Tactical advantage in battle of wits

Just as in the oriental game of "Go", where the placement of the first few stones often dictates the development of the entire game, in teaching mathematics the initial selection can affect the entire learning process thereafter. Judicious selection of topics will allow the student to short-circuit the long and tedious process of learning the detail.

What is apparent to me, from surveying these four books, is just how much of a long-winded meal can be made of the task of becoming proficient at using mathematics as a language.

It is not necessary to practise writing all the phrases one might use in written English; in the same way, neither is it appropriate to be taken through every possible mathematical development step by step.

If detail of this order is required, it is best achieved by interaction with an adept maths guru who can suggest exercises for a particular student's needs. These books, therefore, may not be appropriate for unguided self-study. Mathematics is best acquired by working in small groups under the guidance of an experienced teacher.

The first three of these books adopt the method of instruction by solving real examples, chosen at random from the infinite set of possible problems, as a training aid for the student. As such, they are proficient and useful books, with much content. If students have a problem similar to the one worked through as a template, they will be able to obtain considerable help and guidance from the worked example, providing they can locate it.

Basic Engineering Mathematics will be appreciated by first-year students, who will be used to the teaching methods and problem-solving exercises used in schools. Advanced Engineering Mathematics is a development of a well-liked series that has been around for some years and has been used by many students. It is entirely fit for purpose. Engineering Mathematics through Applications is also more advanced; it professes to teach mathematics "in an engineering context right from the very start". All three books will be of great help to busy lecturers who wish to expose students to worked examples without having to invent any themselves. Such books provide a very useful resource for an academic guide or mentor, who can select the appropriate example for the purposes of the moment and refer students to it.

For some of the more advanced examples, which many people have never had the time or inclination to work through (in Advanced Engineering Mathematics , for example, there is a nicely worked example of the solution of a two-dimensional Laplace equation in rectangular coordinates, and then in plane polar coordinates), the examples provide a useful "validation" of the student's view of whether or not he or she may be right.

Again, this substitutes for expert guidance from an experienced tutor or mentor. As access by the student to staff time is somewhat limited in most modern UK universities, the book provides a useful resource for such validation.

One of the features of these three books, all from authors based at English universities, is their repetition. Clearly, in their own exposure to students, the authors have not encountered students who think on their feet; rather they have found that repetition is necessary to get the points across.

The fourth book, Advanced Mathematics for Engineering and Science , is by North American authors. C. F. Chan Man Fong et al have written a splendid book, which benefits from having a large type size. There are some salutary examples of contour integration, with useful diagrams. They say: "We have been generous in writing down all the steps involved in solving the example problems."

The topics covered are more advanced than in the other three books and appropriate to a UK MSc course rather than to introductory undergraduate courses.

I come away having read a combined total of about 3,000 pages with the feeling that my mathematical training has equipped me to understand and recognise all this material. That is odd, as I do not recall having done as much work as a superficial assessment of these books might indicate would have been necessary.

David Jefferies is senior lecturer in engineering, University of Surrey.