourfactor solution was then performed, which revealed a structure with items clustered into four factors: spatial visualisation and reasoning related to 3D-geometry fundamentals; professional learning consisting of learning to teach geometry, evaluation and learning support, which encompasses items on mediating learning, and practice concerned with developing skills. Possible explanations of the findings are discussed, and their implications for further research suggested.

#### 6 Assessing the structure and sensitivity of the belief systems yielded by the revised mathematics-related beliefs questionnaire

Jose Diego-Mantecon^{1}, Paul Andrews^{1}, Peter Op’t Eynde^{2}, María José González-López^{3}

University of Cambridge^{1}, University of Leuven, Belgium^{2}, University of Cantabria, Spain^{3}

This is the third of three papers in which we describe how the mathematics-related beliefs questionnaire (MRBQ), developed at the University of Leuven (Op’t Eynde and De Corte, 2003), can be adapted for use in Spanish and English educational contexts. In earlier papers we showed that the MRBQ could be refined to yield four reliable scales and ten subscales for both Spanish and English Students (Diego-Mantecón et al., 2007), and that the four scales highlighted a number of differences related to culture, age and gender (Andrews et al., 2007). In this paper, we examine the structure of student belief systems as reflected by the interrelations of the subscales. In so doing we present further evidence of the revised MRBQ’s sensitivity to nationality, age, and gender.

#### 7 Pupil choice of tool

Patricia George, John Monaghan and John Threlfall

University of Leeds

We examine a year 6 lesson where pupils were told to use paper and a calculator and also to use a spreadsheet. Pupils, in pairs used/chose different tools. We examine influences on their choices.

#### 8 Mathematics content knowledge of pre-service primary teachers: developing confidence and competence

Brenda Hamlett

Edith Cowan University, Perth, Western Australia

This paper examines the extent to which a group of first year pre-service teachers enrolled in Bachelor of Education courses in primary and early childhood education at a Western Australian university can be considered as mathematically literate when it comes to teaching the content of the WA primary mathematics curriculum, and describes how both confidence and competence have been improved through the introduction of a multiliteracy unit.

#### 9 Mathematics teacher development with ICT: towards an international GeoGebra institute

Markus Hohenwarter and Zsolt Lavicza

Florida Atlantic University, USA; University of Cambridge

Research indicates that despite the numerous benefits of using ICT in mathematics education, the process of embedding ICT in classrooms is a slow and complex process. Most teachers need more than just being provided with technology if the benefits of ICT are to be substantially realised. GeoGebra is free open-source dynamic software for mathematics teaching and learning that offers geometry and algebra features in a fully connected software environment. In this paper, we outline the emergence of the GeoGebra software as well as ideas and plans for establishing an International GeoGebra Institute to provide training and support for teachers and to coordinate research in relation to GeoGebra.

#### 10 Use of mathematical software in pre-service teacher training: the case of DGS

Vlasta Kokol-Voljc

Faculty of Education, University of Maribor, Slovenia

Use of mathematical software in pre-service teacher training for preparing students to become mathematics teacher has two aspects:

- Using mathematical software as a support for pre-service teacher training
- Preparing the future teachers for using mathematical software for their teaching

In the presentation theory and practice of the second item will be discussed: what are the pedagogical benefits of the use of mathematical software in mathematics teaching.

Different types of mathematical software will be introduced and discussed towards their use in mathematics teaching.

#### 11 Teacher-pupil dialogue in mathematics lessons

Chris Kyriacou and John Issitt

Department of Educational Studies, University of York

This paper reports the findings of a systematic review of the literature looking at what characterises effective teacher-initiated teacher-pupil dialogue to promote conceptual understanding in mathematics lessons in Key Stages 2 to 4. The review was based on an in-depth analysis of 15 studies. Eight key characteristics were identified: going beyond IRF (Initiation-Response-Feedback); focusing attention on mathematics rather than performativity; working collaboratively with pupils; transformative listening; scaffolding; enhancing pupils’ self-knowledge concerning how to make use of teacher-pupil dialogue as a learning experience; encouraging high quality pupil dialogue; and inclusive teaching.

