Developments in Primary Mathematics Teaching

Ann Sawyer

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367444396
• eBook ISBN: 9781000769630
• Edition: 1
• Topic: Education
• Subtopic: Education General

This book, originally published in 1993, addresses the issues surrounding the teaching of mathematics in primary school at the time. The author considers the issues that had arisen through the introduction of the National Curriculum, both in terms of the current "state of the art" and new developments.

Chapter 1
National Curriculum Implications for Teaching and Learning Mathematics

In this chapter, the recent history of mathematics teaching and the research evidence are considered in the light of the changes which the National Curriculum has introduced. The introduction of the 1989 Statutory Orders, the subsequent decision to revise these in 1991, and the likely effects upon schools are discussed briefly as these issues are taken up in other chapters in more detail.

The story of the pressure for change in mathematics teaching goes back to the Great Debate, which began with James Callaghan’s speech at Ruskin College in the autumn of 1976. Since then there has been wide public debate about what and how children should be taught, not just in mathematics, but across the whole curriculum. The public concern was about standards of performance in school work. The curriculum was said to pay too little attention to the basic skills of reading, writing and arithmetic and, in mathematics, was not preparing children for the world of work. It should be remembered that Aims into Practice, the Schools Council project (Ashton et al., 1975), identified six areas of human development and saw the curriculum as developmental through these areas. However, Richards (1985) notes that from the mid-1970s greater emphasis was put upon the needs of society and how these could be met through the curriculum, rather than the needs of the individual; the beginning of this can be noted in Primary Education in England (DES, 1978), which was organized with a subject-based focus, and encourages a return to subject differentiation.

In 1978 the Cockcroft Committee was set up to report upon the teaching of mathematics in schools, with particular regard to the mathematics needed in post-16 education, employment and adult life. The report of the committee was published (Cockcroft, 1982) and made some specific recommendations, including the following:

• schools should encourage the involvement of parents in their children’s mathematics particularly through explaining new processes
• children need to understand their mathematics rather than just learn by heart useful processes
• the range of styles of teaching which children should experience should be promoted to teachers
• teachers should encourage children to develop the processes necessary for mental calculations
• skills of estimation should be developed
• children should experience the practical work essential for the development of mathematical understanding
• the development of mathematical language skills is necessary as part of the learning process
• the use of the calculator as an aid to teaching and learning should be encouraged.

The report expressed concern that, in general, girls performed less well than boys in mathematics. Possible reasons for this were cited: teaching methods which favoured boys; boys were encouraged to be more independent than girls; textbook illustrations tended to favour boys; boys demand more attention from their teachers than girls. At conferences around the country, members of the Cock-croft Committee discussed the main issues in the report, raising awareness in the teaching profession. LEA advisory teacher teams were set up to encourage schools to take account of the Cockcroft Report’s recommendations.

Primary Practice, a Schools Council working paper (1983) encouraged a subject-based approach to the management of the curriculum. This was followed with the Curriculum Matters series from HMI which considered the curriculum subject by subject; this had an impact upon teachers and how they viewed the curriculum. In The Curriculum from 5-16 (DES, 1985a) HMI discussed the curriculum through the five issues of breadth, balance, relevance, differentiation and progression. These issues are now embedded in the National Curriculum.

The mathematics document (DES, 1985b) identifies five main categories of objectives for successful mathematics teaching and learning. These are facts, skills, conceptual structures, general strategies and personal qualities. The sensible use of the calculator is mentioned, and it argues that teaching standard algorithms for long division is now redundant, as is the use of logarithms. The document also lists some objectives for children to have achieved at the ages of 11 and 16. This was an important step along the road to an agreed curriculum for mathematics.

The Schools Examination and Assessment Council (1990) published the results of a survey into children’s performance in mathematics at age 11 and 15. Their conclusions, based on the evidence which was gathered by the Assessment of Performance Unit (APU) in 1987, are interesting. Areas discussed include the use of the calculator, mental skills, estimation and measurement, the use of Information Technology and problem-solving.

