February 14, 1901] 



NATURE 



367 



the influences of two very different sets of conditions, 

 while the latter has been led by it to a single fixed form 

 suitable to a single set of conditions. This is only a 

 suggestion, and might require modification after a special 

 study of the circumstances of the two species ; but it is 

 sufficient to show that we require far more evidence 

 before it can be conceded that such transmission had 

 been made in any way probable. 



The book is well and clearly printed. A portrait of 

 Lamarck forms the frontispiece. E. B. P. 



THE RATIONAL TEACHING OF 

 MA THE MA TICS. 



The Teaching of Elementary Mathetiiatics. By David 

 Eugene Smith, Principal of the State Normal School 

 at Brockport, New York. Teachers' Professional 

 Library. P. xv + 312. (New York: The Macmillan 

 Company. London : Macmillan and Co., Ltd., 1900.) 



IN many training colleges for primary school teachers 

 there are elaborate courses of study on psychology 

 and ethics. Surely a knowledge of morals and of the* 

 mental machinery of boys and girls would be more 

 certainly and more easily acquired incidentally during 

 other studies, such as the natural sciences ; but at these 

 colleges there is seldom any attempt to educate through 

 the natural sciences. We have, though not to the same 

 degree, the same feehng about courses of instruction in 

 mathematics. There is a cold-blooded formality about 

 the mere name which tells all children truly that they 

 are being offered stones for educational bread. But if 

 there must, unfortunately, be separate courses of instruc- 

 tion in mathematics, we should, if we were children, 

 dearly love to be taught by Mr. Smith. He is well read 

 in his subject, and teachers who are also well read 

 will take pleasure in seeing the best views so clearly put 

 forward ; teachers who are not learned in the subject 

 will benefit greatly by reading this book. Short sketches 

 of the histories of arithmetic, algebra and geometry are 

 woven into the text in such a pleasant fashion that one 

 reads and understands without much effort. The merits 

 and demerits of various systems of teaching mathematics 

 to very young children are clearly stated, but we cannot 

 help thinking that too much is made of the philosophy 

 of the numerous German exponents of pedagogy. There is 

 no system which will give good results in the hands of a 

 fool ; there are many systems which will work fairly well 

 in the hands of the average teacher ; a thoughtful man 

 who is in sympathy with his pupils will succeed with any 

 method that he is likely to adopt. 



Philosophers are too fond of distinguishing between 

 teaching for utility and teaching for culture. We 

 take it that even if we teach mathematics for its 

 " bread-and-butter-value," if we teach so that a pupil 

 really understands what he does, then we are really 

 training his logical powers and giving him help in his 

 ethical, religious and philosophical ways of thinking. 

 The more we try to teach merely for culture the more 

 do we make the reasoning obscure and difficult. As if a 

 good teacher could possibly give sordid notions to his 

 pupils I What we really want is that all teachers shall 

 know their business, and then, however quickly they|may 

 NO. 1633. VOL. 63] 



make their children cover the ground of elementary 

 mathematics, and we say the quicker the better, the 

 children will be taught as rational beings. Much of the 

 arithmetic taught in schools is really the teaching of a 

 trade. A particular rule like Practice is merely the 

 application of arithmetic to the trade of a grocer. So 

 also rules like Interest or Discount are labour-saving 

 rules, useful when one has thousands of calculations of 

 the same kind to make, easily learnable by a boy after he 

 leaves school if he has a knowledge of simple arithmetic 

 and if his common sense has had a fair chance of 

 development. Children may be kept for years at 

 "rules" of arithmetic which they never understand, by 

 an unscientific teacher, and this is what the philosophers 

 condemn as utilitarian teaching. There is as little utility 

 about such teaching as there is culture in that of the 

 equally unscientific follower of the greatest psychologist. 

 Of the two, however, the unscientific utilitarian does least 

 harm, for he makes least pretence ; he only stupefies the 

 brain, the other destroys the soul. 



Indeed, the man who aims exclusively at culture 

 always hurts the soul of his pupil, for he teaches that 

 what is useful must be low, and that the study of it must 

 lead to sordid thought. We can no longer afford to laugh 

 when men assure us that they scorn the results of their 

 studies when these results prove to have useful applications. 

 So long as these men were few in number they might be 

 laughed with ; we laughed because they were paradoxical 

 and because we did not fear that the utility of a study 

 could really be lost sight of We are always grateful to 

 philosophers who discover new truths, whatever their 

 notions as to their utility may be. But when the stupid 

 admirers of these men erect their paradoxes into articles 

 of belief ; when headmasters with much capital invested 

 in teaching machinery find that such articles of belief 

 give a fictitious value to their invested capital ; when 

 as a result, 98 per cent, of the boys leaving school at 

 seventeen to nineteen years of age know no mathematics, 

 although they are supposed to have been studying mathe- 

 matics for many years; when we have overwhelming proof 

 from the fields of war and commerce and manufacture that 

 the best race of men in the world is held by want of 

 education, as if by enchantment, from exercismg its 

 natural powers — then we feel that the time has come when 

 a crusade ought to be preached against the pestilent 

 heresy. 



We are very glad to think that Mr. Smith gives great 

 weight to the opinions of Profs. Henrici and Minchin 

 about mensuration and geometrical teaching. Lacroix 

 expressed them clearly, so did Clairaut and Voltaire and 

 Hoiiel and Spencer and Langley, and many another 

 educationist. Laisant says, " But just as there must 

 be a prehminary preparation for arithmetic — namely, 

 practical calculation — so theoretical geometry should be 

 preceded by the practice of drawing." Rousseau said 

 that for young pupils " geometry is merely the art of 

 handling the rule and compasses." Mr. Smith describes 

 the use of shears and cardboard, and he suggests how 

 to follow Galileo's experimental and inductive methods 

 in mensuration, even with boys of intermediate grades. 

 As for demonstrative geometry, Mr. Smith says that in 

 America it usually begins in the tenth or eleventh school 

 year. 



