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 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 choos8, calling them men, sheep, horses, &c.

Next to Addition, as illustrated above, should come Subtraction. Haying counted ten, let the teacher take away one, and the class be made to Bay, one from ten leaves nine one from nine leaves eight, &o. In Subtraction the same variations may be introduced as in Addition. No furthes 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 re versed order the numerical words they have employed, counting from on 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 b learned at one exercise. It is only a small part of it that children will t first have sufficient power of attention to go over with profit. The same remark may be made respecting the following Introductory Lessons.

• and say,

LESSON II.

Let the teacher call the attention of the class, and require them to couns, and then lay down, one by one, a small number of counters, say, for ex ample, five; then let him separate them into two parts, as one and four, thus, one and four are five," and require the class to say the same. Then let him divide the number into different parts, aa 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:

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 aix 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: Onc."-Teacher: "Four." "Class: "Troo."-Teacher: "Tarcs."

Glasy: "Taroo."Teacher: "Ons." Class: "Five," &o

This exercise should not be pressed too fast, but carried on gradually ad 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 inte 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, "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: "Seventeen." Class: "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 com. pound word presents them; let the teacher say four and ten, and not ten and four. 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, 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 tena, 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 bens? 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?

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 numerical name; that thirty-seven, for example, means three tens and 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 glass repeat the same process while the teacher removes 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. Then let the process be reversed, subtracting two successively, which gives, beginning with sixteen, the following, sixteen, fourteen, twelve, ten, &c.

Again the teacher may say to the class, "When I made those marks, how many did I make at a time?" Class: "Two."- Teacher: "Did f make two more than once?" Class: "Yes, sir, a good many times." Then the teacher, covering up all but two: "Now look, how many times two are there?" Class: "Once." Teacher: "Once two are how many?" Then, after the class have answered, showing two more, "How many times two do you see?" "Twice two are how many?" Then go on In the same way with three twos, four twos, &c., to the end.

At this point the pupils may be taught the distinction between even and odd numbers, and be trained to repeat rapidly the even numbers, from two up to twenty.

The pupils may derive important aid in adding and multiplying, by group ing the numerical names with the voice, in something like the following manner. Teacher: "Listen now to me; one, two-three, four-five, ix. How many twos did I count?" Class: "Three twos." Teacher: "Coux three twos just as I did." Then let the teacher ask, "Three Hue two are

how many?" Then require them to count four twos, five twos, &o. This method may be brought in aid of all their earlier attempts in adding and multiplying.

Care should be taken in this exercise that the words be pronounced in a quick and neat manner, with a distinct pause between the groups. If rightly conducted, this exercise furnishes one of the best aids for overcoming the habit of counting, instead of adding, that fatal clog which keeps back many scholars in arithmetic from making any rapid progress.

The successive additions of two, beginning with one, furnish the series of odd numbers. This and the preceding exercise exhibit all the results made by the additions of the number two.

II eleven. Il nine. seven. five.

II three.

I one.

The pupil should, in connection with this exercise, be taught to group the numerical names by twos, beginning from one, so as to furnish the series of odd numbers, thus: one two, three four, five, &c.

Finally, let him count the even series, up to twenty, and then down, til he can do it readily.

Let him count the odd series up to twenty-one, up and down.

LESSON VI.

To teach the pupils to add the number three, let the teacher make three marks on the board, and require the class to name the number; then three more, and so on. Then, covering all but the lower three, let the class name the number in view, as the teacher exhibits successively three more at Bach remove, or three less.

III fifteen
III twelve
III nine
III six
III three

The pupils may then be led to count in groups of three; thus, one, two, three-four, five, six — seven, eight, nine, &c.

From this they may be readily led to name the multiples of three; three, six, nine, twelve, &c.

III fourteen
III eleven
III eight
III five

The first variation from the above method in the addition of three may be made by beginning with two, which gives the following numbers. In order to secure the necessary amount of repetition in these exercises, and yet to avoid weariness, it will be well often to change the manner of adding. Thus, in the accompanying example, the pupils I two may simply name the series made by the successive additions; as, two, five, eight, eleven, fourteen; or, they may state the process more fully, thus: two and three are five, and three are eight, and three are eleven, &c. A still faller statement of the operation would be thus: two and three are five, five and three are eight, eight and three are eleven, &c. The only remaining variety in adding three will be exhibited by beginning with one, which may be illustrated in the same way as the others, giving the numbers, one, four. coven, isn, &c.

LESSON VII.

After the numbers have been mastered in the manner above suggested they should be combined in a variety of ways by means of marks on the board. The columns of marks subjoined are given merely as examples and hints on this point.

A single column may be made to furnish exercise sufficient for a recitation; for, by adding or erasing a single mark at the beginning, a change is made through the whole.

LESSON VIIL

The addition of the number four may be illustrated in the same way as the preceding numbers. It presents the following variations:

The number four may then be combined with the preceding numbers in a great variety of ways. This will serve as a review of what has gone before, and at the same time will extend the discipline on the new element which the lesson contains.

The same mode of illustration may be pursued in relation to the larger numbers, five, six, &c., if the teacher shall find it necessary.