4th. The division by either of the factors of all the products learned in the multiplation table. 5th. The squares and cubes of all the digit numbers. Among the oral drills or exercises for attaining these results, the following will be found useful: 1st, the counting in concert up to 100, by twos, threes, fours, &c., forwards and backwards. This will be, in effect, to make all the possible additions and · subtractions of these numbers. The series produced by suc cessive additions of twos, will be as follows: 2, 4, 6, 8, 10, 12, &c.; 1, 3, 5, 7, 9, 11, &c.; by the additions of threes, 3, 6, 9, 12, 15, &c.; 1, 4, 7, 10, 13, 16, &c.; 2, 5, 8, 11, 14, &c; by the addition of fours, 4, 8, 12, 16, &c.; 1, 5, 9, 13, &c.; 2, 6, 10, 14, &c.; 3, 7, 11, 15, &c. The series which will be produced by the numbers above four will be easily formed from a careful consideration of those given. The successive subtractiong, will reverse the several series. In use, the drill should begin with only two or three of the first numbers in any series; and when these are somewhat familiar, then others may be succesa sively added. 2nd. Reciting the multiplication table in concert, forwards and backwards, 3d. A most useful class drill, and one admitting of many van riations, can be made by the use of a long line of figures taken indiscriminately, and written across the blackboard. As exercises in addition, the teacher may point rapidly to each two figures in succession, and require the class to pronounce promptly, the sum of each couplet. In the same manner he may point out the figures in groups of three or four figures in each, the class giving rapidly the sums found by adding the numbers in each group. The addition of the whole line may also be made and repeated till it can be done with great rapidity. Or taking the lines by couplets, counted as tens and units, add the successive couplets. With this magic line of figures, a great variety of exercises also may be given in multiplication, subtraction and division. Other drill exercises are in use, but cannot be here described for want of space. But other and more complicated operations, in pure numbers, , both integral and fractional, may also be familiarized to a considerable extent. So, too, should the peculiarities of our common system of notation be conquered by systematic drills till every process involved in it is reduced to the ease and certainty of a habit. Here, in these drill exercises is the great battle of arithmetic to be fought and won. These once mastered, the onward course of the pupil is rapid and sure. The rational, or logical processes in arithmetic vary with each problem, and cannot, therefore, be reduced to the precision of an art. But much can be done to make the more common ones familiar. A recognition of their real character and distinct ,existence, by both teacher and pupils, will make their study much more definite and successful. The study of mental aritbmetic, when the logical steps in the solution of each problem are carefully given, is one of the best drill exercises in these rational processes. If the classes in written arithmetic were exercised in such an analysis of the problems as is given in the example in the beginning of this article, their progress in this departinent of the study would be greatly enhanced. • It should be remarked, in conclusion, that there are three distinct periods in the study of arithmetic. These may be denominated the Primary, the Practical and the Scientific. The Primary period begins with learning to count, and embraces all the exercises in numbers in the primary grade in school, These exercises should always be based upon the use of material objects, to be counted by the children. It is the period of concrete arithmetic. The Practical period embraces the study of mental arithmetic and that earlier study of written arithmetic, which has for its object the acquisition of skill in its practical processes. Tho Scientific period embraces the discussion of the pliilosophy of numbers, and of their properties and relations. I had purposed, also, to describe methods of teaching the other branches of common school study, but the space already occupied forbids additional discussions at the present. And, perhaps, enough has been done in showing, as I have attempted to show, how a true method of teaching any branch of science is only to be reached by a careful analysis of the work to be done, and a practical adaptation of the process of teaching to several parts of this work. It is hoped that the plans already given for teaching, will help to produce a much needed reform in our school rooms, and to make the instruction in the several branches discussed more rational and successful. If, without any undue crowding of the minds of children, the branches ordinarily studied in our common schools can be taught thoroughly, as I believe they can, in one-half the time now occupied in their study, then shall we save to our children years of unnecessary toil, and make it practicable to add new and valuable gifts of learning to their school attainments. He who saves me a year of unnecessary study, by improving my methods of learning, not only gives me an additional year for improvement, but also renders that year doubly valuable, by the increased intelligence and power which I bring into its labors and studies. PROGRAMME OF SCHOOL WORK. Order is the first and fundamental law of every true school. System itself is the most impressive of all lessons. Day by day, its silent recurrence of regular times and regular toil presses down upon the soul, like some massive mould into which thoughts and feelings and faculties gradually crowd themselves, until they take upon them its own beauty of proportion, and harmony of parts. The fitful feelings settle down to a smooth and steady flow; the wayward impulses learn to mark time and keep step with the daily march of regular duties. Nor is order in the arrangements of the school of less impor tance to the easy and successful government of the pupils. No school is well governed where order does not mark the entire range of school work and movement. In nothing does the quality of the true teacher shine out more conspicuously than in the organizing of the school-the orderly arrangement of its exercises. In making a programme for the school it should be remembered that there are two classes of work to be done-the teacher's work and pupils' work—the recitations and study. The programme should therefore mark both the time for getting lessons, and the time for reciting them. It is not wise nor safe to leave pupils to their own discretion as to the time to be occupied in studying their several lessons. They need the wisest judgment of the teacher to determine both when to take up each lesson, and how long a time to spend upon it. To aid teachers in preparing a programme for their schools, the following for an imaginary school is here given. Each teacher will need to vary it, of course, to meet the differing character and condition of his school. Let us suppose a school having four general classes of pupils. 1st. The beginners, reading on cards or the alphabet. They should also be taught to print letters and simple words on slates or blackboards, to draw lines and simple figures, and have daily object lessons on color, form and common things. This may be called the A class. A second group read in 1st or 2d Reader, and have lessons in spelling and mental arithmetic, chiefly oral. These constitute the B class. A third group read in 3d Reader, and study Nat. Philosophy and arithmetic. This is the C class. A fourth group, embracing the older pupils, read in 4th or 5th Reader, and study geography, grammar and arithmetic. This is the D class. The C and D classes have also writing lessons, and, to economize time, constitute a single class in spelling. In the following programme the third space gives the work of the preparation of their lessons, &c. When the study of any class is once indicated, it is to be understood that this work continues till some other study is indicated. It is best ordina rily that pupils shall get their lessons within the time appointed. INSPECTORS' REPORTS. The Department of Public Instruction has no general means of getting returns from the primary districts except the Annual Reports of the township School Inspectors. These Reports are made up of abstracts of the Reports made by the several. district Directors to the Boards of Inspectors. To make these Reports uniform and reliable, carefully prepared blanks, both for Directors and Inspectors, are furnished annually by the Department to the several Counties. From year to year I have revised and remodeled these blanks, in order to make them still more precise and plain in form, in the hope of eliminating every source of misunderstanding and error. A very considerable improvement in the fullness and accuracy of the Reports is already apparent; and it is now believed that a large majority of the Inspectors' Reports are full and trustworthy exhibits of the Public School interests of their respective townships. But there are still many Reports in which half the blanks are not filled, and in which there are other evidences of carelessness |