40 



SCIENCE 



[N. S. Vol. XXX. No. 758 



as to its value and pi-actieability, are given 

 in what follows. 



THE LABORATORY METHOD 



The works of Klein and Perry mark the 

 beginning of a movement to improve on 

 present methods and make a more direct 

 and pleasant path for the average student 

 in the field of mathematics. The essence 

 of the laboratory method consists in the 

 performing of the bulk, if not all, of the 

 work in the mathematical class room, which 

 should be equipped with laboratory appli- 

 ances for the graphic, the experimental and 

 the concrete phases of the work. The 

 teacher acts as the director of the labora- 

 tory, the pupils work individually or in 

 small groups, and analogies with the work 

 in the physical laboratory are emphasized. 



The word laboratory undoubtedly came 

 to be used largely from the suggestions re- 

 ceived from analogous work in the physics 

 laboratory. In 1886 Safford in his "Math- 

 ematical Teaching" said that a mathemat- 

 ical laboratory, although not often men- 

 tioned, was a necessity, and should contain 

 such things as relate to ordinary, not purely 

 scientific, measures.^ Young calls attention 

 to the fact that in the physics laboratory 

 students work singly or in small groups 

 under the general supervision of the in- 

 structor, but with direct contact with him 

 for only a few minutes, and that this is a 

 limitation of the physical laboratory, and 

 not an advantage.' 



The advantage of the class recitation 

 over individual or private instruction has 

 been pointed out by W. T. Harris, U. S. 

 Commissioner of Education.' The class is 

 the most potent of all instruments in the 

 teacher's hand. He so manages the recita- 



' " Mathematical Teaching," Safford, 1886, 

 Heath & Co., 1896. 



' " The Teaching of Mathematics," J. W. A. 

 Young, Longmans, 1907. 



" W. T. Harris, 1897. 



tion or class exercise that each pupil learns 

 to see the lesson through the minds of all 

 his fellows, and likewise learns to criticize 

 the imperfect statements made by them 

 through the more adequate comprehension 

 of the teacher. But because in mathemat- 

 ics the instruments are so simple, the benefit 

 of laboratory work may be obtained under 

 class direction, thus getting the good fea- 

 tures of the laboratory system while avoid- 

 ing its defects. 



The procedure which physicists find best 

 pedagogically suggests a plan for mathe- 

 matics; namely, not that the mathematical 

 class exercise be supplanted by a mathe- 

 matical laboratory exercise, but that it be 

 supplemented thereby. The mathematical 

 class exercise should be conducted by some 

 good method as at present, with the usual 

 time allotment. This should then be sup- 

 plemented by work in a well-equipped 

 mathematical laboratory, either under the 

 direction of the teacher or one or more 

 assistants or both. The pupils should do 

 substantially the same work in the labora- 

 tory, and the class exercise should prepare 

 directly for it. 



The feeling that mathematics must be 

 made more concrete and must come into 

 closer touch with the realities about the 

 pupil, is growing in Germany, France, 

 England and America; and the influence 

 of the work of Perry can be distinctly 

 marked in the current thought on the Euro- 

 pean continent.^" 



Professor Moore has defined pure mathe- 

 matics as a language for the convenient 

 expression and investigation of the most 

 diverse relations of life and nature. The 

 principles of the language are not arbi- 

 trary, but are imposed by the phenomena 

 demanding expression." From this it fol- 



" " Teaching of Mathematics," Young. 



" " Cross Section Paper as a Mathematical In- 

 strument," Moore, School Science and Mathe- 

 matics, June, 1906. 



