4. The author has omitted all representations of objects by means of pictures. Many reasons might be offered for this, among the most prominent of which is the fact that they are not needed, since the objects themselves are preferable to mere pictures of objects. The pupil should be so thoroughly drilled with Oral Exercises that by the time he can read a book on Arithmetic he may be able to compute without the assistance either of objects or their pictures.

5. Much care has been exercised throughout the entire work that the arrangement should be systematic, the lessons carefully graded, and the whole be in accordance with the principles of Analysis and Induction.

The entire work is the result of much thought and observation in primary instruction, and is presented to a discriminating public, with the earnest desire that it may do much for the education of the youth of our country.

EDWARD BROOKS.

State Normal School, June 16, 1878.

SUGGESTIONS TO TEACHERS.

THE following suggestions are made to the younger an less experienced teachers who may use this work:

1. It is respectfully suggested that the Oral Exercises receiv that attention which their great importance demands. The pupils should be constantly drilled on exercises besides those found in the book. With young pupils, lessons with the numeral frame will be found of great value.

2. The problems in Mental Arithmetic should be assigned promiscuously, pupils not being allowed to use the book "uring recitation. The pupil selected should arise, repeat the problem, and then give the solution; at the close of which those who have observed mistakes may indicate it by raising the hand, and then some one selected by the teacher may arise and give the criticism.

3. The exercises in Written Arithmetic should be solved upon the slate as a preparation for the recitation, and upon the blackboard during the recitation. The same problem may be given to the whole class, or each member may receive a different problem, as the teacher prefers; the author thinks with beginners the first method is preferable. At first perhaps it is better to teach them the mechanical operations, showing them the reasons for these operations, but not requiring them to tate these reasons in recitation until they have acquired considerable readiness in the different processes. This last suggestion is founded upon the natural order of the unfolding of the young mind, and also upon the experience of some of the most successful teachers of youth.

4. In the Mental Exercises of Multiplication, it will be well to have the pupils solve the problems, which derive the tables of results, upon the slate or blackboard, after which they should be required to commit the tables to memory. The teacher may also show the pupils that any product in the table can be derived by adding the multiplier to the preceding product; thus, since 6 times 4 is 24, 6 times 5 is 24 plus 6, or 30. 5. Care should be taken that the pupils' language be free from all those awkward expressions so common to learners; each sound should be enunciated distinctly, each word correctly pronounced, and the habit of ready and accurate thought be developed-thus securing that combination so admirable in scholarship,-promptness, accuracy, and elegance.

NEW

PRIMARY ARITHMETIC.

SECTION I.

NUMERATION AND NOTATION.

INTRODUCTION.

Suggestions to the Teacher.

UR first ideas of numbers are derived from visible objects, hence the child's first lessons in numbers should be given with such objects. These objects may be books, pencils, grains of corn, beans, etc. Dr. Hill suggests that arithmetic may be taught with a pint of beans. The arithmetical frame is the most convenient for general use.

NAMING NUMBERS.-The names of numbers are usually acquired with the ideas of numbers; and both are given by a process called counting. Children should therefore be taught to count. Be careful that they do not use the names as mere words; see that they know what the words mean. Children can often count as far as a hundred, and yet are unable to select twelve objects from a collection. Have the pupils count with the numeral frame and with other objects.

Beside the common method of counting, I would teach pupils to count, using the expressions one and ten, two and ten, three and ten, etc., two tens and one, two tens and two, etc. It will teach them the principle of naming numbers, and prepare them to understand the method of writing numbers.

A counting exercise may be made lively by increasing or diminishing the number by several at the same time. Little counting games, with beans or grains of corn, will also be found interesting to children. Have children count backward, as well as forward.

WRITING NUMBERS.-AS Soon as a child can name numbers, it should be taught to write them. It might be well at first to write the words one, two, etc., and then introduce the figures, that they may see their advantage in brevity.

Characters.-First give the nine digits, and drill children in naming and writing them until they are entirely familiar with these characters. If they have learned a little addition and subtraction, they may use the characters in solving simple problems. Combination.-When the pupils are familiar with these charaoters, they should be taught to combine them. There are two dis

tinct methods of doing this.

18T METHOD. By this method we give the combined characters without explaining the principle of the combination. Thus we teach that 10 represents ten, 11, eleven, 12, twelve, etc., without any reference to tens and units. This method is not quite so philosophical as the 2d method, but is usually preferred with young learners in oral instruction.

We would give these expressions as far as twenty, and then drill the pupils in reading and writing them until they are quite familiar with them. We would next give the expressions from twenty to thirty, and drill in like manner, and thus continue as far as one hundred.

After the pupils are familiar with this method of writing numbers as far as 100, the teacher may then show them the principle of the combination, that the figure in the first place represents units, in the second place, tens, etc. When this is understood we would require the class to analyze these expressions as follows:PROB. Analyze 25 (twenty-five).

ANALYSIS. In 25, the 5 represents 5 units, and the 2 represents 2

tens.

2D METHOD. The other method begins by explaining the principle of the combination, that is, that 10 represents 1 ten; 11, 1 ten and 1 unit; 12, 1 ten and 2 units, etc., afterwards showing that 11 (1 ten and 1 unit) is the same as eleven, etc.

This may be done by making ten marks on the board, and then commencing a second row with one mark; and showing them that as the one is expressed by 1, the one ten may be expressed by writing a 1 at the left of the first 1, and that 11 represents one and ten; that two and ten may be expressed by 12, etc. The 0 may then be introduced, as necessary to show that the 1 is in the second place, when there is only one ten.

The pupil should be drilled in reading and writing numbers until he is entirely familiar with the subject. Haste here is "bad speed." A thorough knowledge of Notation and Numeration will dispel the usual difficulties of Addition, Subtraction, Multiplication, and Division.

NOTE.-We have suggested that the teacher give this instruction rather than attempted to present it on the printed page. It is simple and easily done; and the teacher, with the numeral frame in the hand, can give a life to it that it cannot possess when put in formal questions in the book. Remember that all good teaching requires a TEACHER.