# BSRLM Proceedings: Vol 23 No 3 November 2003

**Proceedings of the Day Conference held at the University of Birmingham, November 2003**

## Contents

#### 1 Ambiguity in mathematics classroom discourse

Richard Barwell

University of Bristol

Ambiguity is generally seen as problematic in mathematics and this view may also arise in mathematics classrooms. The national numeracy strategy, for example, advises teachers to 'sort out' any ambiguities in students' mathematical language. In this paper, I offer a discursive analysis of a transcript in which ambiguity can be seen as an important resource for making sense of the concept of dimension. This in turn raises questions about the role of ambiguity in doing mathematics.

#### 2 Errors and misconceptions in KS3 'Number'

Chris Bills

University of Central England

In June 2003 Y7 pupils in five schools completed a test based on questions first used in the CSMS and APU studies. The aim was to collect data in order to inform new teachers about pupils' common errors and misconceptions. The data may also be used for purposes of comparison with results collected by teachers in other schools. The study indicates that in this sample a high proportion of pupils gave correct answers but that there were significant numbers of pupils with misconceptions. In a separate study Y7 pupils in one school were asked to perform written calculations in order to identify the strategies used. Accuracy levels were low for multiplication and division and there were a wide variety of non-traditional strategies.

#### 3 Limits - a secondary school view

Bob Burn

University of Exeter (retired)

This paper starts by examining the central part of the definition of a convergent sequence, namely, the use of a pair of inequalities to identify a unique number. This use will be called the 'carpenter's vice'. The secondary school curriculum (including single A level mathematics) is surveyed for those parts in which a limiting process is either explicit or implicit. Signs of the 'carpenter's vice' are sought. While the 'vice' occurs in school mathematics, it is never the centre of attention. This 'vice' is not part of the mathematical tool-kit of a typical beginning undergraduate.

#### 4 Just a bit thick - or is there more to it?

Harry Grainger

University of East Anglia

Whilst research giving voice to student perspectives on Mathematics is fairly widespread and, in conjunction with this, much research has focussed on student difficulties in the learning of Mathematics, very little if any research has asked the average student who "just failed" for their perspective. The work described in this paper details a small project aimed at gaining insight into some of the student perspectives and issues associated with failure to gain a grade C in GCSE. Mathematics. The national average grade at GCSE is a D (in the maintained non-selective sector.)

#### 5 Transformation of functions: learning processes

Athina-Maria Katalifou

School of Education, University of East Anglia

The topic 'transformation of functions' is commonly introduced, at least in the context of secondary Greek education where the study reported here is being conducted, in terms of the effects that the changes of parameters of functions have on their graphs. However, in my experience as a teacher, students, even at the later stage of upper secondary and further education, have substantial difficulties with the subject, especially in the lab courses. The focus of this paper is on students' construction of meanings concerning the structure of mathematical concepts, such as invariancy, while working in an IT-based environment of multiple representations. Two groups of students, engaged with a mathematical activity concerning the concept of transformation of functions and using a newly introduced piece of software were interviewed. Qualitative analysis of the interviews is currently in progress. The research reported here is part of a larger doctoral study.

#### 6 Interpretation of graphs: reading through the data

Carlos Monteiro and Janet Ainley

Institute of Education - University of Warwick

Several studies investigated the interpretation of graphs as pedagogical issue. The studies of Curcio (e.g. Curcio, 1987) presented three levels of students' responses: reading the data, reading between the data and reading beyond the data. Watson's (e.g. 1997) studies suggest a hierarchical schema of classification of interpretation based on three tiers. We presented the idea of Critical Sense in graphing as a skill to analyse data and its interrelations rather than simply accepting the initial impression given by the graph. This paper discusses about convergent and divergent aspects among the authors referred above.

#### 7 'Pre-eighteen students have lost something major': mathematicians on the impact of school mathematics on students' skills, perceptions and attitudes

Elena Nardi, Paola Iannone and Mark J. Cooker

University of East Anglia

It is not surprising to hear and read mathematicians' consternation about the form and content of school mathematics. The high dropout rate at the AS phase (age 17) of A'level mathematics, seems to support the argument that we need to rethink school mathematics. In doing so mathematicians' views may merit more notice. We conducted a series of themed Focus Group interviews with mathematicians from six UK universities. Pre-distributed samples of mathematical problems, typical written student responses, observation protocols, interview transcripts and outlines of relevant bibliography were used to trigger an exploration of pedagogical issues. Here we elaborate the theme "On the impact of school mathematics on students' skills, perceptions and attitudes" that emerged from the data analysis.

#### 8 Some undergraduates' experiences of learning mathematics: (how) can narrative form enable us to create knowledge?

Hilary Povey and Corinne Angier

Sheffield Hallam University

One theme of current research about higher education students of mathematics concerns those who fail. At our institution, some of the entrants are students who have previously failed in mathematics; others come to us with a comparatively weak mathematical background - perhaps through an access course, perhaps through a foundation course, occasionally with only a grade E at Advance Level. Most of these students go on to become confident and effective mathematicians, some even achieving first class honours. WE believe that understanding something of their perceptions of this experience may contribute to the current debate about who succeeds and who fails in higher education mathematics study and why. This paper has two purposes: it begins a discussion about how this success is achieved; it also raises questions about methodology.

#### 9 Written assessments: evidence of professional learning

Stephanie Prestage and Pat Perks

School of Education, University of Birmingham

Written assignments are an expected part of any pre-service professional training for teachers, but do such assignments offer evidence for appropriate professional learning? This paper discusses one assignment, a case study of target setting for an individual pupil, from one cohort of 37 students on a Post-Graduate Certificate of Education course. The analysis of the scripts considers evidence for learner-knowledge, practical wisdom and professional traditions, and any interactions between these from student reflections to identify opportunities for professional learning and a developing teacher-knowledge of assessment in mathematics.

