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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.I
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 Euclid.
+ 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 kuowledge. It must caltivate 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 bis night-clothes, ran to de Laspe'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 "Hl. Pestalozzi," by A. D., (A. Diesterweg;) p. 23.
makes this foolish despair the subject of an appeal to God at an evening-prayer? And do not the question, What was the result ?" and the answer, that a boy discovered the solution-do not these constitute a pietistical statement of a providential answer to prayer! The “honor of the institution, which seemed at stake,” is rescued, it is true; but what honor ? So far as this story goes, I can see in de Laspé only a restless pedagogical zealot,* who urges his pupils to an unnatural mental over-exertion, by especial use of the spur of vanity; who makes fanatics of them. No more monitory warning could be given of an over-excitement calculated to destroy all childlike character. Let the reader only transplant himself in imagination into the despairing, brooding, and study—the abominable fourteen days' restlessness and excitement—of these teachers and poor children, thus hunted to the death, as it were, by their own vanity.
All this seeking ended at last in the Eureka of a boy. But the efforts made by the teachers and pupils together show clearly how the inventive method ought never to be abused; or, rather, they show no particular method was used here at all. The teacher of a science or art ought to know, and be able to do, what his pupils are placed under his care to learn; how otherwise could he teach them? No blind man is calculated to be a guide.
Diesterweg visited de Laspé in 1817, and accompanied him and his pupils in a pedestrian excursion to the Johannisberg in the Rheingau. In passing through that region, whose beauty, famous from ancient times, has attracted to it such a multitude of travelers, to view the mighty stream, its vineyards and peaceful towns, with the wooded mountain in the background, the reader will fancy how delighted teachers and scholars must have been. But he will deceive himself.
They had to take much more care not to get lost while they were at work upon some lessons that required their whole attention. Diesterweg relates in particular the following: “In walking, algebraic problems were given out and solved, for several hours at a time; scholars as well as teachers proposing them. At evening, at the inn, after supper, they made language,' to use the technical term; that is, de Laspé discussed the laws of language with the pupils for several hours, no one showing fatigue or weariness. What would our boys say to this? I must publicly confess that I never saw any where so much enjoyment, so much pleasure in independent thinking and investigation."
Such "enjoyment” reminds me of the Dance of Death at Basle. • I judge only by this story, for I know nothing further of de Laspé sufficient to found an opinion.
NOTE. COUNTEPS IN ELEMENTARY ARITHMETIC.—I nsed white and yellow counters, of different sizes. The smallest white ones were units, larger ones tens, and still larger ones hundreds. To these I added four yellow sizes; the smallest for thousands, and larger ones for ten thousands, hundred thousands, and millions. I did not immediately go any further. * The units served all the purposes for which beans, marks, &c., have been used; as, practice in counting, division into equal and unequal parts, &c.
In teaching written arithmetic, I found the following use of counters very convenient. The children of from six to eight years old usually knew as much about money as that, for instance, four pfennigs made a kreuzer, and six kreuzers a Rechser. I took advantage of this actual experience of theirs to base my instructions. After they had learned sufficiently well to count with the unit counters, I said, “ Just as the large sechser is worth six little kreuzers, so is one larger counter worth ten small ones; so we will call the large one a ten. Then I put with the ten nine more ones, successively, and so taught them to count from ten to nineteen; then I added a tenth one, and changed the ten ones for a second ten, and called the two tens twenty. In the same way I went on to ten tens. Now, just as ten ones is a ten, so are ten tens a hundred; which is again represented by a larger counter. On these exercises there may be constant exercises ; such as, How many ones in two, three, &c., tens? How many ones, or tens, in one hundred ? Then count out ten times ten ones, and then substitute ten tens, to the same value.
By using the counters on the table, the writing and reading of figures will be easily learned. It must only be remembered that the ones stand in the first place to the right, the tens next, &c. Then two ones may be laid down, then three tens, then a hundred, and lastly, at the extreme left, a thousand. Then the pupils may be taught to read them off, thus :—Two; thirty; thirty-two; one hundred; one hundred and thirty-two; one thousand; one thousand one hundred and thirty-two.
