  1. Use each of the numbers (1, 2, 3, 4, &c.) one at a time, and devise many different ways of illustrating and using each objectively. First, the perception of the number as a whole—then, the analysis of the number. Part of the work should involve subtraction. 2. Each number may be illustrated in many ways by large dots variously grouped on cards. With these cards drill the perception in quickness. Let each pupil arrange a certain number of “counters” in several ways of regular form. 3. On each number, ask every possible variety of question. Let the pupils make problems. Let some be made that are to have a certain given answer. 4. As each number is used, let its script form be learned and made by the pupils. After progressing in this way as far as 4 or 5 (some say to 9) teach the figures. Practice counting objects as far as 20. 5. The exercises for slate work should progress very gradually. A higher number should be introduced only after the pupils can use, with readiness, those below it, in their many and varied combinations. Let there be oral work consisting of easy objective problems illustrative of the slate work. 6. In the black-board work the teacher should use a pointer and call for ready and correct mental recitations, as he points to the various problems. 7. The exercises for slate work may be of several different kinds: as, (a) 1 and 1 are . 2 and 3 are . &c. (b) 5 less 1 are . 6 less 4 are . &c. (c) 2 and are 6. and 1 are 5. (d) 6 less are 5. less 2 are 2. (e) 1 1 1 2 2 1 1 2 2 2 1 2 3 2 3 _ _ _ _ _ The columns of (e) may contain from three to nine figures. The teacher must not lengthen them at any time beyond the ability of the pupils. (f) 1 1 1 1 1 1 1 1 1 2 1 3 9 4 5 6 8 7 _ _ _ _ _ _ _ _ _ The upper figure is the same in each: the lower figures are different and are arranged miscellaneously. In the advancement, increase the upper row a unit at a time, as far as 11. 8. Teach the use of the signs ×, -, and =, and let the pupils have slate work similar to the following:— 8 × 5 = . 9 - 6 = . &c. 12 - = 4. 12 - = 8. &c. + 6 = 9. + 3 = 9. &c. 9. Practice counting objects as far as 100, after which drill frequently in writing and reading the numbers, from the black-board, as far as 100. (a) 4 4 4 4 9 19 29 39 &c. _ __ __ __ (b) 7 7 7 7 8 68 18 88 &c. _ __ __ __ Let every possible combination be learned so well that the result can be given instantly. 11. For variety, along with the preceding, there may be used exercises similar to the following:— (a) 2 3 2 1 4 0 1 3 0 4 3 9 8 6 4 0 6 7 6 8 9 5 7 5 _ _ _ _ (b) 2 + 8 + 3 + 7 + 5 = . 4 + 9 + 6 + 1 + 3 = . (c) 21 41 22 32 63 33 64 63 53 __ __ __ “Carrying” may now be taught. 12. Practice writing and reading numbers of three, and four, figures. The pupils at the same time may be given exercises similar to the following:— 213 321 132 413 234 ___ 769 758 897 786 594 ___ Take the last example: the pupil should be taught to think through it rapidly, as follows:—4, 10, 17, 25, 34—write the 4 and carry the 3; 3, 12, 20, 29, 34, 40,—write the 0 and carry the 4; 4, 9, 16, 24, 31, 38; write the whole result. 13. Let the pupils learn to read numbers as high as millions. For a few examples, at first, in subtraction, let the numbers in each order of the minuend be greater than the corresponding ones in the subtrahend; as, 98 45 __ 1364 631 ____ 9842 3512 &c. ____ Use practical problems. 14. Next, those examples necessitating “borrowing” or “carrying” may be given; as, 137092 72348 ______ 6235 4879 ____ The method involving “carrying” is the better one. If equals be added to two numbers, their difference is not changed. In the last example, if 10 is added to 5, to equalize it add 1 to 7, for 10 units of one order equal one unit of the next higher. Adding the 1 to the 7 is called “carrying.” ··2 × 1 = 2 ::2 × 2 = 4 :: :2 × 3 = 6 :: :: 2 × 4 = 8 &c. &c. Let the pupils recite the tables orally. Use for drill the following problems:— 987654321 2 _________ 123456789 2 _________ With the problem on the board let the pupil recite without the aid of the answer. Similarly use the 3’s, 4’s, 5’s, &c. Along with this part of the work, how to multiply by a number of two or more figures may be taught. Placing the multiplication table in the compact rectangular form found in some arithmetics will be profitable and interesting work. 16. Teach the Roman notation to C; how to tell the time of day; how to make change with money; and how to solve easy exercises in pt., qt., pk., and bu.,—gi., pt., qt., and gal.—and in., ft., and yd. 17. The teacher, using a pointer, should drill the pupils thoroughly on the following table. (Try to acquire speed and correctness).
These constitute the multiplication table with the duplicate combinations cut out, leaving but 36 products to learn in the entire field of the common multiplication table. 18. Let the division tables now be learned. 2 into 2 one time . 2 into two times . 2 into three times . 2 into four times . 2 into five times . 2 into six times . 2 into seven times . 2 into eight times . 2 into nine times . 2 into ten times . Let the pupils fill the blanks. Let them learn how often 2 is contained in 5, 7, 9, 11, 13, 15, 17, and 19. Also, when the 3’s, 4’s, etc., are learned, use the intermediate numbers that give remainders. Drill in mental work. Give examples after each table is learned; as 2)563480 ________ 2)7104239 _________ Show how to write the remainder fractionally. Teach the meaning of ½, ?, and ¼. 19. Teach long division using easy graded examples. 15)180( 25)625( 13)168( 50)1150( 25)400( 115)32467( The pupils, with their knowledge of the multiplication table, by experimental work, and from suggestions by the teacher,—prepare their slate work as follows: The divisors of 24 are 2, 3, 4, 6, 8, and 12. The divisor of 25 is 5. The divisors of 26 are 2 and 13. The divisors of 27 are 3 and 9. The divisors of 28 are 2, 4, 7, and 14. 29 has no divisors. In the oral recitation, the first pupil, without referring to his slate, recites as follows:— The divisors of 24 are 2, 3, 4, 6, 8, and 12; 2 twelves are 24, 3 eights are 24, 4 sixes are 24, 6 fours are 24, 8 threes are 24, and twelve twos are 24. The next pupil recites as follows: The divisor of 25 is 5; 5 fives are 25. The third recites: The divisors of 26 are 2 and 13; 2 thirteens are 26, 13 twos are 26. The fourth recites: The divisors of 27 are 3 and 9; 3 nines are 27, 9 threes are 27. The fifth recites: The divisors of 28 are 2, 4, 7, and 14; 2 fourteens are 28, 4 sevens are 28, 7 fours are 28, and 14 twos are 28. The sixth recites: 29 has no divisors; it is a prime number—a number that can be exactly divided only by itself and unity. |
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