**Proceedings of the Day Conference held at the University of Southampton in June 2008**

## Contents

### Special contribution

#### 1 Brian Griffiths (1927–2008): A tribute to a pioneer in mathematics education

Keith Jones and Tim Rowland

University of Southampton and University of Cambridge

Brian Griffiths (Professor Emeritus, University of Southampton) was one of the pioneers of mathematics education and played a significant role in developing the field from the mid-1960s onwards. As well as being remembered for his work on what continue to be known as ‘Griffiths-type’ topological spaces, Brian’s most profound contribution to the establishment of mathematics education as an academic field was, with Geoffrey Howson, in offering a conceptualisation of the relationship between mathematics, society and curricula.

### Research reports

#### 2 Teachers researching their own practice: Evidencing student learning using TINspire

Alison Clark-Wilson

University of Chichester

It is generally accepted that the introduction of interactive technologies into secondary mathematics classrooms presents new challenges to teachers and students both mathematically and pedagogically. This paper reports a case study approach as a teacher began to use the TI-Nspire handheld and software in secondary mathematics classrooms with pupils aged 11-16 years as part of a pilot project. The focus will be on the methodology employed by the researcher in eliciting the teacher’s perceptions of the ways in which the technology was impacting on teaching approaches and learning outcomes.

#### 3 The impact of ICT on mathematical concepts

Tandi Clausen-May

NFER

The rapidly increasing use of ICT in the mathematics classroom clearly influences the ways in which teachers teach and students learn. Computers can offer dynamic visual images that may open up some areas of mathematics to a much wider audience. But can ICT affect not just how we teach, but also what is taught and what is learnt? This paper considers aspects of the possible impact of the use of computers on the mathematical concepts that students develop.

#### 4 Mathematics trainee teachers’ attitudes to computers

Mustafa Dogan

University of Selcuk, Turkey

This study explores primary mathematics trainee teachers’ attitudes to computer in mathematics education with an open-ended question which is a part of a survey. Survey was conducted with a self constructed “Using Computer in Mathematics Education (UCME)” questionnaire. Piloting, reliability (á=0.88) and factor analyses have been performed. Sample is a total of 134 trainee primary mathematics teachers. Explorations of the responses to the question revealed different aspects of attitudes and give insights. Results show that the trainee teachers usually enjoy working with computer although they are able to do considerably minor works about mathematics. They stated that they can learn and teach mathematics effectively if they use computer better, even if they do not feel confident studying mathematics with computer.

#### 5 The assessment of newly qualified teachers’ beliefs about the teaching and learning of mathematics

Ruth Forrester

University of Edinburgh

Beliefs are hard to pin down. Whether or not they are successfully translated into practice, they give important indications of intentions for the future. The “IMAP web-based beliefs survey” was developed in order to assess beliefs about the teaching and learning of mathematics held by prospective elementary teachers (IMAP 2003). This survey gathers data by asking teachers to respond to video clips and teaching scenarios in their own words. The study reported here investigated beliefs held by three newly qualified secondary teachers. In particular the aim was to evaluate the IMAP survey in the secondary context by comparing it with the alternative data collection methods of interview and observation. Results indicate the general effectiveness of the IMAP survey for assessing the beliefs of such teachers.

#### 6 Learners’ Understanding of the Hierarchical Classification of Quadrilaterals

Taro Fujita

University of Plymouth

The hierarchical classification of quadrilaterals might be regarded as an area of study which would help to promote the development of geometrical thinking. This paper reports our investigations in this topic discussed in our former study (Fujita and Jones, 2007, RME vol. 9). In particular, by synthesising past and current theories in the teaching of geometry (van Hiele’s model, figural concepts, prototype phenomenon etc.), I propose a theoretical model and method to describe learners’ cognitive development of their understanding of hierarchical relations of quadrilaterals. I also consider how the model could be utilised to describe and analyse learners’ understanding of the hierarchical classification of quadrilaterals by using pilot data collected in 2008.

#### 7 A constructionist approach to mathematical generalisation

Eirini Geraniou, Manolis Mavrikis, Celia Hoyles and Richard Noss

London Knowledge Lab

The work presented here arises from the MiGen project, which aims at designing, building and evaluating a pedagogical and technical environment for improving 11-14 year-old students’ learning of mathematical generalisation. This paper presents the preliminary analysis of the initial data collected in 24 sessions with year 6 and year 7 students. During these sessions, a prototype microworld was used to explore the range of functionalities we require, and collect initial data regarding students’ strategies and potential prompts that the system could ultimately provide to both students and teachers. A key design objective is to develop an environment that makes it as natural as possible for students to express generality, and a means to do so, rather than simply encourage them to consider special cases, or spot patterns.

