**Proceedings of the Day Conference held at King’s College, London, February 1998 and University of Birmingham, June 1998**

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## Contents

#### 1 A Computer-Based Mathematics Laboratory for Undergraduate Chemistry Students: A Preliminary Evaluation Report

Phillip Kent, Imperial College, London

Ian Stevenson, Institute of Education, London

This is a preliminary report from an evaluation of a first-year mathematics course for undergraduate chemists which is designed around a combination of motivatory lectures and assignments in a “mathematics laboratory”, where students use the computer algebra package, Mathematica. A student questionnaire was administered and a half-dozen follow-up interviews were done. We report here on the findings from those, using summary data from the sample of 57 students, and a comparison of the interview data from two typical students (one with, and one without, A level mathematics).

#### 2 A Survey of Public Images of Mathematics

Urn Chap Sam and Paul Ernest

University of Exeter

This paper reports initial findings of a survey that aims to explore the range of public images of mathematics. Over 500 adults aged 16+ from all walks of life responded the short questionnaire given. Initial findings show that public images of mathematics and learning mathematics were given in the forms of propositions expressing opinions and views or in the form of metaphors. Five main categories of responses emerged from the analysis. They are (a) attitudes towards mathematics and its learning; (b) beliefs about respondents’ own mathematical abilities; (c) descriptions of the process of learning mathematics; (d) epistemology and views of the nature (?f mathematics; and (e) values and goals in mathematics education. Some methodological issues and examples of each category are given and discussed in the paper.

#### 3 Modelling, Strategic Questioning and the Laws of Student Reasoning in A-Level Mechanics

Stuart Rowlands, Ted Graham and John Berry

University of Plymouth

The Newtonian concept of force is a functional quantity that explains changes in motion – force as a relation between two bodies. Many A -level students, on the other hand, conceive force ontologicaliy – as a property that the body possesses. There is much research that suggests that these ontological beliefs are not acquired through experience of the physical world prior to learning mechanics, but may be formed when the student is asked to consider qualitative examples of force and motion for the first time. This paper reports on two possible laws of student reasoning in mechanics based on the results of a pilot-study.

#### 4 Investigatory Approaches in Mathematics Teaching – Implications for Teacher Training in India

Dr. Vijayalakshmi Chilakamarri

King’s College London

In the present age of Science and Technology it is very much essential to develop the process skills, experimental skills, thinking skills and the skills of exploration among the school children. Mathematics is an appropriate subject to inculcate and develop these skills through Investigatory approach. So it is necessary to know to what extent Investigatory Approach is adapted while advocating it in teaching mathematics, in the Initial Teacher Training Institutes and its impact on teaching mathematics in the Secondary Schools. An attempt is made to study in detail how the Investigatory Approach is adapted in the Initial Teacher Training Institutions at the University level and how is it practised at the Secondary school level to improve the quality of mathematics education. My observations,findings and their implications to the Teacher Training in India are put before you for discussion. Any suggestions offered are highly appreciated.

#### 5 Geometry Working Group: Theoretical Frameworks for the Learning of Geometrical Reasoning

Convenor: Keith Jones

University of Southampton

With the growth in interest in geometrical ideas it is important to be clear about the nature qf geometrical reasoning and how it develops. This paper provides an overview of three theoretical frameworks for the learning of geometrical reasoning: the van Hiele model of thinking in geometry, Fischbein’s theory offigural concepts, and Duval’s cognitive model of geometrical reasoning. Each of these frameworks provides theoretical resources to support research into the development of geometrical reasoning in students and related aspects qf visualisation and construction. This overview concludes that much research about the deep process q{the development and the learning of vi sua lis at ion and reasoning is still needed.

#### 6 Advanced Mathematical Thinking Working Group

Stephen Hegedus

University of Southampton

Initially, this group will want to assess the interest in Advanced Mathematical Thinking (AMT), with the possible aim of creating links with other international working groups in the field (e.g. P ME). The group will be introduced to some of the major issues in AMT research today, and possible future developments. Themes of interest at the moment include: students’ conceptions offunction, limits, proof, and students’ conceptions of differentiation and integration. It will look at the relevance of social, pedagogical and methodological issues, such as the use of algebraic/geometric software or methods for analysing (meta-)cognitive behaviour, in developing a more suitable model of AMT. It is a major aim that the group will not concentrate on purely psychological issues but will discuss the nature of mathematics, per se.

