are passed across, we may say, “ There is one ten.” Then, passing one across on the second wire, There is oneteen ; another will make twoteen ; three, threeteen ; four, fourteen ;' &c., to nineteen; and passing the last one across,“ Now we have twoty.” Again, by passing the beads of the third wire singly across we shall have twoty-one, twoty-two, &c., to threety; and, continuing the operation, fourty, fivety, sixty, &c., to ninety-nine, the last bead giving tenty, or a hundred. When the class has thus become familiar with the nomencla. ture, the abbreviations, which are few and simple, should be explained. The most difficult are the first two, oneteen, twoteen, which somehow have been changed to eleven, twelve. The rest are mere abbreviations. Thus, threeteen and threety have been shortened to thirteen and thirty; fiveteen and firety to fifteen and fifty; lastly, twoty (originally twainty) is now twenty, and fourty is forty. This exercise is chiefly intended for those who have no knowledge of arithmetic. But it would be well for the whole school to go over the frame once or twice, as few children have such a clear and abiding idea of the meaning of teen or ty as may be thence derived. When a child is at a loss for a reply to a question, it is not advisable to give him direct assistance. Such a course is apt to foster a habit of leaning on the teacher instead of his own It is far better to aid him by a few suggestive questions. Suppose, for instance, he is at a loss for the amount of 52 and 27, it would be appropriate to ask, “How many are 50 and 20 ?" “Well, then, how many are 52 and 27 ? " Again, if he cannot answer, “Twelve are four times what number ?” he might be asked, “How many fours in twelve ?” Well, then, twelve are four times what number ?” Such instances, however, will rarely occur where the attention of a class is kept fully engaged during these exercises. Every class in oral arithmetic should clearly understand that, during recitations, no question is to be repeated; a rule, indeed, that should never be broken. At the commencement of the study it might be proper for one of the class by turns to repeat each question after the teacher. But even this practice should not be continued too long. It is more important that children should be good listeners than apt arithmeticians. Let them learn, then, to listen with sufficient attention to hold on resources, to what they hear, and they will acquire a habit that will exert a beneficial influence on the whole of after life. * The questions should not be addressed to an individual, but to a class, so that every mind may be steadily engaged during the whole recitation. Children should never be fatigued with long lessons, but, while they are engaged, they should be thoroughly engaged. Let no habits of dreamy wandering of mind be acquired in school. If already acquired, they cannot too soon be broken up. Every week's delay will add to the difficulty. Let each question, then, be solved mentally by every individual in the class, and let the teacher wait till all have given the signal, before any one is called on to state the result, and this call should be made promiscuously, not in regular order, the more certainly to ensure the attention of all. When an answer is given, the class should be asked if all agree; if not, every one who differs should give the result of his solution. Let the teacher then ask the first who spoke, “How do you know it to be so and so ? " requiring an explanation of the manner in which it has been solved; those who differ being called on for the same purpose. The children should be required to give these explanations in a distinct and lucid style. Suppose, for instance, the question to be this : John has 6 nuts, Mary has 2, and James 4; how many nuts have the three children in all? The answer, “ 12 nuts,” being given, and a child called on to state why it is so, may say, “If John has 6 nuts, and Mary 2, John and Mary together have 8, because 6 and 2 make 8; and, if we add James's 4, there will be 12, because 8 and 4 make 12.” Such minuteness of detail may to some seem tedious and unnecessary; but, if the inappreciable * At a convention of teachers, where the subject of oral arithmetic in schools was in discussion, a clergyman rose to give his experience in the matter. All are aware, said he, of the difficulty of maintaining the undivided attention of an auditory, even on subjects on which they feel a deep interest. The opening of a door, the restlessness of a child, the fall of a book : either of these is sufficient to distract the thoughts of a whole congregation. Now a course of oral arithmetic, properly conducted, presents a sure remedy for this serious evil. To classes thus trained, he continued, he could talk or read for half an hour at a time, and receive unwavering attention, although persons were going out and in, backwards and forwards, nearly the whole time. Surely, when we consider how prone are mankind to wandering thoughts, how common it is for all “ to hear a little, and guess the rest,” we can scarcely overvalue so simple a cure for this vicious habit, which obstructs education alike in the primary school and the college, and exhibits its baneful effects in every grade of social life. value of habits of close attention and of clearness of diction be taken into view, the labor will not be considered in vain. Let this system, then, of questioning by the teacher, and demonstration by the class, be steadily persevered in wherever practicable. Finally, let this plan be thoroughly executed, without hurry, and success may be relied on. What man has done, man can do. A child taught thus will possess an immense, an