448 • REPORTS ON THE STATE OF SCIENCE 



instruction in practical arithmetic should be carried further than has been 

 generally the case hitherto, with the object especially of encouraging the 

 use of contracted methods and operations in mental arithmetic ; and of 

 encouraging also the expression of results with only such a degree of 

 (numerical) precision as is consistent with the known degree of certainty 

 of the data on which they are or may be supposed to be based.' ^ 



Views of the Board of Education. 



Evidence is not wanting that some educational authorities and 

 examining bodies desire to encourage the rational and practical methods 

 of teaching arithmetic in schools proposed in the foregoing extracts. 

 The ' Notes for Teachers ' issued by the Commissioners for National 

 Education in Ireland contain valuable hints on the rational teaching 

 of arithmetic, and the Scotch Education Department has adopted the 

 new methods as the basis of work for the Leaving Certificate exami- 

 nation. The latest instance of this tendency is afforded by a Blue-book 

 (Cd. 2638, price 8d.) prepared by the Board of Education in 1905, and 

 containing ' Suggestions for the Consideration of Teachers and others con- 

 cerned in the Work of Public Elementary Schools.' This publication 

 includes a chapter on the teaching of arithmetic and three suggestive 

 schemes of work for the various years of primary-school instruction. In 

 the first of these schemes, addition and subtraction are applied to com- 

 pound rules (money) before multiplication and division are taught ; in the 

 second all the four simple rules are taught to begin with ; and the third 

 scheme is based upon number as a means of measurement, whole numbers 

 being taken first, while common and decimal fractions are afterwards in- 

 troduced as successive refinements of a precise system of measurement. 

 Though these schemes differ in detail as to the method of approaching the 

 elementary processes of arithmetic, they have all been designed with the 

 object of giving children clear ideas about number, a practical acquaint- 

 ance with simple measurements, and confidence in the application of 

 arithmetical operations to problems of everyday life. 



The subjoined extracts from the chapter on arithmetic in the volume 

 referred to indicate the principles upon which the Board of Education 

 suggests the teaching of arithmetic should be based : — 



' The simpler rules of arithmetic should be regarded by the teacher and 

 by the children as being the result of the application of common- sense to 

 operations with number as applied to concrete objects, money, weights, 

 lengths, and areas. 



' The teaching should, however, from the very first embrace problems 

 and examples that require special methods for their solution, and scholars 

 will thus be trained at an early age to use their intelligence, and not to 

 place undue reliance upon the mechanical application of general methods. 



' No difficult examples should be attempted by the scholars until the 

 process to be applied has been thoroughly worked out by the class ; the 

 necessary training in mechanical skill should be acquired rather by re- 

 peated practice in carrying out pi'inciples which the scholars thoroughly 

 understand, than by attempting to work examples in rules of which the 

 underlying reasons have not been firmly grasped. 



' Extract from the Report of a Committee of the Institution of Civil Engineers 

 on the Education and Training of Engineers (April 190G), 



