Mathematics may be seen as the science of magnitude, number, shape, space, and their relationships and also as a universal language based on symbols and diagrams. It involves the handling (arrangement, analysis, manipulation and communication) of information, the making of predictions and the solving of problems through the use of a language that is both concise and accurate.
Mathematics education provides the child with a wide range of knowledge, skills and related activities that help him/her to develop an understanding of the physical world and social interactions. It gives the child a language and a system through which he/she may analyse, describe and explain a wide range of experiences, make predictions, and solve problems. Mathematics education fosters creative and aesthetic development, and enhances the growth of reasoning through the use of investigative techniques in a mathematical context. It is also concerned with encouraging the child to be confident and to communicate effectively through the medium of mathematics.
The mathematics curriculum
Mathematics encompasses a body of knowledge, skills and procedures that can be used in a rich variety of ways: to describe, illustrate and interpret; to predict; and to explain patterns and relationships in Number, Algebra, Shape and space, Measures and Data. Mathematics helps to convey and clarify meaning. Its language provides a powerful and concise means by which information may be organised, manipulated, and communicated. These characteristics make mathematics an essential tool for the child and adult. The application of mathematics in a variety of contexts gives people the ability to explain, predict and record aspects of their physical environments and social interactions. It thus enriches their understanding of the world in which they live. Indeed the application of increasingly sophisticated mathematics in a growing range of economic, technical, scientific, social and other contexts has had a profound influence on the development of contemporary society.
Mathematics education should seek, therefore, to enable the child to think and communicate quantitatively and spatially, solve problems, recognise situations where mathematics can be applied, and use appropriate technology to support such applications. If the child is to become an informed and confident member of society he/she must be enabled to deal effectively with the varied transactions of everyday life and make sense of the mass of information and data available through the media.
It should be recognised that mathematics is an intellectual pursuit in its own right, a source of fascination, challenge, and enjoyment. The exploration of patterns and relationships, the satisfaction of solving problems, the appreciation of designs and shapes and an awareness of the historical and cultural influences that have shaped modern mathematics can contribute to the child's enthusiasm for the subject.
This curriculum seeks to provide the child with a mathematical education that is developmentally appropriate as well as socially relevant. The mathematics programme in each school should be sufficiently flexible to accommodate children of differing levels of ability and should reflect their needs. These will include the need for interesting and meaningful mathematical experiences, the need to apply mathematics in other areas of learning, the need to continue studying mathematics at post-primary level, and the need to become mathematically literate members of society. Integration with all the other subjects will add another valuable perspective to the mathematics curriculum.
The structure of the curriculum
The curriculum comprises five strands:
- Shape and space
These strands, although presented in separate sections, are not isolated areas. They should be seen and taught as interrelated units in which understanding in one area is dependent on, and supportive of, ideas and concepts in other strands. Such linkage within the subject is essential. While number is essential as the medium for mathematical calculation, the other strands should receive a corresponding degree of emphasis. The strands are divided into strand units, which give additional structure to the curriculum.
Number starts with a section called Early mathematical activities, in which there are four strand units: Classifying, Matching, Comparing and Ordering. These units develop at infant level to include counting and analysis of number. In first and second classes the development includes place value, operationsand fractions. Decimals are introduced in third class and percentages in fifth class.
Algebra is formally recognised at all levels and covers patterns, sequences, number sentences, directed numbers, rules and properties, variables and equations.
Shape and space as a strand explores spatial awareness and its application in real-life situations. It includes units dealing with two-dimensional and threedimensional shapes, symmetry, lines and angles.
Measures consists of six strand units: Length, Area, Weight, Capacity, Time and Money.
Data includes interpreting and understanding visual representation. Chance promotes thinking, discussion and decision-making and is familiar to children in the form of games and sporting activities.
Spanning the content are the skills that the child should develop:
- applying and problem-solving
- communicating and expressing
- integrating and connecting
- understanding and recalling.
This mathematics curriculum provides opportunities for the child to explore the nature of mathematics and to acquire the knowledge, concepts and skills required for everyday living and for use in other subject areas.
Providing for individual difference
Children in any one class will show a wide range of ability, attainment andlearning styles, and it is difficult to cater for all their needs if a common programme is followed. Children acquire an understanding of mathematical ideas in an uneven and individual way. The issue of readiness is therefore crucial when planning, teaching and assessing the mathematics programme. It is important to build on the child's previously acquired knowledge, and periods of frequent revision are essential.
Continuous assessment is particularly useful for diagnosis and planning in mathematics. It should focus on the identification of the child's existing knowledge, misconceptions and strategies. It should provide information that will enable the teacher to cater for individual differences in ability, previous learning and learning style, and to resist pressure to push the child to premature mechanical mastery of computational facts and procedures. It will be important that a learning environment is created to enable both boys and girls to learn all aspects of mathematics effectively and to provide opportunities for extension work for more able children.
