**Proceedings of the Day Conference held at the University of Durham, February 2002 and University of Bristol, May 2002**

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## Contents Vol 22 No 1

#### 1 Attending to the rhetoric of attention to think about mathematical thinking

Richard Barwell

University of Bristol

The act of attending has interested a number of researchers in mathematics education and has emerged as a key element of my own approach to the analysis of students’ mathematical interaction. This approach draws on ideas from discursive psychology and conversation analysis and sees talk primarily as a form of social action. It is a principle of conversation analysis that what we say reflects what we are attending to in that moment. Discursive psychology suggests that the rhetorical aspect of our words, that is the way we use words, reflects what we are doing. By looking at what students attend to, and then at how this attention is rhetorically deployed, insights emerge into how students think together.

#### 2 Making sense of mathematical language in a primary classroom

Christine Bold

Open University and Edge Hill College

From an analysis of classroom dialogue in three British mathematics lessons and eight structured interventions with Year 5 children (ages nine and ten), my research highlights several issues affecting the shared understanding of meaning. Mathematical language is far from precise in meaning, and communicative processes are often tentative and transient according to the particular situational context. In particular, references to everyday contexts do not necessarily help the communicative process when teaching and learning the concept of probability in the primary school. Few life events have certainty, or an ‘even chance’, gestures and pictorial images are also influential when trying to communicate one’s understanding of meaning.

#### 3 Teacher identity in initial teacher training

Tony Brown

Manchester Metropolitan University

This paper addresses issues of identity among trainee teachers as they progress through college training in to their first year of teaching mathematics in primary schools. We consider how such human beings construct themselves as subjects against the backdrop of the multiple social demands that they encounter during the training process. We examine how we might conceive of the trainees confronting government policy instruments, such as Ofsted inspections and the National Numeracy Strategy, and then mediating their conceptions of themselves as teachers through the demands these instruments entail. We conclude by suggesting that participation in the institutions of teaching results in the production of languages that serve to conceal difficulties encountered in reconciling these multiple demands with each other.

#### 4 A model of students’ simplification of trigonometric expressions

Ali Delice

Centre for Studies in Science and Mathematics Education

University of Leeds

I present an operational model of how students’ simplify trigonometric expressions. The model has three main components: recognising, recalling and doing. This paper describes the interaction between these components and links this model to other models of doing mathematics.

#### 5 Do teachers understand the theory behind CAME?

Maria Goulding

Department of Educational Studies, University of York

The Cognitive Acceleration in Mathematics Education (CAME) project aims to improve children’s thinking in mathematics and to enhance their achievement. The learning theory underpinning it is described as ‘Piaget and Vygotsky in tandem’, together with a theory of professional development involving networks of teachers and direct modelling of lessons by trainers. Twenty-one teachers involved in the project were interviewed to investigate their understanding of the theory and their explanations for learning gains. They stressed some aspects of the learning theory much more than others and identified dispositional changes as the most likely explanation for performance gains.

#### 6 Discussion and developing mental strategies when solving computer generated problems

Michelle Parker

University of Newcastle

One of the central ideas of Realistic Mathematics Education (RME) is that children develop mathematical knowledge and understanding by comparing their own solutions to problems, with those of other children. The Dutch software, which was used in this study. allowed children to work in pairs and use a variety of mental strategies to solve simple problems. Children were encouraged to explain their strategies whilst working at the computer, and during a teacher-led group discussion. This paper reports on the observed variety of solution strategies and the extel1l10 which children adopted and developed their solution strategies following the group discussion.

#### 7 Using small software on graphic calculators

Sue Pope

University of Surrey Roehampton

During Autumn 2000 and Spring 2001 two teachers in each of four secondary schools (two 11-16, two 11-18) were provided with class sets of graphic calculators, a PC link and a view screen. The graphic calculators had been programmed with some small software applications intended to support the learning of mathematics in Key Stages 3 and 4. Training and support was offered to the teachers through the SMILE team. At a series of four one day meetings teachers had the opportunity to debate issues, share experiences and develop teaching approaches. This report describes the range of teaching approaches used, the impact on children’s learning of mathematics and discusses the issues that arose. This work was funded by BeCTa as part of their Handheld Technology Project.

