**Proceedings of the Day Conference held at St Martin’s College, Lancaster University, June 1999**

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## Contents

#### 1 Solving a Non-Routine Problem: What Helps, What Hinders?

Paul Blanc

Saint Martin’s College, Lancaster

This paper reports on research in progress and in particular discusses the solutions of two sixth form students to part of a non-routine problem. Their solutions appear to show potential insights and possible pitfalls due to the specific form of representation adopted in recording results. The precise way that representations are generated as part of each student’s text and how this text is laid out on the page are claimed to have a significant influence on the progress of their solution. The use of video recording as an additional methodological tool enabled the researcher to identify and confirm such effects.

#### 2 Young Children’s Representations of Negative Numbers

Rute Borba and Terezinha Nunes

Institute of Education, University of London

Previous studies (Nunes, 1993) have observed that uninstructed children and adults have a basic understanding of negative numbers but have difficulties with their representation. This study aims to further investigate the conditions under which meanings for negative numbers are understood and represented by uninstructed children. Sixty 7 and 8 year-olds played one of two games and represented the numbers involved in either an oral mode or by means of manipulative material. The children were able to solve more items when requested to do so orally than when the materials were used. The understanding of measures was greater than that of relations. We provide further evidence of young children’s understanding of negative numbers and an effect of number meaning on performance. Attention must be drawn to the different performances when explicit or implicit forms of representation are used and when distinct meanings for negative numbers are involved.

#### 3 Aspects of the Construction of Conceptual Knowledge: the Case of Computer-Aided Exploration of Period Doubling

Soo D. Chae and David Tall

Mathematics Education Research Centre, University of Warwick

This research was conducted using computers focusing on establishing the connection between visual orbits and symbolic theory based on a concept definition. It is suggested that students who make a connection through visualisation via dynamic computer software have an advantage in understanding the concept of period doubling. The role of the supervisor proves valuable in helping students to construct conceptual knowledge by using appropriate directing questions during the experiment. It is proposed that we can help students to develop their conceptual knowledge by connecting visualisation and symbolism through computer-aided exploration guided by the supervisor and mentor.

#### 4 The Interaction Between the Use of ICT and Mathematics Teachers’ Professional Knowledge Base for Teaching

Cosette Crisan

Centre for Mathematics Education, South Bank University

With the increased availability of the new technology] in our schools, it is important to examine how teachers use it in their instruction. This paper describes a study that involved a number of secondary mathematics teachers who had been using the new technology in their teaching for a number of years. A possible theoretical framework is presented and then used in the analysis of the data collected.

#### 5 Mapping Discursive Structure. Working with Habitus, Discourse and Ideology to Explore the Politics of Mathematics Teaching

Peter Gates

University of Nottingham

Much has been written on teacher thinking and teacher education, but which remains locked within an individualistic psychological paradigm and consequent!J draws little from social theory. However, mathematics teaching is an integral component of an education system that seTVes to both produce and reproduce social structures. Consequent!J, to understand mathematics teaching we need to understand how it operates to form and reform pupils into agents. In order to do this we need to grapple with such sociological concepts as agenry, discourse, ideology and power. In this paper I look at how we might identify a teacher’s positioning within a set of discursive fields, and structural!J how we might map such discursive stnIctures in a way that integrates habitus, discourse and ideology into a deeper understanding of the political nature of mathematics teaching.

#### 6 Using NUDIST to Model Mathematics Teacher Perspectives

Peter Gates

University of Nottingham

Qualitative research is a creative process involving categorisation, organisation and theorisation, a cognitive, creative process rarelY made explicit. It is a crucial activity wherein empirical research data is transftrmed into a descriptive, analYtical or theoretical plane Making sense of data might involve index cards, scissors, glue and lots of floor space, as we form categories, allocate text units, reorganise and theorise. It is now possible to use software that offers flexibility, speed and power to do this. One programme, NUDIST (Non-numerical Unstructured Data. Indexing, Searching and Theorising) and is claimed to be one of the best thought-out programs around I describe and discuss my use of NUDIST in developing an embedded cfynamic model of mathematics teacher perspectives based upon the coding of in-depth interviews.

#### 7 Assessing Early Mathematical Development

Ray Godfrey and Carol Aubrey

Canterbury Christ Church University College

The work discussed here is part of an international study involving Dutch, Belgian, German, Greek, Finnish and Slovenian as well as English children. The project is co-ordinated by the University of Utrecht and employs the Utrecht Early Mathematical Competence Test. In the long term our interest lies equally in critiquing the test and the Dutch methodology and in drawing what conclusions may be drawn from the data. This paper looks only at a few aspects of the analysis which has been carried out in the few weeks since the full British data set has been available.

#### 8 Mathematical Anxiety Amongst Primary QTS Students

Stuart Green and Mike Ollerton

St. Martin’s College, Lancaster

This paper investigates mathematical anxiety amongst some primary QTS students. Information for this study has been gathered from students through individual interviews and personal, written, reflections. Analysis of these information sources reveals a number of factors about the seeds of mathematical anxiety, the conditions which foster its growth and, significantly, the depth of emotional responses to mathematics displayed by some students.

If people believe firmly enough that they cannot do math, they will usually succeed in preventing themselves from doing whatever they recognize as math. The consequences of such self-sabotage is personal failure, and each failure reinforces the original belief. And such belieft may be most insidious when held not only by individuals, but by our entire culture Papert, 1980, p42.

