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is enabled to understand everything as he advances, till he acquires a thorough knowledge of the nature and use of the essential numerical operations.
While the general arrangement of the subjects and examples is strictly progressive and logical, the difficulty of the problems is occasionally varied, in order to prevent the weariness of a long, unbroken ascent, and to afford a grateful alternation of effort and relaxation, like that experienced by the traveler in crossing a country diversified by hill, valley, and plain.
The analytical process which this method requires at every step is calculated to develop and strengthen the mental powers, and to form the habit of rapid and accurate thought. Some illustrations of modes of analyzing questions have been presented merely as suggestions to the pupil; but the plan of the work does not embrace set forms of analysis for the various classes of examples, a contrivance little likely to stimulate invention or promote self-reliance. On the contrary, its distinctive feature is its special adaptation to the mode of teaching which leads the learner to ascertain for himself each step to be taken, to think and reason independently, and to rely upon his own powers and resources, thus securing a vigorous and healthful discipline of his intellectual faculties.
Though this work is intended as a connecting link between the Primary and Written Arithmetics of the Author, thus completing the Series on which he has been so long engaged, it is also complete in itself. It presents a mental analysis of Arithmetic adapted to the younger pupils by its easy gradations, and to advanced pupils by its scientific arrangement and its logical development of the art of computation; and yet it has been limited to the true province of Intellectual Arithmetic, which is to serve as an introduction to Written Arithmetic, and not as a substitute for it, as some authors seem to imagine.
In the spirit of the inductive method, concrete numbers numbers applied to physical objects — have been largely employed in treating of each topic, as the only fit preparation for the exercises upon abstract numbers, which are far more difficult for the youthful mind to grasp.
A few pages of Written Arithmetic have been appended, embracing examples in the ground rules and compound numbers, which may be profitably studied in connection with the mental lessons illustrating the same principles.
Fully aware of the difficulty of the task he has undertaken, the Author has spared no pains in its execution, and he gratefully acknowledges his obligations for the numerous valuable suggestions with which he has been favored by several eminent practical teachers.
The favorable reception of the other books of his Series, encourages him to hope that this attempt to perfect and modernize the original Inductive System of Mental Arithmetic, and adapt it to the wants of schools of the present day, will meet with the general approbation of teachers and educators.
THE teacher who would attain high success must study methods, 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 faithfully 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 hy 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 IÌ, 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 of the solution. Nor should pupils he 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 4 of 42 as of 42 is contained times in the number of which of of 36 is J. 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 púpil may be called at random to finish a solution commenced 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.
I. SECTION I, pp. 9-25, contains addition, subtraction, multiplication, 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 Arithmetical signs. The examples are similar to those in Section I, and a little more difficult.
III. SECTION III, pp. 48 - 75, presents the elements of fractions, and operations upon whole numbers are continued and extended.
SECTION IV, pp. 75-92, defines fractions, the fractional form isintroduced, 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. SEOTION 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.