#### 8 Ethics, performativity and decision mathematics

Paul Hernandez-Martinez and Julian Williams

University of Manchester

The spirit behind the incorporation of Decision Mathematics (and Discrete Mathematics) into the A-level Mathematics Curriculum was that of encouraging the creation of algorithms and problem solving. However, some students and teachers tell us it is “boring” and not “proper” maths. In order to have an insight into the role that Decision Mathematics plays in the current programme of mathematics, we observed some Decision Mathematics lessons and interviewed teachers and departmental authorities. We found that Decision Mathematics in the classroom involves the students being programmed to carry out the algorithms, without any sense of “agency” in problem solving, creating or evaluating algorithms, and with few significant connections or applications of their pure maths. The Decision Mathematics option is said by teachers to provide an “easy” way for students to gain marks, and a “relaxing” module for them in contrast to core pure modules. Performativity and the “league tables” culture of the education system in Britain thus ensure that alternatives to Decision Mathematics may not be considered an ‘equal’ option especially for weaker students, whatever the needs of the students or the teachers’ professional beliefs and ethics.

#### 9 Interactive geometry in the classroom: old barriers and new opportunities

Chris Little

University of Southampton

Although computers and calculators have had a massive effect on some areas of school mathematics, some evidence suggests that geometry teaching has been slow to utilise computer software. This paper discussed three barriers to implementing dynamic geometry in the classroom; curriculum scope: teachers need to be convinced that they can teach geometry more effectively; accessibility of computers: teachers and/or students need ready and regular access to computers; accessibility of programs: they must be easy to learn so that the emphasis is on learning the maths, not the program. With the development of internet-based freeware, such as Geogebra, it is possible that these barriers may be being overcome. The greater accessibility to students, and the availability of digital projectors, then presents the issue of who should be in charge: the teacher leading whole-class discussion, or the student engaged in individual ‘guided re-invention’.

#### 10 The role of context in linear equation questions: utility or futility?

Chris Little

University of Southampton

It is common practice in Key Stage tests and GCSE Mathematics to embed mathematics in real-world contexts. However, the practice has been criticised by some researchers on the grounds of artificiality and construct validity. This paper considers the role of context in four linear equation questions, concluding that the purpose of the context is not utility, but concept formation and abstraction.

#### 11 Imagery and Awareness

John Mason

Open University

I am interested in the core or key awarenesses which underpin the various topics, techniques and concepts of school mathematics. I take awareness to be the basis for action, which means it need not be conscious. Every action undertaken in mathematics, whether rotating a figure, reducing a fraction, counting on, or reasoning about a trigonometric identity is informed by and made possible because of previous experience with similar actions in the past, from which grows the awareness. I am also interested in how attention moves, and what its movement reveals about awareness. In this session I offered people some tasks through which to experience some movements of their attention, and so perhaps to enable awareness of their awarenesses to come to articulation.

#### 12 What are the argumentative activities associated with proof?

Juan Pablo Mejia-Ramos and Matthew Inglis

University of Warwick

Our goal in this paper is to identify the different argumentative activities associated with the notion of mathematical proof. Having identified the different activities we present the results of a bibliographic study designed to explore the extent to which each of these activities has been researched in the field of mathematics education. We conclude by arguing that the comprehension and presentation of given arguments are important, but under-researched mathematical activities related to proof.

#### 13 Learning about motion in a multi-semiotic environment

Candia Morgan and Jehad Alshwaikh

IoE, University of London

Students’ intuitive assumptions and arguments about motion are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Traditional approaches to the teaching of mechanics and everyday physical experience do not sufficiently challenge these assumptions and arguments. In this paper, we report and reflect on a teaching experiment conducted with college students learning about motion with MoPiX – a multi-semiotic interactive learning environment. We discuss the potentiality of new forms of representation for learning about motion.

#### 14 The Development of a Semantic Model for learning Mathematics

Mike Peters

University of Plymouth

The semantic model described in this paper is based on ones developed for arithmetic (e.g. McCloskey et al 1985, Cohene and Dehaene 1995), natural language processing (Fodor 1975, Chomsky 1981) and work by the author on how learners parse mathematical structures. The semantic model highlights the importance of the parsing process and the relationship between this process and the mathematical lexicon/grammar. It concludes by demonstrating that for a learner to become an efficient, competent mathematician a process of top-down parsing is essential.

