Proceedings of the Day Conference held at Manchester Metropolitan University, March 2001
Kingsfield School, South Gloucestershire, and University of Bristol
The data for this study is taken from a project  looking into the development in year 7 students (aged 11-12) of a ‘need for algebra’ (Brown and Coles, 1999) in four teacher’s classrooms in the UK. I introduce the notions of evaluative, interpretive and transformative listening, (adapted from Davis, 1996), to analyse three transcripts from lessons of one teacher on the project. The project design and case study were informed by ideas of enactivist research (Varela, 1999, Reid, 1996). A change occurred in Teacher A’s classroom, as shown in the transcripts, and the listening of both students and teacher became transformative. There is evidence that specific teaching strategies were linked to this change in listening. Once the change occurred the students started asking their own questions within the mathematics.
Constantia Hadjidemetriou and Julian Williams
University of Manchester
We report a study of 12 secondary school teachers’ knowledge of their pupils’ errors and misconception in graphical reasoning. A diagnostic test, previously given to their pupils, was used as a questionnaire to these teachers with instructions that they should record their perception of the difficulties of the items on a Likert scale, and suggest misconceptions students might have that would cause difficulty. We built a rating scale and the item-perception-difficulty measures that resulted were correlated with the children’s actual difficulty as estimated by the test analysis. In addition we sought to confirm the teachers’ responses through informal interviews. The teachers’ mis-estimation of (relative) difficulties could be explained by one of two reasons: sometimes teachers apparently misunderstood the actual question themselves thus underestimated the difficulty of the item. At other times teachers overestimated the difficulty because they did not realise that children could answer the question without a sophisticated understanding of some concepts i.e the gradient.
Katrina Daskalogianni and Adrian Simpson
Mathematics Education Research Centre, University of Warwick
The purpose of this paper is to present a categorisation of high achieving upper sixth-form students’ beliefs about mathematics as a discipline, about themselves as learners and when working in mathematics. The analysis of data suggests that there are three coherent categories of macro-beliefs among students and within each one there is a system of micro-beliefs. We will argue that these systems of macro-beliefs act as a potential medium of predicting students’ working habits and approaches in a given mathematical problem and we will discuss possible ways in which coherent systems of beliefs may change over time.
Cheadle Hulme College
This paper introduces some problems encountered during my research, which was based on a Vygotskian, Activity Theory approach to the teaching of mathematics. It attempts to focus attention on the dialectical complexities of practical problem solving activity.
Hazel Denvir and Mike Askew
Leverhulme Numeracy Research Programme
King’s College, University of London
Longitudinal case study data are informing the different ways that pupils engage and manage their participation within the lesson. Drawing on observations of two children in the “mental and oral” starter of the Numeracy Lesson we develop the argument that, within the whole class sessions, pupils appear engaged with the mathematics in the way the teacher expects them to be while in fact they are engaged in other ways and for reasons other than interest in the mathematics.
This paper explores the challenge of developing a whole school numeracy strategy involving all staff (including non-teaching staff) of a secondary school for boys with emotional and behavioural difficulties. The first stage is to test all pupils for numeracy. The second stage is to place all pupils in 3 bands and set targets for each band. Thirdly, the aim is to begin numeracy sessions, initially of 20 minutes twice a week. This paper will outline possible approaches to preparing staff for the delivery of the numeracy programme.
Dietmar Küchemann and Celia Hoyles
Institute of Education, University of London
We report on responses of high attaining thirteen-year-old students to a multiple-choice geometry question (G3) that formed part of a written survey designed to test mathematical reasoning. We describe trends in the choices made and put forward some suggested reasons for these trends.
Razia Fakir Mohammad
The University of Oxford, Department of Educational Studies
The aim of this study was to explore teachers’ learning in their classrooms by trying to create a co-learning relationship between teachers and a teacher educator. The study was conducted with three mathematics teachers in classrooms in Pakistan. Data was collected through maintaining field notes from the classroom observations, audio-recording conversations in pre and post observations and writing comments in reflective journals. The preliminary analysis uncovered issues of practicality and limitations of a co-learning relationship in teacher development in a real classroom context. A major question arose about the extent of equality in working together.
