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the best, some improvement should be suggested. By following this mode, and making the examples gradually increase in difficulty ; experience proves, that, at an early age, children may be taught a great variety of the most useful combinations of numbers.
Few exercises strengthen and mature the mind so much as arithmetical calculations, if the examples are made sufficiently simple to be understood by the pupil ; because a regular, though simple process of reasoning is requisite to perform them, and the results are attended with certainty.
The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to all things with which we are acquainted, we obtain an abstract idea of number. We first make calculations about sensible objects; and we soon observe, that the same calculations will apply to things very dissimilar; and finally, that they may be made without reference to any particular things. Hence from particulars, we establish general principles, which serve as the basis of our reasonings, and enable us to proceed step by step, from the most simple to the more complex operations. It appears, therefore, that mathematical reasoning proceeds as much upon the principle of analytic induction, as that of any other science.
Examples of any kind upon abstract numbers, are of very little use, until the learner has discovered the principle from practical examples. They are more difficult in themselves, for the learner' does not see their use; and therefore does not so readily understand the question. But questions of a practical kind, if judiciously chosen, show at once what the combination is, and what is to be effected by it. Hence the pupil will much more readily discover the means by which the result is to be obtained. The mind is also greatly assisted in the operations by reference to sensible objects. When the pupil learns a new combination by means of abstract examples, it very seldom happens that he understands practical examples more easily for it, because he does not discover the connexion, until he has performed several practical examples and begins to generalize them.
After the pupil comprehends an operation, abstract examples are useful, to exercise him, and make him familiar with it. And they serve better to fix the principle, because they teach the learner to generalize.
From the above observations, and from his own experience, the author has been induced to publish this treatise ; in which he has pursued the following plan, which seemed to him the most agreeable to the natural progress of the mind.
GENERAL VIEW OF THE PLAN.
Every combination commences with practical examples. Care has been taken to select such as will aptly illustrate the combination, and assist the imagination of the pupil in performing it. In most instances, immediately after the practical, abstract examples are placed, containing the same numbers and the same operations, that the pupil may the more easily observe the connexion. The instructer should be careful to make the pupil observe the connexion. After these are a few abstract examples, and then practical questions again.
The numbers are small, and the questions so simple, that almost any child of five or six years old is capable of understanding more than half the book, and those of seven or cight years old can understand the whole of it.
The examples are to be performed in the mind, or by means of sensible objects, such as beans, nuts, &c. or by means of the plate at the end of the book. The pupil should first perform the examples in his own way, and then be made to observe and tell how he did them, and why he did them
* It is remarkable, that a child, although he is able to perform a va. riety of examples which involve addition, subtraction, multiplication, and division, recognises no operation but addition. Indeed, if we analyze these operations when we perform them in our minds, we shall tind that they all reduce themselves to addition. They are only different ways of applying the same principle. And it is only when we use an artificial method of performing them, that they take a different form.
If the following questions were proposed to a child, his answers would be, in substance, like those annexed to the questions. How much is five less than eight ? Ans. Threc. Why? because five and three are eight. What is the difference between five and eight? Ans. Three. Why? because five and three are eight. If you divide eight into two parts, such that one of the parts may be five, what will the other be? Ans. Three. Why? because five and three are eight.
How tnuch must you give for four apples at two cents apiece ? Ans. Eight cents. Why? because two and two are four, and two are six, and two are eight
How many apples, at two cents apiece, can you buy for eight cents ? Ans. Four. Why?' because two and two are four, and two are six, and two are eig
We sha!l be further convinced of this if we observe that the same table serves for addition and subtraction; and another table which is
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 vida voce. These, however, will occupy the instructer but a very short
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 }, }, 4, &c. and 3, 1, , &c. of numbers, which are exactly divisible 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 , , , &c. of a number, to find that nuinber.
These combinations contain all the most common and most useful operations of vulgar fractions. But being applied only io 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
förmed 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 performs, 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 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 inay be performed in the mind. Afterwards be 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 de. gree 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 tlie 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 contracted 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 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 are.
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.