In our second article we made a further plea for the teaching of mental arithmetic in the lower grades of our public schools. We emphasized the importance of teaching but few combinations in addition, subtraction, multiplication and division of all the numbers and especially of the smaller numbers. We also showed that the method of teaching common fractions does not differ from the method employed in teaching the different operations in relation to integers. We also gave a hint as to the best methods of teaching denominate numbers. In this paper we shall make a special effort to present a rational method for teaching that part of arithmetic which may be called written.

It is hoped that the majority of children in our public schools will have no occasion to use written arithmetic as a science or an art before they reach the third or fourth grade. If through the influence of a superintendent or principal, written work has been required in the first or second grade, success in teaching mental arithmetic has been greatly retarded. From the primary grade to high school, the pupil in arithmetic should be taught to use pencil and paper, crayon and blackboard as little as possible. Quickness, accuracy, and thought power can best be secured by the disuse of pencil and paper, blackboard and crayon.

In adding, subtracting, multiplying and dividing large numbers, mental energy is saved by using the pencil. This is true whether these numbers be considered as abstract or in their more common application to quantity in the commercial world. Another use of the pencil or crayon is demanded when a principle is to be illustrated. In consideration of these two propositions the successful teacher of arithmetic will use pencil and crayon very little.

When the text-book is put into the child's hands for the first time, it will be necessary for the pupil to examine in the class-room the first pages of what to him is a strange book. Before assigning any lesson, the teacher should require the pupils to read and examine carefully every page that is to receive their special consideration. The first subject that the teacher will have occasion to consider in teaching is the art of writing numbers, the art of expressing numbers by means of figures. The pupils have already learned incidentally to write numbers to 100, possibly 1,000. In their language and science work they have learned this much of the language of written arithmetic. Nevertheless it is best for the teacher not to assume this, inasmuch as the child at this stage is unconscious of the fact that numbers are written in Arabic notation according to the scale of ten.

The teacher before calling his class should secure a few hundred small sticks of uniform size - toothpicks will do, but kindergarten sticks, five or six inches in length are better. The child should put these sticks in packages; small rubber bands may be used to hold the sticks in bundles. These sticks should be placed on a table upon which are marked off columns for units, tens, hundreds, thousands, tens of thousands, etc. It would be better if the surface of the table were like a blackboard. This horizontal blackboard could be used in teaching geography and other subjects to most excellent advantage. The teacher should drill the child in handling these packages of sticks. They should be required to present to the teacher a given number. In other words, the children should be given number lessons in relation to the number of sticks that the teacher proposes to represent ultimately in the form of figures. When the pupils have thoroughly comprehended the use of the sticks, the number of sticks in any number of packages, he may place them in the different columns on the table, the child being led to place any number of sticks smaller than the smallest package in units' place, any number of tens' packages in the second column or tens' place, any number of hundreds' packages in hundreds' place. After the child has been thoroughly trained in reading the numbers from the packages in position, after he has made numbers by the use of these packages, and after he has represented them correctly by using these packages, the teacher may put in the place of units' sticks a figure representing the number of units and a figure in the place of tens' sticks representing the number of hundreds. In this way, the pupil will master the relation between units, tens, hundreds, etc. When the child has no further use for the packages, the vertical or ordinary blackboard may be used to advantage. The art of reading numbers has grown out of the teaching of the writing of numbers so that the teacher has only the matter of punctuation to teach to the pupils, the use of the comma in point off written numbers in periods of three figures each.

It is unnecessary to give them more explicit instruction in regard to this work. The teacher who appreciates what we have said will adapt his instruction to the special needs of any school he may have charge of.

Nothing has been said about teaching Roman notation because that has ever received sufficient attention in the lower grades in the reading classes, and in classes where the teacher assigns lessons by the number of the chapter. It is a mistake and a waste of time to spend days in requiring pupils to write large numbers in Roman notation.

Weeks and months have been wasted in requiring pupils to explain long problems in addition. Some years ago there was published in the Michigan School Moderator an explanation of a problem in addition. This explanation was by a student in one of the lower grades of the Big Rapids city schools. The explanation involved the larger part of the space of one page of this valuable journal. The student was obliged to use many words in saying that a certain number of units, added to a given number of units, produced a certain number of units. Since ten units equals one ten, the total number of units equals so many tens and so many units. Write the units and carry the tens to the tens' column, etc., etc. We will not consume valuable space by further explanation. The work of adding should be mechanical. Mental discipline is important, but mental discipline cannot be secured by pursuing the method which the Big Rapids pupil illustrated. Life is too short at the longest, to be squandered in such a way. Teach the pupil to add each column rapidly. Under no circumstances allow the pupil to write on paper or on the blackboard the number to be carried. This must be held in his mind. He should be trained to verify every result. To do this the student should be required to add each column from the top downward. Many excellent bookkeepers employ the good old method of "casting out the nines". In case he has been well trained, he can be taught this method very easily.

In this work we wish to emphasize the importance of requiring the pupil to add rapidly. Boys and girls, men and women who say that they are "slow but sure", say what is untrue. The slow speller, the pupil who hesitates, is invariably a poor speller. Likewise students who hesitate in adding, do so because they have failed to master the combinations of numbers in oral adding.

The teacher should give exercises in adding, every week throughout the written arithmetic course. The reason is obvious. Skill in the counting room and in the commercial world lies in simple addition.

In closing this paper, I take great pleasure in making a few quotations from a monograph written by Professor T.H. Safford. He says: "A good many years in teaching college students in astronomy, both in western colleges and at Williams, leads me to think that the average man has been over, as the phrase is, more mathematics than he has digested or can apply. It is rare that I find such a pupil who is even accurate in arithmetic." I have taken great pains to emphasize the fact that primary arithmetic is fundamental. Professor Safford says: "In fact I can forsee the extent of the beneficent effect of genuinely good teaching of primary arithmetic. It seems to me almost as if it would revolutionize the whole mathematics up to quarternion; at least is combined with Pestalozzian teaching of geometry and drawing." I have taken great pains to emphasize the mental or oral arithmetic. Professor Safford says: "In all rational methods of teaching arithmetic, the mental or oral side of the subject is more important. Here again the inexperienced teacher is at a loss. It is comparatively easy to set sums from a book, and see if the answer agrees; and but little more difficult to consult the 'key'; not so easy, however, to make up, and control the solution of oral, extemporaneous examples.