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The teacher who would attain high success must study methuds, and never take it for granted that he is perfect in his art. Why does one teacher accomplish twice as much as another, with no greater expenditure of time and strength ? Because he has twice as much skill.. Skill is acquired. It is gained by experimenting; that is, by experience guided by good judgment, and enlightened by the study of methods and expedients. The following Suggestions, derived from long experience and much study of the subject of teaching Mental Arithmetic, are submitted for your consideration, and not as rules which you are to blindly follow without the exercise of independent thought.

1. Take great pains in assigning the lesson, adapting its length to the capacity of the class, stating explicitly how it is to be learned and in what manner it is to be recited, and giving sufficient time for its thorough preparation.

2. See that the lesson is faithtully studied. Many teachers waste time over lessons which have not been properly prepared. Sometimes study a lesson with the pupils, to show them how.

3. Do not require the pupils to commit the questions to memory. This is a waste of time. Nor should they commit the answers, excepting the answers to that class of examples which involve a single operation upon abstract numbers; that is, such questions as are usually comprised in the tables of addition, subtraction, multiplication, and division.

4. Never require a pupil to analyze questions according to a set form of analysis, but encourage originality in methods of solution. The fewer words in the solution the better, if it is correct and intelligible. By all means avoid long and complicated formulas.

5. Do not demand reasons for answers which require no process of analysis. If the child knows that 4 from 6 leaves 2, what is gained by requiring him to say, Because 4 and 2 are 6? The thing is no better understood, and time is consumed.

6. The teacher will read the questions himself, the class dispensing with the book, or he will allow the pupils to have the book and read the examples, as he may prefer. In questions requiring analysis the pupils should not be called in turn, but promiscuously or by cards, and, if the example is read by the teacher, time should be given, after the reading, for the class to think, before any pupil is designated to answer. Examples like those in Lesson II, page 11, may be recited by the members of the class in rotation, the questions being read rapidly.

7. The answer to a question requiring a process of solution should not be given before the solution, but it should be given at the conclusion cf the solution. Nor should pupils be required, as a practice, to give what may be called an abstract or general answer before the solution, like the following: of 36 is of

how many times of 42? As many times of 42 as of 42 is contained times in the number of which of of 36 is . Such exercises may be good discipline, but there is no need of consuming time on exercises merely for discipline, as the opportunities for it in acquiring useful knowledge are abundant.

8. When practicable, it is best that the whole class should stand at recitation. At any rate, the pupil who recites should stand, and, if the teacher reads the example, the pupil should repeat it after him, before giving the explanation.

9. Never proceed with the recitation unless every member of the class is giving attention, but do not try to keep the atten. tion too long. Many expedients must be employed to keep the attention awake. Sometimes the pupils may “take places,” sometimes they may be permitted to correct each other, and sometimes a pupil may be called at random to finish a solution coinmenced by another.

10. Aim at thoroughness in every step. This is much promoted by frequent and judicious reviews. With every lesson in advance the preceding should be reviewed; and there should be monthly and quarterly reviews beside.

11. If you suspect that a solution has been committed to memory without being understood, give a similar original question with different numbers.

12. Where it is practicable, illustrate problems and principles by sensible objects. Let fractions be illustrated by dividing an apple, a line, a square, or some other object. The tables of weights and measures should be taught according to the method of object-teaching, and not abstractly committed to memory.

13. As an occasional exercise, let each pupil, from memory, propose to the pupil next above him some question embraced in the part of the book which has been studied, the pupil failing to solve the question put to him losing his place; or, where “place taking” is not practiced, let there be a forfeiture of merits for failure, or a gain for success.

14. Original questions similar to examples 32 and 33, page 28, to be answered simultaneously by the class, should be proposed frequently and enunciated rapidly.

15. The learner should seldom if ever be told directly how to perform any operation in Arithmetic. Much less should he have the operation performed for him. Instead of telling the pupil directly how to go on, examine him, and endeavor to discover in what his difficulty consists, and then, if possible, remove it.

16. The recitation should be conducted briskly, and it should be so managed, if practicable, that each pupil shall endeavor to solve every question proposed: but it is not necessary that the whole lesson should be actually recited by each pupil.

