In this series of articles, no effort will be made to give a history of arithmetic. We shall assume that the subject is taught in all rural schools and most of the village schools. According to reliable reports, the aggregate amount of time devoted to arithmetic is sufficient to warrant excellent results. Many successful teachers maintain that altogether too much time is devoted to this subject. We shall attempt to show why some of the results in teaching arithmetic are unsatisfactory. We shall attempt to show conclusively just how certain faults can be remedied.

Beyond question, the primary object to be realized in the study of arithmetic is mental discipline, -- mental power. Broadly speaking, the application of arithmetic to the demands of business are secondary. It is assumed that if the pupil acquires the power to think, the power to reach correct conclusions quickly and accurately, he cannot fail to use the power efficiently in business.

Our first inquiry is concerning the time the child can profitably begin the formal study of this subject. If the reader wishes to answer this question by appealing to the courses of study outlined for our schools, he will find, without exception, that arithmetic must be presented in the first year of the child's school life. In the first year the child masters all "possible combinations" to 10, learns the figures, learns the signs, and on his slate and at the blackboard uses graphically the language of arithmetic. The second year his work is continued to 20. In the third year he deals with the writing and reading of large numbers, does more or less adding, subtracting multiplying and dividing and solves problems which go a long way to convince the skeptical that the age of prodigies is not a thing of the past. Notwithstanding all this the high school graduate utterly fails to convince his employer that he even possesses an elementary knowledge of this important subject. True, this graduate knows large number of rules, knows how to do sums when he has answers for immediate reference, but so far as being able to meet the exacting demands of the cruel business world, he is not infrequently a failure. Beyond a doubt there is little hope of securing rapid, accurate "reckoners" so long as teachers follow the outline of the so-called teachers' manuals. In our attempt to determine when a child should begin the formal study of arithmetic, it might be well to recall not a few examples of the progress of backward boys and girls who at the age of twelve, thirteen or fourteen have begun the formal study of arithmetic.

What teacher of five years experience in district schools has not had two or three of these large boys and girls take up the study of arithmetic and in four months master the elements or arithmetic? These pupils did not neglect reading, spelling, grammar, geography and physiology. They did not possess extraordinary minds. What is the obvious explanation? I answer, that they were prepared to take up the study of arithmetic, in other words were ready for this subject. The child of five or six years of age is not prepared for the formal study of numbers. In his world there are thousands of things that command his attention. In using his five senses number is involved, but not to such an extent as to justify the waste of time that is now so generally indulged in. It is better to teach numbers incidentally, in connection with science lessons, language lessons, etc., giving such graphic representation as his life in the school room demands.

Next let us examine what should be presented when the child is thought to be prepared for taking up the formal study of numbers. When the teachers suggest that in the first year all "possible combinations" to 10 must be mastered they are asking for much that is useless, much that is positively injurious. If college professors of mathematics were questioned upon numbers up to 100 as children are questioned upon numbers to 10, upon numbers to 20, they would fail utterly. What can be done easily, successfully and delightfully? Most of the numbers to 10 can be broken into two parts. For example, four and four are eight, five and three are eight, six and two are eight, eight less one is seven, eight less six is two, etc., two fours are eight, in eight are two fours, and one-half of eight is four. These combinations are learned in relation to things. When the things are of no service to the child the combinations may be considered in the abstract. A large number of practical examples should be offered for solution. The answers will be given without hesitation. Later the manner of reading at sight. In these drills there will be no use of figures and signs. All of this early training will be oral, and the operations mental. The teacher who is never happy unless her pupils are bent upon some task will ask for busy work in numbers, will ask that slates and pencils, tablets and black-boards be used. Let the progressive teacher bear in mind that the first number work done by the child should be done in the immediate presence of the teacher and under her direct guidance. Habits are to be formed in this work which are sure to be life long. At this stage the use of arithmetical language written on slates and tablets is a positive hindrance to the child's progress. Suppose he is taught to represent numbers by the use of figures, suppose he is taught to indicate operations with signs, suppose he is taught to interpret signs, what is gained?

Few teachers can have failed to observe the boy who answers in the class with never failing accuracy, five and four are nine, not infrequently writes 5 plus 4 equals 8. In making this mistake he starts his computation with five and adds by counting ones, as he introduces into his work a paralyzing process. Sometimes he counts from 1 instead of the first number. This slate work does require time, does furnish "busy work", but at a frightful cost. Let us suppose the child has mastered the number 9 in relation to things. If he uses this new knowledge in writing his answers, he will place the answers instantly and consume so little time in the doing that his ten little problems given by his teacher on the board will consume very little time, so little time that the end sought by the teacher is unworthy of mention. This is an age in which the human mind no longer estimates time - man looks at his watch; he no longer estimates weight - he looks at the balances; he no longer estimates distance - he looks at the tape line; he no longer walks - he uses a bicycle, a street car; he no longer looks at the north star, he looks at a compass. Slates and pencils and tablets must be used on which to record his conclusions. If there is to be any important change for the better in arithmetic teaching it must begin along the line of using the mind more and the slates less. It is hoped that mental arithmetic, the only real arithmetic, may be restored to our schools.