Constructivism and guided-discovery methods
A constructivist approach to mathematics learning involves the child as an active participant in the learning process. Existing ideas are used to make sense of new experiences and situations. Information acquired is interpreted by the learners themselves, who construct meaning by making links between new and existing knowledge. Experimentation, together with discussion among peers and between the teacher and the child, may lead to general agreement or to the re-evaluation of ideas and mathematical relationships. New ideas or concepts may then be constructed. The importance of providing the child with structured opportunities to engage in exploratory activity in the context of mathematics cannot be overemphasised. The teacher has a crucial role to play in guiding the child to construct meaning, to develop mathematical strategies for solving problems, and to develop selfmotivation in mathematical activities.
An important aim of the mathematics programme is to enable the child to use mathematical language effectively and accurately. This includes the ability to listen, question and discuss as well as to read and record. Expressing mathematical ideas plays an important part in the development of mathematical concepts. One of the causes of failure in mathematics is poor comprehension of the words and phrases used. Some of the language will be encountered only in the mathematics lesson, and children will need many opportunities to use it before it becomes part of their vocabulary. In other cases, everyday words will be used in mathematics but will take on new meanings, which may be confusing for the learner.
Discussion plays a significant role in the acquisition of mathematical language and in the development of mathematical concepts. The child may be helped to clarify ideas and reduce dependence on the teacher by discussing concepts and processes with other children. Discussion with the teacher is also essential. As the need arises, the teacher will supply appropriate mathematical language to help the child to clarify ideas or to express them more accurately.
In view of the complexity of mathematical symbols, it is recommended that children should not be required to record mathematical ideas prematurely. Concepts should be adequately developed before finding expression in written recording. The use of symbols and mathematical expressions should follow extended periods of oral reporting and discussion.
The use of mathematical equipment
The child's mathematical development requires a substantial amount of practical experience to establish and to reinforce concepts and to develop a facility for their everyday use. He/she develops a system of mathematics based on experiences and interactions with the environment. The experience of manipulating and using objects and equipment constructively is an essential component in the development of both mathematical concepts and constructive thought throughout the strands of the mathematics programme.
The development of arithmetical skills, i.e. those concerned with numerical calculations and their application, is an important part of the child's mathematical education. This mathematics curriculum places less emphasis than heretofore on long, complex pen-and-paper calculations and a greater emphasis on mental calculations, estimation, and problem-solving skills. Rapid advances in information technology and the ready availability of calculators have not lessened the need for basic skills.
The role of the calculator
An understanding of the structure of number can be enhanced by the exploration of patterns, sequences and relationships with a calculator. Calculators help in the development of problem-solving skills by allowing the child to focus on the structure of a problem and possible means of solution. Calculators can be used to check estimates, to perform long and complex computations, and to provide exact results to difficult problems. However, thecalculator cannot be a substitute for practical activity with materials. Moreover, it must be remembered that the child needs a sound understanding of number to make judgements about when it is appropriate to estimate, to calculate mentally, to make a calculation on paper, or to use a calculator for an exact result. For these reasons, this curriculum provides for the use of calculators in mathematics from fourth to sixth classes, by which time the child should have acquired a mastery of basic number facts and a facility in their use.
Information and communication technologies
Computers have a place in the mathematics curriculum but must be seen as another tool to be used by the teacher and the child. They do not take the place of good teaching and extensive use of manipulatives. Computers provide an alternative to pen-and-paper tasks, are stimulating for less able children, and provide interesting extension work for all levels of ability. There is a wide variety of computer applications available. Adventure-type programs, which require the child to solve specific mathematical problems in a meaningful context, offer opportunities for the development of problem-solving skills. Paired or group activities encourage discussion and collaborative problem-solving. Data-handling programs allow children to manipulate and interpret data they have collected. The emphasis must always be on the process, for example collecting information, deciding on the relevance of questions, and interpreting results.
Developing the ability to solve problems is an important factor in the study of mathematics. Problem-solving also provides a context in which concepts and skills can be learned and in which discussion and co-operative working may be practised. Moreover, problem-solving is a major means of developing higher-order thinking skills. These include the ability to analyse mathematical situations; to plan, monitor and evaluate solutions; to apply strategies; and to demonstrate creativity and self-reliance in using mathematics. Success helps the child to develop confidence in his/her mathematical ability and encourages curiosity and perseverance. Solving problems based on the environment of the child can highlight the uses of mathematics in a constructive and enjoyable way.
Integration in mathematics
Mathematics pervades most areas of children's lives, whether they are looking at and responding to structural forms in the visual arts curriculum or calculating how to spend their pocket money. For children to really understand mathematics they must see it in context, and this can be done through drawing attention to the various ways in which we use mathematics within other subjects in the curriculum.
SESE provides ample opportunities for using mathematics, for example recording results of experiments in science or creating maps in geography, while a sense of time and chronology is essential in history. Collecting data for analysis is also an important feature of SESE and provides the child with real-life examples of data with which to work. Physical education offers myriad opportunities for measurement as a natural part of the activities, for example timing races or measuring the length of jumps. Creating symmetrical and asymmetrical shapes in a gymnastics lesson can also offer real use of mathematical concepts. Mathematical language occurs in all areas of the curriculum, for example in long and short notes in music or using the correct words to describe shapes in visual art activities. Many teachers make use of rhymes, songs and games to reinforce concepts of number and shape, and this can be achieved in English, Irish or using a modern European language where appropriate.