(0) ACADEMIES, INSTITUTES, AND HIGH SCHOOLS-Continued. What reforms are needed in the teaching of arithmetic ? 215. Less adherence to and dependence upon text-books; more thorough primary drill. 218. More easy examples. 223. More mental work, more analytical work, greater quickness. 225. Increase in number of problems under each principle, decrease in num ber of "catch problems"; more mental work. 229. There is too much time put on it in all the lower grades. 237. Introduction of quick and labor-saving methods in all business methods. 242. Better use of mathematical language; arithmetic as a deductive science. 251. More practice in rapid calculation. Many of the unimportant rules should be scarcely touched. My pupils waste energy by scattering too much. 255. A more judicious selection of subjects that time be not wasted upon non-essentials. 257. More mental arithmetic. 262. Something to make it more practical and the student better able to apply it. 270. Fundamental operations of arithmetic only should be taught before algebra. 274. Text-books are either so childish as to give no inspiration to work after the primary grades, or so abstruse and dependent upon logical reasoning beyond a child's capacity as to discourage. 275. Insistence upon accuracy in fundamental operations, and alertness of mind everywhere. What reforms are needed in the teaching of arithmetic?-Continued. 276. More thorough work in elementary rules and in common and decimal fractions. 277. Scholars are pushed ahead altogether too fast, allowed to work slowly and incorrectly; should be drilled in quick addition, etc. 281. More attention to accuracy, rapidity, and practical methods. 283. It should be taught as an art rather than as a science. 286. There should be vastly more drill in fundamental processes. 288. Plenty of examples, more oral and "mental" work. 289. More practice. It seems to me that the agitation for reducing time given to arithmetic is a mistake, though greater economy of effort is possible. 294. Fewer subjects, more speed and accuracy in computation. 297. The difficulty (especially with female teachers) is too great subserviency to the text-book—lack of elasticity in accepting methods. 300. Hire competent teachers only. 304. More mental arithmetic. 307. More mental work, greater accuracy and rapidity. Scope of the subject reduced. 312. More practical work; judicious omission from ordinary text-book; better development of principles. 314. More mental, less written work. 317. A diminution in the number of subjects and more independent work by the pupil. 322. Particular attention to thoroughness, and abundant practice on fundamental rules and business methods, with the omission of some rules and methods formerly deemed essential. 331. Keep the keys out of the way and analyze every problem. 335. To return to the old custom of making the pupil do more thinking. There are too many helps and too much "mince-meat.", 338. Many. 340. More philosophy. 341. Return to mental arithmetic, now sadly neglected. More attention to analysis, less to ingenious devices. 344. Do not permit primary teacher to use a figure in presence of children till they know everything about numbers one to ten. 316. The use of the Grube method with beginners, of denominate numbers before abstract, the expansion of the method of analysis in solving problems usually assigned to proportion. 352. A method that will shorten the time, give the pupils the essentials thoroughly. This will come, I believe, only through the experiments in industrial education. 353. More simplicity, less aiming to puzzle, less work that is wholly theoretical. 359. Brief methods of calculation should be insisted on, also independence. 370. Less of it, in much less time than is now given to it (Superintendent of Schools). 382. More attention to mental arithmetic. 386. The use of such books as Colburn's or Venaole's Mental Arithmetic thoroughly at first; and the rejection of such methods as have recently been injected into the new Colburn's Mental Arithmetic. The public schools are teaching for show. 389. Books without answers are needed. 392. We should not go too far in seeking to make all divisions in arithmetio practical. Discipline must be held in mind. To what extent are models used in teaching geometry? The following reported that models were not used: 216, 219, 220, 226, 238, 239, 246, 253, 256, 257, 263, 266, 267, 277, 278, 288, 299, 300, 302, 316, 334, 352, 370, 378, 384, 393. The following reported "occasionally," "not much," ". very little": 217, 218, 222, 231, 233, 236, 240, 241, 242, 243, 244, 245, 251, 255, 258, 265, 269, 276, 282, 293, 294, 295, 296, 307, 309, 318, 324, 326, 332, 335, 337, 338, 339, 341, 348, 350, 351, 354, 356, 358, 360, 364, 366, 367, 372, 375, 385, 387, 390, 391. Nearly all the remaining reports stated that models were used, specifying, in many cases, that they were found particularly serviceable in teaching solid and spherical geometry. Those reports which stated that the models were made by the pupils themselves were classified with the group "using models." To teach plane geometry to very young students, or solid and spherical geometry to students of any grade, without the aid of models, is a great mistake. To what extent and with what success original exercises? All, except about two dozen, reported that original exercises were frequently used, with good success. Some said that one-sixth of the time allotted to geometry was devoted to them, others said one-half of the time; but the large majority of those specifying the relative amount of time given to such work answered one-fourth. Several reporters took occasion to say that the teaching of geometry