**Proceedings of the Day Conference held at Leeds University, June 2004.**

## Contents

#### 1 Student teachers’ research into realistic mathematics in the context of engineering mathematics days for girls. Who learned what?

Corinne Angier

Sheffield Hallam University

This paper will report on a project designed to engage some initial teacher education students in research. The students are in their first year of a two year route into mathematics teaching. In the context of a mathematics education module they have drawn on ideas from the Realistic Mathematic education tradition from the Netherlands to develop and run four ‘Engineering Mathematics days’. Six schools from a disadvantaged area in Northern England have been funded to send groups of 50 girls to attend the activity days. The student teachers were asked to devise a mini-enquiry focussed on some aspect of the girls’ learning.

#### 2 The influence of lecturers privileging different aspects of derivative on students’ conceptions

Erhan Bingolbali

University of Leeds

This paper explores the influences of university lecturers privileging of different aspects of the derivative on mechanical engineering and mathematics students developing conceptions of the derivative. The data are based on interviews with four mathematics and two physics lecturers, observations of calculus courses and students calculus course notes. The results suggest that lecturers: privilege different aspects of the derivative and types of examples; set different questions on examinations; use different textbooks; and perceive the two departments as having distinct mathematical goals and aims. These differences influence students developing conceptions of the derivative.

#### 3 Mathematics teachers’ learning about and incorporation of ICT into classroom practices

Cosette Crisan

Newman College of Higher Education

In this paper I report on a study of secondary school mathematics teachers’ use of ICT. The study adds new dimensions to understanding teachers’ use of ICT by treating the teaching of mathematics and ICT use as interwoven aspects of a teacher’s practice. The analysis of the data collected using a case study research strategy yielded a number of salient factors, of both contextual and personal nature, which were identified as key to the integration of ICT into mathematics teaching. A framework which conceptualises teachers’ learning about the potential and limitations of ICT and teachers’ incorporation of ICT in their teaching of mathematics will be advanced with the aim of contributing to a better understanding of the pedagogy of teaching mathematics with ICT.

#### 4 Sum and difference problems at Key Stage 2

Hilary Evens and Jenny Houssart

Centre for Mathematics Education, The Open University

A comparison is made of the responses of 11 year olds to two questions involving the sum and difference of two numbers. The first question was asked on a Key Stage two National Curriculum test and was presented in algebraic form. The second question was presented as a word problem and was designed to be more accessible to children. Fewer children were successful in solving the problem expressed in words, though more attempted it than the version presented algebraically. We explore the methods used to tackle both questions and the incorrect answers given as well as offering a tentative comparison between the two.

#### 5 Mathematical beliefs of trainee teachers during the first year of higher education

Ray Huntley

Anglia Polytechnic University

Prior learning experiences help trainee teachers develop beliefs about the nature of mathematics and its teaching and learning, which shape the nature of their training experience. Using ethnographic methods a cohort of trainees in one institution were surveyed about mathematical beliefs. Results indicate evidence of both change and stability in beliefs, and that stated intentions for teaching mathematics are challenged by school demands to meet national criteria for pupil performance.

#### 6 Learning communities in mathematics: developing and studying inquiry communities in mathematics learning, mathematics teaching and mathematics teaching development

Barbara Jaworski

Agder University College, Norway

Learning Communities in Mathematics (KUL/LCM ) is a project which aims to design and study mathematics teaching development for the improved learning of mathematics through inquiry communities between teachers and didacticians. This paper introduces the LCM Project briefly with aims and a theoretical account, and an outline of implementation including methodology, data collection and analysis.

#### 7 An aspect of teacher practice in turkish classrooms: differences in time given to students to do mathematics in two different types of schools

M. Kerem Karaagaç

University of Leeds

In this paper I will look at the structure of mathematics lessons in Turkish state schools and private colleges. Although teachers’ practices have similar structural features in teachers’ own accounts, there are different teacher privileging patterns in classrooms. In particular, the time given to student engagement with examples differs. In private colleges, time allowed for students engagement with problems is markedly less than in state schools.

