**Proceedings of the Day Conference held at the University of Bristol on 02 Mar 2013**

## Contents

#### 01 Learning as a Mathematics Teacher Educator through Narrative Interviewing

Laurinda Brown

University of Bristol, Graduate School of Education

Over many years, I have carried out narrative interviews with practising mathematics teachers for a range of purposes: on their first lessons of the school year; on their role as mentors for mathematics student teachers; and focusing on their learning as teachers of algebra to year 7 students (to mention a few examples). I focus on two strands that illustrate my learning as a mathematics teacher educator; the first on what I have learnt about interviewing, particularly narrative interviewing where stories are co-constructed during the interviews; and the second on my learning about mathematics teacher development that gets applied back into my work on a secondary one-year PGCE course.

#### 02 The use of Bloom’s taxonomy in advanced mathematics questions

Ellie Darlington

University of Oxford

Bloom’s taxonomy for educational assessment has been regularly criticised by many in mathematics education for being particularly ill-fitting to mathematics, and yet continues to be used and discussed in this field. However, the Mathematical Assessment Task Hierarchy (MATH) was designed specifically for the development of advanced mathematics assessments in order to ensure that students are assessed on a variety of knowledge and skills. Here, I use MATH to contrast the types of questions posed in English A-level mathematics and further mathematics examinations with those in the University of Oxford’s mathematics admissions test.

#### 03 ‘In this game you don’t stand still’: policy change as a catalyst for professional development

Jennie Golding

King’s College London and Institute of Education, London

Two well-placed mathematics departments were followed through a complete first cycle of a new examination at 16. One achieved a principled enaction consistent with their espoused beliefs and as a result, developed considerably over a wide range of professional competencies. Differential characteristics of that department are considered in an attempt to isolate those features which supported deep change.

#### 04 Solving word problems algebraically in a spreadsheet environment in primary school

José Antonio González-Calero^{a}, David Arnaub and Luis Puig^{b}

^{a}Universidad de Castilla-La Mancha, ^{b}Universitat de València Estudi General

This paper presents some results from an investigation into the teaching of the algebraic solving of word problems in a spreadsheet environment in the sixth grade of primary school in Spain (11-12-year-old pupils). The main aim of the study was to investigate whether the spreadsheet could be a mediator to the teaching of algebraic problem solving. Through the analysis of excerpts from a case study, the core of the paper is focused on two different types of the difficulties that students showed when solving problems algebraically in a spreadsheet environment.

#### 05 Teaching problem solving to mathematics student teachers in a Malaysian context

Norulhuda Ismail

Institute of Education, University of London

Problem-solving is viewed as an important component in mathematics teaching and learning in Malaysia. As part of my research project on student teachers’ recontextualisation of problem-solving, I would like to share how one mathematics teacher training program in Malaysia teaches problem-solving to student teachers. The courses within the program which focus on problem solving are Methods for teaching mathematics, Laboratory in mathematics education, Microteaching and Operational Research. I am using a critical discourse analysis approach on the data, focusing on what counts as a problem, what problem-solving processes are demonstrated and the values portrayed about the problems and problem-solving processes. The similarities and differences in the ways the courses portray problem-solving and its values are emphasised.

#### 06 Learning congruency-based proofs in geometry via a web-based learning system

Keith Jones, Taro Fujita and Mikio Miyazaki

School of Education, University of Southampton, UK; Graduate School of Education, University of Exeter, UK; Faculty of Education, Shinshu University, Japan;

Congruence, and triangle congruence, in particular, is generally taken to be a key topic in school geometry. This is because the three conditions of congruent triangles are very useful in proving geometrical theorems and also because triangle congruency leads on to the idea of mathematical similarity via similar triangles. Despite the centrality of congruence in general, and of congruent triangles in particular, there appears to be little research on the topic. In this paper, we use evidence from an on-going research project to illustrate how a web-based learning system for geometrical proof might help to develop Year 9 pupils’ capability with congruent triangles. Using the notion of ‘conceptions of congruency’ as our framework, we first characterise our web-based learning system in terms of four different ‘conceptions’ of congruency by comparing the online tasks with activities from a Year 9 textbook. We then discuss how the web-based learning system would aid pupils when they are tackling congruency-based proofs in geometry.

#### 07 Primary school teachers develop their mathematics teaching

Jónína Vala Kristinsdóttir

University of Iceland, School of Education

This ongoing study aims at learning to understand how teachers meet new challenges in their mathematics classes. The research is a qualitative collaborative inquiry into mathematics teaching where seven primary school teachers research their mathematics teaching together with a teacher educator. The results indicate that the teachers are slowly adapting to the processes of reflective practice and studying their own practice.

#### 08 A-level mathematics reform: satisfying the requirements of university courses across the range of mathematical subjects

Peter Osmon

Department of Education and Professional Studies, King’s College London

Three groups of subjects have been identified that are mathematical to varying degrees, with requirements for three corresponding levels of foundational mathematical knowledge (Osmon 2009). But effectively only one post-16 mathematics qualification exists in England: GCE Advanced level. This matches the foundational requirement of the most mathematical group and serves as an entry requirement for their university courses but the numbers are far fewer than the total entering courses in mathematical subjects and consequently the other two subject groups’ courses are populated with mathematically underprepared students. In principle, courses leading to the existing AS-level award and a new sub-AS award could meet the pure and applied mathematics needs of the other two groups, and means for specifying them are described. A progression scheme whereby these three courses form a single post-16 mathematics pathway, along which learners travel as far and as fast as their mathematical abilities and ambitions take them, and with the three awards as exit points, is described. It would use scarce mathematics teaching resources parsimoniously. But because of the widespread culture of mathematics aversion take-up would depend critically on universities using the awards as admission requirements.

#### 09 Professional development for problem solving: a dialogue with one teacher

Miroslava Sovičová^{a} and Marie Joubert^{b}

^{a}Constantine the Philosopher University in Nitra, Slovakia; ^{b}University of Nottingham, UK

It is generally accepted that it is important that problem solving is taught in schools. However, it is well recognised that teachers lack confidence in teaching problem solving in mathematics and that professional development may support them in developing their confidence and skills. One approach to providing such professional development is through the use of curriculum materials, such as those produced by the Mathematics Assessment Project (MAP). This paper reports on a small research study, which explored one teacher’s professional development as he used the MAP materials.

#### 10 The study of intuitions in one prospective teacher’s constructions of mathematical objects

Andonis Zagorianakos

Manchester Metropolitan University

The research is a study of the Husserlian approach to intuition, as it is substantiated by Hintikka, in the case of a prospective teacher of mathematics. It is a case study based on data collected from a course where the students were engaged in open-ended tasks and they were free to choose their own ways of exploring them while working in groups, without the teacher’s intervention. A phenomenological approach that takes objects as self-given and focuses on the student’s intuitions reveals mathematical objects that surfaced from her investigation and the particular circumstances that led to these objects. The research exemplifies the two intuitive stages introduced by Husserl, while introducing a method of discerning them, and argues for the essential part that intuitions play in the construction of mathematical objects.

## Working Group Reports

#### 11 Report of the mathematics education and the analysis of language working group

Geoffrey Kent and Alf Coles

Institute of Education, London and University of Bristol

In this report we present key points from the input and discussion at the working group meeting during the day conference in Bristol.