**Proceedings of the Day Conference held at School of Education, University of Leeds, November 1998**

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## Contents

#### 1 Kenneth Lovell (1915 -1996): Methodist and Pioneer in Mathematics Education Research

Anthony Orton

University of Leeds

#### 2 Routine Questions and A-Level Mathematics Grades

This study concerns student performance in A-level examination questions. In particular whether lower attaining students in mathematics examinations generally gain their marks on routine parts of questions? Students’ scripts in a recent mathematics examination were examined in an attempt to evaluate this question. The question is an important one because routine questions could be awarded fewer marks if algebraic calculators are allowed in examinations. The results are not conclusive but indicate that a problem of this type does exist.

John Berry and Wendy Maull, University of Plymouth

Peter Johnson and John Monaghan, University of Leeds

#### 3 Developing Algebra – A Case Study of the First Lessons from the Beginning of Year 7

This is a case study from the start of a project, funded by the Training and Development Agency For Schools (ITA), looking at one mixed ability year 7 class in a Bristol comprehensive, investigating what algebra is taught and what algebra is learnt over the period of their first term at secondary school. The motivation for the study is to explore the challenge inherent in the quote: ‘Can we develop a school algebra culture in which pupils find a need for algebraic symbolism to express and explore their mathematical ideas?’ (Sutherland 1991). We see the role of the teacher as that of setting up a ‘community of practice’ (Lave and Wenger 1991) in the classroom, where the practice is that of ‘inquiry’ (Schoenfeld, 1996). Resultsfrom the first seven lessons with the group are presented and analysed, setting up further research questions.

Alf Coles, Kingsfield School

Laurinda Brown, University of Bristol Graduate School of Education

#### 4 The Role of the Semiotic Representations in the Learning of Mathematics

We argue that mathematics visualisation and the construction of concepts require the coordination of different representations that belong to different semiotic systems of representations. Unsuitable construction of a concept will impede the acquisition of more abstract conceptualisation. In this context, we will discuss about cognitive obstacles which some teachers and students have when facing mathematical problems related to precalculus and calculus topics. Mathematics visualisation in this context deals with an holistic vision that articulates several representations in a problem solving situation.

Fernando Hitt

University of Nottingham

#### 5 Do KS3 SATs Testfor Qualitatively Different Forms of Mathematical Understanding

Here I present case studies of individual pupils who, whilst achieving the same SATs levels, seem to display different forms of understanding of mathematics. This raises the question about whether KS3 SATs distinguish between procedural and conceptual understanding, as was originally intended.

Andrea Lowe

University of Warwick

#### 6 Mental Models of Force and Motion in 11 to 18 Year Olds

Previous studies have suggested that students may compartmentalise knowledge: their everyday intuitions will serve in everyday contexts and their academic knowledge is activated, if ever, in academic contexts. In this study we put forward a case where significant numbers of students of Newtonian theory provide mixed explanations of motion in the same context. They typically switch from one model to another depending on the values of the masses and forces involved.

Peter Mildenhall

University of Manchester and Bury Grammar School

#### 7 The Research of Ideas of Probability in the Elementary Level of Education

Ideas of probability have been investigated in Mexican elementary education.

Two examples are given to illustrate the way in which epistemological aspects are considered in this research. Teaching experiments with 6-7 year old children suggest that pupils’ interpretations of the tasks they were asked about may result in answers which do not inform on their idea of chance, since they tend not to focus on it. Additionally, by using questionnaires and clinical interviews with 10-15 year old children, it was found that their correct handling of fractions does not assure that they can cope with questions about probability for which a quantification would suppose fraction use.

Ana-Maria Ojeda

University of Nottingham

#### 8 Key Stage 3 Statutory Assessment 1998 – Report on an ATUATM/NATE Joint Study

The Key Stage 3 tests in mathematics are now well established, and QCA ‘ s annual evaluations suggest that they are generally well regarded. In this study, initiated by the curriculum committee of the Association of Teachers and Lecturers, the views of teachers suggest that whilst the tests for mathematics are broadly accepted, there are still a number of issues that need to be addressed: compatibility with GCSE, transparency about which level of attainment each question targets, mental mathematics (time for questions, differentiated papers), how returned scripts are used and the status of teacher assessment.

Sue Pope

Roehampton Institute, London

#### 9 A Critique of Relativism in Mathematics Education: The Needfor an Objectivist Perspective ifwe are to Facilitate Cognitive Growth

Many constructivists tag as ‘absolutist’ references to mathematics as a body of knowledge, and stake-out the moral high-ground with the argument that mathematics is not only utilised oppressively but that it is, in-itself, oppressive. With much reference to Paul Ernest’s (1991) Philosophy of Mathematics Education this tag has been justified on the grounds that ifmathematics is a social-cultural creation that is mutable and fallible then it must be social acceptance that confers the objectivity of mathematics. I will argue that mathematics is a body of knowledge the objectivity of which is independent of origin or social acceptance. Recently, Paul Ernest ( 1998) has attempted to include the category of logical necessity in his elaboration of the objectivity of mathematics. I will argue that this inclusion of logical necessity not only represents a V-turn, but that the way in which Ernest has included this category is an attempt to maintain his earlier position that it is social acceptance that confers the objectivity of mathematics.