#### 8 A scale for assessing probabilistic thinking and the representativeness tendency

Thekla Afantiti-Lamprianou and Julian Williams

University of Manchester

We report a study of children’s probability conceptions and misconceptions using a diagnostic instrument developed from the literature on the representativeness heuristic. Rasch measurement methodology was used to develop the 13-item open response instrument with a sample (N=116) of 12-15 year olds. The result is that a hierarchy of responses at two levels is confirmed for this sample, and a third level is hypothesised. Each level is characterised by the ability to overcome typical ‘representativeness’ effects, namely ‘recency’. ‘random-similarity’ (at level 1), ‘baseĀrate frequency’ and ‘sample size’ (at level 2-3). The validity of our interpretations was tested and some anomalies were identified through clinical interviews with children making the errors (n=8). Another Rasch ability measure, which we named the ‘representativeness tendency’, is constructed from 11 multiple-choice errors.

#### 9 Investigating children’s thinking on the topic of ‘ratio’

Christina Misailidou and Julian Williams

University of Manchester

In this paper we present three items from a diagnostic instrument which was constructed with the aim to reveal children’s proportional reasoning. The items for this instrument were selected having as criterion their diagnostic value, that is, their potential as a compilation to provoke a variety of responses from pupils. The instrument contains two versions. One consists of the items presented mainly as written statements and the other consists of the same items, accompanied with models thought to facilitate children’s thinking. We present data from Year 6,78 and 9 children and we comment on the diagnostic value and the models’ influence on children’s strategies.

#### 10 Argumentation and teachers’ role: the general and the particular

Julian Wittiams and Julie Ryan

University of Manchester

Following our previous work on diagnostic assessment (also pursued here in papers by the Manchester colleagues) and argumentation space. we describe our current work with teachers’ inquiry groups working on argumentation practices in mathematics classrooms. A recurring theme in our analyses is the general and the particular in relation to teachers’ pedagogical strategies. The former is appears to be transferable. pedagogical practice. while the latter relates to spectfic content. We anticipate that teachers professional development will require the development of tools. accounts and dialogue appropriate to each.

#### 11 Diagnostic tools and pedagogical content knowledge: a response

Tim Rowland

University of Cambridge

This paper is a critical response to four papers given in a symposium entitled Diagnostic Tools and Pedagogical Content Knowledge at the Durham Conference. The papers (most of which are summarised in these proceedings) were given by Thekla Afantiti, Constantia Hadjidemetriou. Christina Misailidou, Julie Ryan and Julian Williams, all of the University of Manchester.

## Contents Vol 22 No 2

#### 12 “x2 or 2x?”: Experiments on geometric algebra with dynamic geometry

Shinwha Cha and Phil11p Kent

Institute of Education, University of London

“Geometric algebra” is a set of classical geometrical techniques for defining and manipulating variable quantities. We propose that using geometric algebra may be an effective learning path for secondary students to approach analytic geometry, avoiding the “rush to symbolism” and divorce from geometry that frequently occurs. Geometric algebra is sympathetic with students’ geometrical intuitions and, in the context of dynamic geometry software, it is a semi-abstract means to construct and manipulate variables, equations and inequalities without algebraic symbolism. Dragging of geometric constructions becomes a means of solving “equations “, and this keys in with students’ familiarity with trial-and-error techniques in the classroom. We describe some experiments carried out with I5-year old students which involved geometric constructions for squares and square roots, based on Thales ‘ Theorem. In the former case, the students experienced a dis-equilibrium concerning the relative magnitudes of x2 and 2x, which they were able to resolve by conjecturing and dragging.

#### 13 Constraints and freedoms in mathematics and mathematics teaching: descriptions of practice in year 8 secondary mathematics classrooms

Alf Coles and Laurinda Brown

Kingsfield School, South Gloucestershire and University of Bristol, Graduate School of Education, UK

We are working on a small-scale research project tracking the development of a group of four mathematics teachers who are teaching parallel year 8 groups in mixed-ability classes in the same school. We are currently explori1]g. through this project. the view that mathematics. mathematics teaching and mathematics research can be analysed in terms of constraints and freedoms. After setting up what we mean by ‘constraints and freedoms’ we analyse extracts from two transcripts of lessons. Finally we discuss the practice on the project by offering responses to the points raised in discussion during a session at the BSRLM conference in Bristol where we looked at extended versions of the two transcripts included here.

#### 14 The design of geometry teaching: learning from the geometry textbooks of Godfrey and Siddons

Taro Fuijita and Keith Jones

Centre for Research in Mathematics Education, University of Southampton, UK

Deciding how to teach geometry remains a demanding task with one of major arguments being about how to combine the intuitive and deductive aspects of geometry into an effective teaching design. In order to try to obtain an insight into tackling this issue, this paper reports 0/1 analysis of innovative geometry textbooks which were published in the early part of the 20th Century, a time when significant efforts were being made to improve the teaching and learning of geometry. The analysis suggests that the notion of the geometrical eye, the ability to see geometrical properties detach themselves from a figure, might be a potent tool for building effectively on geometrical intuition so as to provide a bridge into deductive geometry.

