Offering 0pen and Honest Professional Advice
Quotations - Cockcroft -
The Cockcroft Report (1982) took five years to research and write. The research it commissioned into mathematical learning was also published separately to it. It was an independent report that had many leading educationalists contributing to it. Submissions were taken from schools and a whole range of other contributors.
There is no convincing evidence that children have bilogically changed since that time. but there is evidence that politicians ignored, failed to implement Cockcroft' findings.
10. Mathematics provides a means of communicating information concisely and unambiguously because it makes extensive use of symbolic notation. However, it is the necessity of using and interpreting this notation and of grasping the abstract ideas and concepts which underlie it which proves a stumbling block to many people. Indeed, the symbolic notation which enables mathematics to be used as a means of communication and so helps to make it 'useful' can also make mathematics difficult to understand and to use.
11. The problems of learning to use mathematics as a means of communication are not the same as those of learning to use one's native language. Native language provides a means of communication which is in use all the time and which, for the great majority of people, 'comes naturally', even though command of language needs to be developed and extended in the classroom.
Furthermore, mistakes of grammar or of spelling do not, in general, render unintelligible the message which is being conveyed. On the other hand, mathematics does not 'come naturally' to most people in the way which is true of native language. It is not constantly being used; it has to be learned and practised; mistakes are of greater consequence.
Mathematics also conveys information in a much more precise and concentrated way than is usually the case with the spoken or written word. For these reasons many people take a long time not only to become familiar with mathematical skills and ideas but to develop confidence in making use of them.
Those who have been able to develop such confidence with relative ease should not underestimate the difficulties which many others experience, nor the extent of the help which can be required in order to be able to understand and to use mathematics.
235 Short term memory plays an important role in all tasks in which several attributes or items of information have to be considered simultaneously, for example in mental calculations, problem solving, the understanding of complex concepts and the construction or following of an explanation or argument; in other words, in most learning tasks. In order to carry out these tasks it is necessary to draw on information stored in long term memory. Research evidence makes it clear that information is stored better in long term memory if it is assimilated in such a way that it becomes part of a network of associated and related items which support one another. An everyday example of this is provided by the fact that some children whose ability to remember number facts appears to be weak are often able, because of their interest in and knowledge of sport, to remember without difficulty the scores in football or cricket matches which have been played weeks or even months earlier.
238 We have received several submissions which have urged that more emphasis should be placed on 'rote learning'. The Oxford English Dictionary defines 'by rote' as 'in a mechanical manner, by routine; especially by the mere exercise of memory without proper understanding of, or reflection upon, the matter in question; also, with precision, or by heart'.
There are certainly some things in mathematics which need to be learned by heart but we do not believe that it should ever be necessary in the teaching of mathematics to commit things to memory without at the same time seeking to develop a proper understanding of the mathematics to which they relate. As our discussion of memory shows, such an approach is unlikely to meet with long term success.
By this we mean that, whereas an 'average' child can perform this task at age 11 but not at age 10, there are some 14 year oIds who cannot do it and some 7 year olds who can. Similar comparisons can be made in respect of other topics. For example, the top 15 per cent of 10 year oIds in England are able to answer the question 'There are 40 children in a class and three fifths of them are girls. How many boys are there in the class?'. By contrast, the bottom 15 per cent of 14-15 year old pupils in Scotland find difficulty in working out 3/4 of £24; the bottom 15 per cent of 14 year olds in Australia find difficulty with the comparable question 'Mixed concrete costs $24 per cubic metre. What would 3/4 of a cubic metre costh?"
There is little evidence to show the attainment of the most capable 11 year olds, because large scale tests do not usually include very many items which will extend these children and so provide the necessary evidence. However, one American study found that there were a number of 12 and 13 year old pupils in Baltimore who performed at the same level as the top 10 per cent of 17 year olds on a mathematics test designed to reveal potential for college study.
343 We believe it is clear from the preceding paragraphs that it is not possible to make any overall statement about the mathematical knowledge and understanding which children in general should be expected to possess at the end of the primary years. However, the test results which we have quoted, and others which we have studied, indicate that even in the primary years the curriculum provided for pupils needs to take into account the wide gap in understanding and skill which can exist between children of the same age.
436 As we have continued our work we have become increasingly aware of the implications for mathematics teaching of the differences in attainment in mathematics which exist among pupils of any given age and of the extent to which these differences increase as pupils become older. We drew attention in paragraph 342 to the 'seven year difference' which exists among 11 year olds. If we relate this to work in the secondary years, it means that the mathematical understanding of some pupils who transfer to secondary school at 11 is likely already to be greater than that of some pupils who have just left school at 16.
On the other hand, some of those who arrive at the same time may not, while at school, attain the understanding which some of their fellow 11 year olds already possess.
When considering work in the secondary years it is also necessary to remember that pupils learn at very different speeds and that, in the sense we have explained in paragraph 228, mathematics is a hierarchical subject.
640 We believe it is essential to do much more than is being done at present to improve the public image of teaching, and of mathematics teaching in particular. Too many potential teachers appear to be put off by widely reported problems of discipline, of the difficulties of teaching or of unhappy schools.
Yet the results of a small survey carried out on our behalf by the National Foundation for Educational Research, to which we refer in greater detail in paragraph 671 [in chapter 14], showed that the group of mathematics teachers included in the survey, who were in their first three years of teaching in secondary schools, were generally happy in their work. A large majority believed that they were well thought of by their colleagues and that they had gained the respect of their pupils.
We hope that both central and local government will respond to our report by affirming their belief in the importance of good mathematics teaching for all pupils, the need to provide good support and facilities for mathematics teachers who are already in post and, whatever the overall teacher requirement may be, the need for many more good teachers of mathematics. Good publicity is necessary in order to improve attitudes towards the teaching of mathematics on the part of the general public and, in consequence, in the minds of potential teachers.
218 There is, however, one major difficulty which exists for those who teach mathematics through the medium of Welsh. This is the great shortage of mathematics textbooks which are written in Welsh. There is at present no complete primary mathematics scheme in Welsh. Such materials as are available are confined mainly to infant level; other materials are fragmentary and deal only with parts of the mathematics curriculum.
This means that teachers have to prepare most of their own classroom materials. This not only takes a great deal of time but also results in a standard of presentation which is less good than that of a published text. For this reason there are some classrooms in which although the teaching and discussion are in Welsh, children work from books which are written in English.