EDITOR'S INTRODUCTION PERHAPS no single subject of elementary instruction has suffered so much from lack of scholarship on the part of those who teach it as mathematics. Arithmetic is universally taught in schools, but almost invariably as the art of mechanical computation only. The true significance and the symbolism of the processes employed are concealed from pupil and teacher alike. This is the inevitable result of the teacher's lack of mathematical scholarship. The subtlety, delicacy, and accuracy of mathematical processes have the highest educational value, both direct and indirect. To treat them as mechanical routine, not susceptible of explanation or illumination from a higher point of view, is to destroy in large measure the value of mathematics as an educational instrument, and to aid in arresting the mental development of the pupil. As long ago as the time of Aristotle it was pointed out that mathematics should not be defined in terms of the content with which it deals, but rather in terms of its method and degree of abstractness. Kant says of mathematics, in the "Critique of Pure Reason," "The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience." He then goes on to point out the ground of the distinction between philosophical and mathematical knowledge, and adds: "Those who thought they could distinguish philosophy from mathematics by saying that the former was concerned with quality only, the latter with quantity only, mistook effect for cause. It is owing to the form of mathematical knowledge that it can refer to quanta only, because it is only the concept of quantities that admits of construction, that is, of a priori representation in intuition, while qualities cannot be represented in any but empirical intuition." 2 Mr. Charles S. Peirce has recently made the criticism that Kant was not justified in supposing that mathematical and philosophical necessary reasoning are distinguished by the circumstance that the former uses construction or diagrams. Mr. Peirce holds that all necessary reasoning whatsoever proceeds by constructions, and that we overlook the constructions in philosophy because they are so excessively simple. He goes on to show that mathematics studies nothing but pure hypotheses, and that it is the only science 2 Ibid., p. 573. 1 Müller's Translation (New York, 1896), p. 572. 8 Educational Review, 15, 214. 3 which never inquires what the actual facts are. It is "the science which draws necessary conclusions." This acute argument is, I think, at fault in its contention that construction is employed in philosophical reasoning, but is otherwise sound. It fails, however, to point out clearly these facts: I. The human mind is so constructed that it must see every perception in a time-relation-in an orderand every perception of an object in a space-relation as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics. In the present book, the purpose of which is to present in simple and succinct form to teachers the results of mathematical scholarship, to be absorbed by them and applied in their class-room teaching, the author has wisely combined the genetic and the analytic methods. He shows how the elementary mathematics has developed in history, how it has been used in education, and what its inner nature really is. It may safely be asserted that the elementary mathematics will take on a new reality for those who study this book and apply its teachings. NICHOLAS MURRAY BUTLER. COLUMBIA UNIVERSITY, NEw York, February 1, 1900. CONTENTS HISTORICAL REASONS FOR TEACHING ARITHMETIC. - Impor- tance of the question. The evolution of reasons. The beginning utilitarian. Early correlation. Utilitarian among trading peoples. Tradition and examinations. The cul- ture value. As a remunerative trade. As a mere show of PAGE WHY ARITHMETIC IS TAUGHT AT PRESENT. - Two general The utility of arithmetic overrated. The culture value. Teachers generally fail here. Recognition of the culture value. What chapters bring out the culture value. |