#### 15 Children using trial and improvement methods:examples from key stage two tests

Jenny Houssart, Hilary Evens

Centre for Mathematics Education, Open University

We consider the use of ‘trial and improvement’ methods by 11 year olds, drawing upon children’s responses 10 two questions in the 2001 Key stage 2 National Curriculum tests. For both questions, the first step was to consider which solutions count as ‘trial and improvement’. 1n doing this we also identified another category, ‘spot and check’ which differed from ‘trial and improvement’ in that only one (correct) solution was offered and tested. The next step was 10 identify different subĀ¬categories within trial and improve for each question. Children using such methods generally used a relatively small number of trials and reached the correct answer. Where this was not the case, this was nearly always because of errors in calculation.

#### 16 Improving children’s flexibility in elementary arithmetic

Hazel Howat

Mathematics Education Research Centre, Warwick University

The first phase of the numeracy lesson aims to rehearse, develop and sharpen mental skills to give children flexibility with the way they deal with numbers. However, it appears that many children have not responded to the initiative in the way desired. Such children do not seem to acquire the benefits of mental approaches but continue to focus on perceptual or figural representations of number and the procedures associated with them. This paper reports on an investigation that presented these children with an alternative focus – the arithmetical symbol. A constructivist teaching programme was designed to examine whether such a change in focus may improve the sophistication of the approach children use for mental arithmetic.

#### 17 Students’ problem solving processes: two distinct categories

Maria G. de Hoyos

University of Warwick

This study looks at the problem solving ideas and experiences of a group of students that took part in a problem-solving course. The aim is to identify the essential issues behind students’ problem solving processes and to build a model of the situation. This paper discusses the two problem solving categories that emerged from the analysis of the data. These categories were labelled ‘panning for gold’ and ‘building a solution’ and it is argued that they represent two different approaches to solving mathematical problems. The data consisted of the ‘scripts’ that students created for their problem solving processes. These ‘scripts’ are written descriptions of the process students went through for solving a non-routine mathematical problem and were created by the students during the course.

#### 18 My practices as a teacher educator: a discussion of pre- and in-service teacher responses to sessions on the use of cultural objects as teaching aids

Janet Kaahwa

Makerere University

This paper is based on action research that I conducted into my practice of training mathematics teachers in use of teaching aids in teaching in particular, the use of cultural objects. The aim was to discover if there was need to improve my practice and how to do so. Teachers at pre- and in-service levels were involved (two cohorts of student teachers and two practising teachers). Two action research cycles were completed with results showing opportunities for improvement in my practice. This paper offers a discussion of results from cycle one that show variations in the responses to the sessions and the classroom practices of the pre- and in- service teachers

#### 19 The quality of students’ reasons for the steps in a geometric calculation

Dietmar Kuchemann and Celia Hoyles

Institute of Education, University of London

We report on the variety of reasons that students give for each step in a three-step geometric calculation. Where these reasons are non-standard this may be partly due to a lack of familiarity with the appropriate conventions, but it may also indicate that students need to articulate such reasons as scaffolding for their explanations.

#### 20 Students, bicycles and the quirks of symbolic language in mathematical logic

Elena Nardi and Paola Iannone

School of Education and School of Mathematics, University of East Anglia

For many learners of mathematics the symbiosis of the language of mathematical logic with the ordinary language of ‘real-life’ problems is an uncomfortable one. Here we draw on the written responses of sixteen Year 1 mathematics undergraduates to a question that asked them to use quantifiers in order to rewrite the sentence “In each year group in the university there are at least ten students with bicycles” and discuss: the question setter’s intentions and proposed preferred solution; typical and untypical student responses; and, alternatives on how the question setter’s intentions could perhaps be more efficiently realised. The discussion aims to demonstrate that students need to be exposed to the necessity and effectiveness of mathematical language, not simply to its authority.

#### 21 The interactive whiteboard: implications for software design and use

Pat Perks

School of Education, University of Birmingham

Changes to technology offer different ways of interacting with software, particularly for whole class teaching in mathematics. This paper looks at some of the issues for designing (and using) software or software files with one computer on its own, with a data projector and with an interactive whiteboard.

#### 22 Y9 booster kit: mathematics, mutton dressed as lamb’!

Pat Perks and Stephanie Prestage

School of Education, University of Birmingham

In this paper we discuss the pedagogy implicit within the Y9 booster kit: mathematics. Within the document, supposedly, are twelve lessons, designed for use in revision classes with Year 9 (13-14 year olds) pupils, to ‘boost’ their performance in national tests. Detailed analysis of one of the lessons reveals a misunderstanding of the relationship between teaching and learning and between learning and ‘revision ‘. The ‘kit’ highlights the inadequacy of distance education materials.