#### 7 Relations Between Teacher’s Representations and Pupil’s Images

Chris Bills

University of Warwick

This paper presents a simple constructivist model of teaching and learning characterised as a mapping between the set of representations that the teacher uses and the set of images that learners form. Data, collected in a phenomenographic study of primary school pupils’ images, formed in consequence of their interaction with representations of two digit numbers, is discussed. Some implications for teaching are considered.

#### 8 Visualisation and Using Technology in A Level Mathematics

Sally Elliott

Sheffield Hallam University

This project seeks to identify and evaluate the ways in which existing technology can be utilised to promote and develop students’ powers of visual is at ion and to encourage the usage of these skills in A level mathematics lessons. At present, a small scale pilot study has been carried out and the resulting data has been analysed. After briefly summa rising current research in the area of visual is at ion and technology in this paper, I will report on the findings of this initial pilot study, in which materials developed for use with the TI-92, aimed at promoting students’ abilities to visualise the graphs of functions, were trialled with a class of thirteen year twelve students. The subsequent consequences for future directions of the research will also be discussed.

#### 9 “What Can We All Say?” Dynamic Geometry in a Whole-Class Zone of Proximal Development

John Gardiner

Sheffield Hallam University

This paper first develops a theoretical background which culminates in a discussion of the promotion and fostering of socio-mathematical norms (Cobb and Yackel, 1996), the significance of local communities of practice (Winboume and Watson, 1998) and the development of a whole-class ZP D (H edegaard, 1991, Lerman, 1998). It then seeks to indicate a way in which classroom aproaches by teacher and pupils to a dynamic geometry package (Cabri 2 as available on the TI 92) might use this background.

#### 10 Cabri as a Cognitive Tool

Bibi Lins

University of Bristol

Amongst others, Bottino and Furinghetti (1994) investigated the ways in which teachers’ belief are related with respect to two domains seen as correlate: Mathematics and Computers. My study, however, instead of examing the perceptions in two domains, I shall concentrate on how a new domain is constitued, namely the use ofICT in mathematical education. The proposed study takes place inside what Cooney (1994) calls a ‘conception of teacher education as a work of investigation’, precisely by suggesting and investigating the constitution of a domain.

#### 11 Primary Children’s Imagery in Arithmetic

Sandra Pendlington

University of Exeter

This paper discusses four ideas about imagery drawn from the domain of psychology and illustrates these ideas using examples from pilot study interviews with primary children.

#### 12 An Activity Approach to Teaching and Assessment

Chris Day

South Bank University

I have briefly indicated some key developments in the history of materialist dialectics and some principles of activity theory which follow from them. I have illustrated these general notions with some references to a program of teaching and dynamic assessment that I introduced in an earlier paper (Day 1998). I have presented video typescripts together with some quantitative and qualitative data from my research to show that differences in ability between children from different socio¬economic areas were increased rather than decreased in the course of the teaching program. This was shown clearly in data from both static and dynamic assessments.

#### 13 Mathematical Support for Engineering and Science Students

Peter Gill

King’s College London

An investigation of the mathematical difficulties experienced by undergraduates starting courses in science and engineering has revealed a range of misconceptions that appears not to have changed greatly over several decades despite changes in school curriculums. Many of these difficulties relate to the understanding of graphs. There is also evidence that the students find the style of learning expected at HE to be greatly different fromwhat they have experienced at school and this too contributes to their problems. A number of reasons for the problems are hypothesised and some solutions are proposed.

#### 14 Tutors’ Reflections upon the Difficulties of Learning and Teaching Mathematics at University Level: A Report of Work-in-Progress

Elena Nardi

University of Oxford

Following a doctoral study on the learning difficulties of20 Oxford first-year mathematics undergraduates in their encounter with mathematical abstraction, a study is currently being carried out in which the mathematics tutors who participated in the initial study have been asked to reflect upon samples of its data and findings in semi-structured interviews. The interviews address three areas: clarifications and explanations regarding the pedagogical and psychological language used in the samples, a validation/critique of the interpretations in the thesis regarding the students’ learning difficulties and a reflection upon the events in a tutorial from a teaching point of view. The focus of the discussion alternates between a specific (sample-centred) and a general addressing of issues relating to the learning and teaching of undergraduate mathematics. Here short extracts from the interviews are presented and commented upon. Transcription and analysis are now in progress.

#### 15 Convention or Reality

Melissa Rodd, Open University

Margaret Barber, Thomas Telford School

This report both introduces to a mathematics education audience some research in contemporary philosophy of mathematics and develops the relevance of such research to mathematics in education. The philosophical theories to be considered are those of the physicalist¬realists, specifically in this paper, Resnik and Bigelow; physicalist-realists offer theories of how mathematics is, in some sense, part of the physical world The notion of an ‘instantiation’ of a mathematical theorem, or object, is distinguished from a more general and familiar ‘representation’ in order to help conceptualise the possibility of non-linguistic experience of mathematical relationships or entities. The question of relevance to mathematics in education is approached by presenting mathematical activities to engage with and school-texts to analyse.

