**Proceedings of the British Congress of Mathematics Education in collaboration with BSRLM, July 2001.**

## Contents

### Plenary sessions

#### 1 Crossing Boundaries in the study of mathematics teaching: An uneven odyssey of learning and unlearning

Hyman Bass, President of ICMI, University of Michigan

Deborah Ball, University of Michigan

This presentation examines our work in the borderlands between mathematics and education. We trace across our six-year collaboration the development of one strand of our work: the reasoning of justification. We identify four phases of the work that have been significant in the development of our research, and discuss the challenges and benefits of crossing boundaries between our fields for the study of mathematics teaching and learning.

#### 2 Removing boundaries to Mathematics learning: can psychology help?

Terezhina Nunes

Oxford Brookes University

Different communities interested in mathematics education can work together to remove boundaries that prevent access to mathematics for certain groups and individuals. This presentation will focus on the role of psychology in describing children’s learning processes. Following Piaget, I will argue that the origin of mathematical concepts is in the logic of actions. But this logic must be expanded and reshaped in the mathematics classroom for students to become successful mathematics learners. Examples will be presented to illustrate the process and case studies of groups and individuals with difficulty in learning will be discussed.

#### 3 Plenary Panel: Beyond Maths Year 2000

Barbara Young, Celia Hoyles, Robert Hunt, John Bibby

The panel was chaired by Barbara Jaworski who introduced the panelists and promised to keep them to their allotted time. She hoped that they would not hesitate to be controversial and that the audience would make good use of their opportunity to contribute to the debate.

### Discussion groups

#### 4 Language and Mathematics Education

Richard Barwell, Constant Leung, Candia Morgan, Brian Street

The discussion group was designed to bring together mathematics education and applied linguistics. To this end, the panel included two participants from applied linguistics, both attending their first mathematics education conference. Discussion was stimulated by texts introduced by members of the panel. These included a paragraph of guidance from the NNS booklet Mathematical Vocabulary (DfEE, 1999: 2) entitled ‘How do children develop their understanding of mathematical vocabulary’; a transcript from a KS2 classroom of a discussion about dimensions; and a transcript from a KS1 classroom featuring some work on money. What follows is a (very!) brief summary of some of the input of the panel as well as our impressions of some of the issues which emerged during the discussions at various stages.

#### 5 Developing a taxonomy for delivering and leading CPD

Derek Woodrow

It is true to say that the discussion group was generally pessimistic of CPD development, seeing political ‘intuition’ rather than rational decision planning as the likely process. There were, however, developments that could be potentially valuable and needed supporting. It does need, however, a coherent and co-operative development plan from all the stakeholders if the forces of dogma are to be resisted.

#### 6 Assessment

Annie Gamon, Catharine Darnton, Jan Winter

The following notes were scribed as shared feedback from a discussion at BCME on assessment. This discussion was the second session on assessment. We worked in three groups of three on some statements that were important for us. We had stimuli for discussion in the shape of an article by Clare Lee on assessment in edition 175 of Mathematics Teaching.

#### 7 Transition from GCSE to AS and A level

Sue Cramp and panel

The theme for this discussion group arose from research, conducted by Sue Cramp and Elena Nardi, into the introduction of the new AS and A Levels in Mathematics. The findings of this preliminary study have prompted Sue and Elena to submit an ESRC proposal to focus on transition in teaching and learning styles from GCSE to A Level. They hoped that the discussion group would explore general issues arising from the new AS and A Levels as well as providing guidance for possible questions and issues which could be incorporated into their new research.

### Research papers

#### 8 Hungary and its characteristic pedagogical flow

Paul Andrews and Gillian Hatch

University of Cambridge and the Manchester Metropolitan University

This paper reports on almost one hundred observations of mathematics classes in Budapest, Hungary. Undertaken with regard to the survey of mathematics and science opportunities team’s framework, the observations confirmed a characteristic pedagogical flow. Lessons tended to follow a well-defined pattern: a public review of homework was followed by a “warm-up” period which, in turn, was followed by several episodes involving the posing, individual working on and public sharing of solutions of problems before homework was set again. Significantly the mathematics studied and the problems posed presented a perspective on the subject which appeared to privilege mathematical problem-solving, complexity, generality, coherence, topic integration and mathematics as a cultural artefact. Some implications in respect of raising attainment in England are discussed.

