452 REPORTS oy the state of science. 



may be profitably introduced by measurement ; ratio then presents no 

 difficulty and proportion is merely the equality of two ratios. Ifo other 

 rules are required in simple arithmetic ; in particular, the ' rule of three ' 

 should be avoided, as an obstacle to clear thinking and an offence against 

 the common-sense of a child. 



Methods of Approach. 



Attention should be given to local value of digits at the commence- 

 ment of the use of numbers as symbols. In teaching local value, use 

 should be made of bundles of small sticks in tens, pigeon-holes with 

 separate compartments for bricks or other things representing units, 

 tens, and hundreds, or separate rods upon which nine or ten beads can 

 be placed but no more. When the principle is understood, the expres- 

 sion of it by figures in difiFerent columns should be taught but not before. 

 Subtraction can be taught at the same time as addition by treating it 

 as complementary addition. In performing simple multiplication, it is 

 doubtful whether children at the beginning should be taught to multiply 

 first by the left hand or highest digit of the multiplier instead of follow- 

 ing the usual method. The ordinary process appears to be simpler for 

 young children to undei'stand, and it follows naturally from short multi- 

 plication. Pupils who learn contracted multiplication will do so later, 

 when the change from multiplying first by the right-hand digit to the 

 left-hand digit of the multiplier will present little difficulty. 



After acquiring familiarity with the processes of short and long 

 division, bright children may with advantage use the Italian or com- 

 pendious method of working long division, by which remainders only 

 are written down, these being found by complementary addition. The 

 study of formal vulgar fractions should not be commenced until woi'k has 

 been done with decimals ; and there should be no sums involving fractions 

 until clear ideas as to their meaning have been obtained from concrete 

 examples and measurements. By the time children begin written work 

 with fractions they should be so familiar with subdivisions of the inch 

 and centimetre scales that rules for the addition, subtraction, multiplica- 

 tion and division of fractions can be comprehended without any difficulty. 

 With squared paper divided into tenths it is easy to show graphically 

 the products of such numbers as 2 x 3^ or 3? X 2^ ; while by similar 

 graphic exercises pupils will readily see the absurdity of adding mixed 

 numbers by converting them into fractions, and understand that the 

 product of two proper fractions is numerically less than either. The 

 great advantage of work of this kind is that it enables the pupils to 

 insiialise simple principles. 



Practical Work Essential. 



Familiarity with the use of a rule divided into inches and tenths or 

 centimetres and millimetres enables pupils to acquire a sound knowledge 

 of decimals unconsciously ; and at thesame time they learn the significance 

 of place value. Work of this kind can be introduced quite early without 

 any difficulty. There is no need for any elaborate explanation of units 

 and subdivisions if pupils are given sufficient opportunity for practical 

 M-ork with decimal scales and squared paper divided into tenths ; while 

 the difficulty as to the place of the decimal point in later calculations 

 disappears, because clear ideas have been obtained of its meaning by 