#### 16 Mind the ‘Gaps’: Primary Trainees’ Mathematics Subject Knowledge

Tim Rowland, Caroline Heal, Patti Barber and Sarah Martyn

Institute of Education, London

Recent changes in the curriculum for Initial Teacher Training incorporate a stronger focus on trainees’ subject knowledge. (DtEE, 1997) Some evidence would seem to support this shift of emphasis. In the US, Kennedy’s research (1991) suggested that teachers’ mathematical understanding is frequently limited, whilst in the UK Alexander et al (1992) called for improvement of the knowledge base of teachers in order to improve the teaching of mathematics. Inspection evidence identifies teachers’ lack of subject knowledge and confidence in mathematics as being a contributory factor in low standards of mathematics attainment of pupils (Ofsted, 1994).

#### 17 Influences on Student Teachers of Mathematics

Jim Smith

Sheffield Hallam University

This paper reviews work in progress on a study of the nature o.linfluences upon student teachers of secondary mathematics. A practitioner research model has been employed in three phases to date, involving three cohorts {istudent teachers on the one year post-graduate certificate in education course (PGCE) at Sheffield Hallam University. Thefindings suggest that mathematics teachers were motivated to advise student teachers most frequently about aspects of class management. Other aspects of guidance offered within this structure were mainly focused on explanation, examples and exercises. There was some attempt to exhort student teachers to use a range o.l activities, but little guidance was offered about specific activities.

#### 18 Possibilities in Pierce’s Existential Graphs for Logic Education

Adam Vile and Simon Polovina

South Bank University

In our experience students find learning logic difficult, this view is supported in the literature. Charles Sanders Peirce, logician, semiotic ian, teacher, envisaged a representation that would provide tools to enable everyone to reason with formal logic. To this end, and basing his work on his own semiotic principles, he developed a system of graphical reasoning. In this session we will present Peirce existential graphs and consider this form of reasoning as providing possibilities for improving logic teaching and learning. We will also outline our plans for a future study to evaluate the Percian approach.

#### 19 What Can be Learnt by Selecting Anecdotes from a Range of Data? Exemplifying “Noteworthy” Mathematics with a Small Number of Examples

Anne Watson

University of Oxford

Analysis of data collected for qualitative research purposescan often generate further research questions, particularly if an open-minded approach is taken. In this paper I examine the use of anecdotes for a purpose other than that for which they were collected. I suggest that this is a justifiable research procedure so long as the questions asked of the anecdotes are appropriate, and the use of the results is sensitive to the mode of collection.

#### 20 Teacher Trainee Students’ Understanding of Operation Signs

David Womack

University of Manchester

Following the invention of operation signs by young children working m a tramj’ormationally-focussed problem scenario, a similar transfonnational model was presented to 4th year teacher trainee students in which they also were asked to invent signs. The signs referred to the strategies of counting-on, counting-back and counting-up. The students’ responses and their understanding of the difference between counting-back and counting-up will be discussed, and participants invited to give their own opinions on the mathematical status of such ‘counting signs ‘.

#### 21 Geometry Working Group

Convenor: Keith Jones

University of Southampton

This report focuses on some aspects of the nature and role of visualisation and imagery in the teaching and learning of mathematics, particularly as a component in the development of geometrical reasoning. Issues briefly addressed include the relationship between imagery and perception, imagery and memory, the nature of dynamic images, and the interaction between imagery and concept development. The report concludes with a series of questions that may provide a suitable programme for research and lays the foundation for further work of the BSRLM geometry working group.

#### 22 Semiotics and Mathematics Education Working Group

Convenors:

Adam Vile, South Bank University

Paul Ernest, University of Exeter

The theme of this meeting was “applied semiotics” and its role in mathematics education research. There were presentations from two researchers working in the context of semiotic methodologies and perspectives specifically aimed at discussion of the tools and techniques that they were using for their data collection and analysis. Dedrie Cook (Derby) spoke about her work in studying children at play with mathematical artefacts and the tools from linguistics that she had used for analysis of data. Corina Silveira (Southampton) shared her theoretical perspective and research design for analysis of children’s development of counting. The work of Dedrie and Corina exemplified the great variety in work that is considered semiotic, and presented two approached to data collection and design that fit within a semiotic framework.