August 28, 1908] 



SCIENCE 



265 



dents. Another unfair criticism is some- 

 times heaxd from the professor of engineer- 

 ing who says that students can not use their 

 mathematics, when the truth is they have 

 simply forgotten some particular fact, for- 

 mula, or process, which is a fad of that 

 professor. It is unfair to test mathemat- 

 ical training by tenacity of memory or 

 mere quickness in reasoning. 



I have said that we must teach our stu- 

 dents to use their mathematics. Now in 

 the application of mathematics to a con- 

 crete problem there may be distinguished 

 three steps: 



1. The interpretation of the data of the 

 problem into mathematical language. 



2. The formal operations upon the ex- 

 pression or equations thus obtained. 



3. The interpretation of the results back 

 into the terms of the original problem. 



The first and third of these steps are 

 really the most important, but there seems 

 to be a popular impression that the second 

 comprises the whole of mathematics. This 

 impression is doubtless responsible for some 

 criticisms of the educative value of mathe- 

 matics. It is true that relatively a great 

 amount of time must be spent in the class- 

 room in teaching the mechanical processes 

 involved in the second step, and many stu- 

 dents in school and college get no farther. 

 To object to the amount of time spent in 

 this way and to demand, as some do, that 

 we confine our time to teaching general 

 principles and applications is to talk as 

 sensibly as a fond mother who objects to a 

 child beginning his musical education by 

 playing finger exercises instead of tunes. 

 The technique of mathematics must be 

 learned first, but the student who never 

 gets beyond the technique has not learned 

 mathematics. 



The teacher of mathematics should, then, 

 use all possible means of teaching the first 

 and third of the above steps and should 

 bring his pupils to think of them as the 



real thing. For that purpose he should 

 seek for applications and illustrations from 

 as wide a range of subjects as possible. 

 He will find himself handicapped, however, 

 in using many problems of real scientific 

 or engineering importance because of the 

 ignorance of his pupils, especially in the 

 first year in the technical school. To illus- 

 trate a new mathematical principle by an 

 application to a science with which a stu- 

 dent is not familiar is to befog and not 

 ilhunine the subject. Hence there is some- 

 thing to be said in favor of some of the 

 much-criticized problems of the older text- 

 books. To my mind a problem is success- 

 ful if it causes the student to take the three 

 steps just enumerated and is couched in 

 terms familiar to the student, even though 

 it may not be "practical." On the other 

 hand, a type of problem lately coming into 

 use, in which the student is given some 

 formula from a science of which he knows 

 nothing, and is asked to find, say, a maxi- 

 mum value, is as fruitless as if the prob- 

 lem were stated in terms of x, y and z, 

 unless it may serve to convince a sceptical 

 student that the matter he is studying has 

 some practical application. 



And this leads me to the most important 

 thing I have to say, and that is that after 

 the mathematical professor has done his 

 utmost to teach the use of mathematics the 

 engineering professor must take up and 

 complete his work. I doubt if any one 

 really learned the use of mathematics in a 

 first course. Facility in using mathematics 

 comes from actual use and not from the 

 solution of illustrative examples. In the 

 course in mathematics the student expects 

 his problem to be solved mathematically 

 and has his mind alert to find the solution, 

 and that too with mathematical principles 

 fresh in his mind. In a course in engi- 

 neering, his point of view has widely 

 changed. The practical problem has now 

 his main interest, mathematical concepts 



