168 



SCIENCE 



[N. s. Vol. XXVIII. No. 710 



relative proportion or emphasis to these 

 three phases of instruction; that the con- 

 tent of the instruction, within the limits 

 of present usage in engineering schools, is 

 of minor importance ; that thoroughness is 

 essential, and that it is better to cut down 

 the extent of the matter gone over if there- 

 by a more thorough grasp of the subject 

 is secured; and that the instructor must 

 always keep in mind that he is training an 

 average boy of average preparation with 

 a view to using mathematical principles 

 and methods of attack and mathematical 

 operations and conceptions in the mastery 

 of his engineering studies and in the treat- 

 ment of the varied problems which will 

 arise in his later engineering experience. 



The great mass of our engineering stu- 

 dents, like the great mass of our engineers, 

 are not mathematical geniuses. In the dis- 

 cussion of the subject we must keep ever 

 in mind that the average engineering stu- 

 dent is not of strong mathematical bent. 

 Many of those with only mediocre mathe- 

 matical ability make successful engineers, 

 and the student of strong mathematical 

 turn may lack in some direction or may 

 have a disproportionate measure of the 

 importance of his analytical powers and 

 drop behind his less mathematical class- 

 mate. I want to make a plea for the aver- 

 age student, the boy whose analytical 

 powers have to be encouraged and devel- 

 oped. The methods of presentation must 

 be made elastic enough to include this great 

 class of students, or we shall fail to do our 

 duty as teachers. 



I have mentioned three phases in the 

 presentation of mathematical subjects. 

 These may be considered in order. It must 

 be understood that these phases are not 

 mutually exclusive. 



1. T/ieorj/.— Analysis, demonstration and 

 the general derivation and presentation of 

 mathematical principles. The derivation 

 and exposition of mathematical principles 



and operations and the appreciation of 

 mathematical concepts are universally 

 accepted as important elements in the 

 education of an engineer. The use of 

 mathematical forms of attack, the training 

 in processes of reasoning, the formation 

 of logical habits of thought, are hardly sec- 

 ondary in importance. And yet much less 

 emphasis is placed on formal demonstra- 

 tion and reasoning than formerly— fre- 

 quently this element is overlooked or 

 treated in a slipshod way. The student 

 comes to feel that he is after facts and that 

 the derivation and proof of principles in- 

 volves useless effort— he is willing to accept 

 their authenticity. It may be that years 

 ago our instructional methods carried for- 

 mal processes to an extreme and that as a 

 result mathematical work became meaning- 

 less lingo or memorized facts to many stu- 

 dents. This does not furnish argument 

 for the abandonment of training in formal 

 reasoning. For the young mind, practise 

 in analysis, in formal demonstration is 

 illuminating and developing. Even the 

 repetitive forms of analysis in the old-time 

 mental arithmetic had great mathematical 

 educational value. The speaker feels that 

 in the effort to avoid barren formalism the 

 pendulum has swung too far the other way, 

 and that both in high school and in tech- 

 nical school, and in the applied engineer- 

 ing subjects as well, the training in an- 

 alytical methods and formal processes is 

 weak. He believes that good results would 

 follow putting greater emphasis on this 

 phase of instruction than now seems to be 

 the trend. 



2. Practise.— The use and applicability 

 of mathematical principles and processes 

 in the solution of problems, drill on these 

 principles, and the acquisition of facility 

 in their use. To the average student the 

 working of examples is illuminating. 

 Without it the concept is but vaguely com- 

 prehended, the derivation only faintly 



