ADVANCED ARITHMETIC BY WILLIAM W. SPEER DISTRICT SUPERINTENDENT OF SCHOOLS, CHICAGO "This law of organic progress is the law of all progress. Whether HERBERT SPENCER. BOSTON, U.S.A. GINN & COMPANY, PUBLISHERS The Athenæum Press 1899 PREFACE. THE purpose of this book is to aid the teacher in making conditions favorable for the contact of the learner with mathematical realities. Since the simplest as well as the most complex relations become known only through mental activity in comparing, attention is given to an environment. which shall continually induce this activity. The comparison of magnitudes includes all the other operations of arithmetic. This is apparent when we reflect that all advance in knowing is by progressive acts of analysis and synthesis. Whether we compare by means of the object or by means of its symbol, the mind should still be free to move from the vague to the definite through its own acts. In the first book of this series it was said: "That quantity is a ratio between terms which are themselves relative is a truth which has often been pointed out, but which the work of the schools shows to be felt by few." In the minds of many bred upon the language of mathematics, the mode of expression has acquired an independence which excludes the reality. The presentation of relative magnitude as a subject of study in the elementary school has, however, met with gratifying response. Some authors, indeed, whose presentations are entirely foreign to the development of ideas of relative magnitude, have recently written urging M305990 relations of magnitude as the objects of study; others have given the word "ratio" a prominent place in new editions. "A religious vocabulary without religious experiences" is of little value; a mathematical vocabulary without mathematical experiences is of no more value. The student cannot advance in any science if his attention is absorbed in the language. Non-mathematical work cannot generate mathematical ideas. Definitions of mathematics and of quantity avail little without the feeling and the intimate understanding which cause us to shape our work in accord with the illuminating idea. The mode of dealing with the greatest common measure, percentage, longitude and time, square root, mensuration is in accord with underlying mathematical ideas. The pupil advances in the indirect establishment of relations of magnitude by living in practical contact with mathematical realities.1 Each subject has its own elementary ideas. The study of relative magnitude does not give a basis for inferences in history or biology, nor does attention to history give a basis for mathematics. The power to say, "The act is brave," or, "The building is fitly proportioned," begins in comparing through the senses, but the observing which leads to one of these judgments does not furnish a basis for the other. But all elementary work should be formative of a mental habit which may be fitly carried into all lines of effort. 1 "One well-known principle underlying the acquisition of knowledge is that an idea cannot be fully grasped. by the youthful mind unless it is presented under a concrete form. Whenever possible an abstract idea must be embodied in some visible representation. . . Should it appear to any one that we thus detract from the generality of algebraic quantities, it is sufficient to reply that the system is the same which mathematicians use to assist their conceptions of advanced algebra, and without which they would never have been able to grasp the complicated relations of imaginary quantities."-SIMON NEWCOMB. Things must be observed in various aspects if they are to be known, but to say that too much attention to mathematics or to particular things weakens is merely to make a specific application of a universal truth.1 The presentation in this series is from the standpoint that any success is dangerous which lessens the susceptibility of the mind. The constant purpose is to promote growing power to act in new circumstances. W. W. SPEER. 1 "The intensity with which any form of exercise is carried on during the growing period leaves its trace, and the absence of it at the proper time is for the most part irremediable. We should hardly expect much appreciation of color in a person brought up in the dark, however good his natural endowments in this direction. Thus any lack of early experience may leave a spot permanently undeveloped in the central system." - PROF. H. H. DONALDSON. Nor can we expect appreciation of mathematical relations if there is no opportunity for attending to them. If we leave the sense of proportion undeveloped, and leave the pupil unaware of the realities of mathematics in the elementary work, we can scarcely expect interest in relative magnitude at a later period. AUTHOR. |