&c., always change, as the nature of the object changes which is thus used; and by this permanence of the one, and constant change of the other, there will be established in the child's mind the abstract idea of number; that is, a definite consciousness of the relations of more or fewer, independently of the objects which are set before him as being more or fewer."* Thus far Pestalozzi adheres to the method in which arithmetic had always been begun, in a manner strictly accordant with nature. Counting had been taught by beans, &c., and especially on the fingers. "You can count that on your fingers" is proverb. an old He now, however, proceeds further, to artificial school-apparatus for intuition. He and his fellow-teacher, Krüsi, prepared some "intuitional tables" for this purpose. In the first, the numbers from one to ten are separated by marks: a I in the upper horizontal row, II below it, and so on, down to ten such marks for ten. And 175 pages were occupied with exercises to be taught upon these marks. The second intuitional table is in the form of a square, divided into ten times ten small squares. The ten squares in the upper horizontal row are not divided; those in the second are halved by a perpendicular line; those of the third are divided into thirds by two such lines; and so on, to the last, which is divided into ten parts by nine perpendicular lines. The second intuitional table is properly followed by the third in the second part of the "Intuitional Theory." It is a large square, divided into ten rows of ten small squares. The first of the first horizontal row is undivided, the second halved by a horizontal line, the third divided into three parts by two horizontal lines, and so on to the tenth. The ten squares of the first perpendicular row are divided in the same way by perpendicular lines, and the other squares are divided both by perpendicular and horizontal lines, (corresponding with a multiplication table,) according to their order, in a perpendicular and a horizontal row. Thus the hundredth small square, diagonally opposite that which is not divided at all, is thus divided into ten times the smaller squares, of which each is thousandth of the large one. The 'second table, preceding this, consists of thirty-six pairs of parallel lines, equal in length but divided differently. The pair A and B, for instance, are divided by points into six equal parts; but, besides this, A is divided into halves and B into thirds; the former into twice three-sixths, and the latter into three times two-sixths. * Pestalozzi, preface to part 2 of his "Intuitional Theory of the Relations of Numbers," (Anschauungslehre der Zahlenverhältniss.) For the method of using these intuitional tables in instruction, I refer to Pestalozzi's "Elementary Books," and to Von Türk's "Letters from Munchen-Buchsee."* I shall here only offer a few observations on them. By means of these tables it was sought to elucidate to the children the four ground rules, fractions, and the rule of three, even algebraically. In particular, every number was considered as composed of ones, and was referred to ones as its elementary parts. And this was done not only at first to facilitate a clear understanding, but in subsequent parts of arithmetic, and even to a wearisome extent. Instead of seven, 66 seven times one" was used; and again, "One is the seventh part of seven." And thus were composed so many strange, wordy problems; as "Three times half of two, and six times the seventh part of seven, are how many times the fourth part of four?" Pestalozzi should undoubtedly have the credit of calling attention, by his "Elementary Books," to the visual element of arithmetic, which had previously been almost entirely neglected in the schools. Since that time, this element has been much used for primary instruction, and as a means of laying a foundation by the use of the senses for subsequent insight. But at present, most of the arithmetics of the Pestalozzian school vary much from this excessive use of the senses, as is shown by their books of examples. It is clear that there are limits to the use of the intuitional faculties. Pestalozzi exceeded these in various ways; as in the line divided into ninety parts, and a square divided into ninety rectangles, which we find in his "Elementary Books." What eye would distinguish, in his third table, between the square divided into nine times ten rectangles, and that divided into ten times ten, next after it? The necessity of actual intuition at the beginning of arithmetic also led Pestalozzi into an error. "When," he says, "we learn merely by rote that three and four are seven, and then proceed upon this seven just as if we actually knew that three and four were equal to seven, we deceive ourselves, and the inner truth of this seven is not in for we have not that foundation in the evidence of our senses which only can make the empty word a truth to us." us; But granting that I can inwardly see the picture of the statement that 3+4=7 in marks, peas, &c., can I have the same sort of visible basis within me when I would add 59+76=135; or, rather, 3567+47398306? Are all such operations as these last then destitute of intuition? that is, are they all actually empty words and unintelligent labor? *Pt. 1, p. 16, &c, p. 51, &c. t Ib.. p. 58. "How Gertrude Teaches her Children." These considerations may enable us to arrive at a correct estimate and application of the use of intuition. It is intended to assist the work of the understanding, by representations which the eye will easily take in and the mind will easily retain; and to facilitate the comprehension of numbers and their relations to each other, and afterward the methods of operating in agreement with the ideas thus received. If the intuitional powers have fulfilled their task, and if a correct understanding has been attained in the small matters at first studied, the pupil may boldly proceed to greater numbers-to numbers so great that intuition can not deal with them at all. Thus, the scholar's intuition on the subject of fractions may carry him, for instance, at furthest, to the subdivision of a line into twenty-four equal parts, and to their designation in their various different ways, as 2x12; 3×8; 4×6; 6×4; 8×3; and 12×2. By means of such a line as this a clear idea can be formed of the mutual relations of fractions of different denominators; as, for instance, that == 43 or that =, &c. But the eye is not capable of taking in Pestalozzi's line subdivided