**Proceedings of the Day Conference held at the Open University, June 2005.**

## Contents

#### 1 Exploring Students’ Understanding of the Relationship between Recursion and Iteration

Mojtaba Ammari-Allahyari

University of Warwick

In this study, I examine students’ appreciation of the relationship between recursion and iteration, including how they perceive the intra-relationship of the components within those processes. In this environment, the students model trees and fractal-shape objects. These early results show that having a clear understanding of the relation between recursion and iteration and of the flow of control are important in understanding the recursion itself. In addition, functional abstraction is a key concept in dealing with recursion. In the light of these results, I am planning to open up the software such that students will be able to engage with recursion at two levels, namely “functionality” and “functionings”.

#### 2 Talking About Mathematics Problems on the Web

Jenni Back

University of Cambridge

This paper outlines an initial investigation into the nature of communication about a mathematical problem on the AskNRICH discussion board on the NRICH website (www.nrich.maths.org.uk). It involves elements of collaborative problem solving but also shows some conventions adopted from other settings including those associated with classroom talk and text messaging. The webboard offer young people the opportunity to engage in dialogue about mathematics without teacher intervention or instigation and the purpose of the analysis is to gain some ideas about the nature of that communication and its value.

#### 3 Undergraduate Students’ Departmental Affiliation and Conceptions of the Derivative

Erhan Bingobali

University of Leeds

This paper is a byproduct of a study of first year undergraduate mechanical engineering and mathematics students’ conceptions of the derivative. The results indicate that mechanical engineering students develop proclivities towards rate of change aspects of the derivative whilst mathematics students develop proclivities towards tangent aspects. In explaining the differences between the proclivities of the two groups of students, I introduce the notion of students’ departmental affiliation. This is followed up by considerations regarding its genesis and implications for students’ learning and participation.

#### 4 The Truth of Initial Training Experience in Mathematics for Primary Teachers

Tony Brown

Manchester Metropolitan University

Mathematics was a subject that filled many trainee teachers with horror in their own schooling. Yet for trainees in this study university training experience soon persuaded them that “maths isn’t just scary numbers” and as their course progressed such anxieties seemed less pervasive, almost to the point of being de-problematised once the trainee had reached qualified teacher status. How had this been achieved? Despite a history of ambivalence towards the subject of mathematics they did not continue to present themselves as mathematical failures. Rather, they offered an account that left out issues that the trainee would rather not confront. This paper documents theoretical aspects of a major study examining how initial training students effect this transition. It concludes that such trainees “story” themselves so as to sideline mathematics but to present their perceived qualities of themselves in a positive light.

#### 5 Making Mathematics Hard: Student Views from a Caribbean Perspective

Patricia George

University of Leeds

This paper reports an initial finding from a study which explored students’ perceptions of mathematics in a Caribbean setting. The most frequent reason given by students for not liking mathematics was because it was hard. This in itself is not surprising, but what was surprising was that there appeared to be an implicit feeling by some students and explicitly stated by others that it was the teacher who was ‘making mathematics hard’. The paper looks at what ‘making mathematics hard’ might mean to students.

#### 6 Patterns of Student Interactions: What Can They Reveal About Students’ Learning of Mathematics?

Marie Gibbs

University of Bristol

This study is set within the context of a secondary school mathematics classroom in which a graph sketching software program is used. The interactions taking place as a small group of students works through the activities set by the teacher, captured on video, are analysed in terms of different ‘modes of production’ and mathematical processes and learning required to complete the task. The patterns of modes of productions leads to a more in-depth analysis of the mathematical learning of the students. … abstract paragraph end

#### 7 Statistical Reasoning in the Workplace: Techno-Mathematical Literacies And Learning Opportunities

Philip Kent, Arthur Bakker, Celia Hoyles and Richard Noss

University of London

The “Techno-mathematical Literacies in the Workplace” project is investigating the needs of employees in a range of industrial and commercial workplaces to have functional mathematical and statistical knowledge that is grounded in their workplace situations and mediated by the technological artefacts that surround them. We present some emerging ideas using examples drawn from an industrial workplace where we have noted a “skills gap” concerning the use of statistical techniques for controlling a manufacturing process. We review our analytical framework, that combines activity theory and semiotic ideas, and we discuss some prototype “learning opportunities” for situated statistical reasoning based on the educational statistical software, TinkerPlots.

#### 8 Differences in Teachers’ Selection and use of Examples in Classrooms: An Institutional Perspective on Teacher Practice

M. Kerem Karaagaç

University of Leeds

The research presented here is an attempt to explore teachers’ classroom practices in mathematics lessons in Turkish state schools and privately owned educational institutions. I will present data on the use of examples in mathematics lessons in two classrooms from different schools teaching the same content: inverse functions. The results indicate that the time given to student engagement on examples differs. In private college, the time allowed for students’ engagement with problems is markedly less than in state schools. Further analysis on the use of examples shows that the examples teachers selected to use and the way they made use of them are also different. I will discuss the results in the light of socio cultural theories.

#### 9 Computer Algebra Related Conceptions and Motivations of University Mathematics Lecturers An International Study

Zsolt Lavicza

University of Cambridge

As a consequence of sizable investments in technology, Computer Algebra Systems are becoming more accessible and widely used in mathematics teaching and learning in universities. However, little is known about the factors influencing the integration of technology at the university-level. To better understand the affect of technology incorporation on mathematics education a number of school-level studies have focused on the relationships between teachers’ conceptions of mathematics, mathematics teaching, and technology as well as on various social and cultural factors. My study investigates the relationships of these factors at the university-level paying particular attention to cultural elements by taking an international comparative approach.

