# BSRLM Proceedings: Vol 22 No 3 November 2002

**Proceedings of the Day Conference held at Nottingham University, November 2003.**

## Contents

### Research Reports

#### 1 Prospective teachers' subject matter and pedagogical content knowledge of variables

Nihat Boz

University of Warwick

The knowledge base of teaching is an amalgamation of different forms of knowledge. In this paper I focus on prospective teachers' subject matter knowledge as a source of pedagogical content knowledge. I illustrate how subject matter knowledge affects prospective teachers' pedagogical decisions in the context of variables.

#### 2 Somatic markers: teachers' decision-making and students' emotioning in mathematics classrooms

Laurinda Brown and David A Reid

Graduate School of Education, University of Bristol and School of Education, Acadia University, Canada

Teachers and students of mathematics continually act in complex situations, often without time for reflection. Damasio (1996) develops what he calls his 'somatic marker hypothesis' to account for how people manage to accomplish this. By 'somatic marker' Damasio means a bodily predisposition that informs our decision making. Here we describe how we have used the idea of somatic markers in our previous work on teachers' complex decision-making (Brown and Coles, 2000) and students' explaining in mathematics classrooms (Reid, 1999, 2002) and share our current thinking in relation to how we are working to locate somatic markers.

#### 3 Limit - a proof-generated concept

Bob Burn

University of Exeter (retired)

This paper examines the components of the modern definition of the limit of a sequence, in terms of their historical genesis. In classical Greek times it was recognised that limiting arguments must be pursued with inequalities. With Newton the use of a quantity 'as small as one may wish' (our modern ) was combined with inequalities. It was Cauchy who recognised that smallness was to be achieved 'with sufficiently large N'. In each case these components of the modern definition emerge in proofs not in definitions. None of these components conventionally play a part in pre-university proofs and this may explain why the formal definition of limit is regarded as an 'epistemological obstacle'.

#### 4 Building a holistic view of mathematical thinking - data evaluation of improving attainment in mathematics project

Els De Geest, University of Oxford

Anne Watson, University of Oxford

Stephanie Prestage, University of Birmingham

This paper describes how the thinking process behind the data evaluation of a research project led to considering a holistic view of mathematical thinking. Improving Attainment in Mathematics Project (IAMP) is funded by the Esmee Fairbairn Foundation (grant number 01-1415) and involves three academic researchers and nine teachers-co-researchers. The aim of the project is to explore and develop ways of teaching and learning of below average attainers, focussing on stimulating mathematical thinking and understanding of key ideas in mathematics.

#### 5 Learning effect as a structuring resource in algebraic problem solving activity

Logeswary Doraisamy

Mathematics Education Research Centre, Institute of Education, University of Warwick

The present study looked at the problem solving actions of a group of high achieving secondary school students on algebra problems. In this paper, I report on the solution activity of Year 12 students (aged 16 years) on non-standard algebra problems, looking in particular at how that activity was shaped by recent experience on similar problems.

#### 6 The sharing of meaning of mathematical words in a bilingual classroom : a semiotic interpretation

Marie T. Farrugia

University of Malta

The teaching and learning of Primary school mathematics in Malta involves substantial use of code-switching between the local language Maltese, and English. Mathematical terms are usually retained in English. A case-study was carried out to explore the various language strategies that a Primary school teacher used in order to share the meaning of such terms with her seven-year-old pupils. The focus of this paper is the word value. The direct translation of this word from the Maltese tiswa is viewed as a chain of signification and a semiotic model is developed in order to interpret this pedagogic strategy.

#### 7 Theoretical-computational conflicts and the concept image of derivative

Victor Giraldo, Universidade Federal do Rio de Janeiro, Brazil

Luiz Mariano Carvalho, Universidade do Estado do Rio de Janeiro, Brazil

David Tall, University of Warwick

Recent literature has pointed out pedagogical obstacles associated with the use of computational environments on the learning of mathematics. In this paper, we focus on the pedagogical role of computer's inherent limitations on the development of learners' concept images of derivative and limit. In particular, we intend to discuss how the approach to these concepts can be properly designed to prompt a positive conversion of those limitations to the enrichment of concept images.

