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the modern mathematicians do, what is the use of investigating the regular solids? And he will find himself much better able to study in the method of the ancients; a method the neglect of which has been lamented by Fermat, Newton, and Montücla. A later writer has described this method as one which speaks to the eyes and the understanding, by figures and copious demonstrations. And he laments that the more recent mathematicians have allowed themselves to be carried to a harmful extreme by the extraordinary facility of the algebraic analysis. "In fact," he says, "the ancient method had certain advantages, which must be conceded to it by any person only even moderately acquainted with it. It was always lucid, and enlightened while it convinced; instead of which, the algebraic analysis constrains the understanding to assent, without informing it. In the ancient method, every step is seen; and not a single link of the connection between the principle and its furthest consequence escapes the mind. In the algebraic analysis, on the other hand, all the intermediate members of the process are in a manner left out; and we merely feel convinced in consequence of the adherence to rule which we know is observed in the mechanism of the operations in which great part of the solution consists."*
Speaking pedagogically, no one can doubt, after the descriptions thus given, whether the geometrical method of the ancients has the advantage, in regard to form, over the analytical one of the moderns. I have shown elsewhere how harmful it is to give the boys formulas, by whose aid they can easily reckon out what they ought to discover by actual intuition; as in the case where a pupil, who scarcely knows how many surfaces, edges, and angles a cube has, computes instantly by a formula, by a mere subtraction, what is the number of angles of a body having 182 sides and 540 edges, without having the least actual knowledge of such a body.
An instance of the predominance of the analytic method is found in the "Mécanique Céleste" of Lagrange, which appeared in 1788. In this, the author says, "The reader will find no drawings in this work. In the method which I have here employed, neither constructions nor any other geometrical nor mechanical appliances are needed; nothing but purely algebraical operations."
[Translated from Raumer's "History of Pedagogy," for the American Journal of Education.]
THE difference between ancient and modern methods of instruction is remarkably clear in the case of arithmetic.
By way of describing the ancient method, I will cite portions of one of the oldest and best reputed of German school-books-the "Elementa Arithmetices" of George Peurbach.* This author was, in his time, the greatest mathematician in Germany; and one of his pupils was the great Regiomontanus.t
Peurbach's arithmetic began with the consideration of numbers. "These," he says, "are divided by mathematicians into three kinds : into digits, which are smaller than ten; articles, (articuli,) which can be divided by ten without a remainder; and composite numbers, consisting of a digit and an integer. Unity is however no number, but the rudiment of all numbers; it is to number what a point is to a line. In arithmetic it is usual, after the manner of the Arabs, who first invented it, to work from right to left. Every figure, when standing in the first place at the right hand, has its own primitive value; that in the second place has two times its primitive value, in the third place a hundred times, in the fourth one thousand times, and so on."
The second chapter is on addition. "To unite several numbers in one, write them so that all the figures of the first place (units) shall stand under each other, and in like manner of the second place, and Having arranged them in this way, draw a line under them, and then begin the work at the right hand by adding together all the numbers of the right column. The sum resulting from this
• "Elements of Arithmetic. An algorithm of whole numbers, fractions, common rules, and proportions. By George Peurbach. All recently edited with remarkable faithfulness and diligence. 1536. With preface by Philip Melancthon." (Elementa Arithmetices. Algorithmus de numeris integris, fractis, regulis communibus, et de proportionibus. Autore Georgio Peurbachio. Omnia recens in lucem edita fide et diligentia singulari. An. 1536. Cum præfacione Phil. Melanth) Peurbach was born in 1423, and died 1461.
↑ "This philosophy of celestial things was almost born again in Vienna under the auspices of Peurbach. This whole department of learning, (astronomy,) after having lain in dishonor for centuries, has of late flourished anew in Germany, under the restoring hands of two men, Peurbach and Regiomontanus. Their very achievements testify that these two heroes were raised up, for the promotion of that branch of learning, by some wonderful power of divine appointment." This is Melancthon's opinion, as given in his preface to the "Sphæra" of Sacro Bosco. Comp. Montücla, "History of Mathematics," part 3, book 2; also Schubert's "Pcurbach," &c.
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."t
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 Tipa 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.) (Whewell, 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 "Handbuch," 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 any effort on the 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 which 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.
† Kries' “Manual of Pure Mathematics," (Lehrbuch der Reinen Mathematik,”) p. 72, &c.