#### 9 Understanding home school relations in numeracy

Dave Baker with Brian Street and Alison Tomlin

University of Brighton and Kings College, London

This is a discussion of research in the ‘Schooled and Community numeracies’ focus within the Leverhulme funded Low Educational Achievement in Numeracy Research Programme. The intentions of the research are to seek explanations for under-achievement in numeracy that derive from understandings of mathematics as social. We want to understand why some children engage easily with community numeracy practices yet struggle with schooled numeracies. We wish to investigate boundaries children face or which are constructed between home and schooled numeracy practices. The paper will consider some of the conceptual and methodological issues that have arisen in the research.

#### 10 Dynamical aspects of mathematical proof

Tony Barnard

Kings College, London

In regarding mathematical thinking as proceeding via operations involving a small number of ‘items’ at any one time, an important feature is the phenomenon in which a section of mathematical structure may be mentally held as a single unit, possessing an interiority that can be subsequently expanded without loss of detail and trigger connections with other parts of cognitive structure. This article discusses the role of this phenomenon of ‘compression’ and ‘expansion’ in the manipulation of statements involved in certain types of mathematical proof. Implications for teaching linked to an awareness of these aspects is also discussed.

#### 11 Difficulties with mathematical word problems

Richard Barwell

University of Bristol

The extensive literature on students’ ‘difficulties’ with mathematical word problems raises many issues for which the prevailing mostly quantitative paradigms and approaches to research appear inadequate. My own research, which draws on discursive psychology (Edwards, 1997) makes use of a word problem task to generate interaction. This discursive approach, which I set out and exemplify in this paper, offers a fresh perspective on some of these issues.

#### 12 Indicators of abstraction in young children’s descriptions of mental calculations

Chris Bills

University of Warwick

The National Numeracy Strategy encourages class discussion of mental calculation strategies but when children respond to “How did you work that out?” teachers may be able to learn more than just what strategies were employed. This paper proposes three ‘indicators of abstraction’ which might provide indications of childrens developing conceptualisations. The indicators related to children’s descriptions of their calculations are ‘metaphor’, ‘expression of generality’ and ‘method’. Each has three categories of abstraction. Data drawn from a study with 7 to 9 year old children suggests that responses categorised as ‘representative’ and ‘symbolic’ are associated with high levels of accuracy whilst lower accuracy is more often associated with higher frequency of ‘concrete’ responses.

#### 13 Capturing or missing moments

Alison Borthwick

University of East Anglia

The nature of a teacher’s response to a pupil answer can have a strong influence on the learner. Yet some answers are not fully explored, and some may even be missed. It is this notion, which I call ‘missing the moment’ that is at the heart of this research and was the focus for my MA dissertation. This paper examines some of these ‘moments’ and begins to suggest alternative ways of teacher response for their existence and also ways in which to respond. While this research was in itself complete, it also stands to serve as the beginnings of further research in this area, my doctorate, now in development.

#### 14 Beliefs overhang: the transition from school to university

Katrina Daskalogianni and Adrian Simpson

Warwick University

In this paper we introduce the idea of ‘belief overhang’ – the continuation of views about the nature of mathematics which students develop in school but which are carried forward (often inappropriately) to mathematics in university. As part of a larger project considering the role of the affective in the transition from school to university, we consider two of the many distinct belief types we have identified – the systematic and the utilitarian. We outline the characteristics of these beliefs as built in school and the consequences of them during the first few weeks of the students’ university mathematics course. We speculate about the ways in which these beliefs can cause difficulties and possible actions that teachers and lecturers might take to ameliorate the problems.

#### 15 Group and individual development in mathematics teaching:a case study

Barbara Georgiadou-Kabouridis

University of Surrey Roehampton

This paper addresses issues derived from an ongoing research which attempts to explore the kind of support that inexperienced primary teachers need in their struggle to embed the development of their mathematics teaching within the demanding school programme. The research is conducted at group and individual level. This study investigates the affects on teachers’ attributes as they work as individuals and as members of a group. This investigation is approached through the case study of one of the participating teachers in the research.

