Chapter VI The Psychology of Drill in Arithmetic: The Amount of Practice and the Organization of Abilities
The Amount of Practice
It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion.
Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive (that is, in the first two books of a three-book series) in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, “1 and 1 are 2,” “1 and 2 are 3,” etc.? Every case of the action of each bond is to be counted.
THE FUNDAMENTAL BONDS
| 1 + 1 | 2 − 1 | 1 × 1 | 2 ÷ 1 |
| 1 + 2 | 2 − 2 | 2 × 1 | 2 ÷ 2 |
| 1 + 3 | 3 × 1 | ||
| 1 + 4 | 4 × 1 | ||
| 1 + 5 | 3 − 1 | 5 × 1 | 3 ÷ 1 |
| 1 + 6 | 3 − 2 | 6 × 1 | 3 ÷ 2 |
| 1 + 7 | 3 − 3 | 7 × 1 | 3 ÷ 3 |
| 1 + 8 | 8 × 1 | ||
| 1 + 9 | 9 × 1 | ||
| 4 − 1 | 4 ÷ 1 | ||
| 4 − 2 | 4 ÷ 2 | ||
| 11 (or 21 or 31, etc.) + 1 | 4 − 3 | 1 × 2 | 4 ÷ 3 |
| 11 ″ + 2 | 4 − 4 | 2 × 2 | 4 ÷ 4 |
| 11 ″ + 3 | 3 × 2 | ||
| 11 ″ + 4 | 4 × 2 | ||
| 11 ″ + 5 | 5 − 1 | 5 × 2 | 5 ÷ 1 |
| 11 ″ + 6 | 5 − 2 | 6 × 2 | 5 ÷ 2 |
| 11 ″ + 7 | 5 − 3 | 7 × 2 | 5 ÷ 3 |
| 11 ″ + 8 | 5 − 4 | 8 × 2 | 5 ÷ 4 |
| 11 ″ + 9 | 5 − 5 | 9 × 2 | 5 ÷ 5 |
| 6 − 1 | 1 × 3 | 6 ÷ 1 | |
| 2 + 1 | 6 − 2 | 2 × 3 | 6 ÷ 2 |
| 2 + 2 | 6 − 3 | 3 × 3 | 6 ÷ 3 |
| 2 + 3 | 6 − 4 | 4 × 3 | 6 ÷ 4 |
| 2 + 4 | 6 − 5 | 5 × 3 | 6 ÷ 5 |
| 2 + 5 | 6 − 6 | 6 × 3 | 6 ÷ 6 |
| 2 + 6 | 7 × 3 | ||
| 2 + 7 | 8 × 3 | ||
| 2 + 8 | 7 − 1 | 9 × 3 | 7 ÷ 1 |
| 2 + 9 | 7 − 2 | 7 ÷ 2 | |
| 7 − 3 | 7 ÷ 3 | ||
| 7 − 4 | 1 × 4 | 7 ÷ 4 | |
| 12 (or 22 or 32, etc.) + 1 | 7 − 5 | 2 × 4 | 7 ÷ 5 |
| 12 ″ + 2 | 7 − 6 | and so on | 7 ÷ 6 |
| 7 − 7 | to 9 × 9 | 7 ÷ 7 | |
| and so on to | and so on | and so on to | |
| 9 + 9 | to 18 − 9 | 82 ÷ 9 83 ÷ 9, etc. | |
| 19 (or 29 or 39, etc.) + 9 |
If estimating for the entire series is too long a task, it will be sufficient to use eight or ten from each, say:—
| 3 + 2 | 13, 23, etc. + 2 | 7 + 2 | 17, 27, etc. + 2 |
| ″ 3 | ″ 3 | ″ 3 | ″ 3 |
| ″ 4 | ″ 4 | ″ 4 | ″ 4 |
| ″ 5 | ″ 5 | ″ 5 | ″ 5 |
| ″ 6 | ″ 6 | ″ 6 | ″ 6 |
| ″ 7 | ″ 7 | ″ 7 | ″ 7 |
| ″ 8 | ″ 8 | ″ 8 | ″ 8 |
| ″ 9 | ″ 9 | ″ 9 | ″ 9 |
| 3 − 3 | 7 − 7 | 9 × 7 | 63 ÷ 9 |
| 4 ″ | 8 ″ | 7 × 9 | 64 ″ |
| 5 ″ | 9 ″ | 8 × 6 | 65 ″ |
| 6 ″ | 10 ″ | 6 × 8 | 66 ″ |
| 7 ″ | 11 ″ | 67 ″ | |
| 8 ″ | 12 ″ | 68 ″ | |
| 9 ″ | 13 ″ | 69 ″ | |
| 10 ″ | 14 ″ | 70 ″ | |
| 11 ″ | 15 ″ | 71 ″ | |
| 12 ″ | 16 ″ |
TABLE 2
Estimates of the Amount of Practice Provided in Books I and II of the Average Three-Book Text in Arithmetic; by 50 Experienced Teachers
| Arithmetical Fact | Lowest Estimate | Median Estimate | Highest Estimate | Range Required to Include Half of the Estimates |
