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SCIENCE. 



[N.S. Vol. III. No. 60. 



ber of things already recognized as distinct, and 

 it is a very simple matter to go ahead and enu- 

 merate them, though even that must have a psy- 

 chology motivation. But the whole tendency 

 of contemporary psychology is to take a psy- 

 chical continuum, as its datum, and find dis- 

 tinctness (the property at the basis of number) 

 as the outcome of a process of differentiation. 

 The identification of this process, the ascertain- 

 ing of the circumstances under which it arises, 

 the mode of its operation — this is the thing which 

 the psychologist wants to know about number, 

 and is the thing the educator must know to secure 

 the conditions under which the child shall form 

 the number concepts easily and efficiently. The 

 theory of the book, 'Psychology of Number,' 

 viz., that the differentiation and enumeration 

 of units arises through the progressively ac- 

 curate adjustment of means to end, may be right 

 or wrong, but its error can hardly be established, 

 I take it,by a mathematical view which considers 

 number only as it is after it is fully developed, 

 and has become so familiar as to be itself a 

 complete object to the mind. Without pretend- 

 ing to a knowledge of numerical theory which 

 I do not possess, I may say that it seems to me 

 that the work done by Gauss is at precisely the 

 opposite pole from that which the educator 

 needs from the psychologist, i. e., Gauss was 

 attempting to reduce to its ultimate simple nu- 

 merical generalizations the developed mathe- 

 matical structure. Dr. McLellan and myself 

 were engaged upon the much humbler task of 

 finding out what sort of a mental condition 

 creates a demand for number, and how it is 

 that number operates to satisfy that demand. 



May I conclude by referring to the prac- 

 tical point involved? The trained mathema- 

 tician as such is, of necessity, interested in the 

 further use of certain finished psychical pro- 

 ducts. As a mathematician any reference to 

 the preliminary development of these products 

 can only disturb and divert him. But the 

 problem for the pupil is how to get the stand- 

 point of the mathematician; not how to use 

 certain tools, but how to make them ; not how 

 to carry further the manipulation of certain 

 data, but how to get meaning into the data. 

 This is ultimately a psychological question, 

 not a mathematical one, although it has to 



be translated over into mathematical terms and 

 processes ; and none is so well fitted to do it as 

 the mathematician, provided only he will pro- 

 ject himself far enough backward in the scale of 

 development to realize the problem. The point 

 does not conclude with primary instruction. 

 Our text-books of algebra, geometry and high 

 analysis are almost entirely written from the 

 standpoint of an elegant and logical exposition 

 of the matter as it stands to the trained mathe- 

 matician. They are very nice for one who 

 doesn't need them any longer. The first books 

 written from the standpoint of one who is still 

 coming to consciousness of the meaning of 

 his concepts will, perhaps, seem foolishness to 

 the trained mathematician, but they will mark 

 the dawn of a new day to the average student. 

 I venture the statement that (putting aside the 

 few with the inborn mathematical instinct) 

 higher and secondary mathematics is to the 

 majority of students a practical riddle with no 

 definite intellectual content in itself. What 

 meaning it possesses it has got by way of at- 

 tained practical facility in solving problems ; or 

 through its applications to other sciences or to 

 engineering. It will hardly be denied that the 

 educational value of mathematics is not realized 

 until its concepts and methods have a definite 

 intellectual meaning and content of their own. 

 Can this be secured, save as the methods of in- 

 struction follow the evolution of the process out 

 of its cruder psychical forms to the more finished? 

 I shall be more than satisfied to have made 

 many blunders on the mathematical side if only 

 I do not offer myself up in vain as a spec- 

 tacle ; if only more competent psychologists 

 take up the matter, and if only mathematicians 

 may descend from their acquired mathemat- 

 ical plane and endeavor to rethink the psy- 

 chical conditions and steps through which 

 their present magnificent apparatus has grown 

 out of primitive, non-mathematical or crudely 

 mathematical forms up to its present high es- 

 tate. If the psychologist will risk some blun- 

 dering around among the mathematical con- 

 cepts, and the mathematician will recognize the 

 relevancy of the psychological demand, and 

 venture a little blundering upon that side, both 

 parties may not only come to an understanding, 

 but mathematical teaching may get what it to- 



