This article is a version, adapted to the written word, of an invited lecture given to the annual conference of Heads of Departments of Mathematical Sciences in universities, at the University of Birmingham, on 20 April 2001. It is offered here in the belief that an indication of the thinking behind the provision of Masterclasses for schoolchildren may also be of interest to schoolteachers themselves. I have been grateful for the very strong support which teachers have given, over ten years, to the Masterclasses which I have organised.