July 9, 1909] 



SCIENCE 



43 



In second-year mathematics geometry 

 holds the center of attention, and arith- 

 metical and algebraic elements are subor- 

 dinated to it. The distinctive feature of 

 the plan of presenting deductive geomet- 

 rical truths consists of five general steps. 

 The figure required by the demonstration 

 is first sketched in the rough, in a way to 

 exhibit clearly the conditions under which 

 the truth in question is to be established. 

 A careful drawing is next made on paper 

 or on the blackboard with ruler and com- 

 pass, under the specified conditions, and the 

 appropriate parts of the figure that are 

 drawn (protractor admitted) are carefully 

 measured. Pupils are then required to 

 make the best possible inferences as to the 

 conclusions which follow from conforming 

 to the imposed conditions. A correct enun- 

 ciation of the principles to be established 

 is next made and finally a deductive proof 

 is given in standard form. 



The mode of conducting the class work is 

 a combination of the laboratory, the experi- 

 mental, the Socratic and the class recitation 

 modes. One of the advantages of the 

 method is that it impresses the novice with 

 the inadequacy of pure metrical means, 

 and with the necessity of demonstrative 

 methods. 



As the result of six years' experience in 

 teaching elementary high-school mathemat- 

 ics. Professor Lennes asserts his belief that 

 graphical work is of great importance in 

 creating interest and promoting a clearer 

 and more satisfactory insight into subjects 

 which too often are mysterious riddles.^" 



Professor Myers supplements this by 

 saying that laboratory work with real prob- 

 lems, in the formulation and handling of 

 which the pupil habituates himself to the 

 transition from the concrete to the abstract, 

 goes far toward supplementing the present 



" " The Graph in High School Mathematics." 

 N. J. Lennes, School Review, 1906. 



isolated and abstract teaching of secondary 

 mathematics.-" 



THE PHILOSOPHIC ATTITUDE 



The ref onns proposed by Professor Perry 

 emphasized the practical features of in- 

 struction. In geometry especially there 

 was a radical departure from Euclidean 

 methods in the direction of the utilitarian. 

 This tendency, however, is not universal. 

 Objection is raised by a certain school of 

 pure mathematicians to any system of 

 mathematical instruction which is not se- 

 verely logical, and which considers the sub- 

 ject as a means rather than an end. The 

 following views of Professor Halsted may 

 be considered as typical of this demand 

 that mathematics be taught from the outset 

 as a formal training in rigorous thinking.^^ 



Halsted asserts that there must be a 

 course in rational geometry which is really 

 rigorous. This course should be founded 

 on a preliminary course which does not 

 strive to be necessarily demonstrative, but 

 should emphasize the constructive phase. 

 The purpose of the preliminary course 

 should be, as Hailmann has said, to de- 

 velop clear, geometrical notions, to give 

 skill in accurate construction, to cultivate 

 a healthy, esthetic feeling, and the power 

 of visualizing creatively in geometrical de- 

 sign, thus stimulating genuine, vital in- 

 terest in the study of geometry. 



This preliminary course must fit the 

 rigorous treatise on rational geometry 

 which Halsted says should be written by 

 some one familiar with the new, pene- 

 trating, critical researches in the principles 

 of geometry. 



Instead of agreeing with Professor Perry 

 that many of the theorems in Euclid might 

 well be assumed as axiomatic, Halsted as- 



" " Laboratory Equipment," Myers, School Re- 

 view, 1903, pp. 727-41. 

 ^ Halsted, Educational Review, Vol. 24. 