|---|---|---|---|---|
| 3 or 13 or 23, etc. + 2 | 25 | 1500 | 1,000,000 | 800-5000 |
| ″ ″ 3 | 24 | 1450 | 80,000 | 475-5000 |
| ″ ″ 4 | 23 | 1150 | 50,000 | 750-5000 |
| ″ ″ 5 | 22 | 1400 | 44,000 | 700-5000 |
| ″ ″ 6 | 21 | 1350 | 41,000 | 700-4500 |
| ″ ″ 7 | 21 | 1500 | 37,000 | 600-4000 |
| ″ ″ 8 | 20 | 1400 | 33,000 | 550-4100 |
| ″ ″ 9 | 20 | 1150 | 28,000 | 650-4500 |
| 7 or 17 or 27, etc. + 2 | 20 | 1250 | 2,000,000 | 600-5000 |
| ″ ″ 3 | 19 | 1100 | 1,000,000 | 650-4900 |
| ″ ″ 4 | 18 | 1000 | 80,000 | 650-4900 |
| ″ ″ 5 | 17 | 1300 | 80,000 | 650-4400 |
| ″ ″ 6 | 16 | 1100 | 29,000 | 650-4500 |
| ″ ″ 7 | 15 | 1100 | 25,000 | 500-4500 |
| ″ ″ 8 | 13 | 1100 | 21,000 | 650-3800 |
| ″ ″ 9 | 10 | 1275 | 17,000 | 500-4000 |
| 3 − 3 | 25 | 1000 | 100,000 | 500-4000 |
| 4 − 3 | 20 | 1050 | 500,000 | 525-3000 |
| 5 − 3 | 20 | 1100 | 2,500,000 | 650-4200 |
| 6 − 3 | 10 | 1050 | 21,000 | 650-3250 |
| 7 − 3 | 22 | 1100 | 15,000 | 550-3050 |
| 8 − 3 | 21 | 1075 | 15,000 | 650-3000 |
| 9 − 3 | 21 | 1000 | 15,000 | 700-2600 |
| 10 − 3 | 20 | 1000 | 20,000 | 600-2500 |
| 11 − 3 | 20 | 1000 | 15,000 | 465-2550 |
| 12 − 3 | 18 | 1000 | 15,000 | 650-2100 |
| 7 − 7 | 10 | 1000 | 18,000 | 425-3000 |
| 8 − 7 | 15 | 1000 | 18,000 | 413-3100 |
| 9 − 7 | 15 | 950 | 18,000 | 550-3000 |
| 10 − 7 | 15 | 950 | 18,000 | 600-3950 |
| 11 − 7 | 10 | 900 | 18,000 | 550-3000 |
| 12 − 7 | 10 | 925 | 18,000 | 525-3100 |
| 13 − 7 | 10 | 900 | 18,000 | 500-2600 |
| 14 − 7 | 10 | 900 | 18,000 | 500-3100 |
| 15 − 7 | 10 | 925 | 18,000 | 500-3000 |
| 16 − 7 | 10 | 875 | 18,000 | 500-2500 |
| 9 × 7 | 10 | 700 | 20,000 | 500-2000 |
| 7 × 9 | 10 | 700 | 20,000 | 500-1750 |
| 8 × 6 | 10 | 750 | 20,000 | 500-2500 |
| 6 × 8 | 9 | 700 | 20,000 | 500-2500 |
| 63 ÷ 9 | 9 | 500 | 4,500 | 300-2500 |
| 64 ÷ 9 | 9 | 200 | 4,000 | 100- 700 |
| 65 ÷ 9 | 8 | 200 | 4,000 | 100- 600 |
| 66 ÷ 9 | 7 | 200 | 4,000 | 100- 550 |
| 67 ÷ 9 | 7 | 200 | 4,000 | 75- 450 |
| 68 ÷ 9 | 6 | 200 | 4,000 | 87- 575 |
| 69 ÷ 9 | 6 | 200 | 4,000 | 87- 450 |
| 70 ÷ 9 | 5 | 200 | 4,000 | 75- 575 |
| 71 ÷ 9 | 5 | 200 | 4,000 | 75- 700 |
| XX | 40 | 550 | 1,000,000 | 300-2000 |
| XO | 20 | 500 | 11,500 | 150-2000 |
| XXX | 15 | 450 | 12,000 | 100-1000 |
| XXO | 25 | 400 | 15,000 | 150-1000 |
| XOO | 15 | 400 | 5,000 | 100-1000 |
| XOX | 10 | 400 | 10,000 | 100- 975 |
Having made his estimates the reader should compare them first with similar estimates made by experienced teachers (shown on page 124 f.), and then with the results of actual counts for representative textbooks in arithmetic (shown on pages 126 to 132).