#### 15 Developing beliefs about the teaching of primary mathematics

Caroline Rickard

University of Chichester

This paper relates to small-scale, mixed methodology research investigating the developing beliefs of students in Initial Teacher Education (ITE) in relation to the teaching of primary mathematics, undertaken as part of a Masters in Mathematics Education. All students were in their first year of a three year degree, studying at the University of Chichester. The major intention was to identify their emerging beliefs, with a subsidiary interest in what had shaped those beliefs. The project was set in the context of a new curriculum mathematics module, which was underpinned by my own beliefs about module content and teaching approaches. A range of questions were asked of the students through simple questionnaires and a small number of interviews.

#### 16 How shall we talk about ‘subject knowledge’ for mathematics teaching?

Tim Rowland and Fay Turner

University of Cambridge

Sustained research into mathematics teacher knowledge over two decades, much of it in the UK, has drawn attention to the complexity of the knowledge base of mathematics teachers at all phases of education. Yet the official discourse of the topic remains rather blunt and simplistic. For example, remit 4 for the recent Williams review asked “What conceptual and subject knowledge of mathematics should be expected of primary school teachers…”. In this paper, we explore whether, and if so how, it might be possible to conceptualise and talk about knowledge for and in mathematics teaching, in ways that are acceptable to, and accessible by, education professionals and policy makers.

#### 17 What is the Nature of the link between Students’ Mathematical Qualification, Subject Knowledge and their Confidence to teach Primary Mathematics?

Margaret Sangster

Canterbury Christ Church University

This study is an exploration of students’ confidence in their mathematical subject knowledge and their ability to teach mathematics in the 3 – 11 age range. A survey of 78 students was carried out by questionnaire after the second placement in Year 2 of a three year degree course in primary teacher training. The study found a close match between students’ confidence and their qualifications and a further link with perceived success in the classroom. The study raised the question of whether actions around improving subject knowledge in the training institution was raising or decreasing student’s confidence.

#### 18 ‘Profound understanding of fundamental mathematics’: a study into aspects of the development of a curriculum for content and pedagogical subject knowledge.

Mary Stevenson

Liverpool Hope University

The purpose of this paper is to review the development of a curriculum for the Mathematics Enhancement Course at a particular university, and in particular to evaluate the extent to which the course can be deemed to be successful in developing students’ subject, and to a lesser extent, pedagogical knowledge.

#### 19 Growth in teacher knowledge: individual reflection and community

participation

Fay Turner

University of Cambridge

Barbara Jaworski (2001) posed the question “In terms of teacher education, do we see a teacher’s growth of knowledge as a personal synthesis from experience or as deriving from interactions within social settings in which teachers work?” (p. 298) Evidence from my four year study suggests that participants’ growth of knowledge for mathematics teaching has been influenced by individual reflection as well as by participation in communities of practice, with the interaction between the two being dependent on individual contexts. In this paper I present some findings from the case studies of Amy and Kate.

#### 20 Generating mathematical talk in the Key Stage 2 classroom

Lyn Wickham

Bidbury Junior School

This action research, based in a mixed ability Year 6 classroom considers why children should talk in mathematics lessons and the conditions for allowing “good” mathematical talk to take place. Within the study a series of PHSE lessons were developed to support the children to arrive at their own understanding of mathematical talk and design their own rules and descriptions of good talk, which are referred to in subsequent lessons. This study ends with a lesson that shows examples of exploratory talk.

#### 21 A comparison of mathematics teachers’ beliefs between England and China

Yu, Huiying

University of Cambridge

This study compares the beliefs of mathematics teachers in England and China. It explores in detail teachers’ beliefs about three themes of mathematics education: mathematics teaching, the nature of mathematics, and the purposes of mathematics education. It also considers briefly whether gender or teaching experience influences these three aspects of teachers’ beliefs. The research method of this study involved a questionnaire and follow-up interviews. Forty-four English mathematics teachers from 10 secondary schools in Cambridgeshire, England and 96 Chinese mathematics teachers from 10 secondary schools in Shanghai, China participated in the questionnaire survey. Furthermore, two mathematics teachers from each country were interviewed. Findings suggest that the beliefs of English teachers may generally reflect the pragmatic understanding of theory in mathematics teaching, while those of Chinese teachers generally reflect the scientific understanding of theory. With regard to the purposes of mathematics education, Chinese teachers place more emphasis on logical and rigorous reasoning than English teachers, though this study also shows that Chinese teachers agree more strongly about its application to other areas than do English teachers. On the other hand, English and Chinese teachers do not have significantly different beliefs about the nature of mathematics. And finally, gender and teaching experience do not in general affect teachers’ beliefs about these three themes of mathematics education in England and China.