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Intered according to Act of Congress, in the year 1897, by
WILLIAM J. REYNOLDB, in the Clerk's Ofice of the District Court of the District of Massachusetts
ADVERTISEMENT TO THE REVISED EDITION.
THB character of Cofàurn's First Lessons is too widely and thoroughly known to make it necessary to give, in this edition, any extended statement of its principles and method. Ideas which were now at the first publication of this work have now, through the “ great change” that has taken place in elementary instruction in Arithmetic, through its influence, beeome the common possession of all intelligent teachers.
The careful revision of the work which has now been made has suggested very low points in which any change seemed to be required. It has been thought that & moro easy and gradual introduction would render the work more useful to the most youthful beginners.*
The use of the book with beginners demands of the teacher considerable labor in the way of proposing original questions, and devising modes of illustration ; and a short course of Introductory Lessons is prefixed, which the teacher may use as materials and hints in the first steps of the study.
* In the city of Lowell, where this book has been used from its first pubication, the School Committee passed & vote in December, 1846, excluding all other Arithmetics in their Primary Schools; thus showing, in the opinion of intelligent men who acted upon their experience, that Colburn's Pirst Lessons is sufficiently easy for the most javenile scholar
GENERAL VIEW OF THE PLAN.
EVERY combination commences with practical examples. Care ha been taken to s lert such as will aptly illustrate the combination, and assist the imagination f the pupil in performing it. In most instances, immediately after the practical, abstract examples are placed, containing the same pumbers and the same operations, that the pupil may the more easily obsandr the connection. The instructer should be careful to make the pupil observo the connection. 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. The pupil should first perform the examamples in his own way, and then be made to observe and tell how he did them, and why he did them so." *
Several examples in each section are performed in the Key, to show tho method of solving them. No answers are given in the book, except where
* It is remarkable, that a child, although he is able to perform a variety of examples which involve addition, subtraction, multiplication, and divislon, recognizes 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 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. 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? Be. cause 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 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 man. per of performing the examples, and of explaining to him the differenog between thom.
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 ex planation given in a book, especially at so early an age. The instructor must, therefore, give the explanation viva voce. These, however, will 00cupy the instructer but a very short time.
The first section contains addition and subtraction; the second, multiplica tion. 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 commen 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, 1, &c., and 4+4,5, &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 ind 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 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 enablo 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 fractrons.
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 first instructions given to the child in Arithmetic are usually given on the supposition that the child is already able to count. This indeed seems a sufficiently low requisition; and if children were taught to count at home in a proper manner, they would have this power in a sufficient degree when they enter the primary school. But it will be found on trial that most children, when they begin to go to school, do not know well how to count. This may be proved by requiring them to count 20 beans or kernels of corn. Few of them will do it without mistake. The difficulty is they have been taught to repeat the numerical names, one, two, three, in order, without attaching ideas to them. They learn to count without counting things. This point then calls for the teacher's first attention to lead the child to apprehend the meaning of each numerical word by using it in connection with objects.
The kind of objects to be employed as counters should of course be simi. lar, as marks on the blackboard, beans, pieces of wood, or of cork, or the balls in a numeration frame. Provided they are similar, and large enough to be seen without effort by all the class, it is of little consequence what they are; the simpler the better, and those which the teacher devises or makes will, other things being equal, be best of all.* Not more than ton should be used or exhibited to the children in the first few lessons.
Let the class have their attention called to the teacher; and when he laye down a counter, when all can see it, let them say one; let the teacher lay down another, and the class say two ; and so on up to ten. If any of the class become inattentive, let the teacher stop at once; and, after the attention is fully centred on him, let him begin again.
* The Numeration Frame should have ten balls on & bar. Three bars will be sufficient for all the necessary illustrations. It is some posed to employ a Frame with only nine balls on a bar: the use of such a Frame, however, would be a great error in the First Lessons of Mental Arithmetic. The Frame with nine balls is designed to illustrate the idea of Incal value in the decimal notation, and has as many balls as there are significant figures. But Mental Arithmetic begins with the numerical words, and requires for its illustration on a Frame as many balls as there are simple namerical words. These are the first ten, those above being compound. Eleven is formed of two obsolete words, signifying, one and ten; so twelve 18 & compound of two words, signifying two and ten. The names above these, thirteen, fourteen, &c., sufficiently indicate of themselves the simple words of which they are formed.
After going through this addition a few times in this form, it may be va ried thus. The teacher laying down the counters, one by one, as before, the class may be led to say, one and one are two, two and one are throa, three and ono are four, &c.
The above mode of adding may be shortened by leading the class to say as follows: One and one are two, and one are three, and one are four, &c.
At any time the word designating the counter may be used along with the number, as beans, balls, pieces, marks, or books, as the case may be.
At times it will be well to give some fictitious designation to the counters, such as the teacher, or still better such as some one of the class, may choose, calling them men, sheep, horses, &c.
Next to Addition, as illustrated above, should come Subtraction. Hav. ing counted ten, let the teacher take away one, and the class be made to say, one from ten leaves nine one from nine leaves eight, &o. In Subtraction the same variations may be introduced as in Addition. No further illustrations of this operation need be given, as the teacher's discretion will supply all that is necessary.
In connection with these exercises, let the pupil be taught to repeat in reversed order the numerical words they have employed, counting from one ap to ten, and then in reverse order from ten to one.
It is not to be supposed that the whole of the foregoing lesson can be learned at one exercise. It is only a small part of it that children will at first have suficient power of attention to go over with profit. The same remark may be made respecting the following Introductory Lessons.
Lot the teacher call the attention of the class, and require them to count, and then lay down, one by one, & small number of counters, say, for ex ample, five; then let him separate them into two parts, as one and four, thus, •
• •, and say, one and four are five," and require the class to say the same. Then let him divide the nuniber into different parts, as two and three, three and two, four and one, one and one and three, &c., roquiring the class with each division to name the parts and make the addi. tion. Let them always begin at the left end of the line of counters as they face them. Having exhausted the combinations of five, let the number six be taken, giving combinations like the following:
It may be found that a lower number than five should be made the first step in this exercise.
After the combinations of six have been exhausted, the number seven may be taken, and then successively, eight, nine, and ten.
As a part of this Lesson, each question in addition should be con rerted Into a question in subtraction ; thus, five and three are eight; then, 'naving put the two parts together which make eight, remove the three, and lead the class to say,
" three from eight leaves five. The following exercise is important in this connection. Let the teacher select some number, and give one part of it, and require the class as quick as possible to name the complementary part. Thus let six be the number, the exercise will be as follows. Teacher. “Now attend, six is the number I am going to name one part of it; when you hear me name it, do you all name the other part as quick as you can; now be ready ; five.". Class : * Ons." - Teacher : “ Four." Class : " Thoo." - Teacher: " Thrus.” Olun : "Thou.'-Tenchor : “One." Class : “ Five," &a