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FOREWORD

THESE books have been planned to meet fully all the requirements of the latest New York State Syllabus in Arithmetic. Full conformity to both the lesson-plan and the spirit of the syllabus has been achieved not only by a close following of the recommended sequence of topics year by year, but also by providing subject matter which is vitalized in content and method. Some of the special features of the books are as follows:

Teaching Methods.—The teaching methods of these books have been evolved through years of classroom experience. The modern methods and recent theories of teaching arithmetic have been carefully tested out in the authors' classes, and such as proved themselves most effective have been incorporated in these books. As a result, teachers using the books may be confident that every method and problem used is not only thoroughly up-to-date but efficient and teachable.

Study Sections.-The general procedure has been to introduce each topic with a study section, primarily written for and addressed to the pupil. These study sections present separately the steps of development which the pupil needs to take in arriving at an understanding of the process or processes involved. Teachers may either assign these sections for individual study or use them for class preparation, as circumstances may require. Each topic is elastic; that is, it is not necessarily to be taught as a single lesson. In some cases two or moro topics may be covered in one lesson, but in others a single topic may require more than one lesson.

An Abundance of Easy Examples.—It has been found that proficiency in the use of mathematical processes is dependent on the solution of an abundance of easy examples, particularly at the outset in each topic. These books, therefore, provide an adequate amount of drill material, which while easily handled by the pupils, makes for accuracy and speed in the fundamentals and for an understanding of basic processes.

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More drill work in the fundamental operations has been provided than will be needed by the average class. The experienced teacher will realize that after a class has reached the desired proficiency (as determined by the timed tests), further drill in any process will do more harm than good, and the rest of the drill examples may and should be omitted. In some cases, of course, it may be necessary to assign all of the drill exercises in order to bring the majority of the class up to "standard " proficiency.

Problems and Problem Solution.—The problems are of vital value and interest to the average pupil. Necessarily some of them will not be needed in certain localities and teachers should feel free to omit such problems. In all cases, pupils should be encouraged to make up problems and projects utilizing local interests and situations. This has been suggested in many places throughout the series.

It is further suggested at frequent intervals throughout the books that problems be solved “mentally.” An oral exercise of telling in advance how each problem of the lesson should be solved is of immense value for this purpose.

No set form for the solution of problems should be required. Pupils should be encouraged to develop original forms of solution, provided only that they use a clear direct method. For this reason, different methods of solution of the same type of problems have at various places been shown in the text.

Approximate Answers and Checks.—Approximate answers may be required frequently in exercises of all kinds and ample suggestions for such work are made in the text. The value of estimating answers in drill work as well as in problem solution can hardly be overemphasized. The pupil should learn that he is not to consider a solution completed until he has compared the answer with an estimate, and found it to be reasonable.

The pupil should be required also to form the habit of checking his results. Various useful methods of checking are described in the text in connection with the model solutions, and the use of checks for accuracy is called for at frequent intervals throughout the text.

Reviews.—Experienced teachers realize that frequent recall is essential for mastery. For this reason, reviews have been made one of the major features of these Arithmetics. The more difficult processes and topics reappear in problem reviews (Everyday Use of Numbers), chapter reviews, cumulative reviews, and general reviews from grade to grade until their retention is assured.

Timed Tests. — The Timed Tests which have been introduced at frequent intervals provide an efficient way of determining whether a class has attained and is retaining its proficiency in the fundamentals. The time limits assigned for these tests have been derived from their extensive use in both graded and rural schools, so that the speeds given are attainable, without strain, by the great majority of pupils who have been correctly trained. It is understood that any pupil who finishes working the examples in a test before the end of the time allowed is to use the rest of his time in checking his work. The teacher, of course, may change the time allowances according to circumstances. If a teacher desires, the timed tests may be used for drill work; if the figures in them are slightly altered, they will provide further practice. Pupils who have difficulty in passing any test should be encouraged to keep their records in graphs or other ways provided for in the texts of this series. These less proficient pupils should be urged to practice the tests or similar drill examples in the text as extra work. Best results will be obtained if the teacher will keep some record of the results of the timed tests, utilizing such record for the purpose of helping backward pupils diagnose the causes of their failures. Pupils should be taught that accuracy is even more essential than speed, but that both are requirements of the business world.

Other Features. Besides those mentioned above, other important features of these Arithmetics are the motivation by appeals to child interests through such activities as games, playing store, etc., the relation of the problem content to daily affairs and to life situations, and the emphasis placed upon the pupil's acquisition of a fund of useful arithmetical knowledge about such things as prices, measures, business customs, and the like.

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