[page 26]
Appendix II The mathematics curriculum
It is in the interests of the country that as many people as possible should achieve acceptable levels of competence in mathematics. It is also in the interests of the individual to reach as high a level as possible in the subject, not merely because mathematics may help a person to get a better job and do it well, but chiefly because it can help in understanding and interpreting very many aspects of the world in which we live.
The details of the approach and content of the course have to be worked out by consultation within each school; but we may take as a starting point the following objectives, which have been widely discussed over the last two years.
Approach
At every level in the teaching of mathematics the formation of concepts should have priority over the acquisition of technical skills. This is not to imply that such skills are to be neglected, but that emphasis on understanding will facilitate the acquisition of those skills which are needed.
The use of language plays a dominant role in the learning of mathematics, and teachers should be constantly asking pupils to speak and write about what they are learning, mostly in ordinary language rather than in specialised words or symbols.
Symbols should be introduced only when the need for them is perceived and pupils understand what the symbols denote. If pupils are to acquire the ability to manipulate symbols accurately and with reasonable speed, and at the same time to focus their attention on some higher purpose (such as the solution of a problem), practice is necessary. But the accurate manipulation of symbols alone is of limited value; pupils need also to see purpose in manipulation and to attach meanings to symbols when necessary.
If pupils are to be aware of the inter-relationships between mathematics and other subjects of the curriculum and between mathematics and the world of experience, and are to be able to apply their mathematics in these areas, it is necessary to stress these inter-relationships constantly. Examples of the applications of mathematics in science, technical studies, geography, economics, and from industry and commerce, not only serve to help pupils make these connections but also enliven the teaching. Some mathematical ideas might best be developed in other subjects of the timetable, but in any case good liaison is essential.
Content
These are some of the attitudes, concepts and skills which pupils might be expected to acquire as they progress through school. Pupils vary greatly in their levels of achievement, but it is to be hoped that some work of appropriate difficulty will be undertaken with all pupils across the full range of ideas described below. It is to be hoped
[page 27]
also that in the process of doing this work pupils will learn to think clearly, argue logically and communicate effectively.
Between the ages of 5 and 8 children should begin work in the following areas:
i. The development of appropriate language; qualitative description, the recognition of objects from description; discriminating, classifying and sorting of objects; identifying objects and describing them unambiguously.
ii. The recognition of common, simple mathematical relationships, both numerical and spatial; reasoning and logical deduction in connection with everyday things, geometrical shapes, number arrangements in order, etc.
iii. The ability to describe quantitatively: the use of number in counting, describing, estimating and approximating.
iv. The understanding of whole numbers and their relationships with one another.
v. The appreciation of the measures in common use; sensible estimation using the appropriate units; the ability to measure length, weight, volume and capacity, area, time, angle and temperature to an everyday level of accuracy.
vi. The understanding of money, contributing to a sense of the value of money, and the ability to carry out sensible purchases.
vii. The ability to carry out practical activities involving the ideas of addition, subtraction, multiplication and division.
viii. The ability to perform simple calculations involving the mathematical processes indicated by the signs +, - , x, + with whole numbers (maintaining rapid recall of the sums, differences and products of pairs of numbers from 0 to 10).
ix. The ability to check whether the result of a calculation is reasonable.
x. The ability to use and interpret simple forms of diagrams, maps and tabulated information.
xi. An appreciation of two- and three-dimensional shapes and their relationships with one another. The ability to recognise simple properties; to handle, create, discuss and describe them with confidence and appreciate spatial relationships, symmetry and similarity.
xii. An ability to write clearly, to record mathematics in statements, neatly and systematically.
Before the age of 8 for some, but between the ages of 8 and 11 for most, children should continue to develop in these directions, and progress to:
i. The appreciation of place value, the number system and number notation, including whole numbers, decimal fractions and vulgar fractions. The ability to recognise simple number patterns (odds and evens, multiples, divisors, squares, etc).
ii. The ability to carry out with confidence and accuracy simple examples in the four operations of number, including two places of decimals as for pounds and pence and the measures as used.
iii. The ability to approximate.