#### 10 Providing feedback to students' assignments

C J Sangwin

School of Mathematics and Statistics, University of Birmingham

Assessment drives learning, and one crucial part of the learning cycle is the feedback given by tutors to students' assignments. Accurately assessing students' work in a consistent and fair way is difficult. Furthermore, writing tailored feedback in formative assessment is very time consuming for staff, and hence expensive. This is a particular problem in higher education, where class sizes are measured in the hundreds. This paper discusses when and how computer algebra, within a computer aided assessment system, can take on this role.

#### 11 Patterns of children's emotional responses to mathematical problem-solving

David Whitebread and Mei-Shiu Chiu

Faculty of Education, University of Cambridge

This study was based upon the responses of 116 Taiwanese primary school children aged nine-ten years to a questionnaire concerning their emotional and motivational responses to mathematical problems. A cluster analysis revealed four distinct patterns of response, which were differentially related to attainment. These patterns of emotional response were subsequently investigated further with a smaller sample of children using a repertory grid technique and an associated interview. The four patterns were found to have differential characteristics and development processes in terms of emotional variables and preferred problem types.

#### 12 Primary teacher trainees' mathematical subject knowledge: the effectiveness of peer tutoring

Patti Barber and Caroline Heal

University of London, Institute of Education

As a matter of course, primary teacher trainees are introduced to theories of learning and to a range of pedagogical approaches as part of their training for teaching. From the outset they are encouraged to become aware of their own learning processes - a focus on meta-cognition - and share insights with their peers, and to do so in a way that might inform their teaching of children.

#### 13 An investigation into the mathematical knowledge of primary teacher trainees

Maria Goulding

University of York

The U.K. government's National Curriculum for initial teacher training and the associated set of assessment standards both include a focus on mathematical subject knowledge. This paper reports collaborative research into the mathematical subject knowledge of primary student teachers by a group of researchers from four English Universities.

#### 14 Minding your Ps and Cs: subjecting knowledge to the practicalities of teaching geometry and probability

Claire Mooney, Mike Fletcher and Keith Jones

University of Southampton

The review of the implementation of the National Numeracy Strategy by Ofsted (Nov. 2003) has highlighted weak subject knowledge as a consistent feature in unsatisfactory teaching. This study looks at the subject knowledge of generalist primary trainees in the areas of geometry and probability and their ability to apply their knowledge to problem solving tasks. The study goes on the raise the question, 'is a profound understanding of fundamental mathematics (PUFM)(Ma, 1999) possible in generalist teachers?'

#### 15 'Filling gaps' or 'jumping hoops': trainee primary teachers' views of a subject knowledge audit in mathematics

Carol Murphy

University of Exeter

Data from one cohort of PGCE trainee primary teachers is used to examine their perceived value of a subject knowledge auditing process in preparation to teach mathematics in primary schools. Questionnaires (n = 96) were used to collect quantitative and qualitative data. Analysis suggests that some trainee teachers see the auditing process as 'filling in gaps' in their subject knowledge, developing their confidence and supporting them in their teaching, whereas other more confident trainee teachers do not see any relevance and may even see the process as 'jumping hoops' to fulfil the requirements of the course.

#### 16 Prospective primary teachers' mathematics subject knowledge: substance and consequence

Tim Rowland, University of Cambridge

Patti Barber, Caroline Heal and Sarah Martyn, Institute of Education, University of London

In the context of the day symposium on prospective primary teachers' mathematics subject knowledge held at the Birmingham day conference, this paper summarises some findings from research at the Institute of Education. These findings have provided part of the backdrop to subsequent collaborative work with researchers in Cambridge and York.

#### 17 The knowledge quartet

Tim Rowland, Peter Huckstep and Anne Thwaites

University of Cambridge

In this paper we describe a framework for the identification and discussion of primary teachers' mathematics content knowledge as evidenced in their teaching. This was the outcome of intensive scrutiny of 24 videotaped lessons. This framework - the 'knowledge quartet' - is then illustrated with reference to a particular lesson taught by one trainee teacher.

#### 18 Recent UK research into prospective primary teachers' mathematics subject knowledge: a response

Jeremy Hodgen

King's College, London

In this paper, I respond and comment on the papers presented at the colloquium, Recent UK Research into Prospective Primary Teachers' Subject Knowledge. I draw out themes arising from the six presentations, in the context of other related research in the UK and elsewhere. I conclude by raising a series of issues and questions for future work in this area.

#### 19 Mathematics education and applied linguistics: Working group report

Richard Barwell

University of Bristol

In this session I invited participants to work on issues concerning discourse analysis in mathematics education research, particularly that of reflexivity. Reflexivity concerns the inter-relationship between the analyst and their analysis. It arises as a methodological issue from questions such as: how do I interpret classroom discourse data? What are these interpretations based on? What kind of claims can I make about my interpretations? To what extent is discourse analysis 'objective'? As the basis for discussion of the issues raised by these questions, participants were invited to work on a transcript of students working on a classroom mathematics task.

#### 20 Induction for secondary mathematics ITE tutors: Working group report

Sue Pope, St. Martin's College, Lancaster

Linda Haggarty, Open University

Keith Jones, University of Southampton

Becoming a mathematics teacher educator with responsibility for the education of trainee teachers is an under-researched area. This report looks at some key issues that effect new mathematics teacher educators, including how the role has changed, whether the emphasis is on 'training' (in consonance with UK Government terminology) or on education (and what the difference might be), about how people learn to become teachers, and about what is known about teacher educators and how people become teacher educators. The report argues that there are huge opportunities for researching all aspects of teacher education.