PRIMITIVE IDEAS OF NUMBER.

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conceiving the formula "one and one." The great facility with which the mind generally follows the repetition of units to a further extent, one and one and one; one and one and one and one; one and one and one and one and one; might go some way to justify the assumption that there are primitive ideas of other numbers, for instance,-exempting the four, which arises out of two, three and five; but if this should be disputed as a mere hypothesis, it is easy to show that three and five also may be obtained simply by means of the idea "two." For if we have "two," and its first derivative, "four," and we divide the difference between them into two parts, one such part, either subtracted from four, or added to two, will produce the intermediate number three; which number itself deducted from eight gives the five; and in a similar way every other number might be obtained merely by the operation of the idea two.

But however this matter may be in theory, whether we assume primitive ideas of other numbers, besides two; or whether we derive all the others from two, which cannot be disputed to be a primitive idea; in practice it is quite certain that the child has in that part of his body which falls most under his notice, exemplifications of the three numbers: two, three, and five; there being two hands, on each hand five fingers, including the thumb, as it must be in the child's view, whatever may be the usage of our language; and each finger consisting of three joints. These three numbers, therefore, two, three, and five, will, at an early period, be known to the child, especially if his attention have been directed to them by the exercises of the mother's manual, which constitute the fourth section in Pestalozzi's arrangement.

The first step, then, in the instruction of number as a distinct branch of knowledge, should be to lead the child to represent those three numbers in visible objects, and to compare them with each other. For this purpose small cubes of wood are preferable to any other objects, and a few gross of them will supersede every sort of apparatus commonly in use. Being moveable at pleasure, they can be made use of for a

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TWO, THREE AND FIVE.

great variety of exercises, and give full scope to the child's own activity, while on the other hand their cubic form renders them best fitted for the illustration of the relation in which the powers of each number stand to each other. As soon as the child is able to handle a pencil, he should be directed to represent on his slate, by small strokes, or dots, the different sets of cubes which he has placed on the table. To give an example, if the child be called upon, to compare the three nnmbers two, three and five, with each other, he will set on the table first two cubes side by side, and then, in another row three. He will represent the same on his slate, as follows:

The first is called two, or two ones; the second three, or three ones; and the teacher then asks:

How many ones more in three than in two?

How many ones less in two than in three?

In the same way the child should compare two and five; and three and five, and lastly the three numbers ought to be placed in one view,

and the child asked all the questions that arise out of them; for instance:

How many more in five than in three?

Take away two from five, how many will remain?

Put two and three together; do they make more or less than five? &c.

Not to detain our readers unnecessarily at this early stage of instruction in numbers, we refer those who wish for further details to the "Christian Monitor, and Family Friend," where they will find a series of model lessons on this subject; and we proceed here at once to the mode of introducing the child

THE DECIMAL SYSTEM.

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to a knowledge of the decimal system, which ought not to be delayed, as the names of numbers in our language are entirely derived from their arrangement in decimal series. At first, every number in succession ought to be compared with the number ten; for instance, beginning with two, the teacher should call upon the child to put down ten, cubes on the table, strokes on his slate.

Next, the teacher asks the child to divide the ten into twos;

The child thus finds that there are five twos in one ten; and having, in the first instance, ascertained the proportion of two to ten, it will be easy for him to proceed with the following exercises:

seven twos, how many tens and ones? &c. And on the other hand,

three fives, how many tens? &c.

These exercises become somewhat more complicated when the number compared to ten is not a factor of ten: as, for instance, three.

three tens, how many threes? &c.

In this manner, all the numbers under ten should be compared to the decimal series as far as ten tens, or one hundred; and the child should be led to make his own observations as to the different sorts of remnants which occur with each number, and the number of tens which are required to make the sum of the remnants equal to the number compared.

After the pupil has, in this manner, become perfectly familiar with the relation which each number bears to the decimal series, he ought to be made acquainted with the relation which the pure and mixed derivatives of the three elementary numbers bear to each other. By pure derivatives we mean what is commonly called powers; and by mixed det.vatives the combinations of the elementary numbers and their powers by multiplication. The following table will illustrate this still better: