July 30, 1896] 



NA TURE 



309 



Mathematics. 



The form of the report of the conference on mathematics 

 dififers somewhat from that of the other reports. This report 

 is subclivideil under five headings: (i) General conclusions; 

 (2) the teaching of arithmetic ; (3) the teaching of concrete 

 geometry ; (4) the teaching of algebra ; (5) the teaching of 

 formal or demonstrative geometrj-. 



The first general conclusion of the conference was arrived at 

 unanimously. The conference consisted of one Government 

 official and university professor, five professors of mathematics in 

 as many colleges, one principal of a high school, two teachers of 

 mathematics in endowed schools, and one proprietor of a private 

 school for boys. The professional experience of these gentle- 

 men and their several fields of work were various, and they 

 came from widely separated parts of America ; yet they were 

 unanimously of opinion " that a radical change in the teaching 

 of arithmetic was necessary." They recommend " that the 

 course in arithmetic be at once abridged and enriched ; abridged 

 by omitting entirely those subjects which perple.x and exhaust 

 the pupil without affording any really valuable mental discip- 

 line, and enriched by a greater number of exercises in simple 

 calculation, and in the solution of concrete problems." They 

 specify in detail the subjects which they think should be cur- 

 tailed or entirely omitted, and they give in their special report 

 on the teaching of arithmetic a full statement of the reasons on 

 which their conclusion is based. They map out a course in 

 arithmetic which, in their judgment, should begin about the age of 

 six years, and be completed at about the thirteenth year of age. 

 The conference next recommend that a course of instruction 

 in concrete geometry with numerous exercises be introduced 

 into the grammar schools, and that this instruction should, 

 during the earlier years, be given in connection with drawing. 

 They recommend that the study of systematic algebra should be 

 begun at the age of fourteen ; but that, in connection with the 

 study of arithmetic, the pupils should earlier be made familiar 

 with algebraic expressions and symbols, including the method of 

 solving .simple equations. "The conference believe that the 

 study of demonstrative geometry should begin at the end of the 

 first year's study of algebra, and be carried on by the side of 

 algebra for the next two years, occupying about two hours and 

 a half a week." They are also of opinion " that if the intro- 

 ductory course in concrete geometry has been well taught, both 

 plane and solid geometry can be mastered at this time." Most 

 of the improvements in teaching arithmetic which the conference 

 suggest "can be summed up under the two heads of giving the 

 teacher a more concrete form, and paying more attention to 

 facility and correctness in work. The concrete system should 

 not be confined to principles, but be extended to practical 

 applications in measuring and in physics." 



In regard to the teaching of concrete geometry, the conference 

 urge that while the student's geometrical education should begin 

 in the kindergarten, or at the latest in the primary school, sys- 

 tematic instruction in concrete or experimental geometry should 

 begin at about the age of ten for the average student, and should 

 occupy about one school hour a week for at least three years. 

 From the outset of this course, the pupil should be required to 

 express himself verbally as well as by drawing and modelling. 

 He should learn to estimate by the eye, and to measure with 

 some degree of accuracy lengths, angular magnitudes, and 

 areas ; to make accurate plans from his own measurements and 

 estimates ; and to make models of simple geometrical solids. 

 The whole work in concrete geometry will connect itself on the 

 one side with the work in arithmetic, and on the other with 

 elementary instruction in physics. With the study of arithmetic 

 is therefore to be intimately associated the study of algebraic 

 signs and forms, of concrete geometry, and of elementary 

 physics. Here is a striking instance of the interlacing of sub- 

 jects which seems so desirable to every one of the conferences. 



