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pound Numbers ; Circulating Decimals ; Duodecimals ; Alligation Alternate; the Progressions ; Annuities; also, many terms, as “local value," “ borrowing and carrying," "improper fractions,” etc. 'Analysis ” and “ Practice” are employed throughout the book, they have no separate treatment. Arithmetical puzzles are carefully avoided.

8. Such new matter and such new methods as are demanded by the circumstances of the present time are introduced. See the Metric System; the treatment of Percentage; a Table of legal rates of interest in the States and Territories at the present time, from official sources ; Gold at a premium; U. S. Bonds; Average of Accounts; Partnership; Evolution, etc.

9. Complete and thorough reviews both of principles and of processes are kept up throughout. See Topical Reviews, with references, a feature not found in any other arithmetic; General Reviews and Dictation Exercises upon an original plan, by means of which additional examples with their answers are furnished to apply to every important subject in the book, without extra labor to the teacher.

By such a method and such a practice, it is believed that a knowledge of principles will be acquired, and the intellectual powers stimulated to healthful activity and to logical precision, and the pupil will be spared the mortifying experience, so common, of having to unlearn in the counting-room what he has learned in the school-room.

The authors' thanks are due to their fellow-teachers for many valuable suggestions, and for kindly assistance in the method of presenting the various topics and in the revision of the work in the manuscript and in the proofs; also to James A. Dupee, Esq., Banker of Boston, for criticisms and suggestions upon business forms, and to J. L. Woods, Esq., of Brimfield, for hints on the method of extracting the roots.

They avail themselves of this opportunity to express their gratitude for the hearty commendation and liberal patronage bestowed upon their former works by teachers and the public generally.

If the spirit with which they have studied anew the subject of numbers shall be communicated by these pages to their fellow-teachers, and especially if the book shall serve to awaken in the mind of the pupil greater love for study, greater activity and precision of thought, and if the true ends of education shall be more fully attained thereby, their labors will be richly compensated.

Boston, August, 1869.

SUGGESTIONS TO TEACHERS.

Though every page of this arithmetic is an illustration of the authors' method of teaching the subject, yet a few additional hints may be of service to the inexperienced teacher.

The plan of the book contemplates the formation of classes in arithmetic, to each member of which is assigned the same daily lessons.

It is recommended that before a lesson is assigned it should be taught orally by the teacher, with the aid of appropriate objects and illustrations.

This teaching, if properly conducted, will so direct the attention of the pupil, that by observing for himself, the right ideas will be excited in his mind.

Having been assisted in this manner to make a full analysis of a subject or of an operation, he should then be required to give synthetically a statement of what he has observed. The general method of teaching is indicated in the following

ILLUSTRATIVE LESSON. (A method of teaching Arts. 1, 2, and 3 of the book.] The teacher being provided with marbles, pebbles, shells, or other small objects, placed in collections on his table before the class, proceeds thus:

Teacher. “John, show me a single pebble ; " "a single shell.” “Show me any single thing.” “What have you now in your hand ?(John. “I have a single thing.") "A single thing is a unit. What is a unit?” (John. “A unit is a single thing.") “Each member of the class may show me a unit.” “What is a unit?(Class. “A unit,” etc.)

Teacher. “James, show me a collection of units." "A collection of units is a number. What is a number?(James. “A number is a collection of units.”) “The class may answer." (Class. “A number,” etc.)

Teacher (holding up five shells). “What have I here?(Class. “You have a number of shells.”) “Name the number.” (Class. “Five.") “ Jane, you may write the word five on the blackboard.” “You have now represented the number five by the word five ; you may represent it by the letter V underneath the word five.” “Do you know any other way of representing the number five?[Charles passes to the board and writes under the letter V the character 5.1 “How many ways have we found of representing the number five ?(Class. “Three ways; by using the word five, by using the letter V, and by using the character 5.”) “Representing numbers by words, letters, or other characters is expressing numbers.”

“What is expressing numbers ?(Class. “Expressing numbers is representing them by words, letters, or other characters.”)

