Teaching Mathematics in Secondary Schools

external conditions over which the teacher has no con- trol, but partly also to the fact that a number of teachers have had little opportunity to become acquainted with the details of modern methods of teaching mathe- matics, and hence largely employ the methods by which they themselves were taught.There can be no doubt that the two excellent American books on the teaching of mathematicsthe one by David Eugene Smith, the other by J. W. A. Youngare of great assistance to every teacher, but these books cannot answer many con- crete questions on account of the wide range of subjects that they treat of.This book covers a much more re- stricted field, but does it in greater detail.All references to elementary school work, to history, to the description of movements in other countries, to the material equipment, to mathematical clubs, etc., are excluded, and the discussion of general methods is restricted to their fundamental and most useful phases.This book is modern in the sense that it attempts to make mathematical instruction less informational and tries to show how to train students in attacking mathematical problems instead of merely making them learn mathematical facts.But it is not modern in the sense that it advocates certain recent fashions which aim to replace the true study of mathematics by applica- tions of doubtful value.While admitting that a certain amount of applied work is very useful and interesting, the author does not believe that the true value of mathematical study lies in its practical utility, and hence cannot admit that the mensuration of parquet PREFACE Vli floors or the construction of window designs forms the true end of mathematical study.In addition to the purely pedagogical discussions, this' volume contains certain topics in pure mathematics which, on account of their bearing upon teaching, should be familiar to every teacher.These topics re- late principally to the modes of attack (Chapter XV), but also to the foundations of mathematics (Chapter V),to the division of the circle (Chapter XVI), and to a few other subjects.These chapters may be omitted when a rapid survey of the purely pedagogic matters is desired.Like any other pedagogic book, this volume must necessarily contain a great deal that is obvious and com- monplace to the experienced teacher.But to write for the latter only would mean to make this book useless for the prospective teacher to whom the study of mathe- matical pedagogy is most important.