Peter Wilder reviews new texts for A-level, starting with course books written to cover specific syllabuses within modular schemes. In the early days of modular A-levels there were few mathematics texts to prepare students for specific modules; those available covered the whole A-level course. Now a group of examiners and moderators for the London A-level have produced Heinemann Modular Mathematics, a series written specifically "to provide thorough preparation for the ULEAC examinations at AS and A-level".
Each of the 11 books in the series addresses the syllabus for one module in the London A-level. The books give the impression that, in spite of modularity and the availability of new technology, A-level mathematics has changed little in the past 30 years. Problems are traditional in tone and in form; in Pure Maths 3 they are devoid of context, while in Mechanics 3 the contexts are impersonal and abstract: "A particle of mass 1kg moves in a horizontal straight line . . ."
Explanations and worked examples are clear, if rather dry, and provide comprehensive coverage of the London syllabus. There are many exercises, including past exam questions and an examination style practice paper; answers are at the back of each book. Each chapter closes with a summary of key points, listing all the key ideas that the student needs to remember for the examination.
Modules of the London modular A-level seem to me to be incomplete; I had hoped that the writers on this project, with their particular knowledge and familiarity with the London syllabuses, might reveal more structure through these texts. Regrettably, after studying these books, there is the same impression of fragmentation.
What a contrast then to see the latest material from MEI Structured Mathematics. Again the structure is modular, with six modules for an A-level instead of only four in the London scheme, but I do not sense the same fragmentation at all. Here is an attempt to relate mathematics to applications, and to develop understanding. For example, a chapter on trigonometry in Pure Mathematics 3 introduces the factor formulae through an activity, followed by a section entitled "When do you use the factor formulae", in which students investigate what happens when two musicians playing together are slightly out of tune with each other.
The immediate application of the technique to a real-world situation will help the many students for whom mathematics at A-level is an essential preparation for a career in science or engineering.
Throughout the MEI books, there is variety of activity, including worked examples, reflection, discussion, investigation and frequent exercises. Some questions invite students to think about when a particular technique is appropriate, and which method is simplest for a particular purpose. Topics are often illustrated by short historical notes, or notes about important applications.
MEI Pure Mathematics 3 follows on from the previously published Pure Maths 1 and Pure Maths 2; together these three books cover the subject core for A-level mathematics, so may be useful in preparation for other syllabuses. Pure Mathematics 4 is for students who are taking their study of pure maths beyond the subject core, either as a final A-level module, or as part of a further maths A-level. It builds on ideas in earlier books, and opens with an interesting chapter on proof.
In Mechanics 5 and 6, two modules are together in one text. There is a welcome emphasis on modelling real situations; experiments and investigations show the application of topics to practical problems in science and engineering.
I was very impressed by the MEI materials. They form part of a coherent modular course seeking to engage students' interest, and to promote deeper understanding.
Discovering Advanced Mathematics reflects the new subject core for A-level maths. Although the title is Pure Mathematics, the book also covers the requirements for students to study mathematical modelling, the mathematics of uncertainty, and the appropriate use of technology. Each new technique is introduced in a context, and chapters conclude with modelling and problem solving exercises.
Graphics calculators are used frequently throughout. There are some excellent worked examples and many exercises and past exam questions.
The opening chapter looks at problem-solving in mathematics. The modelling theme underlying the course is firmly introduced here under five headings: linear, quadratic, other non-linear, exponential and wave models. This provides ample opportunity to build on GCSE skills, and to develop some skills using the graphics calculator.
The range of applications referred to in the book is refreshingly broader than usually found in A-level texts. Specific activities encourage students to make appropriate use of graphics calculators, through plotting graphs and by writing short programs.
This is an interesting book, offering many imaginative ideas, providing structure and support for all students, and challenge for the most able. I certainly intend to use some of the material in my own teaching.
Peter Wilder is a senior lecturerin mathematics education at De Montfort University, Bedford