IV. THE MATHEMATICAL TEACHING AT THE PRESENT TIME. The mathematical teaching of the last ten years indicates a "rupture" with antiquated traditional methods, and an "alignment with the march of modern thought." As yet the alignment is by no means rectified. Indeed it has but barely begun. The "rupture" is evident from the publication of such works as Newcomb's series of mathematical textbooks, recent publications on the calculns, the appearance of such algebras as those of Oliver, Wait, and Jones, Phillips and Beebe, and Van Velzer and Slichter; of such geometries as Halsted's "Elements" and "Mensuration;" of such trigonometries as Oliver, Wait, and Jones's; of Carll's Calculus of Variations; Hardy's Quaternions; Peck's and Hanus's Determinants; W. B. Smith's Co-ordinate Geometry (employ. ing determinants); Craig's Linear Differential Equations. Determinants and quaternions have thus far generally been offered as elective studies, and have formed a crowning pinnacle of the mathematical courses in colleges. It is certainly very doubtful whether this is their proper place in the course. It seems quite plain that the ele ments of determinants should form a part of algebra, and should be taught early in the course, in order that they may be used in the study of co-ordinate geometry. What place should be assigned to quaternions is not quite so plain. Prof. De Volson Wood introduces their elements in his work on co-ordinate geometry. The professors of Cornell have not taught quaternions directly for some years, but are convinced that most students derive more benefit by a mixed course in matrices, vector addition and subtraction, imaginaries, and theory of functions. The early introduction of determinants seems more urgent than that of quaterni ons. We think, however, that great caution should be exercised in incorporating either of these subjects in the early part of mathematical courses. Those universities and colleges which are, as yet, not strong enough to maintain a high and rigid standard of admission, and whose students enter the Freshman class with only a very meagre and superficial knowledge of the elements of ordinary algebra, would find the introduction of determinants and imaginaries as Freshman studies a hazardous innovation. One of the very first considerations in mathematical teaching is thoroughness. In the past the lack of thoroughness has poisoned the minds of the American youth with an utter dislike and bitter hatred of mathematics. Whenever a subject is not well understood, it is not liked; whenever it is well understood, it is generally liked. Valuable papers on least squares have been contributed in this country by G. P. Bond,* of Harvard; Simon Newcomb, † C. S. Pierce, ‡ and Truman H. Safford.§ The text-books on this subject generally used in our schools are those of Chauvenet, Merriman, and T. W. Wright. *"On the use of Equivalent Factors in the Method of Least Squares," Memoirs American Academy, Vol. VI, pp. 179–212. "A Mechanical Representation of a Familiar Problem,” Monthly Notices of the Astronomical Society, London, Vol. XXXIII, pp. 573-'4; "A Generalized Theory of the Combination of Observations so as to Obtain the Best Results," American Journal of Mathematics, Vol. VIII. "On the Theory of Errors of Observations," Report U. S. Coast Survey, 1870, pp. 200-224. § "On the Method of Least Squares," Proceedings American Academy, Vol. XI. IV. THE MATHEMATICAL TEACHING AT THE PRESENT TIME. The mathematical teaching of the last ten years indicates a "rupture" with antiquated traditional methods, and an "alignment with the march of modern thought." As yet the alignment is by no means rectified. Indeed it has but barely begun. The "rupture" is evident from the publication of such works as Newcomb's series of mathematical textbooks, recent publications on the calculus, the appearance of such algebras as those of Oliver, Wait, and Jones, Phillips and Beebe, and Van Velzer and Slichter; of such geometries as Halsted's "Elements" and "Mensuration;" of such trigonometries as Oliver, Wait, and Jones's; of Carll's Calculus of Variations; Hardy's Quaternions; Peck's and Hanus's Determinants; W. B. Smith's Co-ordinate Geometry (employing determinants); Craig's Linear Differential Equations. Determinants and quaternions have thus far generally been offered as elective studies, and have formed a crowning pinnacle of the mathematical courses in colleges. It is certainly very doubtful whether this is their proper place in the course. It seems quite plain that the ele ments of determinants should form a part of algebra, and should be taught early in the course, in order that they may be used in the study of co-ordinate geometry. What place should be assigned to quaternions is not quite so plain. Prof. De Volson Wood introduces their elements in his work on co-ordinate geometry. The professors of Cornell have not taught quaternions directly for some years, but are convinced that most students derive more benefit by a mixed course in matrices, vector addition and subtraction, imaginaries, and theory of functions. The early introduction of determinants seems more urgent than that of quaterni ons. We think, however, that great caution should be exercised in incorporating either of these subjects in the early part of mathematical courses. Those universities and colleges which are, as yet, not strong enough to maintain a high and rigid standard of admission, and whose students enter the Freshman class with only a very meagre and superficial knowledge of the elements of ordinary algebra, would find the introduction of determinants and imaginaries as Freshman studies a hazardous innovation. One of the very first considerations in mathematical teaching is thoroughness. In the past the lack of thoroughness has poisoned the minds of the American youth with an utter dislike and bitter hatred of mathematics. Whenever a subject is not well understood, it is not liked; whenever it is well understood, it is generally liked. There is almost always some one author whose text-books reach very extended