**Proceedings of the Day Conference held at the University of Cambridge in February 2009**

## Contents

### Research reports

#### 01 Diagrams as interaction: The interpersonal (meta)function of geometrical diagrams

Jehad Alshwaikh

Institute of Education, University of London

Diagrams are part and parcel of mathematics. However, the mainstream among mathematician is prejudiced against the use of diagrams in public. In my PhD study, I consider diagrams as a semiotic mode of representation and communication which enable us to construct mathematical meaning. I suggest a descriptive ‘trifunctional’ the framework that can be used as a tool to analyse the kinds of meanings afforded by diagrams in mathematical discourse. In this paper, only the interpersonal function of the diagrammatic mode is considered with illustrations. In specific, I consider labels, neat-rough diagrams and modality as realisations of that function. Concluding remarks with challenges are presented at the end of the paper.

#### 02 Exploring Children’s Attitudes towards Mathematics

Ben Ashby

University of Warwick

This paper explores the behaviour, attitudes and beliefs of primary school pupils towards mathematics in the classroom and the impact that this may have on their mathematical ability. The study focused on year 3 pupils from a local school, some of whom took part in focus groups towards the end of the project. The children completed short worksheets, which were used to stimulate a guided discussion on what aspects of mathematics the children liked and disliked. The aim of this project was to isolate possible causes of negative attitudes towards mathematics and to discuss what their implications might be.

#### 03 Reflecting on practice in early years’ settings: developing teachers’ understandings of children’s early mathematics

Jenni Back and Marie Joubert

University of Plymouth and University of Bristol

This paper presents some of the findings of the Researching Effective Continuing professional development in Mathematics Education (RECME) Project which was set up to investigate, amongst other things, the role of research in ‘effective’ CPD for teachers of mathematics. The focus in this paper is on a CPD initiative that involved a network of teachers and early years practitioners. The Early Years Foundation Stage (EYFS) covers the care and education of children from birth to five years old and the place of mathematics in these settings have historically been problematic (Gifford 2005; Griffiths 1994; Moyles 1994); we suggest this makes this initiative particularly interesting. During meetings, which involved practitioners from a variety of settings, participants carefully considered children’s mathematical work, especially their spontaneous mathematical graphics (Worthington and Carruthers 2003). This focus led the practitioners to consider ways in which they might support the children’s mathematical development in EYFS settings. We suggest that the professional development of the participants occurred through this collaborative work on researching children’s mathematics in the classroom.

#### 04 Supporting professional development for ICT use in mathematics using the TMEDIA multimedia resource

Bowker, A., Hennessy, S., Dawes, M. and Deaney, R.

University of Cambridge

The T-MEDIA (‘Exploring Teacher Mediation of Subject Learning with ICT: A Multimedia Approach’ (2005-2007). Funded by the UK Economic and Social Research Council (RES000230825)) research project produced an interactive CD-ROM containing a video-based case study of teaching and learning with technology (graphing software, spreadsheet and online games using data projector and laptops) in one secondary mathematics classroom. Designed as a tool for teacher-led, collaborative professional development, the resource aims to stimulate debate rather than present a model of best practice. In the follow-up project outlined here, groups of teachers in 3 schools discussed the pedagogical approaches portrayed, planned a lesson in response, observed each other and reflected together on the outcomes and implications for practice. We present the outcomes of these trials and our development of a ‘toolkit’ that might guide other departments’ use of the resource for professional development.

#### 05 Reflecting on practice in early years’ settings: developing teachers’ understandings of children’s early mathematics

Elizabeth Carruthers and Maulfry Worthington

Redcliffe Children’s Centre, Bristol and Free University, Amsterdam

Local ‘grassroots’ Children’s Mathematics Network groups are initiated and ‘owned’ by teachers and practitioners and they explore and develop their understanding of children’s mathematical graphics (Carruthers and Worthington, 2005; 2006; DCSF, 2008) in their own ways. New research findings reveal the effectiveness of this form of ‘ continuing professional development’ (CPD) and its impact on children’s mathematical thinking (NCETM, 2009). This paper explores the philosophy underpinning these groups, and their inter-connectedness with children’s mathematical graphics.

