334 



SCIENCE 



[N. S. Vol. XXVI. No. 063 



This dictum from the great head of the 

 Sibley School of Engineering is impressive 

 and worthy of careful consideration. It 

 fully justifies our presence here to-day for 

 the purpose in hand and confirms the judg- 

 ment of the founder of this institute and 

 the purpose of the generous donors of the 

 Walker Laboratory we here and now dedi- 

 cate to the study of chemistry. 



We may heartily join in congratulations 

 to the administrators of the will of Mr. 

 Van Rensselaer, who have so faithfully 

 carried out the purpose of the founder, and 

 to the graduates, who have so well seconded 

 the efforts of these able and conscientious 

 men who have brought the Rensselaer Poly- 

 technic Institute to that great eminence in 

 the public esteem, from which they may- 

 look backward with pride and forward in 

 earnest and confident hope of an even 

 brighter and more prosperous future. 



Wm. McMurtrie. 



THE RATIONAL BASIS OF MATHEMATICAL 

 PEDAGOGY '^ 



The rapid development of special meth- 

 ods of teaching special svibjeets has drawn 

 attention of late to the hitherto neglected 

 field of mathematical pedagogy. The fact 

 that mathematics is the last to respond to 

 Improved pedagogical methods is due chief- 

 ly to the unusual weight of precedent which 

 attaches to the subject. This inertia of age 

 is, in reality, the chief difficulty to be over- 

 come, for the great antiquity of elementary 

 mathematics and the diversity of the 

 sources from which it originated make it 

 extremely difficult to harmonize the subject 

 with the spirit of modern civilization. 



Various plans have recently been pro- 

 posed for adapting mathematical instruc- 



'■ Read before American Mathematical Society, 

 New York, December 28, 1906. 



tion to modern conditions, but so far they 

 have been without results of special impor- 

 tance, as no general principle of mathemat- 

 ical pedagogy seems as yet to be recognized. 

 Many of these plans are the results of at- 

 tempts to meet local conditions and there- 

 fore have no general application. Others, 

 however, such as the attempt to correlate 

 mathematics and physics, are intended to 

 stimulate interest in mathematics by estab- 

 lishing more points of contact between it 

 and the other subjects of instruction, thus 

 producing a greater organic unity in the 

 curriculum than has hitherto existed. Such 

 efforts are in line with the constructive and 

 synthetic spirit which characterizes modern 

 scientific thought, and for this reason are 

 worthy of special consideration. 



Without going into a detailed analysis 

 of pedagogical methods, it may be well to 

 consider briefly the fundamental principles 

 on which general pedagogy is based, as 

 these principles are well established and 

 apply with peculiar force to mathematics. 



The primary consideration in all branch- 

 es of pedagogy is the aim of education. 

 This has been variously defined, the Her- 

 bartian definition of it being that it is "the 

 cultivation of virtue based on many sided- 

 ness of interest." By the cultivation of 

 virtue in this connection must be under- 

 stood the proper exercise and control of all 

 the faculties. With this understanding 

 the definition fits in, probably as well as 

 any other, with modern ethical and relig- 

 ious ideals. The second half of the defini- 

 tion which bases the development of virtue 

 on the cultivation of wide and varied in- 

 terests fulfills practical requirements and 

 at the same time affords the proper peda- 

 gogical basis for apperception. The most 

 important feature of this definition is that 

 it recognizes both the practical and the cul- 

 tural aims of education. In other words, 

 it implies that no training can properly be 