For calculator skills, APU found that over two-thirds of 11-year-olds claimed to use a calculator at school, and about half of them to have a calculator at home. Some evidence was found to suggest that the use of calculators at school may have caused an improvement in children’s scores since the previous testing some five years earlier, in the APU’s number concepts and skills tests. However, many 11-year-olds did not understand decimal displays on the calculator, and would revert to pencil and paper method if a decimal display occurred. The report makes the point that children need to be taught how to interpret the calculator display.

For mental skills, APU notes that the Cockcroft Report encouraged schools to reintroduce the development of mental skills. More 11-year-olds were using rounding up procedures to help them to calculate than had been the case in earlier testing. However, APU found that some children were attempting to use paper and pencil algorithms in their heads, which were inappropriate. The implication is that mental skills need to be taught through discussion between teacher and children, rather than through the traditional weekly mental arithmetic test.

In estimation, children performed poorly. There was a lack of appreciation of the size of standard units, and confusion between metric and imperial measures. Again, Cockcroft had recommended that schools should place greater emphasis upon estimation tasks; children need to acquire the skill of accuracy in estimation alongside those associated with accuracy in measurement. APU found that this was not happening.

It was noted that schools lacked sufficient hardware and suitable software for their pupils’ needs for IT applications in mathematics. The evidence on group problem-solving showed that children liked this way of working and that one of its strengths was in the sharing of ideas. APU considered that group problem-solving experiences would lead to the development of children’s reasoning and social skills.

The evidence from APU relates directly to the teaching and learning of the National Curriculum for Mathematics as it reports its evidence by Attainment Target title. The report finds an improvement in performance in shape and space and handling data from the 1982 data to 1987 data. There is a reported decline in number and algebra skills across the same time span, which must have implications for teaching contexts for National Curriculum topics. It is possible that the Cockcroft Report’s messages about children’s acquisition of number skills and the use of calculators had been misinterpreted, so that perhaps children had not been encouraged to develop a range of paper and pencil, calculator and mental methods of calculation.

Since 1987 teachers have been inundated with documents from DES, NCC, SEAC and now DfE. Many of these have been consultation documents which have invited responses within a very tight time scale. For the mathematics curriculum, the sequence of events from 1987 to 1990 was as follows:

1987
July	Consultation document for the National Curriculum
July	National Curriculum Working Group for Mathematics set up
December	Interim Report of the Working Group for Mathematics
1988
January	Task Group on Assessment and Testing report published
August	Final Report of the Working Group for Mathematics
October	Responses to the final report to be received by NCC
December	National Curriculum Mathematics consultation report
1989
March	Orders for the National Curriculum Mathematics laid before parliament
June	Non-Statutory Guidance for Mathematics issued
September	Year 1 children began National Curriculum Mathematics
1990
September	Year 3 children began National Curriculum Mathematics

This time-scale was very short. From July 1987, when the consultation document for the National Curriculum appeared, until Autumn 1989, when Year 1 teachers began to use the National Curriculum Programmes of Study for Mathematics, were just two short, hectic years. There was some LEA training for teachers for the implementation of the National Curriculum, but this tended to be about National Curriculum terminology, not about interpretation of meanings within programmes of study or statements of attainment.

In 1989 the Department of Education and Science published National Curriculum. From Policy to Practice (DES, 1989b) which set out definitions of the new jargon vocabulary of the National Curriculum and described, briefly, what was required of teachers and when. In the same year, and before implementation of the National Curriculum in Year 1, HMI (1989) published a review of practice in mathematics. The review describes good practice in mathematics teaching and learning across the whole primary age range, and includes references to work on pattern and number, including mental calculations, the development of pencil and paper methods and the use of calculators. Reference is made to a shortage of hardware in schools so that, although scope for development of the use of the computer is identified as desirable, in practice it is not feasible without the resources being provided from outside the school’s budget. Although the report is critical of about a quarter of the schools inspected, there is evidence of exciting, stimulating work from others. The production of the review using glossy, good quality paper, with a variety of photographic evidence of children at work, all in full colour, gave an added emphasis to the good quality of the work which the report praised.

