**Proceedings of the Day Conference held at the Open University, February 1999**

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## Contents Vol 19 No 1

#### 1 Metacommenting: developing algebraic activity in a ‘community of inquirers’

Alf Coles, Kingsfield School

Laurinda Brown, University of Bristol, Graduate School of Education

We are working on a Teacher Training Agency (ITA) funded project looking at the teaching and learning of algebra with one mixed ability year 7 class. We see ourselves as developing a ‘community of practice’ (Lave & Wenger, J99J) where the practice is not that of the mathematician but of inquirer (Schoenfeld, 1996) into mathematics. The label articulated to the students for this practice was a ‘purpose’ (Brown & Coles, 1996, 1997) for the year of ‘becoming a mathematician’. The teacher in this community acts as role model for inquirer and metacomments (Bateson, 1972) on the practice of inquiry. In this paper we will present evidence for how such metacommenting support algebraic activity. We are also beginning to find evidence of the students metacommenting on the practice of the group.

#### 2 Untaught mental methods of 11-year-olds: data from the 1987 APU survey

Derek Foxman

Institute of Education, University of London

I have been collaborating with Meindert Beishuizen of Leiden University in a reanalysis of the 1987 APU survey data on methods used by 11-yearÂ¬olds to obtain answers to mental arithmetic questions. The methods used by the children would almost certainly have been untaught at that time. The categories used in this analysis have been derived from the work of Beishuizen in the Netherlands and in England The three main methods noted are: manipulating digits (paper and pencil algorithms); splitting numbers into tens and units; and operating on unsplit or complete numbers. So far five of the 12 APU questions have been analysed ;n these terms, with complete number methods generally proving most efficient.

#### 3 A consideration of the way in images are used in primary mathematics texts

Tony Harries, Bath Spa University College

Rosamund Sutherland, Graduate School of Education, University of Bristol

Ian Winter, Graduate School of Education, University of Bristol

This paper results from a study undertaken in 1998 under the auspices of QCA in UK. The study aimed to make a comparison of representative mathematics text books used in primary schools in 5 different countries – France, Hungary, Singapore, United Kingdom, United States of America. In the analysis particular attention is paid to the way in which images are used within the different cultures to represent mathematical concepts and to scaffold pupil learning.

#### 4 What do mathematics tests test?

Caroline Hilton and Tim Rowland

Institute of Education, University of London

Drawing on research that set out, initially, to match Year 7 children with moderate learning difficulties with Year 1 mainstream children, we consider some difficulties with the use of standardised tests. Information gleaned from clinical interviews with these children lead us to question the value of raw test scores in assessing mathematical skills and understanding.

#### 5 Keeping a diary: sharing experiences in the classroom

Susan Hogan

Open University

This report centres on two case studies of Year 7 students. As part of the data collection process, they complete a diary of their experiences at the end of each lesson. The teacher/researcher also keeps a diary, though this is completed later in the day. Each week the three accounts are shared by the students and the teacher in an attempt to gain greater insight into learning and behavioural difficulties. Extracts from the diaries are presented together with the transcript of an interview with one of the students. At the end of a six week study period it was found that the students’ main preoccupation was with the social construction of the classroom rather than the mathematics of the lessons.

#### 6 Teachers’ perceptions of good tasks in primary mathematics

Jenny Houssart

Centre for Mathematics Education, Open University

Key stage two teachers were asked to describe a ‘good task’ which they had carried out with their class or maths set and to consider what makes a good task. A common response was that practical work makes a good task, though few practical tasks were described. Mental work on the other hand, featured in many descriptions but not in lists of factors of a good task. Teachers had some difficulty in discussing mathematical processes.

#### 7 Undergraduate mathematics teaching project: a methodological report of work-in-progress

Barbara Jaworski University of Oxford

Elena Nardi University of East Anglia

Stephen Hegedus University of Oxford

The Undergraduate Mathematics Teaching Project (UMTP) is a one-year study aiming to characterise, and identify issues related to, mathematics teaching in undergraduate tutorials. It builds on earlier research into mathematics learning in undergraduate tutorials and involves a research collaboration between mathematics educators and mathematicians. From participant observation, semi-structured interviewing, and group discussion, it develops a set of qualitative data which is analysed through repeated critical scrutiny to distil characteristics and issues of the teaching experienced which might be seen as germane to a wider variety of settings. Here we discuss the methods of data collection and of the currently ongoing data analysis.

#### 8 Characterising mathematics teaching using tile teaching triad

Barbara Jaworski. University of Oxford

Despina Potari, University of Patras

This paper reports research which attempts to ~ sense of the complexity of mathematics teaching at secondary school leveL The research was conducted in partnership between two teachers and two researchers over one school term in two schools. A theoretical construct, the Teaching Triad, was used both as an analytical device (by the researchers) and as a reflective agent for teaching development (by the teachers).

