No abstract

Britain one of the educators quoted most often is Professor A. N. Whitehead. “Every child should experience the joy of discovery” has been our slogan for more than twenty years. In this respect, America has been at least partly responsible for the revolution which is taking place in British education. Although the changes did not start with mathematics, it is in mathematics that the greatest and perhaps most exciting changes are taking place today.

Learning mathematics, like learning language and the aesthetic subjects, is an active process and should call into play our imaginative and creative powers so that the subject is a delight to pupils of all ages. Investigations of every aspect of mathematics—number knowledge itself and written calculations, as well as measures, shapes, and representations of all kinds—are included. In the title of this article, the word experience could very well be changed to investigations. Investigations, to me, cover a wider range of activities than experiences, which arc often thought of as practical experiences.

This article outlines a series of mathematical tasks given to very able pupils aged 7–12 years. It also stresses the value of research and inservice education which is firmly teacher and classroom based.

The authors present two formidable names, from whom much is expected and by whom, indeed, much is given. Written with skill and enthusiasm, the book should give valuable help in restoring geometry to a more prominent place in a curriculum from which it has too often been banished. (But whether revisited will be a proper description for much of the subject matter is, sadly, doubtful.) The chapter headings are: Points and lines connected with the triangle, some properties of circles, collinearity and concurrence, transformations, an introduction to inversive geometry, and an introduction to projective geometry. It will be seen that the net is wide. The treatment, though brisk and pointed throughout, moves with that leisurely air of browsing that is essential to all good geometry. A book so full of meat can be exhibited only by sampling. The chapter on Transformations serves excellently. After consideration of translation and rotation, the authors apply their ideas to the once familiar Pythagoras figure and then pass to the half-turn and to reflection. This leads to Fagnano's problem of finding the triangle of minimum perimeter inscribed in a given acute-angled triangle. Then follows "the three jug problem" which for these authors requires reflection; this topic, oddly enough, is followed at once by dilatation, and then by spiral similarity. Twenty-seven pages of Hints and Answers, twenty-nine texts for reference, and six pages of glossary bring to a triumphant conclusion a book which any school or college library neglects at its peril and which all interested in the teaching and reading of geometry will insist on obtaining.