Writing the figures connects itself very naturally with these exercises. Supposing the children can write the Arabic figures, they may be told that they must be written exactly as the counters lic on the table; that the first figure to the right represents ones, just as the first counters to the right do; the next tens, &c. The figures should at first be written in the same order in which they are at first explained; beginning with the units.
It can now easily be made clear what is the rise of the cipher in written arithmetic. Let the pupil first lay down twenty-one in counters; two tens and a unit. But, ask him how will he express two tens and no unit? There must be a sign to show that there is no unit. I took, for this purpose, small round white pieces of pasteboard, which I put wherever there was no figure, whether in the place of units, tens, hundreds, &c. If it be required to lay down 302, the child placed two ones, a cipher for no tens, and three hundreds.
The orderly placing of the counters, the reading off of the number, and the writing of it should proceed together. If there are several pupils, there may be a division of labor; some laying down counters and others writing, and then each reading off the work of the other.
In this way the children will gain a knowledge of the decimal system, and of the profound wisdom with which the ancient Hindoos arranged their figures by it. But the counters can be further used in explaining the ground-rules, espe
* It would be well to have 1, 10, 10), 1000 printed on the counters; and on the other side 1, X, C, M, according to their value.
* The Roman letters on the counters can be easily used so as to show the value of a figure, one, for instance, in different places.
;It was not the Arabs, but the Hindoos-as was already stated—who invented the decimal
cially addition and multiplication, Under the columns of counters lay a rule, for the line, ander which to place the sum. If the units add up to 12, change ten of them for a ten, put it with the column of tens, and put the remainder of 2 under the units, and so on. When with the aid of the counters the children have learned to count, the decimal system, writing and reading figures, and a more or less clear knowledge of the four ground-rules, the counters should be gradually disused.* They might be afterward used again in explaining decimal fractions.
EXPLANATION OF THE USUAL ABBREVIATED PROCESSES IN WRITTEN ARITHMETIC.--I will illustrate by a few examples what is said in the text of the means by which our teachers may endeavor to explain written multiplication and division. For instance, the example in multiplication, 6x11356, may be worked in three different ways, as follows:
The first, a., is the common abbreviated form; b. and c. give the solution at length, as it ought to und must be worked, before the abbreviated mode. For the solution of c., we will suppose a case. Six brothers inherit each 11356 florins, What is the entire sum? The multiplicand consists of one ten thousand, one thousand, &c., down to six units. Each heir will have one ten thousand, in all sixty thousand; also one thousand, in all six thousand; and so on; lastly, each will receive sis units, in all thirty-six. Add these products together, and you will have 68136. The example b. is entirely similar to example c., except that here the multiplication begins with the units, as in the abbreviated mode. The latter will become clear by comparison with b. It will readily be seen that the abbreviation consists in this : that the product of each separate place is not written down in full; but that, when for instance the product of the ones furnishes tens, they are kept in mind and added to the product of the tens, &c.; so that the additions in example b. are performed in the mind. Thus, 6X6=36=3 tens and 6 units, which last are put in the units' place in the product. Then, 6x5 tens=30 tens, which with 3 tens from the first product makes 33 tens, or 3 hundred and three ones, which remainder put in the tens' place in the product; and so on.
The pupil can thus be shown that the abbreviated operation in example a. must begin from the lowest place, so that the overplus from each place may be carried to a higher. system and the wrongly.named Arabic figures. What other mathematical discovery can be compared with this? See Whewell, I., 191.
*In the arithmetics of Diesterweg, Stern, &c., other modes of making numbers visible are used. As to counters, the question is, whether they can be used in schools for a large number of pupils. Herr Ebersberger, of the Altorf Seminary, advises to fit up a large blackboard with parallel horizontal ledges or gutters of tin, in which large counters may be set up, as letters, &c., are set up in using the board to teach reading, &c. Dr. Mager remarks, in his treatise "On the Method in Mathematics," (Ueber die Method der Mathematik,) that he has used counters in teaching. He says, (p. xviii.,) " The second stage brings in the decimal sys. tem, first with counters and then with figures. The smallest counters represent units, a larg. er size tens, the largest hundreds. It is a pleasure to see how the children can use the count. ers to add, multiply, subtract, and divide. When they can work both with counters and mentally, nothing is easier thall to work the same problems in figures; the greater convenience of the written method induces the children to learn it quickly.”