obvious superiority over his fellows. In such practice, the mind is continually on the alert. That sluggish mental inactivity, which is the bane of our schools, and which nullifies most of their good effects, is completely banished. The powers of thought are kept constantly awake, steadily in employment. The pupil is not merely made an expert mathematician, but, what is of infinitely more importance, he is preparing the way for self-education, — he is acquiring the control over his own mind, one of the rarest and most valuable of all acquisitions. N. B. None of the questions in this part of the book are to be worked out on the slate. It is to be, strictly speaking, oral arithmetic, with occasional illustrations by the teacher on the Frame and Black-board. CHAPTER I. INCREASE AND DECREASE OF INTEGERS, OR WHOLE NUMBERS. EXERCISES ON THE NUMERAL FRAME. SECTION I. Increase and Decrease by Unity. 1. ONE bead and one bead are how many beads ? Two beads and one bead? Three beads and one bead ? &c. [Continue to nine beads and one bead.] 2. If one bead be taken from ten beads, how many are left ? One bead from nine beads, how many? One bead from eight beads, how many ? &c. [Continue to one bead from one bead.] 3. One bead and one bead, how many ? One bead and two beads ? One bead and three beads ? &c. [Continue to one bead and nine beads.] [Repeat the above till it becomes sufficiently familiar without the frame. Then repeat it again in abstract numbers, by omitting the word bead, till equally familiar. To a dull class, the word bead may be occasionally used in the following lessons.] 4. One and one are how many ? Take one from two, and how many remain ? Two from two ? How many are two times one ? [Here separate the two beads a little.] One time two ? Are two times one and one time two the same number, then? How many ones in two ? Twos in two ? 5. Two and one are how many ? Take one from three, and how many remain ? Three from three? How many are three times one ? One time three ? Are three times one and one time three the same, then? How many ones in three ? Twos in three ? [One two and one over.] Threes in three ? 6. Three and one, how many ? Take one from four, how many remain ? Two from four ? How many are two and two, then? Four from four ? How many are four times one ? One time four ? Are four times one and one time four the same, then? How many ones in four ? Twos in four ? Threes in four ? [Separate the beads, or rather let one of the class separate them, at all such questions as the last two, till the class can separate them by the eye.] Fours in four ? 7. Four and one, how many ? Take one from five, how many ? Two from five? Three from five? How many are two and three, the ? Three and two ? Four from five ? Five from five? How many are five times one ? One time five ? Are five times one and one time five the same, then ? How many ones in five ? Twos? Threes? Fours? Fives? 8. Five and one, how many ? One from six, how many ? Two from six ? How many are two and four, then? How many are four and two ? Are two and four, then, the same as four and two ? Take three from six, how many ? Three and three, how many, then? How many threes in six, then ? Take four from six ? Five from six ? Six from six ? are six times one ? One time six ? Are six times one and one time six the same, then? How many ones in six ? Twos ? Threes? Fours? Fives? Sixes ? 9. Six and one, how many ? One from seven, how Two from seven ? How many are two and five, then ? Five How many many ? and two ? Are two and five the same as five and two, then ? Three from seven, how many? How many are three and four, then? Four and three ? Are three and four, then, the same as four and three ? Take four from seven? Five from seven? Six from seven ? Seven from seven? How many ones in seven ? Twos? Threes? Fours? Fives? Sixes? Sevens ? 10. Seven and one, how many ? Take one from eight, how many ? Two from eight? How many are six and two, then ? Two and six? Take three from eight? How many are five and three, then? Three and five ? Take four from eight? How many are four and four, then? Two fours ? Take five from eight? Six from eight? Seven from eight? Eight from eight? How many ones in eight? Twos ? Threes? Fours? Fives? Sixes? Sevens? Eights ? 11. Eight and one, how many ? Take one from nine, how many ? Two from nine? How many are seven and two, then ? Two and seven ? Take three from nine? How many are three and six, then? Six and three ? Take four from nine? How many are four and five, then? Five and four? Take six from nine? Seven? Eight? Nine? How many ones in nine? Twos? Threes? Fours? Fives? Sixes? Sevens ? Eights? Nines ? 12. Nine and one, how many ? Take one from ten ? Two from ten? How many are eight and two, then ? Two and eight ? Take three from ten? How many are three and seven, then? Seven and three ? Take four from ten ? are six and ur, then? Four and six ? Take five from ten ? How many fives in ten, then ? Six from ten? Seven from ten? Eight? Nine ? Ten ? How many ones in ten? Twos ? How many are five twos, then? Two fives? How many threes in ten? Fours ? Fives? Sixes ? Sevens ? Eights ? Nines ? Tens? 13. Ten and one, how many? One from eleven, how many ? Two from eleven? How many are nine and two, then? Two and nine? Three from eleven? How many are eight and three, then ? Three and eight? Four from eleven? How many are seven and four ? Four and seven? Five from eleven? How many are six and five, then? Five and six? Six from eleven ? Seven from eleven? Eight? Nine? Ten ? Eleven ? How many ones in eleven? Twos ? Threes? Fours ? Fives? Sixes ? Sevens, &c., to Elevens? 14. Eleven and one, how many ? One from twelve ? Two How many |