After the class have caught the analogy between the simple words, and the compounds which they form, so that one instantly suggests to them the other, then the order of the words may be changed, and the ten put arst. The caution here suggested may seem to some nnnecessary; but a careful observation of the menial habits of children will not fail, I think, to show its importance. In the analysis of the compound words from ten to twenty, e.even and twelve should be omitted till the last; for as the simple words of which they are formed are disguised or obsolete, they tend to obscure, rather than elucidate the subject to the mind of a child. Having obtained the idea through the other words in the series, he may take the statement respecting these on trust. LESSON IV. Having counted twenty, and grouped the number in two tens, let the class count ten more, making in all thirty, or three tens. Keeping the tens separate, let the class count ten more, making forty, or four tens. Let the class then answer such questions as the following: Twenty are how many tens ? Thirty are how many tens? Forty are how many tens ? Four tens are what number? Three tens are what number? Two tens are what number? After this, they may proceed with the higher multiples of ten, fifty, sixty, seventy, eighıy, ninety, a hundred. Through the whole of this exercise, each multiple of ten should be presented in groups of ten, so as to aid the idea by the visible representation. The pupils should be led to see the significancy of each numer. ical name; that thirty-seven, for example, means three tens and seven ; fifty-six means five tens and six. In this way the pupils may be led to understand the Decimal Ratio, at this early stage, and no further trouble need be taken in that direction. When in a later stage of study, he comes to the Decimal notation in written Arithmetic, he will find it only a natural mode of expressing ideas already rendered familiar in practice. LESSON V. Let the teacher stand at the board, and call the attention of the class to what he shall write ; then, making two marks, ask, " How many marks on the board ?" When the class have answered, let the teacher write two more, and ask, “ How many now ?" and so on to the number of twelve or more. Then take a writing book or sheet of paper, and covering all but two of the marks, let the class repeat the same process while the teacher removes the book, so as to bring two more into view at each remove; he numbers read by the class being two, four, six, eight, ten, &c. Then let the process be reversed, subtracting two successively, which gives, beginning with sixteen, the following, – sixteen, fourteen, twelve, ten, &c. Again the teacher may say to the class, “When I made those marks how many did I make at a time?” Class : “ Two.”. Teacher: "Did I make two more than once ?" Class : “Yes, sir, a good many times.” Then the teacher, covering up all but two: " Now look, how many times two are there?" °Class : "Once." Teacher: "Once iwo are how many ?” Then, after the class have answered, showing two more, " How many times two do you see?" “ Twice two are how many ?” Then go on in the same way with three twos, four iwos, &c., to the end. At this point the pupils may be taught the distinction between even and odd numbers, and be trained to repeat rapidly the even numbers, from two up to twenty. The pupils may derive important aid in adding and multiplying, by grouping the numerical names with the voice, in something like the following manner. Teacher: “Listen now to me; one, tro— three, four -- five, six. How many twos did I count ? Class: “Three twos.” Teacher: “Count three twos just as I did.” Then let the teacher ask, “ Three times two are how many ?" Then require them to count four twos, five twos, &c. This method may be brought in aid of all their earlier attempts in adding and multiplying. Care should be taken in this exercise that the words be pronounced in a quick and neat manner, with a distinct pause between the groups. If rightly conducted, this exercise furnishes one of the best aids for overcoming the habit of counting, instead of adding – that fatal clog which keeps back many scholars in arithmetic from making any rapid progress. The successive additions of two, beginning || eleven. with one, furnish the series of odd numbers. || nine. This and the preceding exercise exhibit all the || seven. results made by the additions of the number | | five, two. || three. | one. The pupil should, in connection with this exercise, be taught to group the numerical names, hy iwos, beginning from one, so as to Turnish the series of odd numbers, thus : one — iwo, three — four, five, &c. Finally, let him count the even series, up to twenty, and then down, till he can do it readily. Let him count the odd series up to twenty-one, up and down. LESSON VI. To teach the pupils to add the number three, let the teacher make three marks on the board, and require the class to name the number; then three || | fifteen. II | twelve. more, and so on. Then, covering all but || nine. the lower three, let the class name the || | six. number in view, as the teacher exhibits I|| three The pupils may then be led to count in groups of three; thus, one, two, three -- four, five, six - seven, eight, nine, &c. From this, they may be readily led to name the multiples of three ; three, six, nine, twelve, &c. The first variation from the above method in the addition of three inay be made by beginning with 11 | fourteen. two, which gives the following numbers. II | eleven. In order to secure the necessary amount ||| eight. of repetition in these exercises, and yet 11 | five, to avoid weariness, it will be well ofien | | two. to change the manner of adding. Thus, in the accompanying example, the pupils may simply name the series made by ihe successive additions ; as two, five, eight, eleven, fourteen ; or, they may state the process more fully, thus: two and three are five, and three are eight, and three are eleven., &c. A still fuller statement of the operation would be thus ; two and three are five, five and three are eight, eight and three are eleven, &c. The only remaining variety in adding three will be exhibited by beginning with one, which may be illustrated in the same way as the others, giving the numbers, one, four, seven, ten, &c. LESSON VII. After the numbers have been mastered in the manner above suggested, they should be combined in a variety of ways by means of marks on ihe board. The columns of marks subjoined are given merely as examples and hints on this point. A single column may be made to furnish exercise sufficient for a recitation ; for, hy adding or erasing a single mark at the begin. ning, a change is made through the whole., ARITHMETIC. PART I. SECTION I. A.* 1. How many thumbs have you on your right hand ? how many on your left? how many on both together? 2. How many hands have you ? 3. If you have two nuts in one hand, and one in the other, how many in both ? 5. If you count the thumb with the fingers, how many will it make? 6. If you shut your thumb and one finger, and leave the rest open, how open ? 7. If you have two cents in one hand, and two in the other, how many have in both ? 8. James has two apples, and William has three ; if James gives his apples to William, how many will William have ? 9. If you count all the fingers on one hand, and two on the other, how many will there be ? 10. George has three cents, and Joseph has four; how many have they both together? * The first questions in this gection are intended for very young children. It will be well for the instructor co give a great many more of this kind. Older pupils may omit these. the orange 11. Robert gave five cents for an orange, and two for an apple; how many did he give for both ? 12. If a custard cost six cents, and an apple two cents, how many cents will it take to buy an apple and a custard ? 13. If you buy a pint of nuts for five cents, and an orange for three cents, how many cents would you give for both ? how many more for the nuts than for ? 14. If an ounce of figs is worth six cents, and a half a pint of cherries is worth three cents, how much are they both worth? 15. Dick had five plums, and John gave him four more; how many had he then ? 16. How many fingers have you on both hands? 17. How many fingers and thumbs have you on both hands? 18. If you had six marbles in one hand, and four in the other, how many would you have in the one more than in the other ? how many have in both hands? 19. David had seven nuts, and gave three of them to George ; how many had he left? 20. 'Two boys, James and Robert, played at marbles ; when they began, they had seven apiece, and when they had done, James had won four; how many had each then? 21. A boy, having eleven nuts, gave away three of them; how many had he left ? 22. If you had eight cents, and your papa should how many would you have ? 23. A man bought a sheep for eight dollars, and a calf for seven dollars ; what did he give for both ? 24. A man bought a barrel of four for eight dollars, and sold it for four dollars more than he gave for it; how much did he sell it iur? give you five more, |