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GENERAL SUGGESTIONS.

1. There is no royal road to a knowledge of Arithmetic, and in this book no attempt has been made to preclude the necessity for laborious effort without which, it has been wisely said, life gives nothing to mortals.

2. Self-reliance is the basis of action, and “self-activity is the law of growth.” To render the pupil self-reliant, selfhelpful, and self-acting in the face of difficulties, is the object the teacher should keep steadily in view.

3. Two ideas are fundamental: I. Knowledge cannot be successfully built except on knowledge already acquired. The lesson to be learned to-morrow must start its growth in the lesson learned to-day. II. Lessons assigned a class must not be made too easy for some, nor too difficult for others. Careful judgment is, therefore, required that oral explanations and illustrations be neither too ample nor too meagre.

The least active mind must be made to understand, and, at the same time, the most active brain must be required to labor.

4. In each division of the subject, as treated, will be found inductive steps, definitions, principles, processes, explanations, rules, exercises, and problems,—all to be thoroughly learned and intelligently recited. Mastery of the exercises will give facility in performing the operations required by the problems. 5. The solution of a problem requires three distinct steps:

I. The indication in arithmetical language of the operations to be performed.

II. The mechanical performance of the operations indicated.

III. The statement of the reasoning by which the operations as indicated were obtained, and also the elucidation of any merely mechanical step that has been taken to reach the final result.

If in each subject the introduction, including principles, processes, and explanations, be systematically and thoroughly acquired, the exercises and problems that follow will seem not forbidding obstacles, but, as it were, beckoning friends.

6. Too much importance cannot be attached to the method of dealing with a problem, as pointed out above. The frequent suggestions made throughout the book attest the author's belief in the excellence of the system proposed. One advantage is that the first step-the indication in arithmetical language of the work to be done,—really solves the problem, and that here, in many cases, the pupil's work may be considered as satisfactorily closed. Every teacher must determine for himself, and every intelligent teacher will successfully determine, how far his pupils need to work out and recite the details of a solution. He must go far enough to be convinced that they have got within them a conception of the truth, and are able to declare it. But how is this possible, unless he recognizes the great fact that every pupil is an individual, has a distinct individuality, and is, as far as possible,

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