It will be observed in Table 2 that even experienced teachers vary enormously in their estimates of the amount of practice given by an average textbook in arithmetic, and that most of them are in serious error by overestimating the amount of practice. In general it is the fact that we use textbooks in arithmetic with very vague and erroneous ideas of what is in them, and think they give much more practice than they do.
The authors of the textbooks as a rule also probably had only very vague and erroneous ideas of what was in them. If they had known, they would almost certainly have revised their books. Surely no author would intentionally provide nearly four times as much practice on 2 + 2 as on 8 + 8, or eight times as much practice on 2 × 2 as on 9 × 8, or eleven times as much practice on 2 − 2 as on 17 − 8, or over forty times as much practice on 2 ÷ 2 as on 75 ÷ 8 and 75 ÷ 9, both together. Surely no author would have provided intentionally only twenty to thirty occurrences each of 16 − 7, 16 − 8, 16 − 9, 17 − 8, 17 − 9, and 18 − 9 for the entire course through grade 6; or have left the practice on 60 ÷ 7, 60 ÷ 8, 60 ÷ 9, 61 ÷ 7, 61 ÷ 8, 61 ÷ 9, and the like to occur only about once a year!
TABLE 3
Amount of Practice: Addition Bonds in a Recent Textbook (A) of Excellent Repute. Books I and II, All Save Four Sections of Supplementary Material, to be Used at the Teacher’s Discretion
The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times, 22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Total | |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 226 | 154 | 162 | 150 | 97 | 87 | 66 | 45 | |
| 12 | 74 | 53 | 76 | 46 | 51 | 37 | 36 | 33 | |
| 22, etc. | 50 | 60 | 68 | 63 | 42 | 50 | 38 | 26 | |
| 3 | 216 | 141 | 127 | 89 | 82 | 54 | 58 | 40 | |
| 13 | 43 | 43 | 60 | 70 | 52 | 30 | 22 | 18 | |
| 23, etc. | 15 | 30 | 51 | 50 | 42 | 32 | 29 | 30 | |
| 7 | 85 | 90 | 103 | 103 | 84 | 81 | 61 | 47 | |
| 17 | 35 | 25 | 42 | 32 | 35 | 21 | 29 | 16 | |
| 27, etc. | 30 | 23 | 32 | 29 | 24 | 23 | 25 | 28 | |
| 8 | 185 | 112 | 146 | 99 | 75 | 71 | 73 | 61 | |
| 18 | 28 | 35 | 52 | 46 | 28 | 29 | 24 | 14 | |
| 28, etc. | 53 | 36 | 34 | 38 | 23 | 36 | 27 | 27 | |
| 9 | 104 | 81 | 112 | 96 | 63 | 74 | 58 | 57 | |
| 19 | 13 | 11 | 31 | 38 | 25 | 14 | 22 | 11 | |
| 29, etc. | 19 | 17 | 27 | 20 | 32 | 32 | 19 | 18 | |
| 2, 12, 22, etc. | 350 | 277 | 306 | 260 | 190 | 174 | 140 | 104 | 1801 |
| 3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 | 88 | 1406 |
| 7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 | 91 | 1109 |
| 8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102 | 1354 |
| 9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 | 99 | 86 | 994 |
| Totals | 1164 | 921 | 1125 | 972 | 753 | 671 | 687 | 471 |
TABLE 4
Amount of Practice: Subtraction Bonds in a Recent Textbook (A) of Excellent Repute. Books I and II, All Save Four Sections of Supplementary Material, to be Used at the Teacher’s Discretion
| Minuends | Subtrahends | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 1 | 372 | ||||||||
| 2 | 214 | 311 | |||||||
| 3 | 136 | 149 | 189 | ||||||
| 4 | 146 | 142 | 103 | 205 | |||||
| 5 | 171 | 91 | 92 | 164 | 136 | ||||
| 6 | 80 | 59 | 69 | 71 | 81 | 192 | |||
| 7 | 106 | 57 | 55 | 67 | 59 | 156 | 80 | ||
| 8 | 73 | 50 | 50 | 75 | 50 | 62 | 48 | 152 | |
| 9 | 71 | 75 | 54 | 74 | 48 | 55 | 55 | 124 | 133 |
| 10 | 261 | 84 | 63 | 100 | 193 | 83 | 57 | 124 | 91 |
| 11 | 48 | 31 | 50 | 36 | 41 | 32 | 46 | 35 | |
| 12 | 48 | 77 | 57 | 51 | 35 | 80 | 30 | ||
| 13 | 35 | 22 | 40 | 29 | 35 | 28 | |||
| 14 | 25 | 37 | 36 | 49 | 32 | ||||
| 15 | 33 | 19 | 48 | 20 | |||||
| 16 | 16 | 36 | 26 | ||||||
| 17 | 27 | 20 | |||||||
| 18 | 19 | ||||||||
| Total excluding 1 − 1, 2 − 2, etc. | 1258 | 755 | 565 | 713 | 571 | 558 | 327 | 569 | 301 |
TABLE 5
Frequencies of Subtractions not Included in Table 4
These are cases where the pupil would, by reason of his stage of advancement, probably operate 35 − 30, 46 − 46, etc., as one bond.
