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draw a straight line in different directions. She then lets him try to draw a line twice as long, and parallel with the former; the Mother and Child do the same thing, and always proclaim, the Mother what she has done, and the Child what he has tried to do.
In this manner they gradually proceed to lines, three, four, six times as long. Single lines may then be united into the various angles, not however advancing one step, until he can name each sort of angles, and imitate them pretty correctly on the slate. Form and Language must alwaysgo hand in hand, as this will give to the Child the important habit, in which he cannot be too early iuitiated, of expressing himself on all occasions readily and correctly. These operations may possibly be hereafter more minutely described in a series of elementary geometrical exercises.
The hand by the practice of lines having acquired a certain degree of steadiness and dexterity, writing is greatly facilitated. Readiug and writing should not be taught as separate exercises, but should always accompany each other *.
* These exercises should be performed with chalk, on a
Language and Form being thus cultivated, the relation of Numbers ought by no means to be neglected. Small wooden cubes are ranged in a row, and the Mother counts them over, first as far as 4, 6, 8, 10, afterwards to 20. The child repeats the operation forward and backward, till he is perfect. One, two, three, are successively taken away and again added; two or more squares are formed, and compared, to see by what one is greater than the other. A number of cubes is divided into several equal parts; each of these parts is doubled, tripled, quadrupled, &c.
These and many other exercises can be continued for a length of time within the number of twenty, with every variety of application in adding, subtracting, multiplying, and dividing, and will lead to important results. The little pupil will be enabled, first by way of intuition, and hereafter without it, to determine, if one cube has been added, how many more must be added in order to produce a square; or if one has been taken away, how many more must be taken
board, or better still a slate, 3 or 4 feet square, placed upon an easel.
away, if a square is to remain; then if two be added, or two taken away, &c.
By means of this easy and intuitive exercise he will be prepared, and in time enabled to extract the square root by head.
As soon as he can count with cubes or with other small bodies, the Mother may proceed to the series of exercises hereafter to be given; in regard to which a few short observations may be useful.
1st. These exercises are intended as a preparatory step to Arithmetic, making the pupil distinctly conscious of what he is doing when calculating. The power of combining numbers should not be debased to a mere mechanical operation, with little or no exercise of the mind.
2d. These exercises should not be shortened or hurried over; which would totally destroy the end in view; experience having shewn, that a gradual and well connected progress only can give that clear insight, and intuitive perception, which in time will enable the pupil to solve with facility the most complicated problems.
3d. Reasons should be accurately given for each step in the proposition.
4th. Knowledge of numerical combination, knowledge and imitation of forms, and knowledge of a just and correct denomination, or Number, Form, and Language, should be cultivated Harmoniously, and not one in preference to another.
Should a Mother undertake these exercises with Children of a more advanced age, let her still begin from the first elements; in which, however, she may proceed with a quicker step, as children of this age are more conscious of what they are doing; though they never ought to leave off any exercise, until they can give it readily, and wilh precision, to others.
If any one imagine that so many precursory steps, so many repetitions in the introduction to calculation are superfluous, it may be observed, that this is the first fundamental exercise of the power of Thinking*.
The Mother may now proceed in shewing the grammatical difference of words. She explains by degrees in an easy and familiar way
* "Je considere l'arithmetique non seuleinent comme science, mais comme moyen de developper Pintelligence de 1'enfant, de former son jugement et de l'habituer a raisonner juste." Rivail, disciple de Pestalozzi.
the meaning of the terms substantive, adjective, verb, adverb, &c; and desires her pupil to write the description of one of these words on the slate, and below it a series of three or four words corresponding,she doing the same. The words are compared with the description, and the child is led by developing questions to discover where and why he has failed.
This exercise is continued until he can exactly and readily explain the difference between these words. The substantives and verbs are then considered, with regard to their changes; after which, he is encouraged to find out a series of examples. Next to these exercises, the principal rules of grammar are illustrated.
The child may then endeavour to form one or more phrases, by which the rule is put in practice; and should he not succeed, she herself invents a sentence, and lets hiin repeat and write it, omitting the word exemplifying the rule, which the child may find out, and put into its proper place. More difficult exercises may be given to children whose thinking powers have been developed and strengthened to a certain degree; for instance, the Mother lets them read a short but interesting moral tale, and dictates the same