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We know that all over Europe there are teachers who would wish to change their teaching strategies to a constructivist approach, but find that the shift from a familiar instructional practice to an innovative approach is not easily accomplished. It requires experiences with new teaching strategies, different forms of tasks and their students working in different ways. The material of this book meets these needs of the teachers.

The book includes five units: Early Number Sense; Relationships and Functions; Regular Polygons; 3D Geometry – Solids and Patterns leading to Algebra. The units present not only challenging tasks but also include documented trials of the tasks undertaken in schools of the four countries involved in the project. Thus, the reader can gain knowledge of how other teachers realized the presented tasks in their classrooms and the ways their students developed strategies in problem solving and constructed their own mathematical ideas.

This teaching material could also be implemented by mathematics coordinators and teacher trainers to support the change in knowledge and attitudes expected from the teachers in school. This would enrich the present day mathematics classroom in Europe.

FOREWORD

All five units are based on a creative educational approach to mathematics which is shared by all authors.

Like the Pythagorean School, we use the pentagon to elicit the common core and the connections among our units. foreword

Under ‘problem solving’ we mean a broad range of mental activity which comprises:

abstracting, analyzing, arguing, articulating, checking, communicating, constructing, data collecting, evaluating, experimenting, explaining, deciding, generalizing, grasping, hypothesising, interpreting, investigating, linking, modelling, ordering, organizing, reasoning, recording, representing, structuring and restructuring, verifying, …

PREFACE

The work presented in this book results from the collaboration of four universities in four different European countries together with teachers and their pupils from those countries. It is for use by teachers in the real practical setting of the classroom and though based on sound theoretical foundations does not preach those theories to the reader.

The IIATM project (Implementing Innovative Approaches to the Teaching of Mathematics) was funded through the Socrates-Comenius 2.1 programme beginning in the Autumn of 2003 and was completed in Autumn 2006. It was coordinated by members of the Department of Mathematics and Mathematics Education at Charles University in Prague.

The project aims to promote constructivist teaching approaches in mathematics, to change the role of the teacher in the classroom, to make the pupils more responsible for their learning, to enable the pupils to understand the underlying concepts and to make mathematics more meaningful and realistic and thus give purpose and enjoyment in mathematics for the pupils. The tasks were developed and trialed in all four countries involved in the project and all the cooperating teachers met together when they discussed the values of the tasks and different approaches to the same task, thus ensuring that the final tasks met the needs of all schools in the European Community. We hope the outcome of all these approaches will be that the pupils have a more positive attitude towards mathematics.

The book and the DVD can be used by teachers in primary schools, secondary mathematics teachers and teacher educators. They include the underlying philosophy of the project and examples of mathematical topics taught in a non-transmissive way, with accompanying video-recordings on the DVD. Schools can use the work for whole school development or take units within the book for individual or group professional training and for initial teacher training in schools or education faculties.

The universities and their staff involve in the project were:

Charles University in Prague, Czech Republic; staff: Milan Hejný, Darina Jirotková, Marie Kubínová and Na?a Stehlíková.
Kassel University, Germany; staff: Brigitte Spindeler, Bernd Wollring;
Aristotle University, Thessaloniki, Greece; staff: Marianna Tzekaki, Giorgos Barbas
University of Derby, United Kingdom; staff: Graham Littler, David Benson.

The team involved at each university prepared a unit of work based on a particular area of mathematics. Charles University was the exception to this since they produced two units. The units covered the following mathematical areas:

3D Geometry – Solids

Relationships and Functions

Regular Polygons

Early number sense

Patterns leading to Algebra

The Units were developed by the university staff together with cooperating teachers from schools with whom they had previously worked. The schools and staff involved were:

Czech Republic

Základní škola, Školní 900, Neratovice; staff: Jitka Michnová, Irena Kro?áková
Základní škola B?lohorská 174, Praha; staff: Blanka V?trovcová
Základní škola Ostrovní 9, Praha; staff: Klára K?enková, Jana Poláchová
Základní škola U Santošky, Praha; staff: Miroslav Hricz
Gymázium Elišky Krásnohorské, Praha; staff: Zuzana Korcová
K?es?anské gymnázium Kozinova, Praha; staff: Michaela Ulrychová
Gymnázium Jana Keplera, Praha; staff: Klára Horká
Základní škola Campanus, Praha; staff: Marie Kubínová

Germany

Schule am Jungfernkopf, Kassel; staff: Angela Becker, Kathrin Scheuch
Schule Vollmarshausen, Lohfelden; staff: Brigitte Bergmann

Greece

Πειραματικ? Σχολε?ο Πανεπιστημ?ου Μακεδον?ας, Θεσσαλον?κη; staff: Petros Oikonomou
64o Δημοτικ? Σχολε?ο, Θεσσαλον?κη; staff: Olga Kassoti
11o Δημοτικ? Σχολε?ο Αμπελοκ?πων, Θεσσαλον?κη; staff: Michalis Matsoukalidis

United Kingdom

Bishop Lonsdale Primary School, Derby, UK; staff: Sara Bull, Clare Bladon
Ripley High School, Ripley, Derbyshire, UK; staff: Sue Orchard, Sheila Rollinson

Other schools either trialled the tasks for us or allowed us to trial tasks in their schools for which we are extremely grateful, however these schools were not formally part of the project.

Základní fakultní škola Uhelný Trh, Praha
Základní škola Chlupová, Praha

We are immensely grateful to these teachers for their contribution to the project and to the pupils in the classes they teach, for trying out many of the tasks presented in the book.

The age-range for which the tasks are aimed is from 5 to 15 years. The tasks are divided notionally into three levels suitable for three over-lapping age groups:

Level A, from 5 to 8 years of age,
Level B, from 7 to 12 years of age,
Level C, from 11 to 15 years of age,

We describe these levels ‘notionally’ for the age-range designated, but the class teacher knows the ability of their class best and might find that for their 9 year–old pupils that tasks in Group A or C are more suitable than those suggested in Group B.

Similarly not every task is labelled as suitable for group work or for individual pupils, we leave this up to the classroom teacher. However, with our emphasis on constructivist teaching most of our tasks can be used with groups since we firmly believe that pupils can learn from each other and pupil-pupil conversation discussing questions which arise in the solving of a task, encourages this. Again, the tasks are gender free and physically handicapped pupils can attempt most of them. The tasks have been trialed in multi-ethnic classrooms and by the nature of the project across pupils with different cultural backgrounds, so we feel confident that the work will suit any social dimension. It is not intended that the tasks should be used necessarily as written but can be modified by the teacher to suit the language level and context within which the pupils are familiar.

Some tasks are labelled level T and these are intended specifically for teachers to enable them to see the mathematical background necessary for a particular set of tasks. Similarly within the units challenges are set for the teacher to develop the work, produce a similar set of tasks etc.

We do however recommend that teachers try the tasks before they set them to their pupils to enable them to see the diverse ways a particular task might go and to be better prepared for unusual questions related to the task.

The total resource is a book and a DVD. The teacher can use the book or DVD only or both. The content of the book is also on the DVD and in addition it also contains materials, pictures and videos of the tasks being trialed in schools. If you look at a particular task in either the book or DVD, symbols will show whether there is any additional material for the trials on the DVD. Figures, videos, additional texts and worksheets only on the DVD will be referred to by special symbols. Individual units give information of the use of hyperlinks within that unit.

We hope you enjoy using the book and DVD.

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Charles University Prague, The Faculty of Education
Department of Mathematics and Mathematics Education