| Minuends | Subtrahends | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 11 21 etc. | 2 12 22 etc. | 3 13 23 etc. | 4 14 24 etc. | 5 15 25 etc. | 6 16 26 etc. | 7 17 27 etc. | 8 18 28 etc. | 9 19 29 etc. | 10 20 etc. | |
| 10, 20, 30, 40, etc. | 11 | 29 | 16 | 52 | 32 | 51 | 7 | 30 | 22 | 60 |
| 11, 21, 31, 41, etc. | 42 | 14 | 22 | 32 | 12 | 26 | 19 | 52 | 17 | 10 |
| 12, 22, 32, 42, etc. | 47 | 97 | 5 | 13 | 9 | 21 | 11 | 24 | 19 | 17 |
| 13, 23, 33, 43, etc. | 7 | 40 | 7 | 14 | 15 | 13 | 19 | 19 | 22 | 3 |
| 14, 24, 34, 44, etc. | 8 | 28 | 14 | 58 | 13 | 16 | 14 | 26 | 19 | 7 |
| 15, 25, 35, 45, etc. | 21 | 28 | 29 | 54 | 51 | 15 | 21 | 12 | 24 | 8 |
| 16, 26, 36, 46, etc. | 5 | 18 | 12 | 27 | 35 | 69 | 13 | 17 | 19 | 2 |
| 17, 27, 37, 47, etc. | 5 | 9 | 12 | 40 | 32 | 54 | 24 | 12 | 12 | 1 |
| 18, 28, 38, 48, etc. | 2 | 16 | 10 | 23 | 22 | 36 | 18 | 47 | 16 | 0 |
| 19, 29, 39, etc. | 5 | 7 | 7 | 10 | 13 | 28 | 14 | 23 | 16 | 0 |
| Totals | 153 | 286 | 134 | 323 | 234 | 329 | 160 | 262 | 186 | 108 |
TABLE 6
Amount of Practice: Multiplication Bonds in Another Recent Textbook (B) of Excellent Repute. Books I and II
| Multipliers | Multiplicands | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Totals | |
| 1 | 299 | 534 | 472 | 271 | 310 | 293 | 261 | 178 | 195 | 99 | 2912 |
| 2 | 350 | 644 | 668 | 480 | 458 | 377 | 332 | 238 | 239 | 155 | 3941 |
| 3 | 280 | 487 | 509 | 388 | 318 | 302 | 247 | 199 | 227 | 152 | 3109 |
| 4 | 186 | 375 | 398 | 242 | 203 | 265 | 197 | 163 | 159 | 93 | 2281 |
| 5 | 268 | 359 | 393 | 234 | 263 | 243 | 217 | 192 | 197 | 114 | 2480 |
| 6 | 180 | 284 | 265 | 199 | 196 | 191 | 168 | 169 | 165 | 106 | 1923 |
| 7 | 135 | 283 | 277 | 176 | 187 | 158 | 155 | 121 | 145 | 118 | 1755 |
| 8 | 137 | 272 | 292 | 175 | 192 | 164 | 158 | 157 | 126 | 126 | 1799 |
| 9 | 71 | 173 | 140 | 122 | 97 | 102 | 101 | 100 | 82 | 110 | 1098 |
| Totals | 1906 | 3411 | 3414 | 2287 | 2224 | 2095 | 1836 | 1517 | 1535 | 1073 | |
TABLE 7
Amount Of Practice: Divisions Without Remainder In Textbook B, Parts I And II
| Dividends | Divisors | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Totals | |||
| Integral multiples of 2 to 9 in sequence; i.e., 4 ÷ 2 occurred 397 times, 6 ÷ 2 occurred 256 times, 6 ÷ 3, 224 times, 9 ÷ 3, 124 times. | 397 | 224 | 250 | 130 | 93 | 44 | 98 | 23 | 1259 | ||
| 256 | 124 | 152 | 79 | 28 | 43 | 61 | 25 | 768 | |||
| 318 | 123 | 130 | 65 | 50 | 19 | 39 | 19 | 763 | |||
| 258 | 98 | 86 | 105 | 25 | 24 | 34 | 20 | 650 | |||
| 198 | 49 | 76 | 27 | 22 | 30 | 33 | 16 | 451 | |||
| 77 180 69 | 54 91 46 | 36 50 37 | 31 38 24 | 28 17 12 | 27 13 17 | 16 22 16 | 9 16 15 | 278 427 236 | |||
| Totals | 1753 | 809 | 817 | 499 | 275 | 217 | 319 | 142 | |||
TABLE 8
Division Bonds, With And Without Remainders. Book B
All work through grade 6, except estimates of quotient figures in long division.
