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As soon as a child begins to use his senses, nature continually presents to his eyes a variety of objects; and one of the first properties which he discovers, is the relation of number. He intuitively fixes upon unity as a measure, and from this he forms the idea of more and less; which is the idea of quantity.
The names of a few of the first numbers are usually learned very early; and children frequently learn to count as far as a hundred before they learn their letters.
As soon as children have the idea of more and less, and the names of a few of the first numbers, they are able to make small calculations. And this we see them do every day about their playthings, and about the little affairs which they are called upon to attend to. The idea of more and less implies addition; hence they will often perform these operations without any previous instruction. If, for example, one child has three apples, and another five, they will readily tell how many they both have; and how many one has more than the other. If a child be requested to bring three apples for each person in the room, he will calculate very readily how many to bring, if the number does not exceed those he has learnt. Again, if a child be requested to divide a number of apples among a certain number of persons, he will contrive a way to do it, and will tell how many each must have. The method which children take to do these things, though always correct, is not always the most expeditious.
The fondness which children usually manifest for these exercises, and the facility with which they perform them; seem to indicate that the science, of numbers, to a certain extent, should be among the first lessons taught to them.*
To succeed in this, however, it is necessary rather to furnish occasions for them to exercise their own skill in performing examples, than to give them rules. They should be allowed to pursue their own method first, and then they should be made to observe and explain it, and if it was not
* See on this subject two essays, entitled Juvenile Studies, in the Prize Book of the Latin school, Nos. I and II. Published by Cummings & Hilliard, 1820 and 1821.
the best, some improvement should be suggested. By fol lowing 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 assist ed 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 eight 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 so.*
* It is remarkable, that a child, although he is able to perform a variety 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 find 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
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. Three. 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 much 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 eight.
We shall be further convinced of this if we observe that the same table serves for addition and subtraction; and another table which is 1*
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 inust, 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,}, 4, &c. and }, 4, 3, &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 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 arc 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 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 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