9 Domain, co-domain and relationship: three equally important aspects of a concept image of function
School of Education, University of East Anglia
Students’ concept images of function often lack in understanding the concept as inextricably connected with its domain, co-domain and relationship. The impact of this on understanding properties such as 1-1 can be dramatic. Drawing on students’ responses to a task involving the exploration of whether five given functions were 1-1, onto, both or neither, I discuss the following distinctive elements in their responses: A. use of graphs to identify properties of the function (from relying completely on the graph for mere identification of the property through to substantiating what the graph suggests with verbal explanations and uses of B and C) B. problematic uses of the formal definitions of 1-1 and onto C. effective use of properties specific to the functions in question (e.g. uniqueness of cubic root). The emphasis here is on B.
Julie Ryan and Julian Williams
University of Manchester
We investigate how children reveal and develop their understanding of mathematics through collaborative argument in group discussion. Working with eleven-year-old children who had different responses to diagnostic test items, we describe how those children developed argument across conceptual locales. The analyses of the discussions led to a chart of the key elements of argument that arose, as well as general strategies for managing such discussions productively. Such devices and strategies are presented as planning tools for classroom teachers in the next stage of our study.
11 The effect of metacognitive training on the mathematical word problem solving of lower achievers in a computer environment
Su Kwang Teong
Centre for Studies in Mathematics Education, University of Leeds
This study demonstrates that explicit metacognitive training appear to benefit lower achievers’ mathematical word problem solving in a computer environment. 11 to 12- year-old Singaporean students in collaborative pairs were assigned to two word problem solving groups. The first group received explicit metacognitive training before word problem solving with WordMath (treatment); and the second group undertook word problem solving with WordMath (control). Results from the analysis of pair think aloud protocol data suggest that treatment lower achievers appeared to be more successful and elicited better regulated metacognitive decisions than control lower achievers.
National Institute of Economic and Social Research
Levels of pupil mobility are known to vary significantly between areas and types of schools but hitherto there has been only limited evidence on the link between pupil mobility and low attainment. Recently available data from a large-scale mathematics study are used here to examine this relationship. A case study of one school is used to illustrate the considerable problems associated with high pupil mobility, including the link with deprivation and resulting social needs. This preliminary study also points out the dangers of comparing school performance in national tests without considering the impact of changing school populations.
Derek Woodrow and Janis Jarvis
Manchester Metropolitan University, Institute of Education
Applications for entry to Higher Education show marked differences between ethnic minority groups and marked gender preferences, particularly for mathematics. Analysing UCAS data shows that these differences are persistent. Research into learning preferences suggests that these might be one reason for these differential choices. This paper reports a study of over 417 undergraduates and 628 PGCE students, identifying clear subject differences in learning preferences. Mathematics students lie at one end of the scale and English students at the other.
Manchester Metropolitan University
Colleagues will see from a parallel group that CoPrIME and BSRLM are working towards defining a number of questions which research in mathematics education needs to address. Amongst the collections of issues to be identified are a number drawn together under the heading of Mathematics and Society. This paper will hope to explore some of these foci.
FRAME Teaching Development Group Strand
Compiled by Barbara Jaworski
This is a report of a special discussion group held at the BSRLM Day in Manchester on March 3rd 2001, forming a strand of three sessions throughout the day. It was organised by the FRAME Teaching Development Group for discussion of issues arising from a draft document addressing Mathematics Teaching Development and Teachers’ Professional Education and Development (McNamara, Jaworski, Rowland, Hodgen, Prestage and Brown, 2001.)
16 Geometry and proof: A report based on the meeting at Manchester Metropolitan University, 3rd March 2001
Keith Jones and Melissa Rodd
University of Southampton and University of Leeds
Is Euclidean geometry the most suitable part of the school mathematics curriculum to act as a context for work on mathematical proof? This paper examines some of the issues regarding the teaching and learning of proof and proving specifically in relation to Euclidean geometry.