it teaches "rules through processes,” thus observing two important inductive maxims. The work also observes the inductive method in not giving answers to questions and problems. For an illustration, see the method used in developing the idea of a Fraction (p. 94). Printed answers to the questions there asked, would clearly violate the wise maxim: “Never tell a pupil any thing which you can lead him to discover and express for himself.” Attention is also called to the use of visible illustrations (objects or pictures) in developing new ideas and processes. In the fundamental rules this illustrative or perceptive step is omitted, since it is fully presented in the Primary Arithmetic. The engraved cuts in Fractions, United States Money, and Denominate Numbers, are specially designed to be used as a means of developing and illustrating the subjects considered. It is hoped that they may be found as useful as they are beautiful. Two other features, worthy of special notice, are the great variety of exercises, and their preëminently progressive character. Generally, each lesson contains both concrete and abstract examples, and every new process or combination is at once used in the solution of problems involving mental analysis. This arrangement avoids the mechanical monotony which characterizes long drills on a single class of exercises. The problems, all of which are original, are so graded that they present but one difficulty at a time, and all in their natural order. The pupil's progress is thus made easy and thorough. It is hoped that these and other features may commend this work to all progressive teachers, and that it may prove as successful in the school-room as its plan is natural and simple. Columbus, Ohio, May, 1870. SUGGESTIONS TO TEACHERS. In the preparation of this work two facts were kept in view, viz.: (1) that it is to be studied by pupils who must largely depend upon the living teacher for explanations, and (2) that those methods which are most natural and simple, are most successful in practice. Hence, its pages are not cumbered with long verbal explanations and peculiar methods, of little practical use to pupil or teacher. The author has left something for the teacher to do, and that this may be done wisely, he offers the following hints and suggestions: 1. Mental Exercises.—These exercises should be made a thorough intellectual drill. They should be recited mentally, that is, without writing the results, and, since the reasoning faculty is not trained by logical verbiage, the solutions should be concise and simple. See pages 25, 97, etc. They should also be made introductory to the Written Exercises, of which they are often a complete elucidation. The corresponding examples in the two classes of exercises should be recited together as well as separately. 2. Written Exercises.—The pupils should be required to solve every problem of the assigned lesson on the slate or paper, and the solutions should be brought to the recitation for the teacher's inspection and criticism. Since the answers are not given in the book, they should be obtained by a comparison of the pupils' results. From three to five minutes will suffice to ascertain the answers to twenty problems, and also to test the accuracy and neatness of each pupil's work. The time thus employed is more than made good by the increased interest, self-reliance, and study which the absence of answers secures. The mental problems may also be solved on the slate or paper in preparing the lesson, and then recited, not only mentally as above described, but also as a written exercise. This will increase the number of written problems, and, at the same time, it will secure a careful preparation of the entire lesson. 3. Definitions and Principles.—These should be deduced and stated by the pupils under the guidance of the teacher, and usually in connection with the solution of problems. Take for illustration the definition of multiplication. The pupils multiply 304 by 5. The teacher asks, What have you done? "I have multiplied 304 by 5.” T. Do not use the word “multiplied.” (If necessary the teacher shows what is meant by taking a thing one or more times.) “I have taken 304 five times.” T. By what process have you taken 304 fiye times? "By multiplying it.” T. What then is multiplication? “Multiplication is the process of taking a number T. How many times is the number taken in the above example? “It is taken five times, or as many times as there are units in the multiplier.” ?. Now complete your definition. Multiplication is the process of taking a number as many times as there are units in another number.” These steps should be repeated with other examples until the definition is clearly reached and accurately stated. It should then be written and compared with the author's definition, which should be thoroughly memorized. 4. Rules.—These should also be deduced and stated by the pupils. The true order is this : 1. A mastery of the process without reference to the rule. 2. The recognition of the successive steps in order, and the statement of each. 3. The combination of these several statements into a general statement. 4. A comparison of this generalization with the author's rule. 5. The memorizing of the latter. Take for illustration the rule for adding fractions. 7. What is the first step? "Write the fractions, separating them by the plus sign.” (Pupils write an example.) . What is the second step? “Reduce the fractions to a common denominator." 7. What is the third step? “Add the numerators of the new fractions.” 7. The fourth step? “Under their sum write the common denominator." These questions should be repeated until the answers are promptly and accurately given, and then they should be united in a general statement. The first step may be omitted in the rule. 5. Questions for Review.—These are designed as a final test of the pupil's knowledge. Before they are reached, the definitions, principles, and rules should be thoroughly mastered, and the pupils should be able to make a topical analysis of them and recite each in order. 6. Properties of Numbers.—But one method of finding the greatest common divisor and least common multiple is given, namely, by factoring. The other methods are of no practical use to pupils of this grade, and this is introduced mainly to give practice in factoring numbers. 7. Fractions. This section presents only the elements of Fractions, and these in the simplest manner. The subject will be more exhaustively treated in the Complete Arithmetic. The reduction of compound fractions is made introductory to the multiplication of fractions, as the two processes are best taught together. CONTENTS. SECTION III.-SUBTRACTION. SECTION VII.-COMMON FRACTIONS. The Idea of a Fraction developed—Definition Reduction of Integers and Mixed Numbers to Fractions Reduction of Fractions to Integers or Mixed Numbers Reduction of Fractions to their Lowest Terms Reduction of Fractions to Higher Terms Multiplication of Integers by Fractions Reduction of Compound Fractions Multiplication of Fractions by Fractions Division of Fractions by Integers Division of Integers by Fractions Division of Fractions by Fractions SECTION IX.-REDUCTION OF DENOMINATE NUMBERS. |