#### 23 Vygotsky and the ZPD: have we got it right?

Stuart Rowlands

Centre for Teaching Mathematics, University of Plymouth

For Julia Gillen (2000), the present day distortions in the interpretation of the ZP D are a result of deficiencies of earlier translations. Although Gillen is right with respect to the present day distortions she may be wrong in identifying original translations as the source. Based on Rowlands (2000), it will be argued that the distortions are mostly likely the result of ‘academic Chinese whispers’ whereby secondary sources have created an interpretation of Vygotsky that is far removed from what he was actually writing. It will also be argued that, far from being unimportant, Vygotsky’s ZPD is the culmination of a long struggle to develop a scientific method in psychology and that the ZPD can best be seen as the ‘method of double stimulation’ adapted to the teaching of formal academic subjects.

#### 24 Design initiatives for learning: ICT and geometry in the primary school

Rosamund Sutherland, Steve Godwin and Federica Olivero

Graduate School of Education, University of Bristol

Pat Peel, Easton CE V A Primary School, Bristol

This paper draws on an ongoing project whose overall aim is to examine the ways in which new technologies can be used in educational settings to enhance learning. In particular it discusses the process of developing a design initiative for primary pupils to learn about the properties of polygons. The process draws explicitly on the role of the teacher within the context of exploiting the potential of a dynamic geometry environment. Design evolves in a contingent way related to pupils’ developing conceptions and the purpose of the dynamic geometry environment is to help pupils pay attention to the invariant properties of particular quadrilaterals.

#### 25 Metaphors and cultural models bridge the gaps in communication between workers and outsiders

Julian Williams

University of Manchester

This paper develops the argument in Williams and Wake (2002) about the nature of the resources that can support workers and outsiders in ‘bridging the gap’ between mathematics practised in the workplace and College mathematics. We noticed in our Maths at Work project that when workers try to explain their mathematical practices to outsiders, breakdowns arise when crystallised or reified mathematics has to be justified. We present here briefly some examples in which workers spontaneously used metaphors and models which facilitate explanation and communication. We analyse these resources, drawing on Lakoff and Johnson (1999) and Lakoff and Nunes (2000) in substance and approach.

#### 26 Proceedings of the joint meeting of the Primary Working Group and Geometry Working Group of BSRLM, Bristol, 18th May 2002

The Primary Working Group welcomed an opportunity to meet with the Geometry Working Group in an effort to raise the profile of geometry and research in the earlier years of schooling. It was noted that with the introduction of the National Numeracy Strategy, number work has tended to dominate classroom practices in mathematics and that a greater emphasis on the wider curriculum is an important focus for the future.

#### 27 Mathematics education and applied linguistics Working Group report

Richard Barwell, University of Bristol

Constant Leung, Kings College London

Candia Morgan, Institute of Education

Brian Street, Kings College London

There is a long-standing interest within the mathematics education community in the language dimension of teaching and learning mathematics. The four organisers of this working group have been exploring this aspect of mathematics education from our different perspectives as mathematics educators and applied linguists (see Barwell, Leung, Morgan & Street, 2001, forthcoming). We have found that our different perspectives illuminate different aspects of mathematics classroom interaction. We are then able to explore how these different aspects inter-relate. Our collaboration leads us I to feel that wider interaction between mathematics education and applied linguistics would be fruitful for both communities. Our aim in offering this working group I session was therefore to create an opportunity for such an interaction to develop.

#### 28 British Society for Research into Learning Mathematics joint meeting of Geometry Working Group and Primary Working Group

Convenor: Keith Jones, University of Southampton, UK

Convenors: Julia Anghileri, University of Cambridge, and Tony Harries, University of Durham

Trainee primary teachers’ knowledge of geometry for teaching.

A report based on the meeting at the University of Bristol, 18 May 2002 by Keith Jones and Claire Mooney University of Southampton, UK and Tony Harries, University of Durham, UK

One outcome of the implementation of the (UK) National Numeracy Strategy at the primary school level is the privileging of the teaching and learning of number. Yet, as the recent Royal Society report on geometry stresses, it is important to begin the developing of spatial thinking and reasoning at this level. This report reviews what trainee primary teachers might need to know about geometry in order to teach the geometry component of the mathematics curriculum effectively and confidently. Some initial findings are given from research which suggests that, in the UK, geometry is the area of mathematics in which trainees perform most poorly in initial baseline tests and have the least confidence to teach. Hence it is the area in which trainees need to make most progress if they are to gain qualified teacher status.