#### 16 An investigation of mathematics textbooks and their use in English, French and German classrooms: who gets an opportunity to learn what?

Linda Haggarty and Birgit Pepin

The Open University and Oxford Brookes University

In order to refine our understandings of the teaching and learning cultures of the mathematics classroom in different countries, we need to refine our understandings of the teachers, the learners, the materials used for learning and the interactions between them. However, each of these is influenced, and in some cases determined, by the educational and cultural traditions of the particular country in which the teaching and learning takes place. Until we have a richer, more clearly articulated, and more detailed understanding of the ways each of these factors interrelate, educationists are likely to be pulled in inappropriate and ill-judged directions by policy makers intent on short-term, measurable outcomes of performance improvements in a narrow range of areas which are not only opposed to the values held by their own society, but also unworkable within the cultural traditions shaping it.

#### 17 Equity issues affecting mathematics learning using ICT

Sarah Hennesy and Penelope Dunham

University of Cambridge and Muhlenberg College, USA

Inequities may arise from differential access and use of educational technology in mathematics for groups characterised by gender, ethnicity, income level, and ability. As technology access increases in homes and schools, we ask whether previous inequities are diminished or exacerbated. Equity is considered from the perspectives of opportunities to learn (physical access); educational treatment (how technology is used, by whom); social and psychological factors influencing its use; educational outcomes (achievement, attitudes and motivation). Suggestions for removing boundaries between technology “haves” and “have-nots” are presented.

#### 18 Setting tasks and setting children

Jenny Houssart

The Open University

The work described here was carried out in four primary schools which use setting for mathematics. This research forms part of a project on tasks in primary mathematics. The focus of this paper is a consideration of how tasks are influenced by setting. Transcripts of interviews with teachers are analysed for differences in how teachers of different sets talk about the tasks they give children. Findings suggest that there is likely to be a difference in the mathematical tasks children are offered according to the set they are in. Some information also emerges about systems of setting, apparent reasons for setting and how teachers perceive the pupils in their set.

#### 19 On the ‘tail’ of a sequence, the universal quantifier and the formal definition of convergence

Paola Iannone and Elena Nardi

University of East Anglia

Achieving logical coherence and understanding of the precise meaning of a symbol are crucial points in the learning of a first year mathematics undergraduate. Here, in the context of a course in Linear Algebra and Calculus, we examine students’ written work in order to address issues regarding the understanding of the formal definition of convergence for numerical sequences. In particular we focus on the students’ use of the universal quantifier, ? , in ways that suggest that the students may neglect the ‘universality’ in its meaning: in their applications of the definition of convergence, in some occasions, not all ? are covered and, in some others, a finite number of the terms of the sequence, are left out of the argument.

#### 20 Learning preferences in relation to subjects of study of students in higher education

Janis Jarvis and Derek Woodrow

Manchester Metropolitan University

Applications for entry to Higher Education show marked differences between ethnic minority groups and marked gender preferences, particularly for mathematics. Analysing UCAS data shows that these differences are persistent. Research into learning preferences suggests that these might be one reason for these differential choices. This paper reports a study of over 250 undergraduates and 500 PGCE students, identifying clear subject differences in learning preferences. Mathematics students lie at one end of the scale and English students at the other.

#### 21 Symbol sense – the gap between GCSE and A level

Ruth Sharma

University of Surrey Roehampton

For my Masters Dissertation I researched pupils’ symbol sense as defined by Arcavi (1994). Before I started I expected to find that, as pupils’ matured, their symbol sense would improve. What I had not expected was to see such a clear gap differentiating GCSE pupils’ symbol sense from those studying A Level. This paper suggests that, although some A Level pupils still struggle to understand algebra in the sixth form, there is a clear divide in pupils’ understanding of algebra between GCSE and A Level, a gap which new sixth form pupils have to cross.

#### 22 Mechanics and direct manipulation environments: a role for proof?

Ian Stevenson

Institute of Education, London

This paper examines the relationship between explanations, proof and the use of computational resources in the context of developing mathematical models for mechanics. It reports on three workshops with pre-university students using the Direct manipulation Environment, Interactive Physics, to explore connected particle systems, and analyses the style of reasoning used by the participants. The paper concludes with a discussion of issues raised by the workshops.