GENERAL SUGGESTIONS.

PART I. of this book presupposes the occasional necessity of introducing pupils to the first rudiments of the subject.

In developing the idea of number, and teaching the simple numerical operations, real objects are greatly to be preferred to pictures, however artistic and striking they may be. But what is to be preferred to all other agencies, whether pictures or real objects, or even the book itself, is the voice and action of the live and intelligent teacher, without which little of educational value is ever accomplished in school. Object lessons, however, are of great value in illustrating and impressing the teacher's meaning; but no objects should be used except those of simple form and construction, lest the mind of the pupil be diverted by them from the primary object in view, namely, the inculcation of the idea of number and of numerical combinations.

When objects are employed, the youthful pupil should be allowed to take them in his own hands and give proof of knowledge gained by showing without help how and what they explain. This will be to his liking, and liking is a supreme element of learning.

The successful teacher is not he who does both his own and the pupil's work, but he who best directs the pupil's

activities, leads him to love learning, and to overcome difficulties by his own efforts.

In reciting, pupils should be required to give their answers in complete sentences, and that, too, without hesitation or counting. They should be able to say "4 and 3 are 7," or "4 less 3 equals 1," as promptly and with as little apparent effort as they would spell a word of three letters. In sight exercises, where rapidity is the object sought, results alone should be given; thus: """ one."

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seven,

The use of concert exercises should not be constant, but only occasional, as a ready means to give diversity or to revive flagging interest. As far as possible, the members of a class should be drilled individually, for each has a separate and distinct individuality that demands and must receive from the teacher the carefulest recognition. Every exercise should be made so interesting as to engage the undivided attention of every pupil, and to effect this they must of necessity be lively, varied, and above all brief. The very best judgment of the teacher is demanded here. The processes of Addition and Subtraction are so intimately related that they should be taught at first together. In like manner should be taught the closely related operations of Multiplication and Division.

All written work should be as neat and well arranged as the pupils are capable of doing it. Ill-formed figures and careless arrangement are fruitful sources of error in results.

Pupils while in school should be kept constantly employed. They must do if they would learn. Idleness should not be tolerated for a moment. The forward

movement should be frequently halted, and past lessons repeated. It is by repetition that acquisition becomes fixed and sure: the pupil's ignorance never ceases to be the teacher's opportunity.

Pupils should be drilled on all exercises with great care and thoroughness. The primary lessons of the book are presented as suggestive rather than as exhaustive, and the teacher will often find it necessary to supplement them. In such cases the suggestions at the foot of the pages will be helpful.