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addition will be either a unit, or an article, or a composite number. If a unit, write it under the line, immediately under the units; if an article, write a cipher* there, and add the number of tens to the second column; if a composite number, write the units under the units, and add the tens to the second column. Proceed in the same manner with the second column, but do not forget to add in the tens resulting from the addition of the first column. When you
have finished the second column, proceed to the third, fourth, &c. When you add
up the last column, you can, if the addition gives tens, set them down at once."
The instruction in the other ground rules is given quite in the same way; as is the mode of proving examples. For multiplication he especially recommends the multiplication table. "If you have not thoroughly mastered this,” he says, “ I assure you that, if you do not take pains to learn it, you will make no progress in arithmetic.”
This may suffice to describe the style of Peurbach's arithmetic, four hundred years old; the same method has prevailed even down to our own times. It is in this study, as I have said, that the difference comes out most clearly between the ancient and modern styles of instruction. To show this in a single point, let the reader compare Peurbach's recommendation about the multiplication table with an expression of Diesterweg's. The latter says, "The ancient teachers made the famous multiplication table the basis of all arithmetic. They made it the beginning of the study, printed it in the primer, and impressed it mechanically upon the children's memories. Nowadays it plays a more subordinate part; and this single fact may show how far we have left the worthy ancients behind us in arithmetical instruction.
The multiplication table, with us, comes after the addition and subtraction tables, and before the division table; that is all."
The following observations will state the difference between the ancient and modern methods of instruction in arithmetic.
The object of the ancient method was to enable the children to
Cifram or Zyphram ; others say figura nihili, or circulus; as Hudalrichus Regius, in his “Epitome Arithmetices," (1536) p. 41. Maximus Planudes (in the 14th century) has 751$pa for naught. Fibonacci, a Pisan, wrote in 1202 a “Treatise on the Abacus,” (Tractatus de Abaco,) in which he relates that during his travels he learned the Indian art of arithmetic, by which with ten figures all numbers can be written, (Cum his figuris, et cum signo O, quod Arabice Zephirum appellatur.) (Wlewell, 1, 190.) Lichtenberg (6, 272) says, ** Zero (naught) is derived from cyphra and cypher, the Latin and English for naught; and these from the Hebrew sephar, to count." Menage says, “Chifre.—The Spaniards first took this word from the Arabs. It was Zefro." Spaniards change f into h; hence, Zefro, Zehro, Zero. When did the German Ziffer receive its present meaning?
† In the preface to his “Handhuch," Diesterweg says, however, “Any one desirous of multiplying larger numbers together in his head must know the multiplication table by heart. The inferior grade of computation must be facilitated by this great means of assistance, in order to avoid difficulties in the higher grade.” This agrees with Peurbach.
add, subtract, &c.; an art of arithmetic was sought, not an understanding of it, a theory of it. As a foreman shows his apprentice how to do his work by categorical imperatives, First do this and then do that, without any whys or wherefores, just so was arithmetic taught, without
part of the teacher to communicate to the scholar an understanding of the things he did. Nothing was thought of except skill in operating, which was gained by much practice. This mode of instruction was made more natural by the fact that only written arithmetic was taught.
Pestalozzi and his school opposed this method of instruction, and called it mechanical, and unworthy of a thinking being. The child, they said, must know what he is doing; and should not merely perform operations without any understanding of them, according to the teacher's directions. Understanding is the chief object; the training of the intellect as a properly human discipline, without any relation to future practical life. A few of them claimed that, if the scholar acquired nothing but this intelligent knowledge, if it was done in the proper methodical way, his practical skill would come of itself; that, by the knowing about his art in the proper manner, a man becomes a master of it.*
The ancient method, which kept the pupils at unwearied drilling, trained skillful and certain mechanical laborers. The pupils operated according to traditional rules, which they did not understand, and which even the teachers themselves very likely did not understand, any more than the master-mason, when showing an apprentice how to make a right angle with a string divided by two knots into lengths of three, four, and five feet, can also explain to him the Pythagorean problem.
But although by this method the scholar was excellently well prepared for many computations, which he will have occasion for in practical life, yet he will be quite at a loss how to help himself whenever a case shall come up to wbich he can not apply his rule exactly as he learned to use it. This will appear when he enters upon Algebra; even in undertaking to use letters, instead of figures in his much-practiced Rule of Three. Algebra requires every where a clear, abstract knowledge of arithmetical operations and relations—a just distinguishing between the known and unknown quantities which are to be sought or eliminated, and an understanding of the mode of using these in the most varying cases. But all this will be entirely wanting to the mere routinist, whose thinking is done by traditional rules founded on experience. He would in like manner
An error which they subsequently perceived ; and afterward labored at a union of knowledge and practical skill.
find himself unprovided with an intelligent method of mental arithmetic, such as requires independent work by the scholar; for what this school called mental arithmetic was nothing but an inward display of figures, and an inward operation performed upon them.
Three chief adversaries appeared against the ancient mechanical arithmetic, of whom I have just mentioned two.
The first, namely, was Algebra.* This represented special cases in a universal way; and treated special procedures in arithmetic in such a manner that the course of the proceeding—the law according to which the required quantities were found—was clearly expressed. Letters were every where used for numbers—undetermined numbers; for any letter might stand for all possible numbers.t
Thus, in algebra, the understanding and investigating of universal relations and laws appeared as opposed to mere computations, practiced according to a rule not understood, and aiming only at mechanical facility.
In like manner arose the true method of mental arithmetic, which has become so prominent, especially in later and the latest times, in the place of the usual operating upon pictures of figures within the mind. It was seen that upon this intelligent mental arithmetic must be based a right understanding of the mechanical processes of arithmetic. This was, among other reasons, because the mental method obliged the pupil to perform many operations in an order quite different, and even entirely opposed, to that used in written arithmetic.
The third adversary of the old method of arithmetic was the intuition so prominently urged by Pestalozzi and his school. While algebra took the arithmetical laws out of concrete numbers, and established them as ideas, abstractly, Pestalozzi, on the contrary, sought for means of that intuitional instruction which must precede all reckoning with numbers, and without which that reckoning must be without any proper foundation. As algebra developed itself out of concrete arithmetic, so was the idea of number itself, again, to be deduced from the bodily examination of numerable objects of various kinds. “The mother," says Pestalozzi, “ should put before the child, on the table, peas, pebbles, chips, &c., to count; and should say, on showing him the pea, &c., not “This is one, but “This is one pea,' &c." And he proceeds to say, “ While the mother is thus teaching the child to recognize and name different objects, as peas, pebbles, &c., as being one, two, three, &c., it follows, by the method in which she shows and names them to the child, that the words one, two, three, &c., remain always the same; while the words pea, pebble,
I use this word, like Euler, Montücla, kries, &c., in its wider sense. + Kiies' “Manual of Purc Mathematics," (Lehrbuch der Reinen Mathematik,") p. 72, &c.
&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 inind 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 bim 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 an old proverb.
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 a 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 Rcations of Numbers," (Anschauungslchre der Zuhlcnrerhältniss.)
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
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 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 us; for we have not that foundation in the evidence of our senses which only can make the empty word a truth to 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+4739=8306? Are all such operations as these last then destitute of intuition ? that is, are they all actually empty words and unintelligent labor ? * PL 1, p. 16, &c , p. 51, &c. Ibp. 58.
How Gertrude Teaches her Children."