Under the head of teaching algebra, the conference set forth 

 in detail the method of familiarising the pupil with the use of 

 algebraic language during the study of arithmetic. This part 

 of the report also deals clearly with the question of the time 

 required for the thorough mastery of algelira through quadratic 

 equations. The report on the teaching of demonstrative geometry 

 is a clear and conci.se statement of the best method ol teaching 

 this subject. It insists on the importance of elegance and finish 

 in geometrical demonstration, for the reason that the discipline 

 fur which geometrical demonstration is to be chiefly prized is a 

 discipline in complete, exact, and logical statement. If sloven- 

 liness of expression, or awkwardness of form is tolerated, this 



NO. 1396, VOL. 54] 



admirable discipline is lost. The conference therefore recom- 

 mend an abundance of oral exercises in geometry — for which 

 there is no proper substitute — and the rejection of all demon- 

 strations which are not exact and formally perfect. Indeed, 

 throughout all the teaching of mathematics the conference deem 

 it important that great stress be laid by the teacher on accuracy 

 of statement and elegance of form as well as on clear and 

 rigorous reasoning. Another very important recommendation 

 in this part of the report is to be found in the following passage : 

 "As soon as the student has acquired the art of rigorous demon- 

 stration, his work should cease to be merely receptive. He 

 should begin to devise constructions and demonstrations for 

 himself. Geometry can not be mastered by reading the 

 demonstrations of a text-book, and while there is no branch of 

 elementary mathematics in which purely receptive work, if con- 

 tinued too long, may lose its interest more completely, there is 

 also none in which independent work can be made more attrac- 

 tive and stimulating." These observations are entirely in 

 accordance with the recent practice of some colleges in setting 

 admission examination papers in geometry which demand of 

 the candidates some capacity to solve new problems, or rather 

 to m.ake new application of familiar principles. 



Physics, Chemistry, and Astronomy. 

 The members of this conference were urgent that the study of 

 simple natural phenomena be introduced into elementary 

 schools, and it was the sense of the conference that at least one 

 period a day from the first year of the primary school should be 

 given to .such study. Apparently the conference entertained the 

 opinion that the present teachers in elementary schools are ill 

 prepared to teach children how to observe simple natural phe- 

 nomena ; for their second recommendation was that special 

 science teachers or superintendents be appointed to instruct the 

 teachers of elementary schools in the methods of teaching 

 natural phenomena. The conference were clearly of opinion 

 that from the beginning this study should be pursued by the 

 pupil chiefly, though not exclusively, by means of experiments 

 and by practice in the use of simple instruments for making 

 physical measurements. The report dwells repeatedly on the 

 importance of the study of things and phenomena by direct 

 contact. It emphasises the necessity of a large proportion of 

 laboratory work in the study of physics and chemistry, and 

 advocates the keeping of laboratory note-books by the pupils, 

 and the use of such note-books as part of the test for admission 

 to college. At the same time the report points out that lab- 

 oratory work must be conjoined with the study of a text-book 

 and with attendance at lectures or demonstrations, and that 

 intelligent direction by a good teacher is as necessary in a 

 laboratory as it is in the ordinary recitation or lecture room. 



The great utility of the laboratory note-book is emphatically 

 stated. To the objection that the kind of instruction described 

 requires much time and effort on the part of the teacher, the 

 conference reply that to give good instruction in the sciences 

 requires of the teacher more work than to give good instruction 

 in mathematics or the languages ; and that the sooner this fact 

 is recognised by those who have the management of schools the 

 better for all concerned. The science teacher must regularly 

 spend much time in collecting materials, preparing experiments, 

 and keeping collections in order, and this indispensable labour 

 should be allowed for in programmes and salaries. As regards 

 the means of testing the progress of the pupils in physics and 

 chemistry, the conference were unanimously of opinion that a 

 laboratory examination should always be combined with an oral 

 or written examination, neither test taken singly being sufficient. 

 There was a diflerence of opinion in the conference on the 

 question whether physics should precede chemistry, or chemistry 

 physics. The logical order would place physics first ; but all 

 the members of the conference but one advised that chemistry 

 be put first for practical reasons which are stated in the majority 

 report. A sub-committee of the conference has prepared lists 

 of experiments in physics and chemistry for the use of secondary 

 schools, not, of course, as a prescription, but only as a sugges- 

 tion, and a somewhat precise indication of the topics which 

 the conference had in mind, and of the limits of the instruction. 



Natural History. 



The conference on natural history unanimously agreed that 

 the study of botany and zoology ought to be introduced into 

 the primary schools at the very beginning of the school course, 

 and be pursued steadily, with not less than two periods a week. 