#### 12 An inquiry into elementary teachers’ dispositions toward mathematics instruction and their choices of teaching methods

William O. Lacefield

Tift College of Education, Mercer University, USA

This study considered elementary teachers’ dispositions toward mathematics instruction and their mathematics teaching methods. From 492 teachers (grades K-4) in a Southeastern United States school system, a cluster sample of 90 teachers representing six schools (one inner city, four suburban, one rural/semi-rural) was randomly selected. Participants completed Likert scale questionnaires, one designed to measure dispositions and one designed to measure frequencies of use of particular teaching methods.

Pearson correlation coefficients measured relationships between dispositions (anxiety, confidence, enjoyment, desire for recognition, and pressure to conform) and frequencies of use of particular teaching methods (traditional methods, progressive methods, and methods combining traditional and progressive approaches). While significant (p<.05) correlations were found between types of dispositions, there were no significant (p<.05) correlations between dispositions and teaching methods, suggesting that teachers’ dispositions toward mathematics instruction and their use of particular teaching methods are not straightforwardly connected but are likely affected by multiple factors.

#### 13 Mathematics education in Barbados and Trinidad: challenges and progress

Elaine Lam

Bath Spa University

This paper will provide an overview of Mathematics education in the Caribbean region with specific focus on Barbados and Trinidad based on the author’s PhD thesis. Findings from this study are based on 50 interviews and 87 hours of observations. Curriculum documents as well as specific Mathematical problems will be examined. There will also be some discussion regarding the possibility of education borrowing – the attempt to share best practices across the region – as a means of improving the quality of education in the region. The session will focus on the themes of cultural conflicts, teaching styles and implications for Maths education in both the Caribbean and globally as some of the challenges in Barbados and Trinidad are universal. By examining Maths education in a different context, we may be able to learn more from each other.

#### 14 A coursework task in A level Mathematics – a survey of student opinion

Chris Little

University of Southampton

The use of coursework assessment in English and Welsh high stakes mathematics examinations is rapidly declining. This paper reports the results of a survey of student opinions towards a compulsory piece of coursework for A level mathematics. Girls estimated they spent on average 29% longer on the task than boys. Using an aggregated measure oflsquo;positivity’, 77% of females and 64% of males were on balance positive towards the coursework; 14% of females and 29% of males were on balance negative towards it. 24% of open responses which were classed aslsquo;positive’, and 44%lsquo;negative’. These results suggest that these students overall were positive about doing this coursework, with girls significantly more in favour than boys.

#### 15 Being a sumbody: new stories of choosing maths

Heather Mendick, Sumi Hollingworth, Marie-Pierre Moreau and Debbie Epstein

London Metropolitan University and Cardiff University

Neo-liberalism has led to an expansion of market economics, notably into public sector areas including education. Within this world choice is central. These choices are not only acts of consumption; they are also a means of making one’s-self. We look at: What does it mean for students to make subject choices within this framework of compulsory choice and entrepreneurship? And, in particular, how do people choose mathematics in this context? We do this by looking in detail at interviews with 11 people who chose to study mathematics at university.

#### 16 A synthesis of taxonomies/frameworks used to analyse mathematics curricula in Pakistan

Nusrat Fatima Rizvi

Department of Education, University of Oxford

The paper reports the development of a framework for analysing written and tested curricula. Most of the existing taxonomies/frameworks (e.g. proposed by Bloom, 1956; Smith et al 1996; Biggs, 1995; and Porter, 2002) recognize overlapping as well as discrete categories of cognitive processes. Anderson and Krathwohl (2001) have added another dimension, i.e. categories of knowledge. This paper synthesises these existing taxonomies/ frameworks in order to develop an integrated framework which is appropriate for the mathematics curriculum. The paper will also discuss how the new framework is being used in analysing examination curricula of secondary school mathematics in Pakistan. Purpose of this analysis is to develop a commonlsquo;non-routine’ test paper for the students who have studied different curricula at secondary school level.

