**Proceedings of the Day Conference held at the University of Bristol, June 2006.**

## Contents

#### 1 Probing understanding through example construction: the case of integration

Shafia Abdul Rahman, Open University

Activities that reveal something about learners as they become aware of aspects of mathematical concepts that were not previously focussed upon can be very useful for the learners themselves as well as for teachers and researchers. In this paper I consider example construction tasks as research probes to reveal learners’ awareness of the concept of integration. Forty students studying A-level, engineering, mathematics and education have been invited to construct relevant mathematicalobjects meeting specified constraints. What learners choose to change in mathematical examples reveals the dynamics and depth of their awareness and acts to promote and enrich their appreciation of the concept.

#### 2 Using discursive psychology in research in mathematics classrooms

Richard Barwell, University of Bristol

Discursive psychology has emerged asan anti-cognitivist, anti-realist, anti- structuralist perspective on cognition. This approach includes both a theorisation of the role of discursive practice in thinking, and a methodological approach to theinvestigation of psychological questions. Discursive psychology has informed much of my research into the role of multilingualism in the teaching and learning of mathematics. In this paper, I reflect on what the adoption of this perspective has allowed me to see, as well as what it may have obscured.

#### 3 Pedagogical sensitivity and procedural thinking: an uneasy relationship?

Irene Biza, Thedossios Zachariades, University of Athens (Greece)

Elena Nardi, University of East Anglia

We examined the relationship between subject-matter knowledge and pedagogical content knowledge of 53 in-service mathematics teachers in the context of their written responses to a question that involved: solving the equation |x|+|x-1|=0, examining a flawed student solution and providing feedback to the student. Here we focus on a group of scripts characterised by pedagogical sensitivity but constrained mathematically (substantively and meta-cognitively). Through examination ofexamples from the data we demonstrate, and discuss implications of, some of these constraints: insistence on standard procedural methods, inappropriate contextualisation of otherwise commendable pedagogical practices and inadequate reflection on student thinking.

#### 4 Structuring students’ awareness of generality in whole class discussion

Helen Drury, Open University

While leading whole class conversations, as a teacher, I aim to remain aware of students’ powers to appreciate and express generality. In this paper I focus on a whole class conversation that took place during a game of ‘algebra bingo’. I consider the various types of generalities and examine the extent to which students in the class might be aware of each generality.

#### 5 Primary trainee teachers’ knowledge of parallelograms

Taro Fujita, University of Glasgow

Keith Jones, University of Southampton

Considerable research has indicated that amongst the factors which make the most significant contribution to high student achievement in mathematics is secure subjectknowledge on the part of the teacher as this underpins an approach to mathematics in which topics are seen as part of a coherent set of related ideas, with clear progression and links to previous and future learning. This paper reports part of the findings from a study of trainee teachers’ knowledge of basic geometrical figures, in particular focusing on what knowledge they have of parallelograms and how they use this knowledge to solve geometrical problems. The findings indicate that only a minority of trainee primary teacher have a fully sophisticated knowledge of parallelograms and of how to use the properties of parallelograms to solve relevant problems.

#### 6 Observing subject knowledge in action: characteristics of lesson observation feedback given to trainees

Andrew Harris, Canterbury Christ Church University

This paper outlines the main findings of a small-scale study in which the relative frequency of comments about elements of mathematics teaching found in written feedback given to primary trainee teachers by school-based mentors, university mathematics tutors and university tutors of other subjects was investigated. Furtherissues are raised regarding the awareness of some observer groups about’contingency’ elements of trainees’ mathematics teaching.

#### 7 Relationships with/in primary mathematics: identity, emotion and professional development

Jeremy Hodgen and Mike Askew, King’s College London

In this paper we explore primary teachers’ emotional relationships with mathematics. Drawing on the concept of identity as a “defended self”, we describe and analyse thecase of one primary teacher. We argue that emotion as both an individual and a social element. Finally, we consider the implications for teacher education. This discussion is based on an analysis of our interactions with a primary teacher, Ursula, as a researcher (Jeremy) and as a Higher Education teacher (Mike) as she participated in professional development in mathematics. At an earlier day conference, one of us discussed how Ursula’s desire to be a mathematics teacher was a crucial factor in her professional learning (Hodgen, 2004). Here, we focus on how she became drawn to mathematics despite her initial avoidance of the subject and relate this to the social aspects of learning. Our central theme is a narrative that Ursula told and re-told during these interviews concerning Ursula’s relationships with school mathematics and mathematics teachers, one of whom was Mike.

