in the days of extensive common fractions) became obsolete for scientific purposes, and science found a new servant to assist in her vast computations. The third improvement is the invention of logarithms by Napier in 1614.1 One might expect that a scheme which, by means of a simple table, allowed computers to multiply and divide by mere addition and subtraction, would find immediate recognition in the schools. And yet, so conservative is the profession that, even in high schools in English speaking countries, logarithms find almost no place, in spite of the fact that neither in theory nor in practice do they present any difficulties commensurate with many found in the old-style arithmetic. In Germany the schools are more progressive in this matter. The fourth improvement of moment is seen in our modern methods of multiplication and division. A problem in division three hundred years ago was a serious matter. The old "scratch" or "galley" method 2 was cumbersome at the best, and the introduction of the "Italian Method," which we commonly use, was a great improvement. Nor is the day of change in these operations altogether passed, 1 That is, his "Descriptio mirifici logarithmorum canonis" appeared in that year. The best brief discussion of the relative claims of Napier and Bürgi is given in Cantor, II, p. 662 seq. 2 Well illustrated in Brooks, E., Philosophy of Arithmetic, Lancaster, Pa., 1880, p. 55, 59. for just now we have the "Austrian methods" of subtraction and of division coming to the front in Germany, and we may hope soon to see them commonly used in the English-speaking world. The fifth improvement is partly algebraic. Algebra, as we know it with its present common symbolism, dates only from the early part of the seventeenth century. With its establishment there departed from arithmetic all reason for the continuance of such subjects as alligation (an awkward form for indeterminate equations), series (better treated by algebra), roots by the Greek geometric process, Rule of Three (as an unexplained rule), and, in general, the necessity for any mere mechanism. Mathematicians recognize no dividing line between school arithmetic and school algebra, and the simple equation, in algebraic form, throws such a flood of light into arithmetic that hardly any leading educator would now see the two separated. The present status of school arithmetic is one of unrest. We have these inheritances from the Renaissance, and with difficulty we are breaking away from them. Only recently have we seen alligation disappear from our text-books, and slowly but surely are we driving out "true" discount, equation of payments, arbitrated exchange, troy and apothecaries' measures, compound proportion, and other objectionable matter. Such subjects, are, as already suggested, unworthy of a place in the course which is to fit for general citi zenship; for they are practically obsolete (like troy weight), or useless (like arbitrated exchange), or mere mechanism and show of knowledge (like compound proportion), or they give a false idea of business (like "true" discount). Slowly we are opening the door to the simple equation, because it illuminates the practical problems of arithmetic, especially those of percentage and proportion. "It is evident," says M. Laisant, "that all through the course of arithmetic, letters should be introduced whenever their use facilitates the reasoning or search for solutions."1 The present tendency is decidedly in favor of eliminating the obsolete, of substituting modern business for the ancient, of destroying the artificial barrier between arithmetic and algebra, and of shortening the course in applied arithmetic. As the report of the "Committee of Ten" stated the case, "The conference recommends that the course in arithmetic be at the same time abridged and enriched; abridged by omitting entirely those subjects which perplex and exhaust the pupil without affording any really valuable mental discipline, and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems." 2 Three years later, the "Committee of Fifteen" had this 1 La Mathématique, p. 206. 2 For full report of the mathematical conference, see Bulletin No. 205, United States Bureau of Education, Washington, 1893, p. 104. further suggestion: "Your Committee believes that, with the right methods, and a wise use of time in preparing the arithmetic lesson in and out of school, five years are sufficient for the study of mere arithmeticthe five years beginning with the second school year and ending with the close of the sixth year; and that the seventh and eighth years should be given to the algebraic method of dealing with those problems that involve difficulties in the transformation of quantitative indirect functions into numerical or direct quantitative data." 1 In all this present change and suggestion of change, the radical element in the profession is restrained by several forces: the publisher fears to join in a too pronounced departure; the author is also concerned with the financial result; the teacher is fearful of the failure of his pupils on some official examination (a most powerful influence in hindering progress); and the pupil and his parents see terrors in any departure from established traditions. But in spite of all this, the improvement in the arithmetics in America has, within a few years, been marked very than in any other country. more so 1 Report of the Committee of Fifteen, Boston, 1895, p. 24. CHAPTER IV HOW ARITHMETIC HAS BEEN TAUGHT The value of the investigation of the way in which arithmetic has been taught, especially during the nineteenth century, is apparent. Find the methods followed by the most successful teachers, find the failures made by those who have experimented on new lines, and the broad question of method is largely settled. "The science of education without the history of education is like a house without a foundation. The history of education is itself the most complete and scientific of all systems of education.”1 It is impossible at this time to trace the development of the general methods of teaching the subject, up to the opening of the nineteenth century. Already, in Chapter I, the development of the reasons for teaching the subject has been outlined, and from this the general methods employed may be inferred. Only a hurried glance at a few of the more interesting details is possible. The departure from object teaching-Arithmetic, at least in the Western world, was always based upon object teaching until about 1500, when the Hindu 1 Schmidt, Geschichte der Pädagogik, I, p. 9. |