**Proceedings of the Day Conference held at Newcastle University on 13 Nov 2010**

## Contents

#### 01 Investigating the development of prospective mathematics teachers’ technological pedagogical content knowledge with regard to student difficulties: the case of radian concept

Hatice Akkoç, Marmara University, Turkey

This study investigates how two prospective mathematics teachers integrate technology into their lessons to address student difficulties. Prospective teachers took part in a teacher preparation program which aims to develop technological pedagogical content knowledge (TPCK). As part of this program, prospective teachers participated in workshops which aimed to develop TPCK of derivative and function concepts. Following these workshops, prospective teachers conducted their own workshops during which they discussed student difficulties with various mathematical concepts such as limit, continuity, definite integral, probability and radian with their peers. They also discussed how these difficulties could be addressed during a lesson using technological tools. This paper particularly focuses on radian concept and investigates the development of two prospective mathematics teachers throughout the course in integrating technology into their lessons to address student difficulties with radian concept.

#### 02 Investigating the impact of a developmental research project on teachers’ teaching practice: Listening to mathematics teachers’ reflections

Claire Vaugelade Berg, University of Agder, Kristiansand, Norway

The impact of participating in a research project on mathematics teachers’ teaching practice is analyzed. Results presented in this article uncover the influence of the research project on many aspects of the teachers’ practice and contribute to the development of a better understanding of the complexity of the teaching practice.

#### 03 Calculating: What can Year 5 children do?

Alison Borthwick and Micky Harcourt-Heath, University of East Anglia

In 2006 we collected and analysed answers from a Year 5 Qualifications and Curriculum Authority (QCA) test paper to explore the range of calculation strategies used by a sample of children. Two years later in 2008 we repeated this research using the same questions with a new cohort of Y5 children from the same group of schools. In 2010 we carried out a third set of research. This paper reports on the findings from the 2010 data and examines the range of strategies used by the children. We conclude by considering if we are clear about which strategies lead children to success.

#### 04 Measuring Students’ Persistence on Unfamiliar Mathematical Tasks

Sinead Breen, CASTeL, St Patrick’s College, Dublin

Joan Cleary, Institute of Technology, Tralee

Ann O’Shea, National University of Ireland, Maynooth

182 students responded to a number of Likert-scale items regarding their persistence on mathematical tasks. Rasch analysis was then used to construct a measure of persistence from their responses and to assign persistence scores to each student. The same students, all of whom were enrolled in the first year of a third-level programme, also completed a 30-minute test involving mathematics items from PISA. The latter, although commensurate with the students’ level of mathematical education, represented largely unfamiliar tasks to the students and required the transfer of previously learned mathematical knowledge and skills to a new context. The students’ performance on these items was used to construct a second measure of persistence. Initial findings indicate that although the correlation between the self-reporting measure and the evidence provided by the PISA-type test is statistically significant, there are some inconsistencies between the self-reported data and observed behaviour.

#### 05 Snapshots from a classroom: an analysis of patterns of interaction over an academic year in one year 7 mathematics class

Alf Coles, University of Bristol, Graduate School of Education

This case study draws on data collected in one secondary mathematics department in the UK in the academic year 2007-8. I took six video recordings of Teacher A working with her year 7 class. In line with the enactivist methodology that informed the research, in this paper I look at the final recording of the year and trace the patterns that can be observed back through the rest of the data. The analysis of two patterns offers a partial lighting on how a particular way of working developed, and demonstrates how the patterns that can be seen at the end of the academic year are observable at the beginning of September.

#### 06 Benchmarking Mentoring Practices for Effective Teaching of Mathematics and Science

Francis Kwaku Duah, University of Southampton & Loughborough University from Jan 2011

Pre-service mathematics and science teachers’ perceptions of their mentoring experiences were investigated using the five factor model of mentoring practices as a lens through which mentoring practices can be benchmarked for improvement. The Mentoring for Effective Teaching instrument was used to collect data from 68 pre-service mathematics and science teachers on school placements in two Local Education Authorities (LEAs) in the South East region of England. The results of the data analysis indicate that mentors in the two LEAs overwhelmingly exhibit personal attributes for effective mentoring, provide adequate mentoring in pedagogical knowledge development, model effective teaching and professional practices and provide effective feedback to pre-service teachers. Yet, the results also indicate mentors did not provide adequate mentoring on systems requirements in relation to the national curriculum and school policies.

#### 07 Perceptions of symmetry: A window into how 13 year old students appear to understand symmetry

Sue Forsythe, School of Education, University of Leicester

This study describes Year 8 students in England (aged 12-13) using Dynamic Geometry Software (DGS) to investigate triangles and quadrilaterals which can be generated by dragging two rigid perpendicular lines within a shape. The dialogue and the dragging and measuring strategies employed by the students seem to illustrate that they viewed the shapes through the lens of symmetry. On being questioned about the meaning of symmetry their notions of it were process based rather than coming from an esoteric understanding of the meaning of symmetry.

