"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]. .

66

...

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

226

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In the Table of Simple Unity there were ten of each number on a line; so that on the last line there were ten tens. The other numbers were put thus: IIII, IIIII, IIIIII, IIIIIII, IIIIIIII, IIIIIIIII, IIIIIIIIII. The Table of Simple Fractions had ten squares in each line, and ten lines; the last line being ten squares divided into tenths. The Table of Compound Fractions also had ten lines and ten squares in each. In the first line the unit was divided in halves, thirds, etc., to tenths; in the second line halves were divided into their halves, thirds, etc., to tenths; and in the last line tenths were similarly divided.

In teaching units Pestalozzi did not confine himself to the Table of Units, i.e., to visual sense-impressions. He says that, when the pupils were familiar with this, he 'let them find the same relations on their fingers, or with peas, stones, or other handy objects" (How Gertrude Teaches). After the four simple rules had been mastered the learner was taken to fractions; and not until these were known was he allowed to apply his, now complete, number knowledge to practical arithmetic, i.e., sums concerning money, weights, measures,

etc.

Very full and detailed exercises were given for all the numbers up to 100; and for all the small fractions. These exercises had to be thoroughly mastered and known, before what we now call concrete sums were worked. Although the pupils were dealing with some kinds of objects-diagrams, pictures and things—all the time, yet the formal and mechanical elements were largely present, and must have taken up much of the time and energy of the teachers and learners.

for fractional calculation. Experience has proved, that by my method they arrive at this part of arithmetic from three to four years earlier than by the usual mode of proceeding. And it may be said of this, as of the former course, that it sets the child above confusion and trifling guesswork; his knowledge of fractions being founded upon intuitive and clear ideas, which give him both a desire for truth and the power of discovering and realising it in his mind."

Throughout the teaching of number, Pestalozzi's aim is to develop distinct ideas through grouping (addition and multiplication), separating (subtraction and division), and comparing (ideas of more and less) the objects—as to their quantitative (number) elements-of perception. When the ideas of the learner have been perfected through number-teaching, then the learning of the ordinary arithmetical rules is but the application of his trained ideas to the practical affairs of life; and it will be found that he is able to understand the problems and discover the rules, in most cases, for himself.

Pestalozzi had three arithmetical tables which he used in teaching number. We give sections of these to show what they were.

In the Table of Simple Unity there were ten of each number on a line; so that on the last line there were ten tens. The other numbers were put thus: IIII, IIIII, IIIIII, IIIIIII, IIIIIIII, IIIIIIIII, IIIIIIIIII. The Table of Simple Fractions had ten squares in each line, and ten lines; the last line being ten squares divided into tenths. The Table of Compound Fractions also had ten lines and ten squares in each. In the first line the unit was divided in halves, thirds, etc., to tenths; in the second line halves were divided into their halves, thirds, etc., to tenths; and in the last line. tenths were similarly divided.

In teaching units Pestalozzi did not confine himself to the Table of Units, i.e., to visual sense-impressions. He says that, when the pupils were familiar with this, he "let them find the same relations on their fingers, or with peas, stones, or other handy objects" (How Gertrude Teaches). After the four simple rules had been mastered the learner was taken to fractions; and not until these were known was he allowed to apply his, now complete, number knowledge to practical arithmetic, i.e., sums concerning money, weights, measures,

etc.

Very full and detailed exercises were given for all the numbers up to 100; and for all the small fractions. These exercises had to be thoroughly mastered and known, before what we now call concrete sums were worked. Although the pupils were dealing with some kinds of objects-diagrams, pictures and things—all the time, yet the formal and mechanical elements were largely present, and must have taken up much of the time and energy of the teachers and learners.