popularity among the great mass of schools. Such authors were Webber, Day, Davies, and Loomis. If we were called upon to name the writer whose books have met with more wide-spread circulation during the last decennium than those of any other author we should answer, Wentworth. Mr. Wentworth was born in Wakefield, N. H., fitted for college at Phillips Exeter Academy, graduated at Harvard College in 1858, and then returned to Phillips Exeter Academy, where he has been ever since. He had for instructors in mathematics, at the academy, Prof. Joseph G. Hoyt, afterward chancellor of the Washington University in St. Louis; and, in college, Prof. James Mills Peirce. "The characteristics of my books," says Mr. Wentworth, "are due to what I have found from a long experience is absolutely necessary in order that a pupil of ordinary ability might master the subject of his reading. To learn by doing, and to learn one step thoroughly before the next is attempted, constitute pretty much the whole story." In point of scientific rigor Wentworth's books are superior to the popu lar works of preceding decades. It seems to us that the book most liable to criticism is his Elementary Geometry (old edition). He has been greatly assisted in the writing of his books by leading teachers from different parts of our country. Some of the books bearing his name are almost entirely the work of other men. It is to be hoped that the near future will bring reforms in the mathematical teaching in this country. We are in sad need of them. From nearly all our colleges and universities comes the loud complaint of inefficient preparation on the part of students applying for admission; from the high schools comes the same doleful cry. Errors in mathematical instruction are committed at the very beginning, in the study of arithmetic. Educators who have studied the work of Prussian schools declare that our results in elementary instruction are far infe rior. Says President C. K. Adams, of Cornell University: "In the lowest grades of schools our inferiority seems to me to be very marked. The results of the earliest years of the European course, I mean those devoted to teaching the boy, say from the time he is nine years of age until he is fourteen, when compared with the fruits of the courses pursued during the corresponding years in the average American school, are immeasurably superior." President Adams institutes a comparison between Brooklyn and Berlin schools. Speaking of a Brooklyn boy of fifteen, he remarks: "In the first place it must be said that he has had forced upon him six hours a week in arithmetic, during the whole of the seven primary grades. Then on emerging from the primary school, and coming into the grammar school, he is required to take an average of four hours a week in the same study, during all the eight grades. That is to say, during the whole of the boy's career in school, from the * New England Association of Colleges and Preparatory Schools; Addresses and Proceedings at the Annual Meeting, 1888, p. 24. . time he is seven until he is fifteen, he has devoted no less than five hours a week of recitations to the study of arithmetic alone. If we deduct the hours devoted to reading, penmanship, and music, we find that fiveelevenths of what remains is devoted to arithmetic. Making no deductions, and including the hours devoted to the elementary work requiring no preparation whatever, we find that arithmetic occupies in the class-room considerably more than one-fourth of all the student's time, during the whole of seven or eight years." This statement is applicable with equal force to probably all our schools. The fact is that the study of arithmetic has been, in one sense, greatly overdone in this country. The most melancholy thought in this connection is that, after all, our boys and girls acquire only a deficient knowledge of this subject. Persons who had opportunity for comparison assure us that the American boy does not "figure" as rapidly and accurately as the German boy. If the above assertions be true, then it behooves the American teacher to inquire wherein the foreign methods of teaching excel his own. In some circles the study of pedagogy has not been popular. This apathy is, we think, partly due to the influence of some of our nor mal schools. Many of our normal schools have been conducted very efficiently, but others have had teachers in their faculties who lacked breadth and depth of scholarship, and who brought the study of pedagogy into disrepute by their narrowness and their lack of elasticity in the application of methods. This aversion to the study of theories of teaching is now happily disappearing. Our universities and colleges are beginning to establish chairs of pedagogy. Improvements in the teaching of arithmetic might probably be effected by the general introduction of some such method as that of Grube. The first complete exposition of this method was, we believe, published in this country by F. L. Soldan, formerly principal of the St. Louis Normal School. It seems to gain ground here every year. A most valuable and suggestive monograph on mathematical teaching has been written by Prof. T. H. Safford, of Williams College. Professor Safford is an advocate of the heuristic method of teaching. Grube is the representative of this in arithmetic. The method employed by Spencer in his little book on Inventional Geometry is similar to the heuristic, if not identical with it. The heuristic is, in general, the method in which the pupil's mind does the work. It is a slow method. Thus, Grube considers the numbers from 1 to 10 sufficient to engage the attention of a child (of six or seven years) during the first year of school. "In regard to extent, the scholar has not, apparently, gained very much-he knows only the numbers from 1 to 10. But he knows them." The Germans "make haste slowly," but in elementary education they beat us in the race. Geometry, like arithmetic, should be taught sparingly at a time, but for many years in succession. Profes "Grube's method, by F. Louis Soldan, p. 21. |