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The use of the plates is explained in the Key at the end of the book. Several examples in each section are performed ju the Key, to show the method of solving them. No answers are given in the book, except where it is necessary to explain something to the pupil. Most of the explanations are given in the Key; because pupils generally will not understand any explanation given in a book, especially at so early an age. The instructer must, therefore, give the explanation viva voce. These, however, will occupy the instructer but a very short time.
The first section contains addition and subtraction, the second niultiplication. The third section contains division. In this section the pupil learns the first principles of fractions and the terms which are applied to them. This is done by making him observe that one is the half of two, the third of three, the fourth of four, &c. and that two is two thirds of three, two fourths of four, two fifths of five, &c.
The fourth section commences with multiplication. In this the pupil is taught to repeat a number a certain number of times, and a part of another time. In the second part of this section the pupil is taught to change a certain number of twos into threes, threes into fours, &c.
In the fifth section the pupil is taught to find , , \, &c. and š, 4, 6, &c. of numbers, which are exactly divisible into these parts. This is only an extension of the principle of frac. tions, which is contained in the third section.
In the sixth section the pupil learns to tell of what number any number, as 2, 3, 4, &c. is cne half, one third, one fourth, &c.; and also, knowing 3, 4, s, &c. of a number, to find that number.
These combinations contain all the most common and most useful operations of vulgar fractions. But being applied only to numbers which are exactly divisible into these fractional parts, the pupil will observe no principles but multiplication and division, unless he is told of it. In fact, fractions contain no other principle. The exam.ples are so arranged, that almost any child of six or seven years old will readily comprehend them. And the questions are asked in such a manner, that, if the instructer pursues the method explained in the Key, it will be almost impossible for the pupil to perform any example without understanding the reason of it. Indeed, in
formed by addition, serves both for multiplication and division. In this treatise the same plate serves for the four operations.
This remark shows the necessity of making the pupil attend to his manner of performing the examples, and of explaining to him the difference between them.
every example which he perforins, he is obliged to go through a complete demonstration of the principle by, which he does it; and at the same time he does it in the simplest war. possible. These observations apply to the remaining part of ihe book.
These principles are sufficient to enable the pupil to per. fornı almost all kinds of examples that ever occur. He will not, however, be able to solve questions in which it is necessary to take fractional parts of unity, though the principles are the same.
After section sixth, there is a collection of miscellaneous examples, in which are contained almost all the kinds that Ostially occur. There are none, however, which the princi pies explained are not sufficient to solve.
In section eight and the following, fractions of unity are explained, and, it is believed, so simply as to be intelligible to ulost pupils of seven or eight years of age. The operations lo not differ materially from those in the preceding sections l'here are some operations, however, peculiar to fractions. The two last plates are used to illustrate fractions.
When the pupil is made familiar with all the principles con tained in this book, he will be able to perform all examples in wuch the numbers are so small, that the operations inay be · performed in the mind Afterwards has only to learn the
izpplication of figures to these operations, and his knowledge of arithmetic will he complete.
The Rule of Thrce, and all the other rules which are usually contained in our arithmetics, will be found useless. The examples under these rules will be performed upon general principles with much greater facility, and with a greater de gree of certainty.
The following are some of the principal difficulties whic!! a child has to encounter in learning arithmetic in the rsua! way, and which are seldom overcome First, the examples are so large, that the pupil can form no conception of the numbers themselves; therefore it is impossible for him to comprehend the reasoning upon them. " Secondly, the first examples are usually abstract numbers. This increases the difficulty very much, for even if the numbers were so small, that the pupil could comprehend them, he would discover but very little connexion between them and practical cxamples. Abstract numbers, and the operations upon them, inust be learned from practical examples; there is no such thing as deriving practical examples from those which are abstract, unless the abstract have been first derived fiom those which are practical. Thirdly, the numbers are expressed by figures, which, if they were used only as a con
tracted way of writing numbers, would be much more difficult to be understood at first than the numbers written at length in words. But they are not used merely as words; they require oporations peculiar to theniselves. They are, in fact, a new language, which the pupil has to learn The pupil, therefore, when he commences arithmetic, is presented with a set of abstract numbers, written with figures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures signify, and what is meant by addition, subtraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is, how to perform these operations on figures. perhaps translate the figures into words, but this is useless since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has noi the least conception
After the abstract examples a few practical examples are usually given ; but these again are so large that the pupil cannot reason upon them, and consequenly he could not tell whether he must add, subtract, multiply, or divide, even if he had an adequate idea of what these operations
The common method, therefore, entirely reverses the natu ral process; for the pupil is expected to learn general principles, before he has obtained the particular ideas of which they are composed.
