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tered according to Act of Congress in the year 1849, by
TEMPERANCE C. COLBURN, Widow of Warren Colburn,

In the Clerk's Office of the District Court of the District of Massachusetts.

Entered according to Act of Congress in the year 1858, by

TEMPERANCE C. COLBURN, Widow of Warren Colburn.

In the Clerk's Office of the District Court of the District of Massachusetts.

ADVERTISEMENT TO THE NEW EDITION.

THE character of Colburn'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 new 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, become the common possession of all intelligent teachers.

In issuing this edition of the Intellectual Arithmetic, it has been thought desirable to make an addition in the form of a short introduction to Written Arithmetic, consisting chiefly of examples for solution upon the slate.

As this introduction is intended for young children, and to be used in connection with the Mental Arithmetic, the expla nation of the principles of the operations has been left chiefly to the teacher.

Young children understand oral much more readily than they do written instruction. For this reason, long explanations and the use of definitions and terms have been avoided as much as possible.

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WARREN COLBURN.

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INTRODUCTION.

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 mean. ing of each numerical word by using it in connection with objects.

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The kind of objects to be employed as counters should, of course, be similar, 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 ten should be used or exhibited to the children in the first few lessons.

LESSON I.

Let the class have their attention called to the teacher; and when he lays 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.

After going through this addition a few times in this form, it may be varied 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 three, three and one 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 he.

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. Having 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, &c. 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 up 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 sufficient power of attention to go over with profit. The same remark may be made respecting the following Introductory Lessons.

LESSON II.

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Let the teacher call the attention of the class, and require them to count, and then lay down, one by one, a small number of counters, say, for example, 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 number into different parts, as two and three, three and two, four and one, one and one and three, &c., requiring the class with each division to name the parts and make the addition. 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:

•; &c.

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 converted into a question in subtraction; thus five and three are eight; then, having 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; "One," Teacher: Class; "Four." "Thoo.". Teacher:

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"Three." Class: "Three.". "Five," &c.

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This exercise should not be pressed too fast, but carried on grad. nally as the pupil's strength of mind will allow. Special pains should be taken that the number ten be perfectly mastered in this form of combining its parts. This will give the pupil the most important aid in all his calculations in larger numbers.

LESSON III.

For a number of days after beginning the above exercises, the child should not have the book at all in his hands. If the child has the book in his possession, it will be well for the teacher to take it for a few days, and let the pupil employ himself at his seat in writing on a slate, or with other books. In this way the child has awakened within him the idea of calculation in numbers, without having become wearied with the reading of what excites no interest. After a few days, however, the book may be put into the pupil's hands, and he may be directed to get a lesson in Section I. In the meanwhile the Introductory Lessons should be continued, and form a part of each day's exercise till they are finished. In this way, the pupil, in studying his first lesson from the book, will already have learned the use of counters, and will naturally resort to them at his seat, using beans or marks on his slate for this purpose. It will be far better for him to come to the use of counters in this natural way, than to be enjoined to use them before he has been interested in witnessing their application.

The pupil, in the preceding lessons, has become acquainted with all the numbers as far as ten, regarding them either as units, or as grouped into parts of a larger whole. The next step is to carry him through the numbers from ten to twenty.

First, let the class count with the objects before them from one up to twenty; then, removing all but ten, let the ten be grouped in a pile or if they are marks on the board, let them be enclosed by a line drawn around them, and begin to count upward from ten, "One and ten are eleven; two and ten are twelve; three and ten are thirteen ;". here pause, and examine the composition of the word, thirteen-three ten, or three and ten. Show how the three is spelt in thirteen, and also how the ten is spelt. Then proceed, "four and ten are fourteen," examining the word as in the former case; "five and ten are fifteen; six and ten are," perhaps some one in the class will now be able to give the compound word then go on, 66 seven and ten, eight and ten, nine and ten, ten and

ten." When they can give the compound words readily from the simple ones, then give them the compound word, and let the class separate it into its two component words; thus: Teacher: "SevClass: "Seven and ten," &c. Thus far let the teacher be careful to present the name of the smaller of the two numbers first, for that is the order in which the compound word presents them; let the teacher say four and ten, and not ten and four.

enteen."

After the class have caught the analogy between the simple words, and the compounds which they form, so that one instantly suggests to them the other, then the order of the words may be changed, and the ten put first. The caution here suggested may seem to some unnecessary; but a careful observation of the mental habits of children will not fail, I think, to show its importance.

In the analysis of the compound words from ten to twenty, eleven and twelve should be omitted till the last; for as the simple words of which they are formed are disguised or obsolete, they tend to obscure, rather than elucidate the subject to the mind of a child. Having obtained the idea through the other words in the series, he may take the statement respecting these on trust.

LESSON IV.

Having counted twenty, and grouped the number in two tens, let the class count ten more, making in all thirty, or three tens. Keeping the tens separate, let the class count ten more, making forty, or four tens. Let the class then answer such questions as the following: Twenty are how many tens? Thirty are how many tens? Forty are how many tens? Four tens are what number? Three tens are what number? Two tens are what number?

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After this, they may proceed with the higher multiples of ten, fifty, sixty, seventy, eighty, ninety, a hundred.

Through the whole of this exercise, each multiple of ten should be presented in groups of ten, so as to aid the idea by the visible representation.

The pupils should be led to see the significancy of each numer. ical name; that thirty-seven, for example, means three tens aud seven; fifty-six means five tens and six.

In this way the pupils may be led to understand the Decimal Ratio, at this early stage, and no further trouble need be taken in that direction. When in a later stage of study, he comes to the Decimal notation in written Arithmetic, he will find it only a natural mode of expressing ideas already rendered familiar in practice.

LESSON V.

Let the teacher stand at the board, and call the attention of the class to what he shall write; then, making two marks, ask, "How many marks on the board?" When the class have answered, let the teacher write two more, and ask, "How many now?" and so on to the number of twelve or more. Then take a writing book or sheet of paper, and covering all but two of the marks, let the class repeat the same process while the teacher re

moves the book, so as to bring two more into view at each remove; the numbers read by the class being two, four, six, eight, ten, &c.

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