#### 9 Limits, Continuity and Discontinuity of Functions from Two Point of View: that of the Teacher and that of the Student

Fernando Hitt, Cinvestav; University of Nottingham

Hector Lara-Chavez, lnstiuto Tecn6logico de Zacatepec-UAEM

In this study we show that a primitive idea of limit is inducing an obstacle in the construction of the concept of limit, continuity and discontinuity of functions and an unsuitable internal conceptual structure of the idea of infinity (ambiguity between potential and actual infinity). We analyse a particular teacher’s lesson on the concept of limit of functions and relations to continuity and discontinuity of functions, and we explore the ideas one student has on the same concepts through interviews. We show that the teacher’s use of natural language, when introducing the idea of limit, emphasise a primitive idea; also, he tries to induce basic algorithm strategies to influence students’ learning in an inappropriate way. The wrong strategy followed by the teacher will influence the students’ construction of the concept of limit, where the role of infinity is ambiguous, producing a cognitive obstacle as is pointed out in this case.

#### 10 Training and Practice of Teachers of Probability: an Epistomological Stance

Ana-Marfa Ojeda S.

Cinvestav del lPN, Mexico; University of Nottingham

Steinbring’s proposition of the epistemological triangle (1997) and Duval’s work on semiotic registers of representation (1996) are the grounds of this research on stochastics in the elementary level of Mexican education. The training and practice of teachers of probability in secondary school classrooms (12-15 year old pupils) have been explored with a qualitative approach (Alquicira, 1998). Generally, the training of teachers does not seem to correspond to what the teaching of probability in classroom demands. A linear, formal conception of probabil¬ity underlies a teaching using mainly simplified situations and neglecting semiotic registers other than the algebraic register, with a teacher centred exposition strategy. Whereas teachers’ train¬ing referred to the epistemological triangle may result in teaching based on richer situations and on an interplay of semiotic registers of representation for the students to make sense of the activity, in which they become involved.

#### 11 Conjecturing in Open Geomtric Situations in Cabri-Geometre: an Exploratory Classroom Experiment

Federica Olivero

Graduate School of Education, University of Bristol

In this paper I present a description of a classroom experience which I carried out with a class of Year ]0 students]. It consisted of a sequence of activities aimed at introducing students to conjecturing and justifying in geometry, within the Cabri-Geometre environment. In particular I shall focus on how the intervention was planned in collaboration with the teacher and how students reacted. An evaluation questionnaire was given to both students and teacher at the end of the experiment. The issues raised, mainly concerning working in groups, working with Cabri and open problems, need to be taken into account when designing future teaching and learning sequences.

#### 12 Analysing Data on the Relationship Between Teaching and Learning Addition in a Primary Classroom

Alison Price

Oxford Brookes University

In the analysis of my data on the teaching and learning of addition in Reception and Year One classes I am attempting to characterise how the teacher represents mathematics to the children and the children’s responses in understanding and learning. Here I will describe the process I am developing and welcome discussion on its ability to describe the teaching/ learning interface.

#### 13 Two Marks out of Ten for Constructivism

Stuart Rowlands

Centre for Teaching Mathematics, University of Plymouth

I have a candidate for the most dangerous contemporary intellectual tendency, it is…constructivism. Constructivism is a combination of two Kantian ideas with twentieth-century relativism. The two Kantian ideas are, first, that we make the known world by imposing concepts, and, second, that the independent world is (at most) a mere’thing-in-itself forever beyond our ken …. [considering] its role in France, in the social sciences, in literature departments, and in some largely well-meaning but confused, political movements [it] has led to a veritable epidemic of ‘worldmaking’. Constructivism attacks the immune system that saves us from silliness (Devitt, taken from Matthews 1997).

#### 14 Does ‘CAME’ Work? (2) Report on Key Stage 3 Results Following the Use of the Cognitive Acceleration in Mathematics Education (CAME) Project in Year 7 and 8

Michael Shayer, David C. Johnson and Mundher Adharni

King’s College, University of London

The CAME project aims to contribute to pupils’ achievements and teachers’ professional development by basing classroom practice on research and theory – applicable research. Three major sources are drawn upon: a) research on levels of achievement in mathematics; b) Vygotskian psychology and social constructivism as exemplified in the recent literature on classroom cultures; and c) Piagetian/neo-Piagetian theories on levels of reasoning. These sources have been integrated to provide a theoretical foundation for teacher intervention and pupil-pupU interaction in the early years of secondary school mathematics aimed at increasing pupils’ intellectual development. Findings from the research indicate that the intermediate aims have been achieved – an increase in classroom interactions and significant improvements in pupils’ attainmentsin mathematics. An interesting feature is that the value-added effects were even larger in science and English.

#### 15 Student Teachers’ Responses to Influences and Beliefs

Jim D N Smith

Sheffield Hallam University

This article briefly summarises part of an ongoing study of the teaching styles of student teachers of secondary mathematics (Smith, 1996,1998). The aspect reported on in this article is an analysis of the effects of the influences brought to bear upon four individual student teachers of secondary mathematics as they progress through a one year postgraduate course of teacher training (PGCE) based at Sheffield Hallam University. The students chosen have differing initial beliefs about teaching, learning and mathematics. As anticipated in the literature, the students’ initial beliefs survive virtually intact throughout the year. However, the study suggests that the link between initial beliefs and teaching style is not causal. The study points to ways of encouraging student teachers to employ a range of teaching styles.

#### 16 Geometry Working Group Report: Providing the Motivation to Prove in a Dynamic Geometry Environment

Report by Catia Mogetta, University of Bristol, Federica Olivero, University of Bristol and Keith Jones, University of Southampton

Convenor: Keith Jones, University of Southampton

The use of dynamic geometry software may provide opportunities to improve the teaching and learning of mathematical proof within the context of plane geometry. Yet, it seems, if the approach to proving continues to emphasise a standardised linear deductive presentation, little improvement in student conceptions may result. This paper considers the design of geometrical tasks that could provide the motivation to prove.