When placed between two numbers it denotes that the number before it is to be diminished by the number after it. Thus 7 4, shows that 7 is to be diminished by 4, and is read 7 minus 4, or 7 less 4.

§ 91. The principal sign of DIVISION is a short horizontal line placed between two dots, thus, and shows that the number before it is to be divided by the number after it. Thus 246 shows that 24 is to be divided by 6, and is read 24 divided by six. Division is also, very often, expressed by writing the divisor or number to divide by, under the dividend or number to be divided, in the form of a fraction, thus 24; and signifies the same as 246, and is read in

6

the same manner.

or a pa

§ 92. A line, called a vinculum renthesis (), is used to denote that the same operation is to be performed on two or more numbers. Thus 3 + 7 × 5, or (3 + 7) × 5, shows that the sum of 3 and 7 is to be multiplied by 5. If the expression was 3 + 7 × 5 it would denote that 7 only was to be multiplied by 5, and the product added to 3.

§ 93. Example 1. What whole number is equivalent to the following: 3+ 18 X 11

12 - 5 X 9 + 1 + (3 + 4) X 7?

54 ÷ 3+

2.5 X 615 1+ 356 X 4 + 12 + 16
8 + 3 − 1 + 14
1 + 14 ÷ 5 × 4 + 1/4
X

7

how many? 3. 13. X 15+244 18 10+ 201÷÷11 + 3 X 12 + 18+ 7 4+24+ 36 = how many?

96

8

4. 14

3 + 33 X 5 + 88 × 2 + 60 111 +

(3 × 4) + 18

11 +222 how many?

5. What is the difference between 369 +17

and 36 X 9 - 17?

6. What is the sum of 95, 84, 73, 62, 51, 40, 39, 28, and 17?

7. State the sum of the following numbers, viz: 13, 24, 35, 46, 57, 68, 79, 80, and 91.

8. What is the sum of all the numbers from 1 to 60, inclusive?

9. What is the difference between all the numbers from 40 to 50, inclusive, and 50 to 60, inclusive?

10. What is the sum of all the numbers from 1 to 100, inclusive?

PART II.

TO TEACHERS-MODE OF CONDUCTING CLASSES.

NOTHING is more ungenerous in an author, than to fix arbitrary modes of solving any difficulties that may be interwoven into the subject being treated; for no one has a right to resort to any mode of demonstrating truths, that he does not clearly comprehend.

A number of excellent modes of teaching Intellectual Arithmetic are in almost daily use among a large number of teachers, and some few have been given to the world. It is no part of my design to dictate any prescribed rules at this time for teaching this science, but simply to point to a few of the most common errors that have fallen under my observation, which every intelligent teacher will seek to avoid; and with a few general hints on better modes, leave the subject with you to modify and improve as circumstances shall seem to dictate.

ERRORS IN TEACHING MENTAL ARITHMETIC.

1st Error. The practice of allowing each scholar to pursue the study on his "own hook," without giving recitations, and receiving no instruction from his teacher, save now and then the solution of a problem, without knowing the " why and wherefore" of a single point in any solution, is a prominent evil with many instructors. This mode of teaching, or rather want of teaching, has been justly reckoned as belonging to days gone by; and, says Prof. Thomson, "it is prima facie evidence that those who practise it, are behind the spirit of the times."

2nd-Another error is, allowing pupils to use the slate and pencil, which are emphatically not the instruments for procuring a knowledge of Intellectual Arithmetic. Teachers allowing this error a place, are worthy only of being likened to the first transgressors.

3rd.-A disposition to discourage inquiry on the part of pupils, is another crying evil that many teachers are guilty of. That we may be forgiven many sins is true; whether he who breaks down the rising energies of an inquiring, youthful mind, is a question.

These are perhaps the most glaring errors, that need a rebuke in this place. We pass to notice a very few errors in the language of solutions. Avoid such expressions as the following, viz:

1. 3 times 15 are 45, 4 times 6 are 24, &c. The verb in these and similar cases is singular, and should have the singular form. (See Bullion's Analytical and Practical Grammar, page 39; also Webb's 2nd Reader, page 20.)