#### 8 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.

#### 9 “Winging it!&rdwuo;: control, structure and freedom in mathematics teaching

Hugh Kidd

King’s College, University of London

Ideas such as “effective teaching&rdwuo; are an educational currency which impinge on the practice of teaching in expected and unexpected ways. They shape what is taken as ‘good’ and ‘bad’ classroom teaching of mathematics. Good teaching follows from articulating clear learning objectives and having a well thought-out lesson plan. Such an environment is perceived to be the best for learning mathematics. But is good teaching always dependent on prior and clear plans? In this paper I explore, through the notion of a mathematics teacher &ldwuo;winging it&rdwuo;, relationships between control, structure and freedom in mathematics teaching.

#### 10 Have daily mathematics lessons enhanced pupil confidence and competence?

Chris Kyriacou and Maria Goulding

University of York, Department of Educational Studies

A systematic review group for mathematics education funded by the DfES was established at the University of York in October 2003. Its first review question was “Has the Daily Mathematics Lesson, in the context of the National Numeracy Strategy for primary schools in England, helped pupils to develop confidence and competence in early mathematics?” This paper reports on the processes and dilemmas involved in conducting a systematic review, and reports early findings.

#### 11 Using the plenary to develop reflective and critical thinking and to enhance metacognitive awareness: student teachers’ perceptions and school-based experiences of the daily mathematics lesson plenary

Fiona Lawton

St. Martin’s College, Ambleside

This paper describes how practitioner research was used to investigate use of the plenary as a vehicle to develop reflective and critical thinking and to enhance metacognitive awareness with undergraduate mathematics specialist student teachers. Students’ perceptions and experiences of the Daily Mathematics Lesson Plenary were explored to assess the impact of the research on their perceptions and/or practice in school. The research findings suggest that student teachers who are introduced to plenaries to promote reflective and critical thinking may have the confidence to take these ideas forward into their own practice in the primary school. There was also evidence of students’ practice impacting on the practice of teachers in school.

#### 12 Thinking together – using ICT to develop collaborative thinking and talk in mathematics

Frank Monaghan

The Open University

In this paper I describe some recent research taking place in primary schools in Milton Keynes and London using the ‘Thinking Together’ approach and SMILE software to promote collaborative thinking and talk in the maths classroom.

#### 13 Measuring learning in situations which attempt to link school mathematics to out-of-school mathematical activities

John Monaghan

University of Leeds

This paper explores how learning, in situations which attempt to link school mathematics to out-of-school mathematical activities, may be measured. I present three possible measures which focus on students’ motives, the mathematics employed and on the material resources. For each measure I consider its relevance, possible problems and how the measure may be constructed.

#### 14 Scaffolding, abstraction, and emergent goals

Mehmet Fatih Ozmantar

School of Education, University of Leeds

This study examines the role of scaffolding in the achievement of a mathematical abstraction by focusing on emergent goals. An activity-theoretic approach to abstraction in context is taken. The examination is carried out with regard to the verbal protocols of two 17-year-old students working on a task related to the graphs of y=f(|x|). This examination suggests that abstraction is likely to be achieved through satisfaction of several emergent goals. These goals are observed to be contingent upon four parameters: the scaffolder’s interventions, students, tasks, and prior emergent goals. Dynamic and dialectical interrelationships amongst these parameters are discussed with regard to the students’ verbal protocols.

#### 15 The discourse of ‘subtraction as difference’

Tim Rowland

University of Cambridge

For more than 30 years, UK Early Years discourse has referred to ‘subtraction as difference’ in contrast to ‘subtraction as take away’ (see, e.g. the Teaching Programme for Year 1 in the National Numeracy Framework). An offshoot of a study of videotapes of trainee primary school teachers’ lessons has been clearer thinking (on my part) about the rationale for my long-standing unease with this use of ‘difference’. I explain that objection in this paper, and plea for a change in the way UK practitioners refer to this aspect of subtraction.

#### 16 Parents and mathematics in the primary school

Margaret Sangster

Sheffield Hallam University

A small scale study of the way a community primary school promoted knowledge about the taught mathematics of the National Numeracy Strategy with parents whose children were entering their final year (10-11 year olds). A range of strategies were explored and evaluated.

#### 17 The exercise as mathematical object: dimensions of possible variation in practice

Anne Watson and John Mason

University of Oxford, Open University

By treating collections of questions as mathematical objects, that is ordered sets containing individual questions as elements, we gain insight into the potential role of exercises in learning mathematics. We use the notion of ‘dimensions of possible variation’, derived from Ference Marton, to discuss some exercises. There are implications for the design of question sets, for pedagogical decisions in the use of question sets, and for reflective questioning by learners.