| Dividend | 2 | 3 | 4 | 5 | ||||||||||||||
| Divisor | 1 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 5 | ||||
| Number of Occurrences | 41 | 386 | 27 | 189 | 240 | 26 | 397 | 66 | 185 | 23 | 136 | 43 | 53 | 135 | ||||
| Dividend | 6 | 7 | ||||||||||||||||
| Divisor | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||||
| Number of Occurrences | 21 | 256 | 224 | 68 | 43 | 83 | 23 | 72 | 55 | 38 | 46 | 32 | 54 | |||||
| Dividend | 8 | 9 | ||||||||||||||||
| Divisor | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| Number of Occurrences | 17 | 318 | 30 | 250 | 22 | 28 | 39 | 91 | 19 | 50 | 124 | 49 | 25 | 15 | 18 | 30 | 38 | |
| Dividend | 10 | 11 | ||||||||||||||||
| Divisor | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 258 | 38 | 46 | 120 | 19 | 9 | 24 | 24 | 32 | 21 | 16 | 3 | 7 | 11 | 14 | 3 | ||
| Dividend | 12 | 13 | ||||||||||||||||
| Divisor | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 198 | 123 | 152 | 29 | 93 | 9 | 16 | 7 | 45 | 16 | 15 | 11 | 7 | 4 | 5 | 3 | ||
| Dividend | 14 | 15 | ||||||||||||||||
| Divisor | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 77 | 20 | 13 | 5 | 8 | 44 | 8 | 6 | 69 | 98 | 16 | 79 | 8 | 8 | 4 | 6 | ||
| Dividend | 16 | 17 | ||||||||||||||||
| Divisor | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 180 | 19 | 130 | 14 | 6 | 9 | 98 | 3 | 61 | 9 | 15 | 14 | 6 | 6 | 12 | 3 | ||
| Dividend | 18 | 19 | ||||||||||||||||
| Divisor | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 69 | 49 | 13 | 6 | 28 | 7 | 7 | 23 | 21 | 6 | 10 | 5 | 3 | 4 | 10 | 4 | ||
| Dividend | 20 | 21 | ||||||||||||||||
| Divisor | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||
| Number of Occurrences | 24 | 86 | 65 | 11 | 3 | 23 | 5 | 54 | 12 | 8 | 5 | 43 | 10 | 5 | ||||
| Dividend | 22 | 23 | ||||||||||||||||
| Divisor | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||
| Number of Occurrences | 17 | 16 | 15 | 8 | 13 | 6 | 15 | 7 | 8 | 11 | 8 | 6 | 3 | 2 | ||||
| Dividend | 24 | 25 | ||||||||||||||||
| Divisor | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||
| Number of Occurrences | 91 | 76 | 18 | 50 | 5 | 61 | 1 | 11 | 13 | 105 | 5 | 6 | 5 | 3 | ||||
| Dividend | 26 | 27 | ||||||||||||||||
| Divisor | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||
| Number of Occurrences | 5 | 6 | 3 | 3 | 4 | 6 | 3 | 46 | 8 | 10 | 4 | 2 | 6 | 25 | ||||
| Dividend | 28 | 29 | ||||||||||||||||
| Divisor | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||
| Number of Occurrences | 4 | 36 | 8 | 3 | 19 | 3 | 7 | 6 | 8 | 0 | 5 | 11 | 2 | 3 | ||||
| Dividend | 30 | 31 | ||||||||||||||||
| Divisor | 4 | 5 | 6 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| Number of Occurrences | 21 | 27 | 25 | 6 | 7 | 13 | 4 | 3 | 1 | 1 | 4 | 2 | ||||||
| Dividend | 32 | 33 | ||||||||||||||||
| Divisor | 4 | 5 | 6 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| Number of Occurrences | 50 | 11 | 3 | 6 | 39 | 5 | 8 | 7 | 7 | 2 | 6 | 1 | ||||||
| Dividend | 34 | 35 | ||||||||||||||||
| Divisor | 4 | 5 | 6 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| Number of Occurrences | 8 | 3 | 5 | 2 | 1 | 1 | 10 | 31 | 5 | 24 | 5 | 3 | ||||||
| Dividend | 36 | 37 | ||||||||||||||||
| Divisor | 4 | 5 | 6 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| Number of Occurrences | 37 | 16 | 22 | 2 | 6 | 19 | 12 | 8 | 7 | 5 | 3 | 9 | ||||||
| Dividend | 38 | 39 | ||||||||||||||||
| Divisor | 4 | 5 | 6 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| Number of Occurrences | 7 | 8 | 7 | 1 | 1 | 5 | 4 | 3 | 7 | 4 | 3 | 1 | ||||||
| Dividend | 40 | 41 | 42 | |||||||||||||||
| Divisor | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | |||