without introducing original exercises was necessarily more or less of a failure. Is the metric system taught? Nearly every report showed that this is taught, though in many schools but little attention is given to it. We observed only one instance in which it was "dropped," after having been taught for some years. How long will it be before this country will wheel in line with the leading European nations and adopt this system to the exclusion of the wretched systems now in use among us? Which is taught first, algebra or geometry? How far do you proceed in the one before taking up the other? Excepting a number less than a dozen, all answered that algebra was taught first. The following complete a course in elementary algebra, before taking up geometry: 215, 218, 225, 226, 230, 235, 237, 238, 239, 245, 246, 247, 260, 263, 266, 267, 272, 273, 278, 282, 283, 284, 289, 290, 292, 294, 298, 299, 301, 307, 314, 316, 318, 321, 323, 324, 326, 327, 328, 335, 337, 338, 344, 345, 353, 354, 355, 359, 360, 369, 370, 372, 376, 377, 380, 382, 383, 385, 387, 389, 394. The following take up geometry after having carried the student through quadratics: 214, 223, 236, 244, 252, 255, 256, 258, 263, 274, 275, 280, 286, 291, 295, 297, 300, 303, 305, 306, 311, 317, 334, 336, 339, 343, 350, 351, 356, 363, 378, 381, 384, 391. The following, after having carried the student to quadratics: 216, 217, 233, 250, 251, 257, 264, 302, 357, 367, 373, 386. Through radicals: 228, 243, 262, 322. Through equations: 224, 268, 330. To simple equations: 219, 231, 288, 374, 379. Through factoring: 276, 325. Through L. C. M. and G. C. D.: 220. To fractions: 232, 248, 249. To involution: 229. Of those who take up geometry before algebra, 222 teaches Hill's Geometry for Beginners, 234 teaches the simpler parts of geometry, 242 teaches mathematical drawing, involving about sixty geometric problems (without demonstrations), 315 teaches geometry one year, 293 observes the following order of studies: (1) Beginning geometry; (2) algebra; (3) geometry. In the two institutions, 269 and 270, algebra and geometry are taught together. Is this scheme not worthy of more extended trial? Are percentage and its applications taught before the rudiments of algebra or after! The following answered that in their institution it was taught "after": 217, 224 In most, if not all these cases, the elements of percentage had been taught to the In 325 the two subjects are taught "together." Is it not desirable to introduce the rudiments of algebra earlier than has been the Are pupils permitted to use "answer-books" in arithmetic and algebra ? "Yes," "yes, but not encouraged": 214, 216, 217, 218, 219, 226 (with younger classes), "No": 215, 220, 221, 222, 224, 226 (with older classes), 230, 237, 239 (in algebra, but "Some of the answers: "214, 215, 221, 223, 224, 225, 228, 235, 236, 237, 238, 239, 240, Are students entering your institution thorough in the mathematics required for admission ? The following answered " What are the requirements in mathematics for admission to the institution! "Cube root in arithmetic and equations of the second degree in algebra,” 217. Arithmetic and elementary algebra: 222, 230, 273, 288, 303, 357. Three books in geometry, Brook's Algebra, and arithmetic, 268. Arithmetic and algebra as far as factoring, 370. To ratio and proportion in Olney's Practical Arithmetic, 394. Arithmetic to percentage: 218, 306, 328, 379, 383. V. HISTORICAL ESSAYS. HISTORY OF INFINITE SERIES. The primary aim of this paper is to consider the views on infinite series held by American mathematicians. But the historical treatment of this or any similar subject would be meagre indeed were we to confine our discussion to the views held by mathematicians in this country. We might as well contemplate the growth of the English language without considering its history in Great Britain, or study the life-history of a butterfly without tracing its metamorphic development from the chrysalis and caterpillar. A satisfactory discussion of infinite series makes it necessary that the greater part of our space be devoted to the views held by European mathematicians. Previous to the seventeenth century infinite series hardly ever occurred in mathematics; but about the time of Newton they began to assume a central position in mathematical analysis. Wallis and Mercator were then employing them in the quadrature of curves. Newton made a most important and far-reaching contribution to this subject by his discovery of the binomial theorem, which is engraved upon his tomb in Westminster Abbey. Newton gave no demonstration of his theorem except the verification by multiplication or actual root extraction. The binomial formula is a finite expression whenever the exponent of (a+b) is a positive whole number; but it is a series with an infinite number of terms whenever the exponent is negative or fractional. Newton appears to have considered his formula to be universally true for any values of the quantities involved, no matter whether the number of terms in the series be finite or infinite. The binomial theorem was the earliest mathematical discovery of Newton. Further developments on the subject of infinite series were brought forth by him in later works. He made extensive use of them in the quadrature of curves. Infinite series came to be looked upon as a sort of universal mechanism upon which all higher calculations could be made to depend. Special methods of computation, such as contin *This article was read before the New Orleans Academy of Sciences in December, 1887, and printed in the "Papers" published by that society, Vol. I, No. 2. Some alight changes have been introduced here. |