#### 08 A week with secondary mathematics through history and culture

Ioanna Georgiou, Institute of Education, University of Warwick

This paper is about the mathematics lessons I delivered during the last week of the previous school year at a secondary school, years seven to ten. The lessons involved history of mathematics as well as sociocultural elements. The material chosen are described and briefly examined. The students’ reactions to the idea and whether they engaged with the lesson is also looked at. The kind of questions students asked in class and what these revealed is discussed. Another area of discussion is students’ answers as well as their interpretations of the issues raised in class. As each of the four classes was of different achievement (from bottom to top sets), I comment on whether achievement seemed to have any effect on students’ reactions. Finally, some teaching issues I encountered when attempting to teach with this approach are raised.

#### 09 The affordances and constraints of turn-taking in the secondary mathematics classroom

Jenni Ingram, University of Warwick

This paper discusses the affordances and constraints of the rules of turn-taking in classrooms. The study draws on conversation analytic studies of both classrooms and ordinary conversations as well as data from a collection of sixteen secondary mathematics lessons with 12-13 year old pupils. Examining McHoul’s (1978) rules for turn-taking in the classroom alongside examples from the data in this study where interactions deviate from these offer alternative ways of interpreting the actions of both teachers and pupils in whole class interactions.

#### 10 Engineering Students’ Understanding Mathematics ESUM

Barbara Jaworski, Loughborough University

ESUM is a developmental research project concerned with innovation designed to improve the teaching of mathematics to first year materials engineering students in a UK university. The main aim of the project is creating a culture to foster students’ more conceptual engagement with mathematics. Here I mean “more” in the sense of more than they have previously had in their earlier studies; more than just an instrumental understanding (Skemp 1976) and more than previous cohorts of students that I have taught in the past. An important question here is how such understanding can be seen and recognized and this is part of the study. In this short paper, I focus on organizational, theoretical and methodological aspects of the study.

#### 11 Why do GCSE examination papers look like they do?

Ian Jones, University of Nottingham

GCSE mathematics examinations have been criticised for being too structured and not adequately assessing process skills. Exam papers are produced by private awarding bodies working to government regulations. A given paper and its mark scheme is usually written by one individual, and then reviewed and revised by a Qualification Paper Evaluation Committee (QPEC). As such, QPEC meetings have a significant role in how the final published exam paper looks. Over the past year I have observed several QPEC meetings across the three awarding bodies that publish GCSE papers. I describe how what gets said in QPEC meetings bears on the structure and process skills assessed in exam papers.

#### 12 Measurement: everywhere and nowhere in secondary mathematics

Keith Jones, University of Southampton

School mathematics is commonly structured into number, algebra, geometry and statistics. This raises the issue of where to place ideas within the topic of measurement since some aspects of measurement (such as measuring length or area) have a geometrical component, while other aspects of measurement (such as time or money) are about number. Furthermore, when actual measures are unknown, relationships between measures can be expressed – and this is one of the roots of algebra. Additionally, probability can be thought of as a form of measure (of uncertainty) and the various measures of data variation, such as standard deviation, can also be viewed as a form of measurement. All these considerations mean that the placing of measurement in the mathematics curriculum can be problematic for curriculum designers and policy makers; and equally tricky for teachers to teach in the most effective way. Informed by a review of the research basis for teaching key ideas in secondary school mathematics, this paper argues that measurement is both everywhere and nowhere in secondary mathematics; that is, measurement occurs across the topics that comprise secondary school mathematics, but the ideas of measurement are so scattered that the teaching of measurement in secondary school mathematics may lack some focus that might store up problems for learners as they progress with mathematics.

#### 13 Level 3 mathematics: a model for the curriculum

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

A model for a reformed Level 3 mathematics curriculum is derived, assuming its prime purpose is to provide the mathematical foundations for several types of Level 4 courses. Then issues and practicalities along the way are explicitly identified and proposals made for their resolution. Mechanisms for determining detailed content, based on earlier work, are also proposed. The model is structurally similar to current GCE Advanced Level, however routes preparing for the different types of Level 4 course are explicit and the model is intended to be more encouraging for students outside the mainstream and also more economical in its use of scarce teaching resources.

#### 14 Conceptions of ‘Understanding mathematics in depth’: What do teachers need to know and how do they need to know it?

Mary Stevenson, Liverpool Hope University

In recent years there has been much debate about the preparation and supply of mathematics teachers, e.g. Williams (2008), Smith (2004). There has been a corresponding growth of interest in what constitutes subject knowledge for mathematics teaching, and how this is developed. Much research has focused upon primary teachers, whereas the nature of subject knowledge required by secondary mathematics teachers has been relatively under-researched. In this paper I report on work in progress in an investigation into what characterises ‘deep understanding of mathematics’ as understood by two specific groups of secondary pre-service and serving mathematics teachers. Additionally I comment upon data collected on degree classification and outcomes of postgraduate initial teacher education.

#### 15 Analysing Secondary Mathematics Teaching with the Knowledge Quartet

Anne Thwaites, Libby Jared and Tim Rowland, University of Cambridge, UK

This paper describes how the Knowledge Quartet (KQ), which was developed with mathematics teachers in primary schools, has been tested in a secondary mathematics context. Aspects of this research are illustrated with reference to a lesson on completing the square. First we exemplify the mapping of episodes in the lesson to the KQ, then we report how one of these episodes, concerning the choice of examples, was subsequently used in a secondary mathematics PGCE teaching session.