Introduction
Mathematics is recognised as one of the sciences and has been described and defined in many different ways. It is a creative activity and is one of the most useful, fascinating and stimulating divisions of human knowledge. It is a process of managing and communicating information and has the power to predict and provide solutions to practical problems as well as enabling the individual to create new imaginative worlds to explore. We use mathematics in everyday life, in science, in industry, in business and in our free time.
Mathematics education is concerned with the acquisition, understanding and application of skills. Mathematical literacy is of central importance in providing the child with the necessary skills to live a full life as a child and later as an adult. Society needs people who can think and communicate quantitatively and who can recognise situations where mathematics can be applied to solve problems. It is necessary to make sense of data encountered in the media, to be competent in terms of vocational mathematical literacy and to use appropriate technology to support such applications. This curriculum will be a key factor in preparing children to meet the demands of the twenty-first century.
Mathematical applications are required in many subjects. The ability to interpret and handle data is of particular relevance in history and geography. Measures and Shape and space relate to the visual arts, physical education and geography. Integration gives the child a reason and motivation to develop mathematical skills and concepts that can be used in all subjects. Numeracy and estimation are particular to mathematics, while evaluating findings, reporting back, predicting and reasoning are used both in mathematics and throughout the curriculum.
Mathematics in a child-centred curriculum
The child learns from the people and materials around him/her. It is experience of the social and physical world that is the source of concepts, ideas, facts and skills. Integration of these experiences is the vital ingredient. If the child is given the chance to manipulate, touch and see objects that help him/her to acquire an understanding of concepts, he/she will understand more effectively than if words and symbols are the only learning tools. Often discovery learning alone will not be enough. The child needs guidance in formulating theories about what it is he/she is discovering. The child also needs help in developing the language for describing accurately what it is he/ she is doing. The teacher and the child's peers have a vital role to play in his/her educational experience. Yet ultimately it is the child who creates the balance between his/her knowledge and the knowledge of those around him/her.
Constructivism
Constructivist approaches are central to this mathematics curriculum. To learn mathematics children must construct their own internal structures. As in reading and writing, children invent their own procedures. We accept that children must go through the invented spelling stage before they begin to develop a concept of the structures of spelling. The same is true of mathematics.
Young children attempt to count or order things in the environment and they develop rules for themselves to do so. They should be encouraged to try out these personal strategies, to refine them by discussion and to engage in a wide variety of tasks. It is in the interpersonal domain that children can test the ideas they have constructed and modify them as a result of this interaction.
When working in a constructivist way children usually operate in pairs or small groups to solve problems co-operatively. Tasks that are written on one sheet can be given to groups of two or more children. This makes consultation, discussion and cooperation essential. Children work at their own pace but are encouraged to complete the task as fully as possible within the set time. They are expected to respect one another's solutions, not to discredit partners' reasoning, and to discuss the train of thought used in the process.
This sociocultural theory sees cognitive development as a product of social interaction between partners who solve problems together. It acknowledges the importance of the home and family in the child's learning and focuses on group interaction. It is a process approach rather than a step-like, incremental one. One form of instruction used is scaffolding. Here the teacher modifies the amount of support according to the needs of the child by modelling the behaviour, for example possible methods of approaching a problem. The teacher breaks down the task and makes the task manageable for the individual child, thus supporting the development of the child's own problem-solving skills.
Through discussion the child becomes aware of the characteristics of a task. He/she must be encouraged to use the correct vocabulary needed for a particular task. Young children are egocentric, and it is through social interaction that they can begin to appreciate the points of view of other people. Sequences of instruction involve discussion, hands-on experience and practical exploration. As adults we expect objects to behave in a stable and predictable manner. Children have to learn to recognise these attributes. They need to handle and use a variety of objects in order to form their own rules and structures for dealing with the world. This is of particular importance in mathematics.
Children need to work out when to use a particular plan, what they want to achieve and the actual procedure needed to complete the task. Through experiencing many different types of problems they become more efficient. The wider the range of problems they encounter the more likely they are to generalise the rules and use them in new situations.
While direct instruction is very important in mathematics, children also need to develop their own learning strategies. We need to teach children to look at how they arrived at a result rather than just concentrating on the answer as an end in itself.
Children need training in the skills of collaboration and co-operation, in listening to, accepting and evaluating the views of others. These skills are applicable throughout the curriculum. Work on open-ended problems, where the emphasis is placed on using skills and discussion rather than seeking a unique solution, is recommended. Many methods may be used in solving a mathematical task.