In other words, writing includes composition. We learn to write so that we may have another means of expressing our thoughts. Learning to make letters is not writing, but only getting possession of the means for writing. The art will "enable the children . . . to express themselves clearly about every possible thing, whose form and substance may be made known to them, whether by word of mouth or by writing; and firmly impress the knowledge of it. . . . Writing is [to be] perfected not only as an art, but also as a profession" [e.g., the work of a literary man].

...

It is worth while to notice how modern views on the teaching of writing are returning to the Pestalozzian standpoint: writing being taught through drawing, and in connection with composition. In one point Pestalozzi was much in advance of the present method of teaching penmanship, in that he taught the combination of the letters already learned into syllables and words, before mastering the writing of all the letters in the alphabet.

III. Number Teaching.

In dealing with this subject Pestalozzi is at the very opposite pole to that which marks what is usually understood by the teaching of arithmetic. He did not set out to teach his pupils how to do sums, but how to understand numbers. On his plan the learner was able to do sums, easily and accurately, because he understood numbers; on the other plan children learn to do sums but may never understand numbers. His pupils were able to discover the ordinary rules of arithmetic from their study of the principles of numbers; pupils under the other system learnt the rules by rote

and worked the sums unintelligently. As a matter of fact his pupils were the most acute and rapid of practical arithmeticians, amazing every one by their speed and accuracy. He made no use whatever of figures until his scholars knew the numbers themselves perfectly, up to ten; and he taught no tables of weights and measures, nor what may be called business arithmethic, until the pupil had mastered the theory and art of numbers, and then only such tables and calculations as the scholar was likely to want in his future calling.

Number knowledge must, like all other knowledge, start from, and develop through, sense-impressions. Here is Pestalozzi's own theory of number: “This science arises altogether out of the simple composition and separation of units. Its fundamental formula is this: one and one are two'; 'one from two leaves one'. Any number, whatever be its name, is nothing else but an abridgment of this elementary process of counting. Now it is a matter of great importance, that this ultimate bases of all number should not be obscured in the mind by arithmetical symbols. The science of numbers must be taught so that their primitive constitution is deeply impressed on the mind, and so as to give an intuitive knowledge of their real properties and proportions, on which, as the groundwork of all arithmetic, all further proficiency is to be founded. If that be neglected, this first means of acquiring clear notions will be degraded into a plaything of the child's memory and imagination, and its object, of course, entirely defeated.

"It cannot be otherwise. If, for instance, we learn merely by rote 'three and four make seven,' and then we build upon this 'seven,' as if we actually knew that

three and four make seven, we deceive ourselves; we have not a real apprehension of seven, because we are not conscious of the physical fact, the actual sight of which can alone give truth and reality to the hollow sound. . . .

"The first impressions of numerical proportions should be given to the child by exhibiting the variations of more and less, in real objects placed before his view . . . in which the ideas of one, two, three, etc., up to ten, are distinctly and intuitively presented to his eyes. I then call upon him to pick out in those tables the objects which occur in the number one, then those which are double, triple, etc. After this I make him go over the same numbers again on his fingers, or with beans, pebbles, or any other objects which are at hand. . . .

"In this manner children are made perfectly familiar with the elements of number: the intuitive knowledge of them remains present to their minds while learning the use of their symbols, the figures, in which they must not be exercised before that point be fully secured. The most important advantage gained by this proceeding is that arithmetic is made a foundation of clear ideas; but, independently of this, it is almost incredible how great a facility in the art of calculating the child derives from intuitive knowledge. .

"A square [tablet] is put up, and the teacher asks: 'Are there many squares here?' Answer: 'No, there is but one'. The teacher adds one, and asks again: 'One and one; how many are they?' Answer: 'One and one are two'; and so on, adding at first by ones, afterwards by twos, threes, etc.

"After the child has in this manner come to a full understanding of the composition of units up to ten,

and has learned to express himself with perfect ease, the squares are again [used] in the same manner, but the question is changed: If there are two squares, how many times have we one square?' The child looks, counts, and answers correctly: If there are two squares, we have two times one square'.

"The child having thus distinctly and repeatedly counted over the parts of each number up to ten, and come to a clear view of the number of units contained in each, the question is changed again, the squares being still put up as before. Two: how many times one is it? Three: how many times one?' etc.; and again: How many times is one contained in two, three?' etc. After the child has in this manner been introduced to the simple elements of addition, multiplication and division, and become conversant with their nature by the repeated representation of the relations which they express, in visible objects, subtraction is to be exercised upon the same plan, as follows: the ten squares being put up together, the teacher takes away one of them, and asks: 'If I take one from ten, how many remains?' The child counts, finds nine, and answers: If you take one from ten, there remains nine'. The teacher then takes away a second square, and asks: One less than nine: how many?' child counts again, finds eight, and answers: 'One less than nine are eight'; and so on to the end.

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'This exemplification of arithmetic is to be continued in successive exercises, and in the manner before described. For example:

"As soon as the addition of one series is gone through, the subtraction is to be made at the same rate, thus: having counted together one and two make three, and two make five, and two make seven, and so on up to twenty-one squares, the subtraction is made by taking away two squares at a time, and asking: 'Two from twenty-one: how many are there left?' and so on.

"The child has thus learned to ascertain the increase and diminution of number, when represented in real and movable objects; the next step is to place the same successions before him in arithmetical tables, on which the numbers are represented by strokes or dots."

Such a training in real number will, Pestalozzi asserts, enable the child "to enter with the utmost facility upon the common abridged modes of calculating by figures. His mind is above confusion and trifling guesswork; his arithmetic is a rational process, not mere memory work, or mechanical routine; it is the result of a distinct and intuitive apprehension of number, and the source of perfectly clear ideas in the further pursuit of that science." As he says in another place, his method "was to develop the internal power of the child rather than to produce those results which, nevertheless, were produced as the necessary consequences of my proceedings. . . . The effect of my method was to lay in the child a foundation of knowledge and further progress, such as it would be impossible to obtain by any other. . . .

"The increase and diminution of things is not confined to the number of units; it includes the division of units into parts. This forms a new species of arithmetic, in which we find every unit capable of division and subdivision into an indefinite number of parts.