the necessity of always giving it. When the question is proposed, the scholar should answer it, and give the process imme diately, without being called upon so to do. The extent to which the question "How do you know?" should be repeated, must be left to the discretion of the teacher. It should, in all cases, be repeated until the pupil clearly perceives and is familiar with his reasons. A neglect properly to train the mind to systematic thinking, has operated, probably, more than any other one cause to enfeeble the mental energies of our whole race. Teachers, would you secure the improvement of your pupils? seek to improve yourselves. Be students in mental philosophy. Nor can you avail yourselves of a more favorable opportunity to pursue the study of this most interesting science, than by devo ting a portion of your time to the instruction of the young. Those innumerable questions of children, which to others are unmeaning, will to you be full of interest. They are the effect of a cause, and that cause you will seek to know. They are questions which the wisest of men would ask, ignorant of the same facts. The little boy who says he is not five years old, but knows that he is six, would be as much pleased as any one else to hear the remark made by another, did he himself know the order in which the names of numbers are applied to things. He does not know that he cannot be six until he is five; for he has not learned but that he can commence at the other end of the series, in which case he would come to six first. Again: the little boy stretches himself a little above his ordinary height, and asks if he is not-six years old. This child evidently confounds age with height. Nor is it certain that his previous instruction has not led him into this mistake. The father had, undoubtedly, told the child, that when he was six years old, "a big boy," elevating his hand at the same time to a certain height, that he would make him some present: thus virtually telling him, that height and age were synonymous terms. It is not unfrequently the case, that if you require the child, when he has learned to repeat the names of numbers, to count your fingers, that he will place his finger on unity, or one, and count five: thus showing that he does not know that the names of numbers apply to distinct units placed in a certain order. And even when he has learned to count, he not unfrequently supposes that the number which he calls five, for example, would be five if standing alone, as John is the name of a boy. This mistake arises from his misapprehending the names of numbers for the names of things. Nor can the child always tell, even when he has learned to count, how many are one and one. This is a new mode of combining numbers; it contains a new idea, and is a thing to be learned. The foregoing facts, and many more which might be mentioned, are well known to those familiar with children. Hence, when the teacher has written the formula upon the blackboard, he will direct the attention of the class to the following particulars: Teacher, The class will notice these short lines, which I have drawn upon the board. We will call them units, or ones. They may be used to represent number, and this number may be of things of any kind, as chairs, tables. If I wished to inform you that I had two apples, I might do so by drawing two lines, which would represent their number; and these same lines might be used to represent two things of any other kind. These lines here show how many units the figures written under them stand for. We make use of the figures for the same purpose as we do the lines, to represent number. It is a shorter process to write a figure than to make as many lines as a single figure would represent. You will also notice, that as you count, every new name applied, or unit counted, adds one unit to the preceding; this is one mode of adding. You will observe, also, that if I commence at the left hand and count five, and then count the same things back, from right to left, the names of the numbers are changed; the unit which I first called one, I now call five. The names, therefore, which we use in counting, are names of numbers, and not names of things; and it depends upon the place a thing oocupies in relation to other things, what number it shall be called. The number two comes after one, and three after two, etc. John, how many more than one is two? How many, then, are one and one? Then one from two, how many remain? You must remember what I have now told you, for it will aid you in pre paring your next lesson. The teacher will do well, at the commencement of each les son, to review the leading thoughts of the previous lesson; for it is the design of this work to appropriate to immediate use every item of knowledge gained. Teachers, you will have need of patience and perseverance; your labor is arduous, and often your prospects of success are discouraging. But your calling is high, it is noble. Your Great Exemplar was He by whom the worlds were made. Be encouraged, therefore; your reward is before you. Let the assistance which you render your pupils, be an aid to their own mental efforts, and not a substitute for them. Come down to the child, and lead him up higher; accompany him in his upward progress, and do not undertake to force him to come up alone. Employ suitable illustrations on subjects suited to his capacities. Let him behold truth in pleasing variety, robbed of none of its intrinsic beauty from a want of interest on your part. Feed him with knowledge; lead him daily by still waters and in green pastures. MENTAL ARITHMETIC. SECTION I. Teacher. Well, my little scholars, you are about to commence a new study, and although it is the study of Arithmetic, yet you are not to make use of pencil and slate, as older scholars do, but you are to work out the questions in your mind. George, do you know what I mean by working them in your mind? S. In our thoughts. T. Very well, and when you have worked them out in your thoughts, you will then express, to your teacher, those thoughts in words, and this will be a recitation. James, when I say the class in mental arithmetic may recite, what will you understand me to mean? S. That we may express to you, in words, our thoughts upon our lesson. T. And what is the object of expressing to your teacher your thoughts upon your lesson ? S. That we may know whether our thoughts have been correct. T. We will then proceed to our FIRST LESSON. Teacher. George, can you count? Scholar. I can. T. Let me hear you. S. One, two, three, four, etc. T. Do you know the names of figures? S. I do. T. What is the name of this figure or character, 1? S. One. T. What of this, 2? S. Two. 7. What of these, 3, 4, 5? S. Three, four, five? T. What does the figure two stand for? S. Two ones; as two chairs. T. What does the figure three stand for? S. Three ones. T. What does the figure five stand for? S. Five ones. T. As you count from one to five, how many ones do you add for each figure counted? S. One. 7. Very well. I will illustrate this to you on the blackboard. (See explanation of the formula.) 1. 11. 111. 1111. 11111. LESSON SECOND. Teacher. How many more than one is two? T. How many more than two is three? S. One. T. How many more than one is three? S. Two. T. How do you know? S. I know, because two is one more than one, and three is one more than two; therefore three is two more than one. T. How many more than three is four ? T. How many more than two is four ? S. Two. |