17. But the most important requisite to success is to create and to sustain an interest in the study. How can this be done? In the first place you must be really very much interested yourself. In the second place, you must teach well. And if you are deeply interested in the subject, you will be very likely to find out how to teach it skillfully.

CONTENTS.

SECTION I, pp. 9-25, contains addition, subtraction, mu'tiplication,
and division, and numbers are expressed only by words. The very
young pupil will be aided in solving the examples by sensible objects,
such as knives, cents, marbles, apples, fingers, etc.

II.

SECTION II, pp. 26 - 47, introduces the Arabic figures and Arith-
metical signs. The examples are similar to those in Section I, and
a little more difficult.

ΙΙΙ.
SECTION III, pp. 48 – 75, presents the elements of fractions, and op-
erations upon whole numbers are continued and extended.

IV.

SECTION IV, pp. 75 - 92, defines fractions, the fractional form isin-
troduced, and the principles of fractions are still further developed.

v.

SECTION V, pp. 93 - 101, gives the tables of compound numbers and
practical examples on the same

VI.

SECTION VI, pp. 101 - 124, gives a variety of practical examples,
together with the more complicated combinations of fractions.

VII.

SECTION VII, pp. 125 - 138, introduces several notes, giving more or

less minute suggestions with reference to the various operations in

fractions.

VIII.

SECTION VIII, pp. 139-148, develops some of the more important

principles in per centage.

IX.

SECTION IX, pp, 149 - 163, consists of a few Lessons composed of

miscellaneous examples, together with Multiplication and Review

Tables,

X.

In pp. 164 - 176, a few of the more simple principles of Written
Arithmetic are given. These pages may be studied with or after the
preceding pages.

INTELLECTUAL ARITHMETIC.

SECTION FIRST.

LESSON I.

1. HENRY had one knife, and he has found another; how many knives has he now?

2. Charles bought an orange for two cents, and an apple for one cent; how many cents did he pay for both ?

3. Mary gave two peaches to Sarah, and kept two herself; how many peaches had she at first ?

4. If you have four marbles in one hand, and three in the other, how many have you in both ?

5. If you have four fingers on each hand, how many fingers have you on both hands?

6. James found five apples under one tree, and four under another; how many apples did he find under both trees?

7 Addie has five canaries, and Ella has three ; how many canaries have they together ?

8. Robert had three peaches, but he has given one of them away; how many peaches has he now?

9. John having four cents, spent two of them for an orange; how many cents has he now ?

10. David has five cents in one hand, and two cents in the other; how many cents has he in both hands? How many more in one hand than in the other?

11. Frank has six gray squirrels, and Herbert has three ; how many squirrels have they both ?

12. Edwin has six figs, and Philip has four; how many more figs has Edwin than Philip? How many have they both ?

13. William having eight plums, gave five of them to Louisa; how many did he keep? How many less did he keep than he gave away?

14. Edward has six doves, and Charles has eight; how many more has Charles than Edward ? How many have they both ?

15. Lewis bought a pig for eight dollars, and sold it for ten dollars ; how much did he gain?

16. A man bought a sled for ten dollars, and sold it for seven ; how much did he lose ?

17. A man owing ten dollars, paid four dollars ; how much did he still owe?

18. A man owing ten dollars, paid all but four dollars ; how much did he pay?

19. Mr. Adams sold a pig for three dollars, and a sheep for six dollars ; how many dollars did he receive for both ?

20. Albert has eight rabbits, and Arthur has two; how many rabbits have they both ?

21. George found seven eggs in one nest, and three in another; how many eggs did he find ?

22. Frank gave three cents for an orange, and had seven cents left ; how many cents had he at first ?

23. If a melon is worth ten cents, and an orange is worth four cents, how many cents are they both worth? How much more is the melon worth than the orange ?

24. Mary had nine cents, and her mother gave her three; how many cents had Mary then ?

25. Robert had ten peaches, but he has given four of them to David ; how many peaches has Robert now? How many more than David ?

26. Mr. Day bought a barrel of flour for ten dollars, and sold it for two dollars more than he gave for it. how much did he receive for it ?

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