#### 8 Reflections on the role of task structure in learning the properties of quadratic functions with ict - a design initiative

Stephen Godwin, University of Bristol

Rob Beswetherick, John Cabot City Technology College, Bristol

This paper will draw on research being developed within the Teaching and Learning strand of the ESRC InterActive Education: Learning in the Information Age project which is examining the ways in which new technologies can be used in educational settings to enhance learning. It will focus on the learning and understanding of quadratic functions using a graphical software package and includes a discussion of how the structuring of the activities influences the nature of the learning environment and how it might influence student exploration of mathematical concepts.

#### 9 Extending a sequence of shapes: pictures, patterns and problems

Jenny Houssart, Hilary Evens

Centre for Mathematics Education, Open University

We consider children's responses to a sequence question from the 2001 Key stage 2 National Curriculum tests. The most common method of successful solution involved some form of table of numbers. Other methods included drawing and use of a relationship. The idea of a 'best method' proved problematic, as both the apparently sophisticated and reliable methods produced errors.

#### 10 Towards new trends on the role of users of technology: a look at some research fields

Bibi Lins

University of Bristol - CAPES (Brazil)

Nowadays the role of users of technology is being more and more acknowledged and it is becoming crucial for various fields of study to (re)look at it, specifically when concerning working organisations and educational settings. This paper briefly discusses ontological spaces that users of technology are located within the fields of Artificial Intelligence (AI), Human-Computer Interaction (HCI) and Sociology of Technology (ST). It also shows how the awareness of the role of users of technology is gradually changing the focus of such fields. At last, an outline of my PhD studies is presented, which concerns the role of users of technology in the Mathematics Education field.

#### 11 Exploring critical sense in graphing

Carlos Monteiro and Janet Ainley

Mathematics Education Research Centre, Institute of Education, University of Warwick

In current social contexts there are various situations in which people participate in graphing activities. The school has an important role in the teaching of graphing knowledge to citizens. Several researchers have stressed critical sense as an important aspect of the data handling process. This paper reports on a pilot study exploring some tasks in which primary school teachers might approach graphing, using critical sense as an important element. Analysis of the results suggests factors, which may be significant in the design of such tasks.

#### 12 The use of origami in the teaching of geometry

Sue Pope

St. Martin's College, Lancaster

#### 13 Mathematics education and applied linguistics: working group report

Richard Barwell, University of Bristol

Constant Leung, King's College, London

Candia Morgan, Institute of Education

Brian Street, King's College London

The aim of this second meeting of the working group was to engage in issues arising from applied linguistics research. The session focused on an example of data from a project in which Brian Street is involved. Having circulated the data beforehand, Brian invited a discussion around some of the issues a group of mathematics may be able to address in a more informed way, or at least in a different way, from an applied linguist. This short report begins with a brief outline of the data discussed, followed by Brian's reflections on the discussion and how it took his thinking forward.

#### 14 Opportunities for the development of geometrical reasoning in current textbooks in the UK and Japan

A report based on the meeting at the University of Nottingham, 16th November 2002

Taro Fujita, Curriculum Studies, University of Glasgow

Keith Jones, Research and Graduate School of Education, University of Southampton

Developing a good model of the school geometry curriculum continues to be one of the most important tasks in curricular design in mathematics. This paper reports on an initial analysis of current best-selling textbooks used in lower secondary schools in Japan and the UK (specifically England and Scotland). The analysis indicates that, following the specification of the mathematics curriculum in these countries, Japanese textbooks set out to develop students' deductive reasoning skills through the explicit teaching of proof in geometry, whereas comparative UK textbooks tend, at this level, to concentrate on finding angles, measurement, drawing, and so on, coupled with a modicum of opportunities for conjecturing and inductive reasoning. The available research suggests that each approach has its own strengths and weaknesses. Finding ways of capitalising on the strengths and mitigating the weaknesses could prove helpful in formulating new curricular models and designing new student textbooks.