| Number of Occurrences | 38 | 9 | 2 | 34 | 2 | 6 | 6 | 3 | 7 | 5 | 7 | 28 | 30 | 10 | 3 | |||
| Dividend | 43 | 44 | 45 | |||||||||||||||
| Divisor | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | |||
| Number of Occurrences | 7 | 5 | 10 | 3 | 2 | 7 | 6 | 4 | 5 | 0 | 24 | 6 | 7 | 10 | 20 | |||
| Dividend | 46 | 47 | 48 | |||||||||||||||
| Divisor | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 9 | |||
| Number of Occurrences | 3 | 3 | 2 | 2 | 2 | 6 | 2 | 2 | 0 | 3 | 7 | 17 | 4 | 33 | 2 | |||
| Dividend | 49 | 50 | 51 | 52 | ||||||||||||||
| Divisor | 5 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | |
| Number of Occurrences | 4 | 7 | 27 | 9 | 2 | 4 | 6 | 3 | 8 | 2 | 3 | 1 | 2 | 5 | 5 | 5 | 3 | |
| Dividend | 53 | 54 | 55 | 56 | ||||||||||||||
| Divisor | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | ||
| Number of Occurrences | 4 | 3 | 2 | 2 | 12 | 5 | 1 | 16 | 5 | 3 | 4 | 2 | 0 | 13 | 16 | 8 | ||
| Dividend | 57 | 58 | 59 | 60 | 61 | |||||||||||||
| Divisor | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 6 | 7 | 8 | 9 | 7 | 8 | 9 | 7 | 8 | 9 |
| Number of Occurrences | 0 | 3 | 1 | 3 | 2 | 2 | 3 | 1 | 2 | 3 | 0 | 3 | 3 | 9 | 1 | 1 | 2 | 5 |
| Dividend | 62 | 63 | 64 | 65 | 66 | 67 | ||||||||||||
| Divisor | 7 | 8 | 9 | 7 | 8 | 9 | 7 | 8 | 9 | 7 | 8 | 9 | 7 | 8 | 9 | 7 | 8 | 9 |
| Number of Occurrences | 4 | 6 | 1 | 17 | 5 | 9 | 5 | 22 | 0 | 1 | 10 | 1 | 2 | 1 | 4 | 0 | 1 | 1 |
| Dividend | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | ||||||||||
| Divisor | 7 | 8 | 9 | 7 | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 9 |
| Number of Occurrences | 1 | 3 | 2 | 0 | 6 | 1 | 6 | 2 | 1 | 0 | 16 | 10 | 7 | 5 | 3 | 3 | 5 | 3 |
| Dividend | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | ||||
| Divisor | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
| Number of Occurrences | 3 | 2 | 3 | 0 | 4 | 1 | 0 | 2 | 4 | 15 | 2 | 4 | 1 | 2 | 0 | 3 | 2 | 7 |
Tables 3 to 8 show that even gifted authors make instruments for instruction in arithmetic which contain much less practice on certain elementary facts than teachers suppose; and which contain relatively much more practice on the more easily learned facts than on those which are harder to learn.
How much practice should be given in arithmetic? How should it be divided among the different bonds to be formed? Below a certain amount there is waste because, as has been shown in Chapter VI, the pupil will need more time to detect and correct his errors than would have been required to give him mastery. Above a certain amount there is waste because of unproductive overlearning. If 668 is just enough for 2 × 2, 82 is not enough for 9 × 8. If 82 is just enough for 9 × 8, 668 is too much for 2 × 2.
It is possible to find the answers to these questions for the pupil of median ability (or any stated ability) by suitable experiments. The amount of practice will, of course, vary according to the ability of the pupil. It will also vary according to the interest aroused in him and the satisfaction he feels in progress and mastery. It will also vary according to the amount of practice of other related bonds; 7 + 7 = 14 and 60 ÷ 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15 and 61 ÷ 7 = 8 and 5 remainder. It will also, of course, vary with the general difficulty of the bond, 17 − 8 = 9 being under ordinary conditions of teaching harder to form than 7 − 2 = 5.
Until suitable experiments are at hand we may estimate for the fundamental bonds as follows, assuming that by the end of grade 6 a strength of 199 correct out of 200 is to be had, and that the teaching is by an intelligent person working in accord with psychological principles as to both ability and interest.