Stuart Rowlands

University of Plymouth

#### 10 Distributed Cognition, Computers and the Interpretation of Graphs

One aspect of recent research on students’ understanding of graphs in computer and in pencil and paper media is reported on. Over 200 Year 10 students in two comprehensive schools in Leeds participated in a teaching experiment, 100+ with and 100+ without the use of computers. Pre/post-tests and interviews provide data for a comparison of learning with the two media. The wider study examines students’ construction and interpretation of context graphs (related to situations) and context free graphs (x-y graphs). This paper concentrates on students’ interpretation of context free graphs.

Pumadevi Sivasubramaniam

University of Leeds

#### 11 Children’s Understanding of Number Patterns

This article deals with children’s reactions when they are confronted with problems concerning number patterns. In German schools, problems concerning patterns and number patterns are rare, some curricula do not even mention them.

Therefore my documented tests are the first contact with these problems for most children. This is also a big opportunity for my research, because the aim is to show the genuine reactions of the children. The main topic of this article will be some results of a school-project with 9 to 10-year olds.

Anna Susanne Steinweg

University of Dortmund, Germany

#### 12 Are Mental Calculation Strategies Really Strategies?

Some of the approaches used by children when calculating involve a flexible and inferential use of number knowledge. These approaches are often called “strategies”, and it is a reasonable ambition for teaching to enhance children’s skill in this area. But what exactly does effective performance of this kind involve? This paper considers to what extent children’s calculation

‘strategies’ are really strategic, and explores the viability of teaching mental calculation through direct instruction in “strategies”.

John Threlfall

University of Leeds

#### 13 Mental Calculation Strategies Employed by Teacher-Trainee Students

First year student teachers were asked to mentally solve different types of addition and subtraction problems typically found in mental arithmetic practice tests. Subsequently they were asked to describe how they found the answers, in terms of figuring skills such as counting-on, counting-back, counting-up, using number bonds etc. The student replies were analysed and classified according to which figuring skills were applied to which type of problem. Summaries of these figuring skills applied to the decontextualised question numbers were also demonstrated diagrammatically.

David Womack

University Manchester

#### 14 Geometry Working Group

This report focuses on the use of imagery to solve a range of spatial problems. The research projects reviewed in this report offer some insight into the range of strategies used by solvers of spatial problems and point to relationships between spatial and verbal skills.

Convenor: Keith Jones

University of Southampton

#### 15 Advanced Mathematical Thinking Working Group

The last session highlighted an interest in the idea of the procept in elementary mathematical thinking and its possible use in transfer to advanced mathematical thinking (AMT). This session set out to analyse the paper by Gray and Tall (1994), to enlighten the group’s understanding of the idea and its consequences. The group offered mixed reactions towards the paper and advanced mathematical thinking as a genre. This paper highlights the main themes of the paper, the reactions of the group and future ideas for discussion.

Stephen Hegedus

University of Southampton

#### 16 Primary Student Teachers’ Understanding of Mathematics and Its Teaching: A Preliminary Report

The research set out to investigate the ways in which non-specialist student teachers conceptualise mathematics and its teaching and how their views evolve as they progress through an initial training course.

Tony Brown, Una Hanley, Liz Jones, Olwen McNamara

Manchester Metropolitan University

#### 17 Research Report on the Classroom Implementation of the Socratic Method in Mechanics

Following twenty years of international research into student ‘misconceptions’ offorce and motion, there is an ongoing debate about how to improve student understanding in mechanics. Following recent research into the Socratic method for overcoming misconceptions in mechanics, this research is now continuing in the classroom to evaluate the effectiveness of, and to refine, the method. At this stage of the research, the Socratic method is being developed with a view to creating a strategy for handling intuitive responses to concept questions. After an initial classroom session and a series of individual interviews with a particular series of Socratic questions, the responses were considered and the questions refined before further taped interviews. In this session we will report on initial work carried out within the topic of projectile motion. We will then discuss some of the issues raised in attempting to develop this approach.

Kathy Green, Maxine Mercer, Peter McWilliam and Stuart Rowlands

University of Plymouth

#### 18 Cabri as a Cognitive Tool

The main aim of this paper is to analyse the role of Cabri-Geometre (Baulac, Bellemain & Laborde, 1988) as a cognitive tool in the teaching and learning of mathematics.

Catia Mogetta

University of Bristol Graduate School of Education

#### 19 Misconceptions of Modelling in Mechanics: A Review of the Recent A-Level Textbooks in Mechanics

A-level boards have included a modelling approach in the syllabus of their respective mechanics modules, and the recent plethora of textbooks, including those that represent the various exam boards (e.g. AEB, London, MEI), are structured according to this approach. This session will review the many recent A -level textbooks in mechanics that have included a modelling approach, and will attempt to show that these books are confused as to what modelling in mechanics actually is and the possible pedagogical consequences.

Stuart Rowlands

University of Plymouth