#### 9 Investigative and formal approaches to mathematics: disjunction or evolution?

Calia Moletta

Graduate School of Education University of Bristol

First year mathematics undergraduates face a massive introduction of formal mathematics as opposed to the investigative and heuristic approach they mainly use dealing with their GCSE and A Level mathematics. This paper outlines some of the results of a study aiming to explore the features of the (possible) disjunction between an investigative and a formal approach to the elaboration of rigorous justifications within a problem solving context. The analysis focuses on the use of arguments drawing on mathematical as well as experiential knowledge in tilt development of productive thinking enacted within the problem solving processes.

#### 10 Convention or necessity? the impersonal in mathematical writing

Candia Morgan

Institute of Education, University of London

Why does most academic writing appear to exclude the voice of the author and the human side o/the subject matter? This tendency to the impersonal is perhaps even more marked in mathematics than in other subject areas and seems to be one of the sources of disaffection for some students. But is it possible to produce more ‘userÂ¬friendly t mathematical texts? Does the nature of mathematics itself constrain the choices available to writers or are the constraints only conventional? How does the personal narrative style of school ‘investigations t relate to the genres used in more advanced mathematics? This paper makes a start at discussing these questions.

#### 11 Young children’s mathematical imagery: methodological issues

Sandra Pendlington

University of Exeter

Imagery is a mental process, hidden to the researcher. This paper discusses the problems of accessing this hidden world, particularly when the research is being undertaken with young children. A pilot study is described in which these problems were experienced, resulting in the development of a grounded theory approach. Some early results from the first case study are used to illustrate three ways in which imagery has emerged wing this method.

#### 12 Mathematics teachers and the use of computers within the classroom

Miriam Penteado

State University of Sao Paulo, Brazil

Graduate School of Education, University of Bristol

Teachers’ engagement is fundamental in order to get effective use of computers in schools. This paper presents work involving mathematics teachers and researchers whose aim was to organise and carry out computer-based activities for a group of secondary school students. This practical work highlighted the relevance of the relationship between the mathematical content to be taught and the resources provided by software, sources which can be used for preparing the activities, and classroom interaction.

#### 13 Primary trainees’ mathematics subject knowledge: an update report

Tim Rowland, Sarah Martyn, Patti Barber, Caroline Heal

Institute of Education, University of London

At a previous BSRLM meeting, we reported that we had failed to find any connection between the strength of primary PGCE students t mathematics subject knowledge (as audited in the course) and their performance in school after two terms. Our findings for the final school placement were very different, however, when there appeared to be an association between subject knowledge and competence in teaching number.

#### 14 Have socioculturists turned vygotsky on his head?

Stuart Rowlands

Centre for Teaching Mathematics, University of Plymouth

In this session I will argue that Vygotsky’s Zone of Proximal Development (ZPD) is a research methodology of theory and practice that attempts to establish psychology in the ‘scientifically based method’ of Marx. I will also argue that the ZPD has nothing to do with ‘the child in social activity with others that emphasise sociocultural conditions’, or communal funds of knowledge ” or ‘situated cognition’, or ‘activity theory’ or any other sociocultural interpretation that is currently in vogue.

#### 15 Parental involvement in secondary mathematics homework

Lin Taylor, Pinder Singh, Patricia Alexander

IMPACT Project, University of North London

This is a report of a pilot study carried out in two North London schools during 1998. The parents were supported in helping their children with their mathematics homework.. We were interested in the effect this would have on the attitudes towards mathematics and homework of the parents and the children, on the communication between parents, children and teachers, and to what extent the parents felt supported. We were also interested in how it affected the children’s mathematical understanding. The results of this pilot study were encouraging, and we are currently involved in further development and dissemination of the project.

#### 16 Advanced mathematical thinking working group: mathematicians as learners

Convenor: Stephen Hegedus, University of Oxford

During the last meeting of the Advanced Mathematical Thinking (AMT) working group at Leeds, the main issue which arose out of our discussion on the role of symbol was the nature of mathematical thinking between novice and expert mathematicians. It was proposed that Professor Leone Burton (University of Birmingham) would address the group at the next meeting following her recent study on the epistemologies of practising research mathematicians.

#### 17 Language Use and Geometry Texts

BSRLM Geometry Working Group Convenor: Keith Jones, University of Southampton

A report based on the meeting at the Open University of Leeds, February 1999 by Anna Chronaki, Open University (with contributions from Keith Jones, University of Southampton)

Recent research suggest that with classroom tasks that combine spatial experiences, mathematising, and communicating, pupils may reveal the nature of their own spatial images and personal language in describing these spatial contexts, and experience the use of formal terminology in making accurate descriptions of their observations and constructions. This report focuses on issues of language use involved in geometry activities when particular emphasis is placed on encouraging pupils’ practice of informal and formal mathematical vocabulary.