**Proceedings of the day conferences held at the University of Southampton on 16-17 November 2001.**

## Contents

#### 1 A comparison of Primary Mathematics Curriculum in England and Qatar: the SOLO taxonomy

Aziza Alsaadi

University of Surrey Roehampton

The study reported here is part of a wider study, which aims to compare the primary mathematics curriculum in England and Qatar. This paper focuses on the analysis of the expected standards set through educational policy in national curricula in both countries using the SOLO taxonomy. The results suggested that the expectations from English pupils aged 7 and 11 were higher than those from Qatari pupils. This raises important issues related to classroom practice in teaching mathematics.

#### 2 Teaching young children to make explicit their understanding of negative measures and negative relations

Rute Borba and Terezinha Nunes

Oxford Brookes University

A previous study (Borba and Nunes, 1999; 2000) showed that young children solve negative number problems significantly better orally than when requested to provide an explicit representation for negative numbers before solving the problems. This paper presents an intervention study aimed at discussing with young children how they can explicitly represent the understanding they already have about negative numbers. It was also investigated if instruction on one number meaning- measure or relation – is transferred to another. All instructed groups improved in performance in comparison to a control group. Understanding of relation problems was transferred to measure problems but understanding of measures was not transferred to understanding of relations.

#### 3 Student teachers’ understanding of rational numbers

Bill Domoney

Oxford Brookes University

Previous research has shown that secondary school students’ understanding of fractions is dominated by the part-whole concept to the detriment of their understanding of a fraction as a number in its own right. The present paper reports on an investigation into the understanding of intending primary teachers in this area. Four representatives of a cohort of sixty students on a PGCE course specialising in the lower primary age range were asked detailed questions probing their knowledge of fractions. The conclusion was that the part-whole concept dominates. All of the students were familiar with the numerical concept from their work on the PGCE course, but they reverted to the more familiar part-whole ideas in attempting to solve problems.

#### 4 Exploratory talk within collaborative small groups in mathematics

Julie-Ann Edwards and Keith Jones

University of Southampton, UK

This report describes one aspect of a wider research study on exploratory talk within collaborative small groups in secondary mathematics lessons. It outlines students’ views of using collaborative activity to learn mathematics. The fuller research study explores the extent to which exploratory talk occurs in collaborative peer groups in secondary mathematics classrooms.

#### 5 Mental calculation methods used by 11-year-olds in different attainment bands: subtraction questions in the 1987 APU survey

Derek Foxman

Institute of Education, London University

This is a continuation of a reanalysis of data on 11-year-old children’s responses to mental arithmetic questions that were obtained as part of the APU’s (Assessment of Performance Unit) mathematics survey in 1987, using categories developed by Beishuizen in the Netherlands. The sample of 247 children was divided into three bands of attainment as measured independently by their scores on a written test of concepts and skills. Differences in the responses within these bands are illustrated for three subtraction questions.

#### 6 The use of practical and experimental tasks in geometry teaching: a study of textbooks by Godfrey and Siddons

Taro Fujita

University of Southampton

In this paper, I consider the following issues: how were experimental tasks being developed after being introduced in the early 20th Century, and how did people at that time regard the relationship between practical and formal geometry? To address these issues, I focus on the geometry textbooks (1903, 1912) and their revisions by Godfrey and Siddons. As a result of the examination, the following things can be concluded. First, the development of the experimental tasks can be described as a modification of initial introduction of experimental tasks after 1903. Secondly, Godfrey considered that experimental tasks should not be excluded from deductive stages in geometry, because, the ‘geometrical eye’ should be developed through experimental tasks at any stage of geometry.

#### 7 Conceptualising the factors that influence mathematics teaching of a group of newly qualified teachers in Greece

Barbara Georgiadou-Kabouridis

University of Surrey Roehampton

This work inquires into the hypothesis that the newly qualified teachers who have graduated from Greek Departments of Education need support in teaching mathematics in their classrooms. Here we attempt to explore the needs and concerns that a group of newly qualified teachers has about mathematics teaching. These teachers participate in my research project while teaching mathematics in a Greek primary school. The preliminary analysis of part of the data collected in the context of my research project revealed teachers’ beliefs regarding the influence of particular factors in their teaching of mathematics. These factors address issues in the teachers’ university preparation and in the absence of programmes for their induction as teachers in the school environment.

#### 8 Improving teaching methods: how listening to students can make a difference

Peter Hall

Tonbridge Grammar School for Girls, Tonbridge, Kent

There is much work concerning student voice. I have tried to develop questions to ask my students which will improve the teaching and learning within my classroom. This paper sets out a summary of my preliminary work in this area, including some examples of student response.

#### 9 Being emotionally available: teaching mathematics to children with EBSD

Susan Hogan

Open University

This paper presents the results of five years research into my practice, teaching mathematics to children with emotional, behavioural and social difficulties (EBSD). The aim is to explore the possibility of a pedagogic framework for improving the accessibility of mathematics for these children. In order to achieve a balance between caring and learning, the teacher needs to be “emotionally available” to the children while, at the same time, providing opportunities to access mathematics.

