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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 in 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 multiplication. 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, 1, 4, &c. and 3, 1, 4, &c. of numbers, which are exactly visible into these parts. This is only an extension of the principle of fractions, 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 one half, one third, one fourth, &c.; and also, knowing 3, 4, 3, &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 examples 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 dif ference between them.
every example which he performs, he is obliged to go through complete demonstration of the principle by which he does it; and at the same time he does it in the simplest way possible. These observations apply to the remaining part of the book.
These principles are sufficient to enable the pupil to perform 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 usually occur. There are none, however, which the principles 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 most pupils of seven or eight years of age. The operations do not differ materially from those in the preceding sections. There 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 contained in this book, he will be able to perform all examples, in which the numbers are so small, that the operations may be performed in the mind. Afterwards he has only to learn the application of figures to these operations, and his knowledge of arithmetic will be complete.
The Rule of Three, 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 degree of certainty.
The following are some of the principal difficulties which a child has to encounter in learning arithmetic, in the usual 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 examples. Abstract numbers, and the operations upon them, must 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 from 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 dif ficult to be understood at first, than the numbers written at length in words. But they are not used merely as words, they require operations peculiar to themselves. 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. consequence is, that he learns only one of all these things, and that is, how to perform these operations on figures. He can 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 not 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 consequently 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 natural 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. The pupil learns a rule, which, to the man that made it, was a general principle; but with respect to him, and often times to the instructer himself, it is so far from 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 between it and them, either in the sound, or in something else. If he is fortunate enough to discover 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 his rules, 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 some will find the right course themselves, whatever obstacles be thrown in their way. But they apply to the greater part; and it is probable that there are very few who have not experienced more or less inconvenience from this mode of proceeding. Almost all, who have ever fully understood arithmetic, have been obliged to learn it over again in their own way. And it is not too bold an assertion to say, that no man ever actually learned mathematics in any other method, than by analytic induction; that is, by learning the principles by the examples he performs; and not by learning principles first, and then discovering by them how the examples are to be performed.
In forming and arranging the several combinations the author has received considerable assistance from the system of Pestalozzi. He has not however had an opportunity of seeing Pestalozzi's own work on this subject, but only a brief outline of it by another. The plates also are from Pestalozzi. In selecting and arranging the examples to illustrate these combinations, 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
Sir,-This will be delivered to you by my son Samuel, whom I beg leave to commit to your care, hoping that by your well-known skin and attention you will be able to make something of him; which, I am sorry to say, none of his masters have hitherto done. He is now eleven, and yet can do nothing but read his mother tongue, and that but 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 would do nothing in geography for want of memory. In short, if
be 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 him, 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 youth looks like. I suppose he is a linman 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 do not be afraid. Nobody will hurt you. How old are you? Eleven last May, sir.
A well-grown boy of your age, indeed. You love play, I dare say.
What, are you a good hand at marbles?
Pretty good, sir.
And can spin a top and drive a hoop, I suppose?
Then you have the full use of your hands and fingers?
Can you write, Samuel?
I learned it a little, sir, but I left it off again
And why so?
Because I could not make the letters.
No! Why, how do you think other boys do? Have they more fingers wil you?
you not able to hold a pen as well as a marble?
nuel was silent.
et me look at your hand.
Samuel held out both his paws, like a dancing bear.
see nothing here to hinder you from writing as well as any boy in ..e school. You can read, I suppose ?
Tell me then what is written over the school-room door.
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 know any thing of the Latin Grammar?
Have you never learned it?
I tried, sir, but I could not get it by heart.
Why, you can say some things by heart. I dare say you can tell me the names of the days of the week in their order.