46 



SCIENCE 



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



weights and values of familiar things, such 

 as, finding the dimensions of the school- 

 room, the number of square feet in the 

 floor and walls, the number of cubic feet 

 in the room, the weight of the air in the 

 room, the weight of the walls of the whole 

 building, the number of bricks, etc. These 

 would be more interesting than the com- 

 plicated problems given in some of the ad- 

 vanced arithmetics. 



Professor Neweomb deplores the fact that 

 students who have taken a college course in 

 physics can not compute the quantity of 

 water which woiild be evaporated by the 

 heat generated by the combustion of a ton 

 of coal; or the number of cubic feet of air 

 which could be warmed in the same way. 

 He says that there is no good reason why 

 this kind of elementary physics should not 

 form a part of arithmetic, except adherence 

 to traditional customs characteristic of the 

 district school, and the prejudices of the 

 so-called practical men against everything 

 scientific in education. 



In the study of geometry, the pupil 

 should begin with constructive problems 

 solved graphically; in beginning algebra, 

 the pupil should first thoroughly familiar- 

 ize himself with the use of symbolic lan- 

 guage. Algebra is a kind of language, and 

 to be proficient in its use this language 

 must be learned by practise like any other 

 unusual or foreign language. 



Neweomb concludes by saying that it 

 may be true that by adopting these sug- 

 gestions the pupil would not get through 

 any one book more rapidly and would 

 make no better show of his knowledge upon 

 examination. The advantages to be gained 

 would be fewer courses, through fewer de- 

 tails of arithmetical applications being 

 necessary, and a greater facility in the 

 applications of arithmetic, algebra and 

 geometry to practical questions. 



A somewhat similar plea for practical 



mathematics is made by Fitzga in his work 

 on natural methods of instruction.^* The 

 author first emphasizes the fact that in 

 practise the use of mathematics arises from 

 some external cause and that only concrete 

 comprehensible things create a demand for 

 its use, such as the coins, measures and 

 weights in common use. The fact that 

 numbers can not be seen, and that they are 

 only phases of observation {BeohacMungs- 

 momente) makes it necessary to present to 

 the child's mind such objects, from which 

 he can through observation fix numbers. 

 In this selection much is gained if objects 

 are chosen which will awaken interest. The 

 things that interest the child most are those 

 that are used in life, and such things as the 

 child sees handled by adults. In the be- 

 ginning of arithmetical instruction, there- 

 fore, numbers should be derived from dif- 

 ferent parts of the body, such as the fingers, 

 eyes, etc., and later from arithmetical mag- 

 nitudes {Rechnimgsgrdssen) . These things 

 are at first to be used for the purposes of 

 observation lessons, for in this manner 

 number presentations will be grasped with- 

 out difficulty. 



There is no question but that an exact 

 knowledge of the mathematical magnitudes 

 of life is necessary for the child, but the 

 present methods of presentation do not per- 

 mit of the exact observation of them. The 

 child can not, through such a process, grasp 

 the idea of magnitude, nor the relation of 

 measures and their parts, and without this 

 knowledge there can be no understanding 

 of the subject. The observation of arith- 

 metical magnitudes does not depend upon 

 the physical properties of the objects ob- 

 served, but upon the relation which they 

 bear to one another. There are in each 

 observation lesson certain vital features to 



^* " Die naturliche Methode des Rechenunter- 

 riehtes der Volks und Burgerschule," E. Fitzga, 

 Wien, 1898. 



