and to be addressed directly to the teacher; yet it must be carefully borne in mind that these definite instructions indicate only one course of procedure out of many that the teacher possessed of tact and invention may successfully use.

4. Under the heading, "Suggestions to Teachers," there is given a general outline of the course the teacher should pursue in his work, both as regards himself and the pupils. This outline should be fixed firmly in the memory, and applied in the class-room at every stage of the pupil's work.

5. The suggestions and exercises given upon numbers from 1 to 10 inclusive are simply intended to indicate to the teacher the points that should be covered, and the kind of work that should be done. At this stage of the pupil's progress, the only thing that will secure good results is the tact of the teacher in repeating again and again these simple exercises in such a manner, that the child is kept intensely interested, and perceives in every exercise the very thing intended, instead of simply repeating after the other pupils, or the teacher, meaningless words.

6. The articles in the new matter for the use of the teacher are numbered by using letters after the number of the article before which the new matter is placed. Thus, just before Article 13, six pages are introduced, and the articles are numbered 13a, 13b, 13c, 13d, and 13e.

7. The matter which is common to this and the pupil's book is indicated by numbers placed at the bottom of the pages. These numbers denote the pages in the pupil's book, where the same matter is found.

8. The outline of elementary work in fractions and compound numbers in the pupil's book indicates clearly the course of instruction and drill that should be pursued; hence, no suggestions are given for the teacher beyond division.

ARITHMETIC

SUGGESTIONS TO TEACHERS

Art. 1a. The following Propositions apply equally to every stage of the teacher's work and indicate the General Method which he should pursue both as regards himself and his pupils; hence they should be carefully studied and fixed clearly in the mind.

I. Let the work of each day be prepared as follows:

1. Master so thoroughly the lesson assigned to the pupils that you can dispense with the use of the text-book in the recitation, except when dictating examples for the pupils to solve.

Observe, this involves

(a.) Holding all the facts in the lesson in the mind as a unit.

(b.) Perfect familiarity with the connection the lesson sustains to the pupil's previous work and its bearing on his advanced work.

2. Notice carefully the root thoughts or important points in the lesson around which details are naturally grouped. Emphasize thoroughly these points in your teaching.

3. Select with great care all the points in the lesson that need illustration and prepare a method of presenting each of these points before attempting to teach them.

Observe, that to be successful in explaining difficult points,

(a.) Your illustrations must be simple and familiar, yet striking and varied.
(b.) You must use small numbers which the pupils can hold in their minds with-
out such an effort as will distract their attention from the point illustrated.
(c.) You must present each step in your illustration separately, and in its logical
order.

II. Let the work in each recitation be SYSTEMATIC, DEFINITE, and POINTED, and let it be performed in a SPIRITED manner.

To secure this result, the following must be observed:

1. Mark out for yourself a general plan for conducting class exercises, and adhere to it, subject to such modifications as the condition and surroundings of your class may necessarily demand.

Such a plan should include at least the following:

(".) A method of disposing, without interrupting your work, of questions of order that may arise during the recitation.

(b.) A definite time for pupils to present the difficulties they may have met in studying the lesson assigned to them.

(c.) A time and method for illustrating the pupil's difficulties.

(d.) A method of rapidly and correctly testing each pupil's knowledge of the work assigned.

(e.) A systematic plan for blackboard work, including order in going to and from the board, position at the board, form in which the work should be put on the board, and a method of correcting the work when finished.

(f.) A time and plan for oral exercises, and for written exercises upon the slates. 2. Hold your pupils rigidly in class exercises to the work assigned. Let no side or useless details divert your own or the pupil's attention from the real subject under discussion.

This involves the following:

(a.) The work assigned in each advanced lesson includes, without being stated, an explanation when required of any point previously discussed.

(b.) Explanations of difficulties either by the teacher or pupils are class exercises; hence, each pupil should be held responsible to reproduce such explanations at any time after they are given.

(c.) While the strongest encouragement and fullest liberty should be given to the pupils at the proper time and under proper restrictions to ask questions on the point under discussion, all useless and trifling questions and such as do not have direct reference to the subject should be absolutely excluded.

III. Let special attention be given to each of the following points: 1. The pupil's use of language.

(a.) Do not allow the pupil to use contracted statements or forms of expression that are grammatically or otherwise incorrect.

(b.) Encourage him, by suggestions, to use a variety of words to express the same thought; also to observe closely the special meaning of business terms and expressions.

2. Practice in making well-formed figures.

(a.) In primary classes constant practice should be given in making figures

both on the slates and blackboard. The pupil should be required to imitate closely the figures given in the Tables on pages 17, 24, 41, 45, and 65. Each exercise from the Tables should be copied neatly on the slates, and the teacher should carefully examine the work and commend what is well done,

(b.) In more advanced classes exercises should be given frequently in writing on paper the solutions of examples. This will give practice in making figures and in arranging the work in neat form.

3. The arrangement of work on the slate, blackboard, or paper. (a.) When a new subject is commenced, illustrate on the blackboard the way in which the work may be written, and insist that each pupil place his work on the board, slate, or paper at all times in a systematic and neat form. (b.) Let the pupil's work on the blackboard or slate be always carefully noticed. Commend what is well done, and require what is not well done to be rewritten.

4. The oral and written solution of examples.

(a.) Avoid stereotype forms. Encourage originality in the method of analysis
and solution.

(b.) Insist upon both accuracy and expertness. To secure these results, exercise
the pupil frequently upon abstract numbers and simple examples which he
can solve readily, holding him to doing the work in a given time.
(c.) Insist, when reading examples for solution, that the pupils follow you so
closely that short ones need never be read more than once. Insist, also,
that no book be used when a pupil is explaining an example on the black-
board.

IV. In solving examples, let the pupil be required, in every case, to master the thought before attempting the solution. Let him also, in wording his analysis or solution, be required to guard carefully against the use of set forms of expression.

In carrying out the requirements of this proposition, observe the following:

1. To master the thought in an example, let it be read with great care, thus:

(a.) Determine the exact meaning of each technical term used.

(b.) Locate clearly in the mind the nature of the transaction and the relations of the parties concerned, or the peculiar conditions stated.

(c.) Examine carefully the quantities or numbers given, and notice, from the nature of the transaction, the relations of these quantities through which the required result can be determined.

2. In giving either an oral or a written solution of an example, the pupil should be required to state:

(a.) What is given and what is required.

(b.) The relations of the given quantities from which what is required can be found.

(c.) The steps that must be taken in their order and the process that must be used to obtain the required result.

In making these three statements, clearness, accuracy, and brevity should be the only conditions imposed.

APPLIANCES FOR PRIMARY WORK.

16. One or more of the following devices should be in the hands of every teacher of Primary Arithmetic.

This is one of the most simple and practical devices in use. No school-room should be without it.

A large frame with 20 balls on each wire, and arranged as shown in the cut, so that no balls are seen by the class except those which are in use, is to be preferred. In using the frame, the teacher should arrange carefully the course of illustration before appearing in class. The instructions given hereafter for the use of objects can be readily applied to the Numeral Frame.

For example, arranging the balls as shown in the cut, we have the illustration required to present counting as directed on page 14.

To use the Numeral Frame with effect the teacher must carefully examine the lesson to be illustrated, before going into class, and devise the best arrangement of the balls to make each point clear to the pupils and to fix the subject in their minds. Tact must be exercised in the use of all such devices or the result will be a failure.