Tim Rowland reviews a publication of masterclass lectures

Mathematics, like music, is no respecter of age. As Frances Kirwan reminds us in this book, the teenage polymath Carl Gauss proved the constructibility of a regular 17-sided polygon, and so dedicated his life to mathematics. In our own time, Ruth Lawrence had gained her first mathematics degree from Oxford at an age when her contemporaries would barely be considering their GCSE options.

Following the success of Christopher Zeeman’s 1978 Christmas lectures on mathematics, the Royal Institution invited him to inaugurate a series of Mathematics Masterclasses in London, with a view to stimulating pre-cocious mathematical talent. Many provincial groups now provide RI masterclass series.

This book is an outcome of the classes, which began in Reading, Berkshire, in 1992. Each of the 12 chapters is a written account of a two-and-a-half-hour lecture given to a class of 30 or so 13-year-olds. Each follows a standard format, including some exercises. The material covers popular pure and applied topics - from geometry, number theory and probability to dinosaurs, bubbles and weather.

The book encapsulates my love-hate relationship with the Mathematics Masterclass movement, based on 15 years’ involvement with similar initiatives in Cambridgeshire. The enthusiasm of the contributors is palpable, the expertise beyond question. But we have not sufficiently acknowledged that even able 13-year-olds need exposition which is informed by knowledge of how adolescents think mathematically. Notwithstanding their achievement relative to their age, few of the pupils who come to the RI classes think quite like good mathematics undergraduates.

The teachers who really know what able 13-year-olds can do - and what they find difficult - are reluctant to be pressed into giving masterclasses. Liz and Chris Bills are exceptions. Their chapter on chaos theory is a model of careful exposition of advanced material to a general audience, with well-chosen examples to convey generalities. Their brief biographical notes help to humanise the maths.

Professional academic mathematicians who typically give the classes are inclined to overestimate and “lose” their audience. This problem is compounded in the written version of the lectures, as the text cannot interact with the reader. Michael Sewell acknowledges his inexperience with such a young audience, but notes in the preface that “I learned how to write (a masterclass) by studying Sir Christopher Zeeman’s notes.” If only it were that simple.

David Stirling’s chapter on number theory probably tries to cover too much ground. His proof concerning the form of primitive Pythagorean triples is pretty much the one found in undergraduate textbooks - and many of my undergraduate students struggle with it. Stirling is by no means alone in misjudging the appetite of 13-year-olds - albeit capable ones - for deductive argument shrouded in symbolism. They may be precocious, but they are children nevertheless. (Pupil evaluation of our Cambridge classes regularly highlights the interval biscuits as the most popular feature.) On the other hand, in a delightful contribution on polyhedra and weather, meteorologist Andy White is careful to help his readers maintain a global overview of the problem he addresses. In the final chapter, on water waves, Winifred Wood anticipates some of the potential difficulties and tries to circumvent them before they block comprehension and enjoyment.

This book is most welcome, although most of its content will probably need to be mediated to the young reader by a teacher of some kind. I hope parents, teachers (and pupils) will find the work as stimulating as I did.

Dr Tim Rowland is a lecturer in mathematics at Homerton College, Cambridge