#### 10 Hippy Chix and Geek Chic: What do Positive Images of Women Mathematicians Look Like?

Heather Mendick

London Metropolitan University

In this article I explore images of women mathematicians within popular culture (including film, TV and the internet) and how they might help young women to build positive relationships with the subject. There have recently been several films about male mathematicians, all depicting highly gendered images of mathematics and mathematicians. I explored these at BSRLM last year. Since then I have been looking at images of female mathematicians, and trying to be more positive. The images I discuss are Carol Vorderman from Countdown, Willow from Buffy the Vampire Slayer, Seth Cohen from The OC and a cosmetics bag bearing an image of a woman and the words: ‘I’m too pretty to do math’.

#### 11 Aspects of Proof in Mathematics Research

Juan Pablo Mejía-Ramos

University of Warwick

Without having a clear definition of what proof is, mathematicians distinguish proofs from other types of argument. This has become increasingly difficult in the last thirty years, as mathematicians have been able to use ever more powerful computers to assist them in their research. An analysis of two types of proof (mathematical proof and formal proof) and two types of argument (mechanically-checked formal proof and computational experiment) reveals some aspects of proof in mathematics research. The emerging framework builds on the distinction between public and private aspects of proof, and revises the characterization of mathematical proof as being formal, convincing, and a source of understanding.

#### 12 An Authentic Packaging Task in the Classroom

John Monaghan

University of Leeds

Tony Staneff

St Aidan’s CoE High School, Harrogate

We describe the work of one Year 9 class who undertook a two week long activity around designing packaging for shelf-ready tea as part of a project on linking school mathematics to out-of-school mathematical activities. We describe the project and the classwork and discuss issues arising from the work.

#### 13 Student Teachers’ Experiences of Using Spreadsheets

Declan O’Reilly

University of Sheffield

This paper recounts 14 student teachers’ experiences of using spreadsheets to teach mathematics in secondary schools. As part of their formal assessment, they submitted accounts, listing the problems and benefits encountered in using this software. These accounts form the basis of the analysis. The results suggest that knowing where the problems occur may help student teachers to forestall them. Secondly, the results indicate that student teachers incur additional costs through using spreadsheets: in terms of planning and teaching. The benefits accrue to the children in terms of better understanding and attitudes towards learning – and ultimately to the teacher – in achieving these goals.

#### 14 Mathematical Abstraction: A Dialectical View

Mehmet Fatih Ozmantar

University of Leeds

In the classical Aristotelian empiricist view, abstraction is associated with an ascending developmental process from the concrete to the abstract. In this paper, however, I argue that the development coming about in the formation of mathematical abstraction can be best portrayed as a dialectical development to and fro between the concrete and abstract. This argument is exemplified on the basis of the verbal protocols of two students working together on a task connected to sketching the graphs of absolute value linear functions.

#### 15 Using Visual Tools to Promote Mathematical Learning

Sandra Pendlington

University of Bristol

This paper describes work done with low achieving 10-year olds. It explores the use and nature of successful visual tools. It discusses the importance of negotiating the labelling of the structure of tools to avoid the learning paradox and the use of tools as mediators between the children’s ideas and the world of mathematical symbolism.

#### 16 Functional Mathematics: What is it?

Tom Roper, John Threlfall and John Monaghan

University of Leeds

Functional mathematics, introduced in the Tomlinson Report (DfES 2004b), and taken up by two succeeding white papers (DfES 2005a, DfES 2005b), might be described as a ‘charmed phrase’, a phrase that every body can sign up to because of its “penumbra of vagueness” (Apple, 1992). But what is functional mathematics? What problem has it been called into being to solve? Have there been previous solutions to the problem and if so what can we learn from them?

#### 17 “Three Cheers for Derek Haylock!” Primary PGCE Students’ Use of Mathematics Teaching Handbooks

Tim Rowland

University of Cambridge

I was recently asked to review an American mathematics book intended for pre-service primary teachers, from a comparative perspective. It occurred to me that although we routinely recommend books such as Derek Haylock’s ‘Mathematics Explained for Primary Teachers’ to our own primary PGCE students, we know very little, beyond anecdote, about how (or whether) they use such books during their training. The survey reported here was designed to find out.

#### 18 Getting an Insight into How Students Use Their Graphical Calculators

Louise Sheryn

University of Leeds

I report on a data collection tool that I used within my doctoral studies. My study investigates the depth and type of learning that takes place when a student uses a graphical calculator within an AS Level Mathematics course. I collected various types of data during the project: interviews; observations; student journals; key-stroke data from graphical calculators and combinations of all four. The key-stroke data was collected using a piece of software called Key Recorder that runs in the background of a graphical calculator recording all the user’s key strokes. It is then possible to playback the data file to see what the user saw and determine how they used their graphical calculator. I plan to outline some features of the Key Recorder followed by some initial observations made during analysis of the data I have collected.

### Working Group Report

#### 19 Mathematics Education and Policy: Working Group Report

Steve Lerman

London South Bank University

Andy Noyes

University of Nottingham

This was the initial exploratory meeting of a Working Group on policy issues in mathematics education. Colleagues studying/researching the effects of the numeracy strategy, SATs, OfStEd, TTA policies, the Smith Report participated in discussing the implications of DfES and TTA policy for pupils, teachers, parents and schools. The aim of this first session was to gain an overview of what people are doing and also to begin to identify what research needs doing by the maths education research community regarding policy and mathematics. Each participant outlined their research and/or interests relating to policy, as well as identifying what they thought we should consider as a community. Following a short presentation there was further raising of, and discussion around, key issues relating to mathematics education and policy.