Mathematics teachers are at present doing what philosophers urge - asking what we do, why and whether we should. In an age of calculating machines need children know multiplication tables? Should we teach manipulation in algebra? If so, how much? Are logarithms obsolete? Should we teach trigonometry at all? If so, why? What parts of recent mathematics ought to come into the school syllabus? What should go out to make room for new topics? More fundamental, what considerations should determine our choice of syllabus?