Teacher (writing upon the board the characters 1, 2, 3, 4, 5, 6, 7, 8, 9]. “For what are these characters used ?” (Class. “They are used for expressing numbers.”) “These characters, used for expressing numbers, are figures. What are figures ? (Class. “Figures are certain characters,” etc.) “The expression of numbers by figures or by letters is Notation. What is notation?" (Class. “Notation is the expression,” etc.)

For a class of quite young pupils, this lesson is sufficient. For older pupils, additional topics, as in Arts. 4, 5, etc., may be given.

The teacher should examine his class thoroughly as he proceeds, to see if they understand his teaching. After the above lesson is thoroughly taught, the following topics should be assigned to be illustrated and defined by the class at the next recitation : - A unit ; a number ; expression of numbers ; figures ; notation (more or fewer topics, as the teacher has previously illustrated), also an exercise in writing figures neatly upon the slate or paper, for inspection.

A part of each lesson assigned should consist of such exercises and examples to be performed upon the slate as are found in the book.

In preparing for recitation, the pupil will make the text-book his guide, where his teacher has already been his travelling companion.

At the recitations the pupil should be required to take the pebbles, shells, or other objects in his hand, and teach the lessons to his class in the manner and in the order observed by the teacher in assigning the topics.

The slate work of the pupils should be examined by the teacher. Neatness and order in arranging the work, in making figures, etc., should be required.

REVIEWS. — Each recitation should consist in part of a review of the topics of previous lessons.

THE TOPICAL REVIEWS at the close of each section of the book will be a guide to the pupil in preparing for his reviews. Ordinarily a few topics of one section should be reviewed with each advanced lesson of the succeeding sections.

The topics should be so prepared that they can be illustrated by the pupils of the class as they are called upon, without questions from the teacher. (See Illustrative Exercise above, also suggestions in the teachers' KEY AND MANUAL.)

This form of review is a much more thorough test of the pupil's proficiency than the ordinary mode of questions and answers. It has the advantage also of occupying far less time, and, above all, of cultivating the active powers of the pupil's mind. The teacher may rest assured that it is

entirely practicable for pupils in all stages of development, less being required of younger than of older pupils.

A part of the time of each recitation should be devoted to performing examples upon the slate or board, in the presence of the teacher. In this exercise the teacher will be greatly aided by

The DictatION EXERCISES which are contained in the KEY AND MANUAL. These consist in part of miscellaneous examples ; but mainly of the examples of the Arithmetic, so modified as to require an entirely different answer from that given in the Arithmetic, and to which the pupil has no clew. They are dictated thus :

The teacher, turning to the article in the KEY AND MANUAL corresponding to any article in the Arithmetic which he wishes to review, assigns an example or examples, and announces the modification. A pupil having performed the example or examples assigned, announces his result to the teacher, who compares it with the result given in the Key.

It is only by practice of this kind that the pupil secures that self-reliance without which he may spend months in the counting-room before he is fitted to perform with accuracy and with facility the commonest arithmetical operations.

The pupils should have abundant practice in the fundamental operations, especially in addition. To facilitate this practice, Walton's ARITHMETICAL TABLE, or something similar, should be in the hands of every student of written arithmetic.

It is with a view to securing accuracy in these operations, that so large a number of examples without answers are given in the first part of the book. The pupil should be made as ready in performing these operations as he is in reading simple sentences. The teacher can excite a lively interest in his pupils by giving them occasionally examples drawn from the transactions of daily life with which they are familiar.

F r further suggestions on the method of teaching the various topics, the teacher is referred to the KEY AND MANUAL. It is recommended that he should examine that before teaching any topic.

The above are but hints; they apply in the main to all teaching : they are merely suggestive, and will, of course, be adopted only so far as they can be incorporated with the methods of individual teachers : though the book is adapted to the method illustrated above, it does not require any peculiar method of teaching.

Though great care has been bestowed upon the book to secure accuracy, errors are almost unavoidable in the operations of so large a number of examples. By pointing out to the authors or publishers any errors which may be discovered, a benefit will be conferred upon them, also upon teachers and pupils using the book.

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