2. Three is the one third of three times itself; for, three is one third of three times three.

3. In questions like those to be found in Sec. 22 of this work, (example 22 for instance,) most teachers allow the question to be repeated in the midst of the solution. This not only weakens the power of the problem in promoting mental discipline, but at the same time mars the beauty of the solution. (See Sec. 22, Example 22, SOLUTION.)

4. Many teachers, desirous of having their pupils "show off" well, are in the habit of drawing the solutions of the questions out of them by littles, taking care to tell the pupil the answer to the question before asking it. This practice cannot be too strongly condemned. For an illustration of this mode of teaching, see Page's Theory and Practice of Teaching," under head of "DRAWING OUT PROCESS."

5. A fifth error is that of passing over the questions, requiring the answer simply, without a single reason for it-without the least shadow of a demonstration.

6. The sixth and last error we shall give in this connexion, is that of allowing pupils to leave one subject and pass to another, before the first is understood and learned. The practice of this error has probably made more superficial students, than all the others combined. Parents and teachers cannot too carefully avoid it.

SUGGESTIONS ON THE BEST MODE OF TEACHING ARITHMETIC.

In order for a person to become a successful teacher of Arithmetic, as well as any other branch of science, he must have a thorough knowledge, both of the general principles, and of the different modes of illustrating those principles to the young. To secure both of these means, he must be possessed of a natural love for the branch.

Pupils of nearly the same degree of advancement, should be formed into a class--and regular lessons assigned, to be recited each day at some particular hour. The advantages of having a school properly classified, are numerous and important. 1. It is a great saving of time to the teacher.

2. A more powerful stimulant to exertion can be awakened, and explanations given to the whole class as easily, as to each individual separately.

3. As much, and oftentimes much more, is learned in thirty minutes on the recitation seat, as could be learned by the pu pil, apart from such recitation, in whole hours, and perhaps days. By hearing the different modes of solution, the teacher himself often gets new modes of illustration, which may be of great value to him in the future practice of his profession.

The prime object in every recitation, is to bring something to the minds of the pupils that they have not found in the book, and which will not only give the class increased confidence in the teacher, but will at the same time secure the attention of every member of the class.

Every question and example should be fully and clearly analyzed, the reason for every step explained, until such time as each scholar is able to go through with the analysis without difficulty or mistake.

Whatever may be the condition of a school, the motto of all teachers should be perfection; and unless this motto is lived up to, they must unavoidably fail of success.

Every pupil, however young, should be fully impressed with the idea that he must depend upon himself. He must do his own learning. Whenever it becomes necessary to render any scholar assistance, it should be done indirectly, and not directly. The teacher's business is not to do the work of the pupil, but to teach him how to do it.

BOOKS IN RECITATION.

In the recitations, the pupils should generally be allowed to have their books open before them, the readiness and accuracy with which they perform the examples, being made the test of qualification.

MODES OF SOLUTION.

The forms of solution given, are not designed to be used in all cases, but may at any time be discarded if a better mode can be given to supply its place. The language of the solutions used should, in general, be uniform, in the same school. A pupil should in no case be allowed a form of solution that he does not himself clearly comprehend.

The tables should be recited with the books closed. If more questions are desirable than are contained in the several sections, they can be added under each head by the teacher.

Note. A smaller work for children is in process of preparation, and will be ready for the press soon; which, it is hoped, taken in connexion with this, will render pupils who pursue them systematically and thoroughly, fully prepared to enter the study of written arithmetic, and gain a speedy and exten sive knowledge of it without difficulty.

An Elementary and Practical Arithmetic, in which have been made VARIOUS IMPROVEMENTS in arrangement and nomenclature, as well as the means of securing THOROUGH DISCIPLINE in the principles and applications of the science, by JAMES B. DODD, A. M., of Transylvania University, Ky., is designed to follow this work.