#### 17 Some reflections on the philosophy of mathematics education

Stuart Rowlands

Centre for Teaching Mathematics, University of Plymouth

Much reference has been made to Paul Ernest’slsquo;philosophy of mathematics education’ to legitimise current trends in mathematics education. This session presents the argument that 1. Thelsquo;philosophy’ is more a sociology than it is philosophy. 2. The very basis of thelsquo;philosophy’ contains a contradiction – that mathematics cannot be separated from its social origins, yet mathematics has a logical necessity that is independent of origin. 3. Thelsquo;philosophy’ downplays mathematics as a formal, academic discipline in the attempt to promote a childcentred pedagogy. 4. Thelsquo;philosophy’ makes unwarranted assumptions that have been taken aslsquo;given’. For example, thatlsquo;absolutist’ orlsquo;Platonist’ views of mathematics necessarily implies the transmission model of teaching mathematics.

#### 18 Beginning teachers’ use of representation

Fay Turner

University of Cambridge

The effectiveness of representations used by elementary school teachers, in the first three years of their teaching, is the focus of this paper. I report on findings, from a study which makes use of the Knowledge Quartet framework. I draw on data from the first two years of the study which indicates that the use of representations is a key feature of lessons. Beginning teachers recognised that their use of representations were not always appropriate. There is some evidence that focused reflections have facilitated the development of the teachers’ pedagogical knowledge in relation to the use of representations.

#### 19 Adolescence and secondary mathematics

Anne Watson

University of Oxford

In this paper I outline an argument that the intellectual demands of the secondary mathematics curriculum are fully compatible with adolescence, as a particular phase of life concerned with identity and cognitive development.

#### 20 Observing students’ use of images through their gestures and gazes

Tracy Wylie

Kingsfield School, South Gloucestershire

In this paper I report on a study observing six year 13 students (18 years of age) working in pairs on a set of mathematical problems, to test the hypotheses related to students’ use of imagery. The research was done from a constructivist perspective, looking through a socio-cultural lens. It is concerned with the relationship between mental imagery, thought and action. The results have showed that it is possible to observe students’ use of imagery through their gestures and suggest that those students who have access to and are able to manipulate mental images are more successful problem-solvers.

#### 21 Teachers’ beliefs about mathematical problem solving, their problem solving competence and the impact on instruction: the case of Ms Electra, a Cypriot primary teacher

Constantinos Xenofontos

University of Cambridge

This paper refers to the case of Ms Electra, a Cypriot primary teacher, with respect to her beliefs about problem solving, her problem solving competence and the teaching of a problem solving activity. Ms Electra was interviewed about her problem solving beliefs. After the interview, she was given a mathematical problem and asked to solve it and explain her thought. Ms Electra prepared a lesson based on that problem and taught the problem solving activity in her classroom.

#### 22 Ways of linking geometry and algebra: the case of GeoGebra: BSRLM Geometry Working Group

Markus Hohenwarter and Keith Jones

Florida Atlantic University, USA; University of Southampton

This paper discusses ways of enhancing the teaching of mathematics through enabling learners to gain stronger links between geometry and algebra. The vehicle for this is consideration of the affordances of GeoGebra, a form of freely-available open-source software that provides a versatile tool for visualising mathematical ideas from elementary through to university level. Following exemplification of teaching ideas using GeoGebra for secondary school mathematics, the paper considers current emphases on geometry and algebra in the school curriculum and the current (and potential) impact of technology (such as GeoGebra). The paper concludes by raising the implications of technological developments such as GeoGebra for the pre-service education and inservice professional development of teachers of mathematics.