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CHAPTER IV.

ARITHMETIC (STANDARD 1.). It is a golden maxim in teaching Arithmetic that the teacher should never begin with enunciating a rule, but with a problem to be worked mentally. The writer thinks that this should be observed even with Standard I. That is, easy problems should be given in Simple Addition, and as these become harder they should be worked on slates, until the Government limit of 999 is reached. It is assumed that the children have abandoned, in the Infant School, the vile unitary method of addition (4 and 1 are 5, 5 and 1 are 6, etc.); and have been taught to add together such numbers as 4 and 5 directly, and without counting on the fingers.

In the first stages the numbers given out should be regarded by the children as wholes; and the analysis into units, tens, and hundreds, should only be commenced at a later date. The child knows of 20 as a whole; not as of two tens, and no units. There is no subject that will promote the intelligence of the school more than Arithmetic, if it is taught continuously and from the earliest stages, through concrete problems; there is none so utterly deadening to the intellect as mechanical abstract Arithmetic. Nearly one-half of the Arithmetic in Standard I. should be Mental; and wherever this is the case, the slate and paper Arithmetic are accurately done. Moreover, the teacher should reflect that most of the actual Arithmetic of the working man's life is mental.

A child should never be set to perform two mental operations at the same time, if it can be avoided. Before using slates, therefore, the class shonld have acquired facility in performing the processes of Addition of several figures; of Subtraction; and of Multiplication up to 6 times 12.

Arithmetic trains the reasoning powers; unconsciously to the child it furnishes him with the Laws of Mind, and trains him to be accurate, persevering, and patient to work towards an unseen, but hoped-for, end. Working problems also strengthens the inventive faculty, and assists the imagination.

In Addition encourage the children to add the figures in groups: thus 4, 5, 4, 5, 6 should be mentally grasped as two fours = 8, two fives = 10, 8 and 10 = 18, 18 and

24; instead of 4 and 5 are 9; 9 and 4 are 13; 13 and 5 are 18; 18 and 6 are 24. Again, instead of allowing the child to count 4 and 5 are 9, 9 and 4 are 13, etc., let him acquire the habit of saying to himself the result only, as (taking 4 and 5 together) 9 (taking 9 and 4 together), 13, and so on to the end.

6 =

MODEL LESSON IN ARITHMETIC (NEW RULE). (1) Commence with a few concrete exercises in Mental Arithmetic, leading up to the rule to be introduced.

(2) Work out on the blackboard similar exercises, but rather harder than the preceding.

(3) From this working show the necessity and processes of the new rule to be learnt.

(4) Illustrate the rule by copious exercises given to the children to work out on the blackboard with the teacher.

(5) Give similar exercises for slate practice by the class.

N.B.—Be careful to break up the work into Stages logically depending on each other, and proceeding from the simple to the more complex. In many instances one stage only can be taught in a lesson. When a succeeding stage is attacked, connect the lesson with the preceding.

(6) While the quicker children are doing additional exercises, call out the duller ones, and repeat the explanatory processes on the blackboard to these.

(7) To keep up the Notation, dictate the numbers; and to keep up the Numeration, let the children read their

answers.

(8) In Subtraction let the children provetheir

answers.

NN

SIMPLE ADDITION : ON SLATES (ALL IN PROBLEMS). First Stage.--Adding up numbers when the total amounts to less than 10, as1 1 2 2

5 6 1 2 2 3

4 3 etc., up to

etc. 2 3 4 5

9 9 The same with three lines.

[Subtraction should be taken with the Addition, within the same limits, and without borrowing, as4

5 2

2

etc.] 2

3 Second Stage.Up to the limit of 19 in the total; in two and three lines, as 6 + 4 = 10; 3 + 4 + 5 = 12, etc.

Third Stage.--To the limit of 99 in the total. "Carrying " is here introduced.

Fourth Stage.To the limit of 999 in any one line.

These stages are carried on by means of Notation and Numeration, the 0 being more and more frequently introduced as the class advances.

Fifth Stage.-Analysis of numbers into units, tens, and hundreds, according to the headings herewith appended

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Sixth Stage. The same with H. T. U. on the top of the columns, without lines.

As some of the children will work their addition exercises more rapidly than others, it is useful for the teacher to have a means of telling at sight the answers to additional sums given individually to the quicker children. For this purpose let any line, say 342, be taken;

342 add a second line, which will convert this into 000

658 with 1 before it, viz. 658. This couple will there

27 fore make 1000; do the same with another couple,

973 say 27 and 973; the four lines now make up

2000.

216 Add a proof line, say 216, and the answer will be found at sight, 2216; or two proof lines may be 2216 added at sight, provided there is no carrying, as 212 and 312. Then the answer will be 2524 told at sight.

SIMPLE SUBTRACTION. First Stage.—Mental exercises in subtracting numbers not greater than 20; as 6 from 12, 7 from 11, 9 from 20, etc. Second Stage.--Here there is to be no borrowing, as679

482 321

61

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Third Stage.-The same with H. T. U. affixed, as

H. T. U.

3 4 5
231

1 unit from 5 units = 4 units
3 tens

4 tens

1 ten
2 hundreds 3 hundreds = l hundred

114

on.

Answer: 1 hundred, 1 ten, 4 units = 114. Fourth Stage. First Method : by equal additions.

Explanation. If one boy has 7 marbles, and another 4, the difference will be 3; if we give each one marble, the one has 8 and the other 5, and the difference is still 3; and so on if we give each 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.

We may therefore add any number to the top line without altering the difference, if we add the same number to the bottom line. We agree always to add 10, if we add at all. But we only add to the top line if it is less than the bottom; thus in 6 from 5, we find we cannot get

But if we add 10 to the 5 this becomes 15; then 45 we can take away 6, and there is 9 left. But we have 26 not yet added 10 to the bottom line. But we will do This might be done by calling the 6, sixteen, or

19 we can add 1 to the 2, and call that 3, as the 2 is two tens, and the 3 will mean three tens, or one ten more than before. Having thus added 1 to the 2, this becomes 3, and 3 from 4 leaves 1. So the answer is 19.

Plenty of exercises should be given in which only one addition thus takes place, before we repeat the process ; so that the children may become accustomed to one of the most difficult feats in Arithmetic.

Second Method: by Decomposition. Take the same example as before

45
26

SO.

19

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