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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. The 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 tearns 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 Hook, his sagacity is exercised, not in discovering the operariona necessary to solve it; but in comparing it with the exampoleas which he has performed before and endeavouring to di
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 somo 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 ex. perienced more or less inconvenience from this mode of proceeding Almost all, who have ever fully understood arithL'etic, have been obliged to learn it over again in their own W
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 summer vacation, received a new scholar with the following letter:
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 skill 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 clev. an, 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 ihat he made nothing of it. He was tried at accounts, but it appeared that he had no genius for that either. He could do sorbing the goography for wont
of memory. Loops
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 vour 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 thai 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-10 uncommon error! Let us see, however, what the youth looks like. I suppose he 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 logged.
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.
No! Why, how do you think other boys do ? Have they more fingers than you ?
Are you not able to hold a pen as well as a marble ?
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 ? Yes, sir.
Well-taking more pains will enable you to read better. Do you know any thing of the Latin Grammar ?
Yes, sir, I know them.
And you could probably repeat the names of your brothers and sisters, and all your father's servants, and half the people in the village besides,
I believe I could, sir.
If you were to have a penny a day, what would that make in a week ?
Right. Why here you have been practising the four great roles of arithmetic, addition, subtraction, multiplication, and division. Learning accounts is no more than this. Well, Samuel, I see what you are
I you what able do; but your school-fellows.
Samuel went away, glad that his examination was over, and with more confidence in his powers than he had felt before.
The next day he began business. A boy less than himself was call. ed out to set him a copy of letters, and another was appointed to hear him in grammar. He read a few sentences in English that he could perfectly understand to the master himself. Thus by going on steadily and slowly, he made a sensible progress. He had already joined his letters, got all the declensions perfectly, and half the multiplication table, when Mr. Wiseman thought it time to answer his father's letter ! which he did as follows:
observe, you must do it. We have no it can't Here are now go Camong
I now think it right to give you some information concerning your son. You perhaps expected it sooner, but I always wish to avoid hasty judgments. You mentioned in your letter that it had not yet been discovered which way his genius pointed. If by genius you meant such a decided bent of mind to any one pursuit as will lead to excel with little or no labour or instruction, I must say that I have not net with such a quality in more than three or four boys in my life, and Your son is certainly not among the number. But if you mean
only the ability to do some of those things which the greater part of mean
kind can do when properly taught, I can affirm, that I find in him na peculiar deficiency; And whether you choose to bring him up to trade or to some practical profession, I see no reason to doubt that he may in time become sufficiently qualified for it. It is my favourite maxım, sir, that every thing most valuable in this life may generally be acquired by taking pains for it. Your son has already lost much time in the fruitless expectation of finding out what he would take up of his own accord. Believe me, sir, few boys will take up any thing of their own accord but a top or a marble. I will take care while he is with me Niat he loses no more time this way, but is employed about things that are fit for him, not doubting that we shall find him fit for them.
I am, sir, yours, &c.
Though the doctrine of this letter did not perfectly agree with Mr. Acres' notions, yet being convinced that Mr. Wiseman was more likely to make something of his son than any of his former preceptors, he continued him at his school for some years, and had the satisfaction to find him going on in a steady course of gradual improvement. In due time a profession was chosen for him, which seemed to suit his temper and talents, but for which he had no particular turn, having rever thought at all about it. He made a respectable figure in it, and wen. through the world with credit and usefulness, though without a germing