[page 28]
iv. A sound understanding of place value applied to the decimal notation for numbers. The ability to carry out the addition and subtraction of numbers with up to two decimal places and the multiplication and division of such numbers by whole numbers up to and including 9.
v. The multiplication and division of numbers with up to two decimal places by 10 and 100.
vi. An appreciation of the connections between fractions, decimal fractions and the most common percentages.
vii. The ability to use fractions in the sequence 1/2, 1/4, 1/8, 1/16, or 1/3, 1/6, 1/12, or 1/5, 1/10, including the idea of equivalence in the discussion of everyday experiences.
viii. An appreciation of the broader aspects of number, such as bases other than 10 and easy tests of divisibility.
ix. An ability to read with understanding mathematics from books, and to use appropriate reference skills.
A number of children of this age will be capable of more advanced work, and they should be encouraged to undertake it.
The above items are taken from Mathematics 5 to 11 (HMSO, 1979), No. 9 in this series, where the ideas are discussed in greater detail.
After the age of 11 pupils should continue to consolidate the above work and to develop their knowledge further. We give below, for ease of reference by teachers, a self-contained list of objectives for pupils of the ages 11 to 16. This list appeared in the HMI discussion document Curriculum 11-16 (DES, 1977).
i. Feel familiar with and at ease among the whole numbers and their relationships one with another.
ii. Perform with understanding the four operations of arithmetic.
iii. Maintain rapid recall of the sums, differences and products of pairs of numbers from 0 to 10; this will be achieved by continual application to questions which require it.
iv. Apply with understanding the knowledge, concepts and skills of ii and iii to larger numbers.
v. Perform with understanding straightforward operations on simple fractions and decimals.
vi. Understand percentages and use them in simple problems.
vii. Be able to estimate number and approximate.
viii. Appreciate that pocket calculators do not make arithmetic unnecessary; use calculators efficiently and apply checks to ensure accuracy.
ix. Know enough about computers to have no irrational fear of them, and have an appreciation of how logical processes are applied to the manipulation of data.
x. Be able to read tabulated information, as in price lists and timetables, and work out the probable interpretations of unfamiliar information presented in this form.
xi. Know enough about diagrams, charts and graphs to be able to interpret those commonly used for communication.
xii. Know enough about simple statistics to be able to interpret them correctly and not be deceived by them.
[page 29]
xiii. Be able to perform such calculations about money as are useful in everyday life.
xiv. Be able to estimate and use a variety of instruments to make measurements in mass, length, time, angle and measures derived from these - for example, velocity; appreciate what they are doing when they measure and, in particular, understand approximation; be able to perform with confidence and understanding calculations depending on the measures, particularly those encountered in science and technical studies.
xv. Be able to read and understand clocks and other combinations of dials.
xvi. Solve correctly many real problems in real situations (for example, cut a dress from three metres of cloth, order timber for shelving, use a pocket calculator to tell which size packet of detergent is the 'best buy').
xvii. Handle, create, discuss, write about three-dimensional objects and solve some problems about them physically as well as by calculation and by scale drawing; interpret diagrams, plans and maps; appreciate the abstractions made in all two-dimensional representation.
xviii. Experience and understand pattern in shape and number.
xix. Have some understanding of proportion, both in shape and in number.
xx. Do simple algebra; they should, for example, learn to generalise patterns in arithmetic, be able to understand and use symbols in the context of mathematical statements, and carry out straightforward manipulation of symbols in simple formulae and equations when the need for this is appreciated.
A number of pupils of 16 are capable of more advanced work and should be helped to undertake it.
Mathematics has not hitherto been included as an essential component in the course of every pupil staying on at school after the age of 16, but the proportion of pupils studying mathematics has tended to increase over the years. Furthermore, the feeling that pupils should continue mathematics, at some level, as a preparation for a wide variety of fields of employment and for a wide range of courses in higher and further education continues to grow.
On the appropriate form of mathematics courses for those for whom A level mathematics is unsuitable, there is little agreement. The problem is under discussion in many places, and the various responsible bodies will in due course make their recommendations.
Meanwhile schools can confidently encourage pupils who are capable of studying A level mathematics profitably to do so, as it will continue to be a widely valued qualification. Schools need also to consider what other mathematics courses they should provide, bearing in mind the specific needs of their pupils, local circumstances, and the constraints imposed by limited staffing. When planning mathematics courses at this level, it is important to consider to what extent they should incorporate elements of statistics and computer studies.