For one of the easier bonds, most facilitated by other bonds (such as 2 × 5 = 10, or 10 − 2 = 8, or the double bond 7 = two 3s and 1 remainder) in the case of the median or average pupil, twelve practices in the week of first learning, supported by twenty-five practices during the two months following, and maintained by thirty practices well spread over the later periods should be enough. For the more gifted pupils lesser amounts down to six, twelve, and fifteen may suffice. For the less gifted pupils more may be required up to thirty, fifty, and a hundred. It is to be doubted, however, whether pupils requiring nearly two hundred repetitions of each of these easy bonds should be taught arithmetic beyond a few matters of practical necessity.
For bonds of ordinary difficulty, with average facilitation from other bonds (such as 11 − 3, 4 × 7, or 48 ÷ 8 = 6) in the case of the median or average pupil, we may estimate twenty practices in the week of first learning, supported by thirty, and maintained by fifty practices well spread over the later periods. Gifted pupils may gain and keep mastery with twelve, fifteen, and twenty practices respectively. Pupils dull at arithmetic may need up to twenty, sixty, and two hundred. Here, again, it is to be doubted whether a pupil for whom arithmetical facts, well taught and made interesting, are so hard to acquire as this, should learn many of them.
For bonds of greater difficulty, less facilitated by other bonds (such as 17 − 9, 8 × 7, or 12½% of = 1⁄8 of), the practice may be from ten to a hundred percent more than the above.
Underlearning and Overlearning
If we accept the above provisional estimates as reasonable, we may consider the harm done by giving less and by giving more than these reasonable amounts. Giving less is indefensible. The pupil’s time is wasted in excessive checking to find his errors. He is in danger of being practiced in error. His attention is diverted from the learning of new facts and processes by the necessity of thinking out these supposedly mastered facts. All new bonds are harder to learn than they should be because the bonds which should facilitate them are not strong enough to do so. Giving more does harm to some extent by using up time that could be spent better for other purposes, and (though not necessarily) by detracting from the pupil’s interest in arithmetic. In certain cases, however, such excess practice and overlearning are actually desirable. Three cases are of special importance.
The first is the case of a bond operating under a changed mental set or adjustment. A pupil may know 7 × 8 adequately as a thing by itself, but need more practice to operate it in
285
7
——
where he has to remember that 3 is to be added to the 56 when he obtains it, and that only the 9 is to be written down, the 5 to be held in mind for later use. The practice required to operate the bond efficiently in this new set is desirable, even though it is excess from a narrower point of view, and causes the straightforward ‘seven eights are fifty-six’ to be overlearned. So also a pupil’s work with 24, 34, 44, etc., + 9 may react to give what would be excess practice from the point of view of 4 + 9 alone; his work in estimating approximate quotient figures in long division may give excess practice on the division tables. There are many such cases. Even adding the 5 and 7 in 5⁄12 + 7⁄12 is not quite the same task as adding 5 and 7 undisturbed by the fact that they are twelfths. We know far too little about the amount of practice needed to adapt arithmetical bonds to efficient operation in these more complicated conditions to estimate even approximately the allowances to be made. But some allowance, and often a rather large allowance, must be made.
The second is the case where the computation in general should be made very easy and sure for the pupil except for some one new element that is being learned. For example, in teaching the meaning and uses of ‘Averages’ and of uneven division, we may deliberately use 2, 3, and 4 as divisors rather than 7 and 9, so as to let all the pupil’s energy be spent in learning the new facts, and so that the fraction in the quotient may be something easily understood, real, and significant. In teaching the addition of mixed numbers, we may use, in the early steps,
11½
13½
24
—— rather than 79½
98½
67
——
so as to save attention for the new process itself. In cancellation, we may give excess practice to divisions by 2, 3, 4, and 5 in order to make the transfer to the new habits of considering two numbers together from the point of view of their divisibility by some number. In introducing trade discount, we may give excess practice on ’5% of’ and ’10% of’ deliberately, so that the meaning of discount may not be obscured by difficulties in the computation itself. Excess practice on, and overlearning of, certain bonds is thus very often justifiable.
The third case concerns bonds whose importance for practical uses in life or as notable facilitators of other bonds is so great that they may profitably be brought to a greater strength than 199 correct out of 200 at a speed of 2 sec. or less, or be brought to that degree of strength very early. Examples of bonds of such special practical use are the subtractions from 10, ½ + ½, ½ + ¼, ½ of 60, ¼ of 60, and the fractional parts of 12 and of $1.00. Examples of notable facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like 2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9.
In consideration of these three modifying cases or principles, a volume could well be written concerning just how much practice to give to each bond, in each of the types of complex situations where it has to operate. There is evidently need for much experimentation to expose the facts, and for much sagacity and inventiveness in making sure of effective learning without wasteful overlearning.
