3. Again, the plan of requiring beginners to illustrate the first steps in numbers by counters, etc., is both simple and efficient in developing their intellectual powers. Abstract explanations, are seldom understood by a child, and therefore fail to strengthen his intellect. Pictorial illustrations excite and amuse his fancy; but it cannot be denied they too often divert his mind from the principle to be illustrated, and leave upon it no sharp, well-defined impression of its truth.

The method here employed is readily understood by the pupil, and is calculated to concentrate his thoughts upon the thing to be done. It also calls his judgment into exercise, and strengthens his memory. The hands help the head. Their agency fixes the attention, and photographs the principle upon the mind.

4. Another advantage of personal illustrations is the pleasing and profitable exercise they afford the pupil. To show by counters or upon the blackboard that five and two are seven, that three times four are twelve, is a great feat for a child; and the thrill of delight he feels at its accomplishment, cannot fail to create an interest in his lessons, and bar the door against idleness and mischief.

5. This method also affords the teacher superior facilities for imparting a knowledge of Numbers to his pupils. Released from the shackles of stereotyped immovable objects and prosy explanations, he is free to choose his tools, free to decide how much and what kind of help to give, and free to determine where to begin and where to stop. Such illustrations are invested with that subtle, mysterious influence of sympathy, which gives them a charm and a propelling power on the progress of pupils which lifeless pictures and written language can never command.

With these brief explanations, and many thanks for the generous reception of his former series, the author submits the work to teachers and the guardians of education, with the hope it may give an impulse to the study of Mental Arithmetic, and prove an auxiliary to intellectual advancement.

NEW YORK, Aug, 1872.

J. B. THOMSON.

SUGGESTIONS.

THE following Suggestions are submitted, not as arbitrary rules, but as practical hints, to be adopted so far only as they meet the wants of the class.

1. The best mode of instruction, is that which most effectually awakens a pupil's attention, and creates in him a determination to educate himself. Any method, therefore, which does not lead him to think and reason, is of little or no value, because it fails to develop the elements of manhood.

2. Beginners in Mental Arithmetic should be furnished with suitable tools, as slates, blackboards, numeral frames, counters, counting boards, etc.,* and be taught to use them in learning to count, and in illustrating the elementary combinations and principles of numbers.

3. After children learn to repeat the names of numbers, particular care should be taken that they comprehend the meaning of the terms; that they have a correct idea of more and less; that eleven, for example, is more than seven and less than thirteen. Their minds should also be carried forward from concrete to abstract numbers, and from particular to their universal application. On being asked the meaning of "three,” a child once showed his third finger-the one which he had been accustomed to count in the third place.

4. Do not attempt to teach too many things at one lesson. The attention of children can be kept active but a short time

*The "Counting Board" is a new and valuable accession to the apparatus of primary schools. Its length depends upon the size of the recitation room, and the number of pupils in the class. When placed against the wall, its width is usually from 15 to 18 inches; the upper surface is divided into parts by distinct marks or strips of wood; the edges are faced with a plain molding, raised sufficiently to prevent the counters from rolling off. If placed in the middle of the room, it should be twice this width, to allow pupils to stand on either side of it.

without fatigue. As soon as they begin to show signs of weariness, the exercise should be closed.

5. The Tables should be carefully illustrated and understood before they are committed to memory. After this the pupil should make them so familiar, that when the combination of any two digits is required, the result shall at once flash upon bis mind.

6. The language employed in verbal explanations should be simple and in point, giving just help enough to lead the pupil to discover the principle in question. More than this is not only useless but hurtful.

7. After a principle has been explained, the pupil should be required to reproduce both the explanation and the principle. This will show whether the point is understood, and if understood, will fix it in his mind.

8. In recitations special pains should be taken to secure the attention of the class, to keep every mind active, and ready to answer any question proposed. Where listlessness or mischief prevails, recitations are useless.

9. Particular attention should also be paid to the method of solving and reciting problems. It is not enough that a pupil gives the right answer. The steps in the reasoning should be legitimate, and the process stated in correct language.

10. It is not advisable to require the same formula of reasoning in every solution. Any form originated by the pupil is admissible, provided it be logical and properly expressed. There is a choice, however, in the modes of analysis, and unquestionably that form is best, which is the most simple, clear, and concise. While latitude is allowable in the modes of analysis, pupils should be encouraged to aim at the best.

II. In the curriculum of studies for primary and intermediate classes, Mental Arithmetic is worthy of a more prominent place than it commonly receives. As an intellectual discipline, it is confessedly unsurpassed by any other branch, and as a preparation for the practical duties of life, it is unequaled. A thorough knowledge of it is a "ready reckoner," which is at once reliable, always at hand, and of universal application.

MENTAL ARITHMETIC.

PRELIMINARY EXERCISES.

LESSON I.

TO TEACHERS.-The following Lessons assume that the class have an imperfect knowledge of counting, and of elementary numbers. The object is to show them practically how many things each number expresses, and impart to them a distinct idea of more and less.

1. How many of these little girls and boys can count? All who can, may hold up a

hand.

2. The teacher puts down a counter*, as a book, and asks, "How many books are here?" The class answer, "One book." Putting down another, "How many now?" "Two books." Putting down another, “How many now?" "Three books."

3. Hand me one book. Hand me two books. Hand me three books.

4. Each may show me one finger. fingers. Three fingers. Four fingers.

Show me two Five fingers. 5. When you say, one, two, three, four, five, etc., what is it called?

Ans. Counting.

The teacher is supposed to be furnished with a Blackboard, a Numeral Frame, and a box of Counters; the pupils with small slates and pencils. For " counters," he may use any convenient portable objects; as, pebbles, bits of paper, pencils, cents, etc. Blocks of one, two, or three cubes, and so on, up to a block of ten cubes, distinctly marked upon it, are also valuable aids.

6. Again, putting down three books and then another, the teacher asks, "How many now?" "Four books." Putting down another, "How many?" "Five books."

7. The first child may hand me "one book." The next "three books." The next "two books." The next "four books." The next "five books.”

8. Each may hold up two fingers. Four fingers. Three fingers. Five fingers.

The thumbs are often regarded as fingers, and in counting may be so considered by the class.

9. Make two straight marks upon your slate; now make another beside these, and count them.

10. Make one more beside these, and count them. 11. Make one more, and count them all.

12. What number comes next after one? (The class answer in concert.) What comes after two? What after three? After four?

13. Two comes after what?

Three after what?

Four after what? Five after what?

14. What comes next before five? What before four? Before three? Before two? Before one? 15. Count from one to five in concert.

16. Count from five back to one in concert.

17. How many of the class can express the numbers one, two, three, four, five, by figures.

18. Make the figure 1 (one) upon your slates (the teacher writing it upon the blackboard.)

19. Make the figure 2 (two.) The figure 3 (three.) The figure 4 (four). The figure 5 (five.) Make each of these figures twice more.