THE NEW YORK 746491 ASTOR, LENOX AND COPYRIGHT, 1895, BY D. APPLETON AND COMPANY. ELECTROTYPED AND PRINTED MAY 1910 22282 EDITOR'S PREFACE. 371.3 m In presenting this book on the Psychology of Number it is believed that a special want is supplied. There is no subject taught in the elementary schools that taxes the teacher's resources as to methods and devices to a greater extent than arithmetic. There is no subject taught that is more dangerous to the pupil in the way of deadening his mind and arresting its development, if bad methods are used. The mechanical side of training must be joined to the intellectual in such a form as to prevent the fixing of the mind in thoughtless habits. While the mere processes become mechanical, the mind should by ever-deepening insight continually increase its power to grasp details in more extensive combinations. Methods must be chosen and justified, if they can be justified at all, on psychological grounds. The concept of number will at first be grasped by the pupil imperfectly. He will see some phases of it and neglect others. Later on he will arrive at operations which demand a view of all that number implies. Each and every number is an implied ratio, but it does not express the ratio as simple number. The German language is fortunate in having terms that express the two aspects of numerical quantity. Anzahl expresses the 2 multiplicity and Einheit the unity. Any number, say six, for example, has these two aspects: it is a manifold of units; the constituent unit whatever it is, is repeated six times. It is a unity of these, and as such may be a constituent unit of a larger number, five times six, for instance, wherein the five represents the multiplicity (Anzahl) and the six the constituent unity (Einheit).* Number is one of the developments of quantity. Its multiplicity and unity correspond to the two more general aspects of quantity in general, namely, to discreteness and continuity. There is such a thing as qualitative unity, or individuality. Quantitative unity, unlike individuality, is always divisible into constituent units. All quantity is a unity of units. It is composed of constituent units, and it is itself a constituent unit of a real or possible larger unity. Every pound contains within it ounces; every pound is a constituent unit of some hundredweight or ton. The simple number implies both phases, the multiplicity and the unity, but does not express them adequately. The child's thought likewise possesses the same inadequacy; it implies more than it explicitly states or holds in consciousness. This twofoldness of number becomes explicit in multiplication and division, wherein one number is the unit and the other expresses the multiplicity-the times the unit is taken. Fractions form a more adequate expression of this ratio, and require a higher consciousness of the nature of quantity than simple numbers do. Hence the difficulty of teaching this subject in the ele * Hegel, Logik, Bd. I, 1st Th., S. 225. mentary school. The thought of demands the thought of both numbers, 7 and 8, and the thought of their modification each through the other. The methods in vogue in elementary schools are chiefly based on the idea that it is necessary to eliminate the ratio idea by changing one of the terms of the fraction to a qualitative unit and by this to change the thought to that of a simple number. Thus halves and quarters and cents and dimes are thought as individual things, and the fractional idea suppressed. In the differential calculus ratio is most adequately expressed as the fundamental and true form of all quantity, number included. The differential of x and the differential of y are ratios. The authors of this book have presented in an admirable manner this psychological view of number, and shown its application to the correct methods of teaching the several arithmetical processes. The shortcomings of the "fixed-unit" theory are traced out in all their consequences. The defects of a view which makes unity a qualitative instead of a quantitative idea are sure to appear in the methods of solution adopted. Pupils studying music by the highest method learn thoroughly those combinations which involve double counterpoint. As soon as the hands are trained to readily execute such exercises the pupil can take up a sonata of Beethoven or a fugue of Bach, and soon become familiar with it. On the plan of the old lessons in counterpoint, the pupil found himself helpless before such a composition. His phrases furnished no key to the compositions of Bach or Beethoven, because the latter are constructed on a different counterpoint. So the methods of teaching ar unit" system do not lead toward matics, but away from it. They training in thinking the ratio invo of number. The psychology of number req ods be chosen with reference to the mind of the pupil into this ratio idea. The steps should be s gradual; but it should be continuo constantly gains in his ability to ho the unity of the two aspects of qu the discrete and the continuous, the plex and the simple unit. Measurement is a process that m clear. The constituent unit becomes and vice versa, through being measure the measure of others. This, too, is the decimal system of numeration, a ing the different orders of units, each cludes constituent units and is include of a higher unit. The hint is obtained from this the in arithmetic should be based on the uring in its varied applications. Again, since ratio is the fundamen how fallacious are those theories whi basis for mathematics by at first produ vivid idea of unity-as though the were to be built up on this idea. It is abstract unit is not yet quantity nor an ti y, but simply the idea of individuali |
