| 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.