The National Curriculum for Mathematics in primary schools was introduced in stages: Year 1 in 1989, Years 2 and 3 in 1990, Year 4 in 1991, Year 5 in 1992 and Year 6 in 1993. By September 1993 the whole primary phase will be taught within the framework of the programmes of study. In 1994 statutory assessment at the end of Key Stages 1 and 2, that is for years 2 and 6, will be in place. However, as the timetable for implementation of the National Curriculum took four years to complete, some schools decided to introduce the programmes of study earlier than required throughout the school, in order to help support the Key Stage 1 teachers and to aid familiarity with the process and content of the National Curriculum.

The National Curriculum was introduced in order to raise standards. In the National Curriculum consultation document (1987b) it states, ‘The Government wants attainment targets and the content of what is taught to reflect current best practice and achievement’. Within the programmes of study and statements of attainment for the 1989 National Curriculum there is reference to using calculators and computer databases. (The programmes of study within the Final Report of the Working Group contained further guidance, including the use of Logo, but this was ‘lost’ in translation into Statutory Orders.) The examples in the 1989 Statutory Orders suggest using Logo, calculators, mental calculations, simple Basic computer programs and databases. Although there is no legal requirement to use these examples as learning activities, they highlight to teachers the recommendations of the Cockcroft Report, and evidence of good practice from HMI and APU reports.

For primary schools the National Curriculum has brought about a widening of the traditional mathematics curriculum. Attainment Target 1, Using and Applying Mathematics, was devised to take a central role in the mathematics curriculum. The programmes of study offer opportunities for children to apply their knowledge, skills and understanding in problem-solving and enquiry situations, to communicate the results of that enquiry and to demonstrate their achievements with reasoning, logic and proof. These were not part of the traditional curriculum for many schools and have been aspects of mathematics teaching which needed to be included within a school’s curriculum plan. For some teachers, this has meant a new approach to their teaching and the need to develop assessment strategies for using and applying mathematics.

There is an expectation that children will be encouraged to develop personal qualities that will stimulate positive attitudes towards mathematics. Within the Statutory Orders no mention is made of equal opportunities, of girls’ traditionally poor image of themselves as mathematicians, nor of the needs of children from different ethnic groups and cultures, such as support for bilingual learners. No mention is made of the advisability of encouraging parents to be involved in their children’s mathematics. All children in state schools have an entitlement to mathematics teaching which reflects the programmes of study.

The structure of the number system, place value, developing mental and paper and pencil methods of calculation, are included in the programmes of study. As has already been mentioned, the use of calculators is implied within the Orders. Nowhere within the Orders are algorithms for calculation prescribed and this empowers teachers to encourage children to develop their own methods of calculation, based upon mental, paper and pencil, calculator and a mixture of these methods and then to develop methods which do not require the aid of a calculator.

The introduction of algebra at Key Stage 1 has raised the status of pattern work. Traditional activities which involve copying, repeating, extending patterns using beads, coloured rods and interlocking cubes, are an essential part of the algebra curriculum at Key Stage 1. This work is extended within Key Stage 1 and beyond, to develop an appreciation of the importance of pattern in number relationships, and to the use of symbolic and graphical representations to express those relationships. However, teachers’ and student teachers’ own experiences of algebra at school were not always positive; as one returner to teaching put it, algebra ‘was about x and y and I never understood it’. The detail of algebra in Attainment Target 3 has not been part of all primary mathematics curricula. The emphasis upon algebra in AT3 implies the need for teachers to receive support and INSET training in order to help them to develop their ‘comfort zone’ for algebra. As will be seen in Chapter 5, some teachers have used this opportunity to encourage children to explore pattern in a wider field of study, incorporating music, science and art and thus to encourage children to explore mathematical concepts and to develop their mathematical thinking through other curriculum areas. In the world outside school, mathematics is needed as a means of communication within ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Contents
• Introduction
• 1 National Curriculum Implications for Teaching and Learning Mathematics
• 2 Starting Points for Using and Applying Mathematics: Maths Trails
• 3 Logo. Some Case Studies of Implementation in the Classroom
• 4 Using Concept Keyboards and Calculators
• 5 Using Themes and Topics
• 6 The Management of Mathematics Teaching and Learning in Schools
• 7 Assessment and Record-keeping
• 8 Looking Forward
• Bibliography
• Resources
• Index