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. CHAPTER XII. PESTALOZZI'S METHODS OF TEACHING VARIOUS Geography. In the Swan's Song Pestalozzi says that the accurate observation of the different conditions of water, at rest or in motion: its changing into dew, rain, vapour, steam, hoar-frost, hail, etc. and its action on other objects of nature; and the expressing of the results of such observations in clear and fitting language, give the beginnings of physical geography. The pupil must first be taught to observe the country around his own home; not studying it through a map, but by actually walking about the land itself. must learn to make a map-correcting any mistakes in his first attempts from fuller and more accurate knowledge gained from later visits-before he is allowed to see, much less to make use of, a school map. The maps used in school teaching should be blank maps. He One of the Yverdon pupils, Professor Vulliemin, thus describes the actual teaching in geography: "The first elements of geography were taught us from the land itself. We were taken to a narrow valley not far from Yverdon, where the river Buron runs. After taking a general view of the valley, we were made to examine the details, until we had obtained an exact and complete idea of it. We were then told to take some clay, which lay in beds on one side of the valley, and fill the baskets which we had brought for the purpose. "On our return to the castle, we took our places at the long table, and reproduced in relief the valley we had just studied, each one doing the part which had been allotted to him. In the course of the next few days more walks and more explorations, each day on higher ground, and each time with a further extension of our work. Only when our relief was finished were we shown the map, which by this means we did not see till we were in a condition to understand it." From the very beginnings geography is to be correlated with the other sciences, such as natural history, agriculture, geology, etc.; not only because these are directly connected with each other, but also because greater and continuous interest is thus aroused. Dr. Biber, after describing, in glowing terms, the pre-eminent advantages of the surroundings at Yverdon, for teaching geography to the pupils there, says: "He taught them to watch the gathering up of the morning mists, and the shadows of the early clouds, which passing over the glittering lake hid for a moment, as with a veil of gauze, its streams of undulating gold; he directed their eyes to the flaming characters with which the sun writes the farewell of day on the traceless surface of eternal snow; he stood listening with them to the majestic voice of nature, when the autumnal gale howling on the floods, rolled billow after billow to the bleak shore; he guided their steps to the mountain caves from whose deep recesses the stately rivers drew their inexhaustible supplies. "Wherever he found a leaf in the mysterious book of creation laid open, he gave it to them to read, and thus, within the narrow sphere of their horizon, taught them more of earth and earthborn beings, than they could have learned by travelling, in the pages of a heavy volume, all round the globe. This was indeed 'intuitive' teaching, and experience proved that, independently of the moral effect which such an intercourse with nature can never fail to produce, the reality and vivacity of the ideas awakened in the children, concerning the relations of the great elements to each other, and to the beings whose existence they support, ensured a permanent and lively attention to whatever ulterior instruction in the science of geography it was deemed expedient to impart. "The simple features by which the hand of nature has distinguished the different countries, were presented to the mind long before the artificial mould into which man has cast them. Physical and mathematical geography, founded upon the ideas acquired by self-observation, formed the ground-work of this branch of the method, and statistical facts were superadded at the end, arranged in concise tables so as to facilitate their recollection." History. Pestalozzi held that it was unwise to attempt to teach historical incidents, and their causes and effects, to young children. Not only are children unable and unfitted to judge of the doings and motives of men and nations, but their moral and intellectual progress is hindered and hampered by attempts to do this, and by so early an acquaintance with the wickedness and violence they have to learn about in the study of history. Dr. Biber says: "The historical lessons laboured under still greater imperfections. Pestalozzi, from a |