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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 2×12; 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 == 24 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.†

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 instance,

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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

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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. Το 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.

oughness in the theory of the powers and roots in progressions, and also in the elements of algebra and geometry,* plane and solid; knowledge of the theory of combinations and the binomial theorem ; facility in managing equations of the first and second degree, and in the use of logarithms; a practiced knowledge of plane trigonometry; and especially a clear comprehension of the connection of all the propositions in the whole system of lessons."

A hundred years before, in a Prussian ordinance of 1735, no methodical knowledge was required, even of gymnasium graduates.† On the question whether the gymnasium course should also include conic sections and spherical trigonomety, opinions differ. Only the teachers of two gymnasia declare for instruction in the infinitesimal calculus, while others are decidedly opposed to it, and certainly with entire propriety. Pupils of distinguished mathematical talents should follow their mathematical course further, at the university or at the polytechnic school.

There is no study where so urgent a warning is needed against the overstimulus of the scholars as in mathematics. It is known that, in Pestalozzi's institution, Schmid's influence caused this department to occupy a disproportionate space, and pushed every thing else into the background. The children were also experimented on; and were encouraged to exercise exhibitions of arithmetical skill, in the same manner as injudicious gymnastic instructors quite go beyond the limits of their art, and instruct their pupils in rope-dancing, for the sake of exhibiting their own skill in the skill of their scholars. To teach the infinitesimal calculus in a gymnasium is a similar excess.

No teacher should ever seek, by excessive stimulation, to spur on his pupils to an unnatural point of attainment, which most of them can never reach. If a few of them reach the desired summit, they usually retain their place on the peak of their intellectual Mont Blanc only a very short time, and by the most violent exertions. When the teacher ceases his efforts, or they leave school, they throw aside the study in disgust; and, according to the fixed law of nature, the excitement is succeeded by a relaxation. The teacher should be contented and pleased, if his pupils attain to some little excess of knowledge, doing so under healthy natural incentives, not too great

The ordinance of 1812 prescribed the first six and the eleventh and twelfth books of


See Prof. Lentz, in the "Annual Report on the Royal Frederic-College, at Königsberg,” (Jahresbericht über das Konigl. Friedrichs-Kollegium, in Königsberg.) 1837.

The mathematical instruction at the schools of arts and trades, and polytechnic schools, is meant to determine the future practical ability in mathematics; that in the gymnasia, rather the formal knowledge of it. The former, therefore, requires a higher degree of skill in the pupil, which also must be based upon a scientific knowledge. It must cultivate the roots of the study to develop it.

for their faculties; if they gain an entirely clear understanding and entire facility in the study up to this point. What has been thus acquired is not easily forgotten after the school-years; and, even if he goes no further with that study, he will always retain a certain degree of knowledge, which, if his teacher was intelligent and judicious, can not easily fail him.

I can not resist quoting a case given by Diesterweg, to illustrate what I have said about excessive stimulation of scholars. In speaking of de Laspé, principal of a private institution at Wiesbaden, he calls him a natural genius in didactics, who "accomplishes extraordinary things by the help of enthusiasm. For," he continues, "is it not praiseworthy and instructive, even if on other accounts to be disapproved of, to see girls of twelve occupying themselves, with genuine delight, with mathematical constructions, and, without assistance, solving problems which any one would admit to be difficult for that age. Many instances," Diesterweg continues, "have occurred in de Laspé's school, to show with what enthusiasm an energetic teacher can fill his scholars. I will relate one. High Mining Councilor K.,* during a visit to the institution, at the invitation of de Laspé, gave out to the boys and girls a geometrical problem. All, great and small, teachers and scholars, went to work on it. No one discovered the solution. Thus passed the first day. On the next, all went early to work on it again, but in vain. De Laspé endeavored to renew the enthusiasm of the school, but no one found out the solution. A dull feeling of weariness and despair came over the whole institution. Nothing could be accomplished in this way. The honor of the institution seemed to be at stake; de Laspé worked, and begun and ended his efforts in bad humor. On the fourteenth day he held an evening devotional exercise for encouragement, and prayed that God would strengthen him and the members of his institution for the solution of the problem. What was the result? At about three in the morning, a boy, in his night-clothes, ran to de Laspé's bedside; he had discovered it. De Laspé sprang up and struck a light; the boy went through his operation. It was right! The whole house was called together on the instant, and the triumph made known. De Laspé was a pedagogical genius." So far Diesterweg.

But does de Laspé, according to this account, really deserve the name of a pedagogical genius? Does a teacher deserve that name, who inspires girls of twelve with a truly unnatural passion for mathematics? a man who, when his whole institution has fallen into a dull weariness and despair because neither he nor any body else in it can solve a problem which a stranger has happened to propose to them,

* Kramer. See "H. Pestalozzi," by A. D., (A. Diesterweg,) p. 23.

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