#### 10 Mathematics as a constructive enterprise

John Mason

Open University, Milton Keynes

Seeing mathematics as a constructive enterprise could open the possibility for learners to contact a creative aspect of mathematical thinking. Asking learners to construct mathematical objects seems initially to require great creativity, yet every mathematical task learners attempt has a creative element, however obscured by memorised and trained procedures.

The aim of this session was to report on research undertaken with Anne Watson, by offering task-exercises designed to draw attention to aspects of mathematical thinking and mathematical awareness which are sometimes obscured by standard tasks perceived in standard ways. They were intended to draw attention to how people construct mathematical objects which satisfy constraints. They were also intended to highlight the mathematical theme of ‘freedom and constraint’ through experiencing a shift in perception concerning different possible dimensions of variation, and within those, of permissible range of change. This applies both to the mathematical content of the task-exercises, and to their structure.

#### 11 Gender and mathematics revisited

Heather Mendick

Goldsmiths College, University of London

This paper is an outline of the approach I am taking in my PhD research on gender and maths. This issue has been widely debated since the 1980s when second wave feminisms started to have an impact on education, and concern was expressed about the relative underachievement and under-participation of girls in maths. In this paper I situate my work in the context both of the previous research and of the current educational and political climate, in particular the boys’ underachievement debate and the marketisation of education. I argue that looking at students’ choices to do or to reject maths and the nature of student identifications with maths through their narratives is a productive way of moving forward thinking in this area.

#### 12 Symbol sense: Teacher’s and student’s understanding

Sue Pope and Ruth Sharma

University of Surrey Roehampton

During 1999-2000 research into student’s symbol sense following Arcavi’s definition (1994) and using common errors and misconceptions as a basis for a questionnaire was undertaken with high attaining female students aged 15 to 18. The same questionnaire has been used with experienced teachers and trainee secondary teachers.

The findings have some important implications for the teaching of algebra that might help to develop symbol sense in both students and teachers – reading sense into symbols and understanding the nature of symbols, functions and variables and relationships between representations.

#### 13 Developing a framework for analysing children’s learning of mathematics inside and outside the classroom

Dr Alison J Price

Oxford Brookes University

Analysis of classroom data collected while observing children learning addition in Reception and Year One classes identified the need to describe the relationship between children’s understanding and learning of mathematics inside and outside of the mathematics classroom. In this paper I show the development of a framework which allows me to make sense of such relationships, which in turns allows more detailed analysis of the children’s learning. This analysis shows young children making sense of their mathematics learning in and out of the classroom, challenging an understanding of learning as ‘situated’.

#### 14 Safety in numbers? Number crunching and making sense

Tim Rowland

University of Cambridge

Quantitative and qualitative research methods offer us different kinds of insights. Whilst statistical analyses might tell us ‘that’, only by getting ‘inside’ the data can we begin to understand ‘why’. In the session at the BSRLM conference I presented some ‘thats’ which had emerged from statistical analysis of the relationship between subject knowledge variables and teaching performance variables, in the context of primary initial teacher education. One purpose was to invite conjectures about why these relationships might hold (or not).

#### 15 The contributions of non-cartesian philosophies of mathematics to western mathematical education instruction

Barbara J. Savage

Department of Educational Studies, University of Oxford

Cartesian mind/body dualism is foundational in Western mathematics teaching and learning. My dissertation explores the possible existence of non-Cartesian ontologies and how these affect mathematical conceptualization. Are there other ways of perceiving and understanding mathematics? And, if so, how can these be used to restructure mathematics’ teacher education and, therefore, classroom learning?

#### 16 Investigating students’ understanding of locus with Dynamic Geometry

Shinwha Cha and Richard Noss

Institute of Education, University of London

Dynamic Geometry (DG) offers new possibilities for the investigation of loci. The two functions “Locus” and “Trace” provide complementary tools for thinking about loci: “global” in the Euclidean sense, “local” in the pointwise sense of analytic geometry. By using Trace the concept of functional relationship can be explored, an idea which is neglected in the current curriculum. We report on a small experiment with 15 year old students, in which we compared the students’ understandings of the ‘local’ and ‘global’ structure of loci under the different cognitive and cultural influences of working with conventional tools (compass and straight-edge) or DG.

#### 17 Developing a new pedagogy for geometry

Keith Jones and Taro Fujita

University of Southampton, UK

Major improvements in the teaching and learning of geometry will only come, argues a recent report from the Royal Society and Joint Mathematical Council, through the development of a completely new pedagogy for geometry. This report examines existing models of pedagogy for geometry and considers what research might have to contribute to the development of new approaches. New pedagogic approaches for geometry need to give greater emphasis to work in 3-D, incorporate the effective use of computer technology, especially dynamic geometry, and focus on discursive methods of engagement and methods of assessment so that the pressure on pupils is not solely to rote learn.