November 14, 1907 J 



NA TURE 



27 



SCHOOL ARITHMETIC. 



(i) Arithmetic for Schools. By the Rev. J. B. Lock. 

 New edition, revised and enlarged with the assist- 

 ance of V. M. Turnbull. Pp. vii + 480. (London : 

 Macmillan and Co , Ltd., 1907.) Price 4s. 6d. 



{2) Arithmetic, chiefly Examples. By G. W. Palmer. 

 Pp. .K + 339 + xlii. (London: Macmillan and Co., 

 Ltd., 1907.) Price 35. 6d. 



(3) .4 Modern Arithmetic, with Graphic and Practical 

 Exercises. By H. Sydney Jones. Part i. Pp. 

 xii + 361. (London: Macmillan and Co., Ltd., 

 1907.) Price 3s. 



WE are g-enerally accustomed to consider that the 

 teaching of arithmetic has gained its promi- 

 nent place in all modern educational systems for two 

 reasons, viz., on account of (i) its utilitarian value, 

 and .(2) its culture value. The writer of a general 

 text-book on arithmetic, as well as the teacher of 

 arithmetic, should have both these ideas promi- 

 nently before him, and so far as either of them fails 

 to consider the subject from these two points of view, 

 so far does he fail in its presentation. .\ little con- 

 sideration shows that at different points in the study 

 of the subject the relative importance of the two 

 reasons for teaching it varies considerably ; but 

 generally, its utility must give way to the culture 

 training. 



L'ntil within the last twenty years the tendency of all 

 works on arithmetic was to build up a system which 

 paid no attention to either reason, but treated the 

 subject more as a series of detached methods — the 

 more mechanical the better — of solving problems set 

 by the author, by the teacher, or by the examiner. 

 We had, in fact, a period which reminds us of the 

 paid problem-solvers of the fifteenth century. Since 

 then wc have had two parallel movements of reform 

 in arithmetical teaching. The first was chiefly con- 

 fined to the public elementary schools, where the cul- 

 ture value was considered as the only value. Arith- 

 metic was to be treated as a " series of problems," 

 and inspectors, examiners and teachers went problem- 

 mad. The other movement was found at its height 

 in the evening continuation schools and in the tech- 

 nical schools. In that case, all teaching of arithmetic 

 was considered of importance as it bore on the daily 

 life of the pupils. Fortunately, we now see signs of a 

 combination of these two movements, and provided 

 their relative importance is correctly gauged, the 

 results from the teaching of arithmetic will show a 

 corresponding improvement. 



It is interesting to note that each of the three books 

 under review claims to have treated the subject from 

 both points of view, and they all show traces of the 

 two movements referred to above. At the same time 

 they all reveal and will help to promote a weakness 

 in the teaching of arithmetic in the preparatory 

 schools and in the secondary .schools. In the 

 public elementary schools of the country the arith- 

 metical education of the pupils until about the 

 age of seven consists of a thorough knowledge of 

 the numbers i to 10. This includes the complete 

 analysis and synthesis of each number. Succeed 

 NO. 1985, VOL, '/7 ] 



ing years are taken up with the extension of these 

 principles to 20, and to 100, and this is followed 

 by the four rules. Such work is also taken up in 

 certain preparatory schools. These books seem to be 

 intended for secondary schools, i.e. for pupils from 

 the age of 12 and upwards, and yet they take up 

 the four simple rules. The only satisfactory reason 

 for doing so would be a more rigid and scientific 

 treatment of the subject, with full details as to the 

 development of the processes used. None of the books 

 can be considered satisfactory in this respect, though 

 both Mr. Jones's and Messrs. Lock and TurnbuU's 

 books show an advance on previous text-books. It 

 would seem, therefore, that the authors, while recog- 

 nising that pupils of twelve ought to be ready to take 

 up a more scientific study of arithmetic, are not pre- 

 pared to look for that improvement in the earlier 

 work which would permit of this step being taken. 



All three books show signs of the former school 

 of arithmetic. This is specially seen in the multipli- 

 cation of money. There is no justification for the 

 retention of the old method of tens vi^hich is simply a 

 disguised form of practice. The objection to the 

 direct method usually is that the working has to be 

 roughly done with no definite place in the setting 

 down. That can be easily overcome by setting down 

 the working of each unit directly below that unit. 

 The direct method corresponds closely to the method 

 always used in division. .-Xgain, in multiplication and 

 division by decimals, we find one of the many 

 mechanical rules given in each book. These should 

 only be necessary in the case of beginners, and a pupil 

 who thoroughly understands " place values " should 

 not require any of these mechanical aids. 



(i) Mr. Lock's arithmetic has now reached its fifth 

 edition, and we can congratulate the author on the 

 improvement of that edition. The omissions from and 

 re-arrangement of the text are all for the benefit of 

 the teaching of elementary mathematics. There are 

 still some traces of the older methods to be found. On 

 p. 67 the old extended method of finding the H.C.F. 

 is given, and alongside is a contracted method (the 

 quotients are not necessary) with the remark, "The 

 work is best arranged thus." On the succeeding 

 page, however, the authors give the working of two 

 examples in the old style. On pp. 96 and 97 we find 

 a series of examples of continued and complex frac- 

 tions which we hope the authors will remove from 

 the next edition The importance given to vulgar 

 fractions generally takes away from the value of the 

 book as a modern text-book of arithmetic. 



(2) Mr. Palmer's book is an example of a type of 

 text-book which has become common during the last 

 few years. 



" It is chiefly made up of examples. The explana- 

 tory matter consists almost entirely of worked-out 

 examples, except in certain parts of the book where 

 explanation seemed necessary in consequence of some 

 departure having been made from the ordinary method 

 of treatment." 



There is no doubt that such a book is the best that 

 one can put in the hands of pupils, but it requires a 

 teacher who is an expert in his subject. \\'e can 



