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, The child how many times have we one square?' 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. The "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. "In the course before described, a stroke representing the unit was made the intuitive basis of instruction; and it is now necessary, for the new species of calculation just mentioned, to find a figure which shall be divisible to an indefinite extent and yet preserve its character in all its parts, so that every one of them may be considered as an independent unit, analogous to the whole; and that the child may have its fractional relation to the whole as clearly before his eyes as the relation of three to one, by three distinct strokes. The only figure adapted to this purpose is the square. By means of it the diminution of each single part, and the proportionate increase of the number of parts by the continued division and subdivision of the unit may be made as intuitively evident as the ascending scale of numbers by the addition or multiplication of units. A fraction table has been drawn up [to show this]. "Now as the alphabet of forms is chiefly founded upon the division of the square into its parts, and the fractional tables serve to illustrate the same division in a variety of manners, the alphabet of forms, and that of fractions, prove in the end the same; and the child. is thus naturally led to connect in his mind the elements of form with those of number, both explaining and supporting each other. My method of arithmetic is therefore essentially founded upon the alphabet of forms, which was originally intended only for the purposes of measuring and drawing. "By means of these fractional squares, the child acquires such an intuitive knowledge of the real proportions of the different fractions, that it is a very easy task, afterwards, to introduce him to the use of figures |