#### 06 Assessing the digital mathematics curriculum

Tandi Clausen-May

Department for Research in Assessment and Measurement, National Foundation for Educational Research, Slough, UK

In their literature review of e-assessment Ridgeway, McCusker and Pead note a ‘ emerging gap between classroom practices and the assessment system’ (2004, 17-18). This gap threatens to undermine the effective development of ICT in the teaching and learning of mathematics. An examination of currently available onscreen assessments indicates that stand-alone instructional programs, designed to teach a specific set of skills or topics, are relatively well supported by tests composed of constrained item-types which can be computer-administered and marked. On the other hand, tool software such as dynamic geometry or computer algebra packages may be neglected in the classroom because their use does not form a focus construct within the current assessment system. In this paper, some of the constraints on test development that have led to this situation are explored, and ways in which tool software usage might be incorporated into an effective mathematics assessment are considered.

#### 07 Comparing Research into Mental Calculation Strategies in Mathematics Education and Psychology

Ayshea J. Craig

Institute of Education, University of London

This paper argues for the importance of re-examining theoretical assumptions in research into mental calculation strategies and strategic thinking in mathematics education. By contrasting research into strategic thinking in mathematics education with that in cognitive and developmental psychology, three areas are identified where important details of the model of strategic thinking are left unexplored in education research while being dealt with more thoroughly in the psychological literature. The areas identified are: the positing of innate processes; the nature of memory; and the relation between conscious and unconscious mental processes. The status and reliability of introspective reports on mental processes are discussed as an illustration of the potential of research in psychology to further inform mathematics education research in this area.

#### 08 Primary pupils in whole-class mathematical conversation

Thérèse Dooley

University of Cambridge, U.K. and St. Patrick’s College, Dublin

Although plenary sessions are common to mathematics lessons, they are often characterised by traditional approaches that endorse the position of mathematics as a kind of received knowledge and the teacher as sole validator of students’ input. A socio-constructivist view of mathematics calls for a more conversational style of interaction among participants. In this paper, an account will be given of a lesson in which children aged 9 – 10 years calculated the sum of integers from one to one hundred. Particular attention will be paid to one pupil, Anne, and her reassessment of a conjecture that she made early in the lesson. I suggest that particular teacher ‘moves’ facilitated engagement of other students with her idea and that this was one factor that led to her new insight.

#### 09 Socio-constructivist and Socio-cultural Lenses on Collaborative Peer Talk in a Secondary Mathematics Classroom.

Julie-Ann Edwards

University of Southampton, School of Education

This paper uses socio-constructivist and socio-cultural lenses to examine transcripts of pupils’ peer talk recorded while they were undertaking open-ended mathematical tasks in a naturalistic classroom setting. I discuss the two theoretical frames and then present episodes of peer talk from pupils between 12 and 14 years old which demonstrate how a socio-constructivist view of the zone of proximal development is enacted, and how a socio-cultural lens offers a window on social aspects of these established working groups which serve to provide the necessary support to enable all members of the group to access the mathematical knowledge being constructed.

#### 10 Lower secondary school students’ knowledge of fractions

Jeremy Hodgen(a), Dietmar Küchemann(a), Margaret Brown(a) and Robert Coe(b)

(a) King’s College London, (b) University of Durham

In this paper we present some preliminary data from the ESRC funded ICCAMS project, and compare current Key Stage 3 students’ performance on fractions and decimals items with students from 1977. We also present some interview data concerning students’ models of fractions, and in particular their use of diagrams to represent part-whole relationships.

#### 11 Linking Geometry and Algebra: English and Taiwanese Upper Secondary Teachers’ Approaches to the use of GeoGebra

Allison Lu Yu-Wen

Queens’ College, Faculty of Education, University of Cambridge, UK

The idea of the integration of dynamic geometry and computer algebra and the use of open-source software in mathematics teaching underpins new approaches to studying teachers’ thinking and technological artefacts in use. This study opens by reviewing the evolving design of dynamic geometry and computer algebra; teachers’ conceptions and pioneering uses of GeoGebra; and early sketches of GeoGebra mainstream use in teaching practices. This research has investigated English and Taiwanese upper-secondary teachers’ attitudes and practices regarding GeoGebra. More specifically, it has sought to gain an understanding of the teachers’ conceptions of technology and how their pedagogies incorporate dynamic manipulation with GeoGebra into mathematical discourse.

