ELTMAPS Page

Contents:

- Leo Rogers: Introduction 

Part I: Background and Principles

- Jarmila Novotná and Graham Littler:

Meaningful Mathematics

- Jarmila Novotná and Marie Kubínová:

Changing Classroom Environment and Culture 

- Leo Rogers:

From Icons to Symbols: Reflections on the Historical Development

of the Language of Algebra 

Part II: Social conditions

- Maria Meletiou-Mavrotheris and Despina A. Stylianou:

Advancing from Elementary to Secondary School Mathematics in

Cyprus: A Step or a Leap? 

Part III: Special topics

- Demetra Pitta-Pantazi and Eddie Gray:

Qualitative Differences in Elementary Arithmetic and the Role

of Representations 

- Nicolina A. Malara and Simona Ponzi:

Intuitive Reasoning of Pupils Facing Proportionality

Situations 

- Nicolina A. Malara:

From Fractions to Rational Numbers in their Structure: Constructive

Classroom Episodes Concerning Operations and Ordering 

- Nicolina A. Malara and Rosa Iaderosa:

On the Difficulties met by Pupils Involved in Didactical Research

on the Plane Direct Isometries through the Computer 

- Nicolina A. Malara:

On the Difficulties of Visualization and Representation of 3D Objects

in Middle School Teachers 

Leo ROGERS, Jarmila NOVOTNÁ (Editors), Theory, principles and research. Effective learning and teaching of Mathematics from primary to secondary school, 2003, pp. 210, € 19.00, ISBN 88-371-1393-5

Introduction:

It is common knowledge that children have difficulties in the transition from primary to secondary school. This problem has been the focus of research into cognitive and affective aspects of learning as well as into pupils’ social and emotional problems. There has also been a wide recognition by teachers of the pedagogical difficulties which accompany this transition, not only of the efficient presentation of content, but also in the changes in teaching styles and classroom organisation experienced by pupils. Nowhere are these problems more acute than in the learning and teaching of mathematics, a fundamental school subject. This situation is often exacerbated by poor communication between primary and secondary school teachers, but a fundamental reason for this is that many National teacher education systems do not attempt to bridge this gap, neither do many local educational systems encourage or support such contacts. The ELTMAPS Project grew out of the concern of a group of European mathematics teachers and researchers who came together in this common enterprise to try to identify some core issues in the cognitive area and also to address the "problematique" of presenting mathematical material and helping learners to develop ways of thinking mathematically. This book contains some of the research papers written over the period of the ELTMAPS project reflecting the contexts and problems involved in teaching aspects of mathematics to pupils in the 9 to 14 age range. In these pages it is not possible to do justice to all the topics and discussions covered in the project, so references at the end of each paper lead to selected readings and supporting examples. The choice of subjects herein partly reflects the research interests of the individuals involved, but also relates to the ongoing problems of teaching mathematics in this age range. The first section concerns general problems of background and context, and the problems of presenting mathematics to pupils so that it has meaning and relevance and at the same time changing classroom practices and organisation so that this approach is more possible. While the development of mathematical language - in particular the language of algebra – is approached from a historical point of view, we can reflect on this as a model for appreciation the difficulties pupils have in representing their ideas in the different modes and understanding the significance of the changes from one mode to another. The social conditions of change are addressed in the paper in the second section where at the time of transfer the physical environment changes for pupils and with it, the expectations, ways of working and attitudes of teachers together with the curriculum content and the way it is expressed in texts. It is clear that without good communication between primary and secondary teachers, together with a respect and understanding of each other’s contexts, it is difficult (though not impossible) to overcome these problems. If the educational system has no structure for enabling primary and secondary teachers to meet and discuss common problems, any private initiatives are much more difficult to sustain. Another problem here is the lack of continuity both in philosophy and pedagogical practice in the school curriculum. So often primary pupils face completely different presentation and attitudes when they first enter the secondary school. The issues addressed in the final section of this book cover one of the most significant of research areas, and concern the way in which we construct and use representations. The problem begins with traditional ways of presenting elementary mathematics to children where the kinds of representations found in textbooks and used by teachers are not always the most helpful or efficient for building further mathematical ideas. One of these concerns is the use of part-whole representations of fractions, which have a practical origin and are useful for demonstration and first ideas, but if employed exclusively can prevent children from achieving more sophisticated concepts such as rational numbers and the use of fractions as operators. An interesting experiment discussed here is the introduction of proportionality problems where pupils are encouraged to discuss and reason with each other, and justify their answers. These problems are introduces without any formal instruction, and it is  interesting to see how pupils develop their ideas and the appropriate language in these situations. The development of appropriate language along with sound mathematical concepts is the key to the development of elementary algebra. This is treated in detail in one of our accompanying volumes, ELTMAPS: The ArAl Project (Arithmetic to Algebra Project) where problem situations are presented to pupils who are then challenged to solve gradually more complex problems in a step-by-step process, supported by the teacher. Language features also in the development of geometrical concepts, and the difficulties pupils have when facing symmetry problems are demonstrated in the last report in this section. Popular myth promotes the computer as the solution to many teaching problems, but a considerable body of research in this area which tells us that this is not necessarily the case. In fact, our problems multiply. There is not a lot of good educational software, and there are a number of reasons for this. The first use of computers outside of pure mathematics was commercial. Specialised systems were built for business, industry and the military, which indicated the direction for software development. At that time, few people were interested enough to try to develop systems especially for education, and in the early days, putting computers into the classroom was beyond practical possibility. Until recently, apart from a few exceptions, most educational software was merely putting the textbook on the computer screen with little thought for realising the true creative potential of the machine. Today however, new educational software like Derive, or the Dynamic Geometry programmes Cabri and Geometer’s Sketchpad demand that we examine many aspects of our educational philosophy, our teaching methods and classroom organisation. When we do this, we find that we also discover hidden problems, and the assumptions we make about pupils’ understanding of concepts. The other volume in this series ELTMAPS – Classroom Contexts offers some glimpses into the classroom from the teacher’s point of view. There are no clear answers, but suggestions for practical ways in which some of these problems may be confronted. We hope you will find at least some of these ideas useful.