into ten times ten portions. In this case the understanding has to assist the eye much more than the eye the understanding. We have seen that instruction in arithmetic has always commenced with visual intuition, and that Pestalozzi endeavored to erect this natural proceeding into a method-a system which should proceed from a right beginning to a right end, in a right manner. With this design he published his "Elementary Books" and Intuitional Tables. And yet, the numerous and even excessive exercises upon these tables had really nothing whatever to do with arithmetic. After the pupil had completed the whole of these exercises, without even knowing the Arabic figures, these last may be made known to him. "in the usual manner," and their value as dependent on their places. After this comes operations with figures. But my experience has been, that it is precisely for the understanding of these operations that intuition is most necessary. The tiresome, inanimate marks of the Pestalozzian tables seem to me peculiarly unsuitable for children, who rather require colored or shining things, such as will easily impress their fancy. And again, if these things are to open the road to operations with figures, they must represent not mere units, but must be adapted to the decimal system-the system of Arabic figures. I made use of counters; which, if properly managed, will afford much assistance.t A difference must be made between numbers and figures. The ⚫ Türk, 101. † See Appendix III. on this point. same number can be indicated by very different figures; as, for * To comprehend the wondrous and almost magic power of the socalled Arabic figures, it is only necessary to work the same example with these and with the Greek or Roman figures. The example in the note is very simple; the difference will appear more evidently on trying even a very moderately large example in "long division" with the Roman figures. And if there is such a difference even in the elementary part of arithmetic, how much greater will it be in more complicated work! In later times this written arithmetic, so far from being an object of admiration, has, on the contrary, been so violently attacked that mental arithmetic has assumed a remarkable predominance over it. A teacher wrote a little work, entitled "Head or Thought-Arithmetic;" in which written arithmetic was almost synonymous with "mindless arithmetic." This reaction, however, was quite natural. We have already seen that in early times pupils were taught only the operations with figures; that they only learned to juggle according to the rules given them, and did not even know how they arrived at the results of their operations. Schiller objects to certain authors that "language did their thinking and wrote their poetry for them." In like manner the wonderful decimal system thought for these scholars, if not even for their teachers themselves. It is at present a source of satisfaction, that by mental arithmetic this juggling business is to be brought to an end. And for certainty's sake it is strictly forbidden to perform the mental operations with the help of imaginary figures, this being really identical with written arithmetic. But a proper regard should be paid to the latter; and it should be remembered how soon we come to the limits of mental arithmetical operations where we become obliged to use figures, letters, or visible representatives of some kind. Many persons are inclined to exceed these limits, even by force; and imagine that by the most complicated examples in mental arithmetic they can develop the scholar's capacity to the utmost extent. But a skillful mathematician of (A) (B) 432)864(2 CCCCXXXII) DCCCXXXXXXIV(II This is but a trivial example of the magic of the decimal system; 100,000 florins are how many each to ten men? Ans.-10,000 florins. The fault is our own if we do not admire such a system. Berlin has asserted, in contradiction to these, that "mental arithmetic is not actually an exercise of the understanding, because it requires the use of the memory exclusively." No one can deny this statement as to the use of the memory; nor that those virtuosos, who are accustomed to exhibit their skill in mental arithmetic, are usually of very trifling capacity in other matters. The correct belief is that of those who, like Diesterweg and Stern, have opposed not merely the earlier mechanical written arithmetic, but have also sought to penetrate the essential principles of the mechanism of it, and to make their pupils understand, so that the latter might make use of written arithmetic with the same clear comprehension as mental arithmetic. It was seen that the difference between mental and written arithmetic consisted chiefly in the abbreviations which are used in the latter. But the pupil readily apprehends the briefer processes of the latter, when explained to him in full by the teacher.* For arithmetical instruction is concerned with the explanation of abbreviations, from the elements up to the infinitesimal calculus; with marks and formulas invented by the most penetrating mathematical minds. To the pupil these appear to be mere magic marks and formulas, until he is made acquainted with the mode of their production. In the higher grades of the study, however, the pupil may be accustomed to the purely mechanical use of many algebraical formulas and of logarithms, in the same way in which the mechanical use of arithmetical figures used to be taught. The question how far arithmetical instruction should be carried in one and another school, is in some cases easy, and in others difficult, to answer. For elementary schools, Diesterweg was right in saying, “Every child should here go so far in arithmetic as to be able to solve readily in writing or mentally such problems as he will meet in common life." In the common schools there should be no prominent efforts after isolated distinction in any department. It is much more difficult to fix a limit for arithmetical instruction in the burgher schools, because these schools are of very various characters, according to circumstances. The general future occupation of the children who attend the burgher schools has particularly great influence in this respect. By examining a large number of school programmes, from various parts of Germany, I have found that at present most of the gymnasia proceed to about the same extent in mathematical instruction. The Prussian ordinance on examinations, of 1834, requires "Thor *For an example see Appendix IV. |