#### 12 A-level mathematics: a qualification for entry to quantitative university courses

Peter Osmon

Department of Education and Professional Studies, King’s College London

Meetings with concerned groups of academics with a particular interest in the mathematics knowledge of students when they arrive at the university are reported. There was general agreement on two immediately tractable issues and the appropriate actions: an A-level mathematics curriculum without options, so as to maximise students’ knowledge in common, and examinations that test understanding and not merely a memory and manipulative skill- so as to encourage deeper learning than at present. The relatively low numbers taking A-level Mathematics is a much tougher issue. The consequences include many university courses in quantitative subjects admitting students without A-level Mathematics, and adapting content and teaching accordingly, so as to survive. The underlying problem is to understand the unpopularity of mathematics after GCSE and what might be done about it.

#### 13 The Validation of a Semantic Model for the Interpretation of Mathematics in an Applied Mathematics Problem

Michael Peters(a) and Ted Graham(b)

(a)Learning Development Centre, Aston University; (b)Centre for Teaching Mathematics, University of Plymouth.

The semantic model proposed by Peters (2008) was developed whilst working with learners of mathematics solving algebraic problems. In order to investigate in more detail the role of the parsing process and its relationship to the lexicon, a different set of questions were devised based on Laurillard’s (2002) work with undergraduate students. These same questions were also given to a set of mathematics tutors so that a comparison could be made between the two groups and to see if their behaviour could be explained using the semantic model. The analysis of these sets of data indeed show the importance of the parsing process, and as predicted by the model, a competent mathematician employs a top-down parsing strategy.

#### 14 Mathematical Knowledge in Teaching: the Nuffield seminar series

Kenneth Ruthven and Tim Rowland

University of Cambridge, UK

Over the last two years, BSRLM members from several universities have contributed to a national seminar series on Mathematical Knowledge in Teaching that has met on six occasions. The final report of the series, supported by the Nuffield Foundation, is now available, and an edited book is in preparation. The seminars have examined current scholarship and research bearing on how teachers’ subject-related knowledge underpins successful mathematics teaching, and on how such knowledge can be assessed and developed. As a consequence, it has been possible to identify areas where there is a need for further research in this important field.

#### 15 ENRICHing Mathematics: Progress in Building a Problem-Solving Community

Cathy Smith and Jennifer Piggott

Homerton College, Cambridge and NRICH, University of Cambridge

The SHINE enriching mathematics project recruited secondary school students in two socioeconomically deprived London boroughs for out-of-school workshops over the course of a year. Students worked collaboratively on open maths tasks, with discussion guided by NRICH leaders and participating school teachers. Here we outline two aspects of the project evaluation: how we analysed progress in collaborative class work and how the students described what they had learnt. Students found Shine maths enjoyable, different and more challenging than school maths. Their teachers observed improvement in problem-solving behaviours. The model of a maths talk learning community offered ways to categorise changing classroom behaviours and helped to identify tensions and effective practices of classroom management.

#### 16 Developing the Ability to Respond to the Unexpected

Fay Turner

Faculty of Education, University of Cambridge

In this paper I present some findings from a four-year study into the development of content knowledge in beginning teachers using the Knowledge Quartet as a framework for reflection and discussion on the mathematical content of teaching. Findings which relate to the participants’ ability to react to pupils’ unexpected responses are discussed. Data from three case studies suggest that the framework helped participants to consider their unplanned actions when teaching mathematics. There was also evidence that over the course of the study the participants become more able to act contingently in relation to the mathematical content of their teaching.

### Working group report

#### 17 BSRLM Geometry working group: Establishing a professional development network to support teachers using dynamic mathematics software GeoGebra

Keith Jones, Zsolt Lavicza, Markus Hohenwarter, Allison Lu, Mark Dawes, Alison Parish, Michael Borcherds

University of Southampton, UK; University of Cambridge, UK; Florida State University, USA; Comberton Village College, UK; University of Warwick, UK; Queen Mary’s Grammar School, UK

The embedding of technology into mathematics teaching is known to be a complex process. GeoGebra, an open-source dynamic mathematics software that incorporates geometry and algebra into a single package, is proving popular with teachers – yet solely having access to such technology can be insufficient for the successful integration of technology into teaching. This paper reports on aspects of an NCETM-funded project that involved nine experienced teachers collaborating in developing ways of providing professional development and support for other teachers across England in the use of GeoGebra in teaching mathematics. The participating teachers tried various approaches to better integrate the use of GeoGebra into the mathematics curriculum (especially in geometry), and they designed and led professional development workshops for other teachers. As a result, the project initiated a core group which has started to be a source of support and professional development for other teachers of mathematics in the use of GeoGebra.