#### 8 Is it ever appropriate to judge an argument by its author?

Matthew Inglis and Juan Pablo Mejia-Ramos, University of Warwick

There is a widespread belief in the mathematics education community that students should be encouraged to avoid basing their level of conviction in mathematical arguments on the authority of the argument’s source. In this paper we report an experiment which investigated the role that authority plays in the argument evaluation strategies of undergraduate students and research active mathematicians. Our data show that both groups were more persuaded by an argument if it came from an authority figure. The implications of this finding are discussed. It is argued that the role of authority in mathematical argumentation – both in terms of actualbehaviour and of normative behaviour – requires deeper scrutiny.

#### 9 Learners’ shifting perceptions of randomness

Peter Johnston-Wilder, University of Warwick

In clinical interviews, learners were invited to talk about their experiences of making sense of the emerging sequence of outcomes from repeated trials using differentgenerators, some of which were biased. Analysis of the interviews revealed distinct ways of viewing the phenomena represented by the interview tasks. Drawing upon the local and global meanings of randomness identified by Pratt (1998), learners were found to shift rapidly between local and global perspectives. In this paper, data from a single interview is presented to illustrate the shifting perspectives.

#### 10 Integer instruction: an experimental comparison

Andreas Koukkoufis and Julian Williams, University of Manchester

Two versions of Linchevski & Williams’ ‘dice games’ method for integer addition and subtraction were experimentally contrasted. We describe the methods, present some statistical analyses and discuss the findings. We finally suggest more attention to situated meanings and intuitions in realistic contexts is needed.

#### 11 Subtraction of fractions through the eyes and ears of fifth grade modellers

Andreas O. Kyriakides, Open University

The central question addressed in this paper concerns the ways in which modelling activities ground the conditions for a group of fifth graders to experience a progression in their awareness of subtraction of fractions. The teacher’s narratives along with students’ written work and transcripts of audio-taped class discussions constitute the primary data source for analysis. My attention is particularly drawn toa close examination of a teaching episode that appears to serve my research query. The study documents strong evidence that students could refine their fractional reasoning when exposed to a learning environment that sensitises them to detect problematicity through confronting impasses publicly, questioning themselves and peers, conjecturing and welcoming broken expectation.

#### 12 Developing on-line questionnaires for mathematicians

Zsolt Lavicza, University of Cambridge

In this paper I will discuss the development of an on-line questionnaire that I have designed for my dissertation research. By employing this questionnaire, I aim to gauge university mathematicians’ use of Computer Algebra Systems (CAS) in undergraduate mathematics courses and to understand their thinking about the advantages and disadvantages of CAS use in university-level teaching. The development of the questionnaire is based on an interview study with mathematicians that I conducted in 2005. Thus, I integrate issues that emerged from this earlier study with concerns described in the mathematics education literature. The complexity of the questionnaire design is complicated by the fact that I am examining mathematicians in three countries, Hungary, the United Kingdom, and the United States, which requires me to consider different aspects of international comparative research and differences in cultures. I will report on the difficulties that I encountered during the design process of this questionnaire and highlight issues which researchers must pay attention to when they decide to use on-line questionnaires.