The usual mode of proceeding is as follows. learns a rule, which, to the man that made it, was a genera. principle; but with respect to him, and oftentimes to the instructer himself, it is so far froin it, that it hardly deserves to be called even a mechanical principle. He performs the examples, and makes the answers agree with those in the book, and so presumes they are right. He is soon able to do this with considerable facility, and is then supposed to be master of the rule. He is next to apply his rule to practical examples, but if he did not find the examples under the rule, he would never so much as mistrust they belonged to it. But finding them there, he applies his rule to them, and obtains the answers, which are in the book, and this satisfies him that they are right. In this manner he proceeds from rule to rule through the book.
When an example is proposed to him, which is not in the book, his sagacity is exercised, not in discovering the operations necessary to solve it; but in comparing it with the examples which he has performed before, and endeavouring to dis.
cover some analogy betweon it and them, either in the sound or in something else. If he is fortunate enough to discovo any such analogy, he finds what rule to apply, and if he has not been deceived in tracing the analogy, he will probably solve the question. His knowledge of the principles of hi: rule is so imperfect, that he would never discover to which of them the example belongs, if he did not trace it by some analogy, to the examples which he had found under it.
These observations do not apply equally to all; for som: will find the right course themselves, whatever obstacles by: thrown in their way. But they apply to the greater part and it is probable that there are very few who have not exp rienced more or less inconvenience from this mode of procees: ing. Almost all, who have ever fully understood arithmeti have been obliged to learn it over again in their own wa.. And it is not too bold an assertion to say, that no man ever. actually learned mathematics in any other method, thanky analytic induction; that is, by learning the principles by the examples he performs; and not by learning principles firs, and then discovering by them how the examples are to Wi performed.
In forming and arranging the several combinations, th: author has received considerable assistance from the system ng Pestalozzi. He has not, however, had an opportunity of seeing Pestalozzi's own work on this subject, but only a brief outlina of it by another. The plates also are from Pestalozzi. ? selecting and arranging the examples to illustrate these coni: binations, and in the manner of solving questions generally, he has received no assistance from Pestalozzi.
THE BOY WITHOUT A GENIUS.
Mr. Wiseman, the schoolmaster, at the end of his sum mer vacation, received a new scholar with the following letter :
Sir,—This will be delivered to you by my son Samuel, whom I brie leave to commit to your care, hoping that, by your well-known skilt and attention, you will be able to make something of him, which, I am sorry to say, none of his masters have hitherto dope. He is now elev en, and yet can do nothing but read bis mother tongue, and that bus indifferently. We sent him at seven to a grammar school in our neighbourhood; but his master soon found that his genius was not turned to learning languages. He was then put to writing, but he set about it so awkwardly that he made nothing of it. He was tried at accounts, but it appeared that he had no genius for that either He could do nothing in geography for want of memory. In sbort, it
he has any genius at all, it does not yet show itself. But I trust to your experience, in cases of this nature, to discover what he is fit for, and to instruct him accordingly.. I beg to be favoured shortly with your opinion about hiin, and remain, sir,
Your most obedient servant,
When Mr. Wiseman had read this letter, he shook his head, and said to his assistant, A pretty subject they have sent us here! a lad that has a great genius for nothing at all. But perhaps my friend Mr. Acres expects that a boy should show a genius for a thing before he knows any thing about it—no uncommon error! Let us see, however, what the you'h looks like. I suppose he is a human creature at least.
Master Samuel Acres was now called in. He came hanging down his head, and looking as if he was going to be flogged.
Come hither, my dear! said Mr. Wiseman. Stand by me, and so not be afraid. Nobody will burt you. How old are you ?
Eleven last May, sic.
No! Why, how do you think other boys do ? Have they more fingers than you ?
I see nothing here to hinder you from writing as well as any boy in the school. You can read, I suppose ?
Samuel with some hesitation read, WHATEVER MAN HAS DONE MAN MAY DO.
Pray how did you learn to read ?-Was it not with taking pains ?
Well-taking more pains will enable you to read better. Do you
Why, you can say some things by heart. I dare say you can tell me the names of the days of the weeks in their order.