The facts of primary importance are:—
(1) The textbook or other instrument of instruction which is a teacher’s general guide may give far too little practice on certain bonds.
(2) It may divide the practice given in ways that are apparently unjustifiable.
(3) The teacher needs therefore to know how much practice it does give, where to supplement it, and what to omit.
(4) The omissions, on grounds of apparent excess practice, should be made only after careful consideration of the third principle described above.
(5) The amount of practice should always be considered in the light of its interest and appeal to the pupil’s tendency to work with full power and zeal. Mere repetition of bonds when the learner does not care whether he is improving is rarely justifiable on any grounds.
(6) Practice that is actually in excess is not a very grave defect if it is enjoyed and improves the pupil’s attitude toward arithmetic. Not much time is lost; a hundred practices for each of a thousand bonds after mastery to 199 in 200 at 2 seconds will use up less than 60 hours, or 15 hours per year in grades 3 to 6.
(7) By the proper division of practice among bonds, the arrangement of learning so that each bond helps the others, the adroit shifting of practice of a bond to each new type of situation requiring it to operate under changed conditions, and the elimination of excess practice where nothing substantial is gained, notable improvements over the past hit-and-miss customs may be expected.
(8) Unless the material for practice is adequate, well balanced and sufficiently motivated, the teacher must keep close account of the learning of pupils. Otherwise disastrous underlearning of many bonds is almost sure to occur and retard the pupil’s development.
The Organization of Abilities
There is danger that the need of brevity and simplicity which has made us speak so often of a bond or an ability, and of the amount of practice it requires, may mislead the reader into thinking that these bonds and abilities are to be formed each by itself alone and kept so. They should rarely be formed so and never kept so. This we have indicated from time to time by references to the importance of forming a bond in the way in which it is to be used, to the action of bonds in changed situations, to facilitation of one bond by others, to the coöperation of abilities, and to their integration into a total arithmetical ability.
As a matter of fact, only a small part of drill work in arithmetic should be the formation of isolated bonds. Even the very young pupil learning 5 and 3 are 8 should learn it with ’5 and 5 = 10,’ ’5 and 2 = 7,’ at the back of his mind, so to speak. Even so early, 5 + 3 = 8 should be part of an organized, coöperating system of bonds. Later 50 + 30 = 80 should become allied to it. Each bond should be considered, not simply as a separate tool to be put in a compartment until needed, but also as an improvement of one total tool or machine, arithmetical ability.
There are differences of course. Knowledge of square root can be regarded somewhat as a separate tool to be sharpened, polished, and used by itself, whereas knowledge of the multiplication tables cannot. Yet even square root is probably best made more closely a part of the total ability, being taught as a special case of dividing where divisor is to be the same as quotient, the process being one of estimating and correcting.
In general we do not wish the pupil to be a repository of separated abilities, each of which may operate only if you ask him the sort of questions which the teacher used to ask him, or otherwise indicate to him which particular arithmetical tool he is to use. Rather he is to be an effective organization of abilities, coöperating in useful ways to meet the quantitative problems life offers. He should not as a rule have to think in such fashion as: “Is this interest or discount? Is it simple interest or compound interest? What did I do in compound interest? How do I multiply by 2 percent?” The situation that calls up interest should also call up the kind of interest that is appropriate, and the technique of operating with percents should be so welded together with interest in his mind that the right coöperation will occur almost without supervision by him.
As each new ability is acquired, then, we seek to have it take its place as an improvement of a thinking being, as a coöperative member of a total organization, as a soldier fighting together with others, as an element in an educated personality. Such an organization of bonds will not form itself any more than any one bond will create itself. If the elements of arithmetical ability are to act together as a total organized unified force they must be made to act together in the course of learning. What we wish to have work together we must put together and give practice in teamwork.
We can do much to secure such coöperative action when and where and as it is needed by a very simple expedient; namely, to give practice with computation and problems such as life provides, instead of making up drills and problems merely to apply each fact or principle by itself. Though a pupil has solved scores of problems reading, “A triangle has a base of a feet and an altitude of b feet, what is its area?” he may still be practically helpless in finding the area of a triangular plot of ground; still more helpless in using the formula for a triangle which is one of two into which a trapezoid is divided. Though a pupil has learned to solve problems in trade discount, simple interest, compound interest, and bank discount one at a time, stated in a few set forms, he may be practically helpless before the actual series of problems confronting him in starting in business, and may take money out of the savings bank when he ought to borrow on a time loan, or delay payment on his bills when by paying cash he could save money as well as improve his standing with the wholesaler.
Instead of making up problems to fit the abilities given by school instruction, we should preferably modify school instruction so that arithmetical abilities will be organized into an effective total ability to meet the problems that life will offer. Still more generally, every bond formed should be formed with due consideration of every other bond that has been or will be formed; every ability should be practiced in the most effective possible relations with other abilities.