#### 13 The company of words: using concordances to develop language in the maths classroom

Frank Monaghan, Open University

Some thirty years ago, the linguist Michael Haliday argued that: The core of the difficulty in the mathematics classroom is that the teacher often understands and takes for granted the whole register of mathematics, and thinks only ofthe mathematical aspects of these items, whereas for the learner they may also be unfamiliar language – they are ‘peculiar’ English. It is therefore desirable that the mathematics teacher should be aware of the register of mathematics as a sub-set of English . To this end, mathematics educators and . English Language teachers should collaborate in the production of guidelines, illustrative descriptions and teaching materials concerned with this problem. (Halliday, 1975: 121) Whilst numerous researchers have tackled the implications of this and sought to explain the ‘peculiarities’ of maths texts (Pimm, Morgan, Sfard, Barwell, Leung, Street, to name but a few) it remains the case that responses have largely been at a very local level with individual teachers or teams of teachers seeking to address the issue in their particular setting rather than there having been a more systematic approach to curriculum design informed by linguistic analysis. In this paper I will tryto outline one possible way forward using a corpus of mathematical materials (takenfrom the SMILE scheme), interrogated using concordancing software, more on both of which later, but first I would like to give an illustration of how we can take things for granted about language when we rely on our intuitions. This based on a study by Tversky and Kahneman (1973).

#### 14 Identity and undergraduate mathematics: a discussion of Baron-Cohen’s systemiser/empathiser dichotomy with regard to mathematics undergraduates and associated gender issues

Melissa Rodd, Institute of Education and Margaret Brown, King’s College London

Continuing previous work on mathematics undergraduates’ identities, we present and critique some of Simon Baron-Cohen’s ideas on ‘systemizing’ and ’empathizing’ and how these notions relate to gender. We are concerned that his association of systemizing with ‘male brains’ could be detrimental to female participation in mathematics.

#### 15 A cultural-historical approach to teaching geometry Part 2: the results of a pilot study

Stuart Rowlands, University of Plymouth

In the last proceedings (Rowlands and Carson, 2006) we discussed a curriculum initiative that aims to ‘bring to life’ the major primary events in the history of Greekgeometry. In particular, the combination of the intellectual act of abstraction and the possibility of formalised, logical proof were discussed. In Part 2 the results of a pilot study to see whether year 9 ‘mixed ability’ and year 10 ‘gifted and talented’ students can be meaningfully engaged with these two primary events are discussed.

#### 16 Mechanics should be integral to secondary school mathematics

Stuart Rowlands, University of Plymouth

Mechanics has never been the most popular subject in A-level mathematics, eitherwith the students, the teachers or educators. The ‘innovative’ attempts to popularise mechanics appear to have failed and it is conceivable that the subject will be dropped from the A-level syllabus within the next two decades. This article argues the importance of mechanics and why it should be integral to secondary school mathematics. Mechanics is the exemplar of mathematical modelling, is the logical point of entry for the enculturation into scientific thinking and provides the means to develop an understanding of the relationship between mathematics, the theoretical objects of science and the way science and mathematics speak of the world.

#### 17 The long-term effects from the use of CAME (Cognitive Acceleration in Mathematics Education), some effects from the use of the same principles in Y1&2, and the maths teaching of the future

Michael Shayer and Mundher Adhami, Kings College London

The CAME[1] project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive developmentof students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention. The RCPCM project[2], an intervention for the first two years of Primary education, doubled the proportion of 7 year-olds at the mature concrete level to 40%, with a mean effect-size of 0.38 S.D. on Key Stage 1 Maths. Yet, instead of the intervention intention, it is now suggested that a better view is to regard CAME as a constructive criticism of normal instructional teaching, with implications for the role of mathematics teachers and university staff in future professional development.

#### 18 Why we are still teaching: a small-scale study of mathematics teacher retention

Jan Winter and Laurinda Brown, University of Bristol

This paper reports on a small-scale study of ex-students from a PGCE course and the progress of their teaching careers. Contact was made with ex-students from seven cohorts, going back as far as 1976 and respondents were asked to complete a questionnaire. It explored their reasons for either remaining in teaching (e.g. working with children) or leaving (e.g. workload). This paper suggests areas in which progress is being made in retention (e.g. pay and conditions).

#### 19 Geometry Working Group: Informing the pedagogy for geometry: learning from teaching approaches in China and Japan

Keith Jones, University of Southampton and Taro Fujita, University of Glasgow

An authoritative report into the teaching and learning of geometry argued, amongstother things, that the most significant contribution to improvements in geometry teaching are to be made by the development of good models of pedagogy, supported by carefully designed activities and resources. This meeting of the Geometry Working Group provided an opportunity to consider approaches to the teaching of geometry developed in China and Japan and to review what research might have to contributed to developing new pedagogic approaches.