This workshop

1. This workshop • Background - why I chose to work in this way - the need for theory • The Authors - George Cuisenaire, Caleb Gattegno and Madeleine Goutard • Putting theory into practice – an example – learning about fractions • Key understanding - 3 phases of working – empirical, systematisation, mastery. Practical activity to illustrate the ‘dangers of empiricism’ • Key principles - the basis of a theory • Why children need to write maths for themselves – samples of children’s writing • Why children need to discover maths for themselves – videos of children working

2. This project – background • A teacher is a researcher • Own a theory

3. The mathematicians • George Cusenaire • Caleb Gattegno • Madeleine Goutard

4. Putting theory in action • Teaching fractions “ …… teaching as little as possible. My practice is to give the pupils one single fraction only, which provides them with the appropriate terminology and the written conventions, leaving them to invent all the rest.” • The naming activity – group skills, articulating comparisons • Adapting the activity for different purposes • Draw out anything which can be generalised • Return to the texts for clarification

5. The Dangers of Empiricism “It is generally agreed that concrete experience must be the foundation of mathematics learning. When children find it difficult to understand arithmetic it is at once suggested that this is because it is too abstract; for small children the study is then simply reduced to the counting of objects. It seems to me that there has perhaps been too great a tendency to make things concrete and that perhaps the difficulties children experience spring from the fact that they are kept too much at the concrete level and are forced to use too empirical a mode of thought.”

6. Empiricism, systematisation, mastery • Empirical – “based or acting on observation or experiment, not on theory; regarding sense-data as valid information; deriving knowledge from experience alone.” “..one starts gleaning facts. This is done by trial and error, the results being accepted or rejected according to the criterion imposed on oneself. These facts are gathered at random, everybody gleaning what he can… . …The children have acquired more a technique than knowledge founded on reasons.” • Systematisation – “.. to organise experience, to clarify facts so as to fill gaps if some are found, to propose groupings of some significance, in a word to invent sure means with which a thorough study of the situation could be undertaken.” • Mastery – a deeper understanding of the structures involved in these situations .. Every element or group of elements is seen to potentially contain the infinite set of which it is part, as soon as the dynamic link between the elements have been noticed.”

7. Key findings – the basis of a theory • Children should write maths for themselves • to discover relationships between operations, • to feel powerful over the subject, • to ensure symbols contain personal meaning, • to learn the ‘manufacturing secrets’ of equations, • to learn to transform • Children should discover maths for themselves • To always have a ‘route’ for rediscovering something when needed • To have a way of verifying new information • To feel in control of their learning • To feel the excitement of discovery