The facts of Dr. Rice's table show that there is a positive relation between the general standing of a school system in the tests and the amount of time devoted to arithmetic by its program. The relation is not close, however, being that expressed by a correlation coefficient of .36½. Within any one school system there is no relation between the standing of a particular school and the amount of time devoted to arithmetic in that school's program. It must be kept in mind that the amount of time given in the school program may be counterbalanced by emphasizing work at home and during study periods, or, on the other hand, may be a symptom of correspondingly small or great emphasis on arithmetic in work set for the study periods at home.
A still more elaborate investigation of this same topic was made by Stone ['08]. I quote somewhat fully from it, since it is an instructive sample of the sort of studies that will doubtless soon be made in the case of every elementary school subject. He found that school systems differed notably in the achievements made by their sixth-grade pupils in his tests of computation (the so-called 'fundamentals') and of the solution of verbally described problems (the so-called 'reasoning'). The facts were as shown in Table 11.
TABLE 10
Averages for Individual Schools in Arithmetic
| City | School | 6th Year | 7th Year | 8th Year | School Average | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Result | Principle | Result | Principle | Result | Principle | Result | Principle | Percent of Mechanical Errors | Minutes Daily | ||
| III | 1 | 79.3 | 80.3 | 81.1 | 82.3 | 91.7 | 93.9 | 80.0 | 83.1 | 3.7 | 53 |
| I | 1 | 80.4 | 81.5 | 64.2 | 67.2 | 80.9 | 82.8 | 76.6 | 80.3 | 4.6 | 60 |
| I | 2 | 80.9 | 83.4 | 43.5 | 50.9 | 72.7 | 79.1 | 69.3 | 75.1 | 7.7 | 25 |
| I | 3 | 72.2 | 74.0 | 63.5 | 66.2 | 74.5 | 76.6 | 67.8 | 72.2 | 6.1 | 45 |
| I | 4 | 69.9 | 72.2 | 54.6 | 57.8 | 66.5 | 69.1 | 64.3 | 70.3 | 8.5 | 45 |
| II | 1 | 71.2 | 75.3 | 33.6 | 35.7 | 36.8 | 40.0 | 60.2 | 64.8 | 7.1 | 60 |
| III | 2 | 43.7 | 45.0 | 53.9 | 56.7 | 51.1 | 53.1 | 54.5 | 58.9 | 7.4 | 60 |
| IV | 1 | 58.9 | 60.4 | 31.2 | 34.1 | 41.6 | 43.5 | 55.1 | 58.4 | 5.6 | 60 |
| IV | 2 | 59.8 | 63.1 | — | — | 22.5 | 22.5 | 53.9 | 58.8 | 8.3 | — |
| IV | 3 | 54.9 | 58.1 | 35.2 | 38.6 | 43.5 | 45.0 | 51.5 | 57.6 | 10.5 | 60 |
| IV | 4 | 42.3 | 45.1 | 16.1 | 19.2 | 48.7 | 48.7 | 42.8 | 48.2 | 11.2 | — |
| V | 1 | 44.1 | 48.7 | 29.2 | 32.5 | 51.1 | 58.3 | 45.9 | 51.3 | 10.5 | 40 |
| VI | 1 | 68.3 | 71.3 | 33.5 | 36.6 | 26.9 | 30.7 | 39.0 | 42.9 | 9.0 | 33 |
| VI | 2 | 46.1 | 49.5 | 19.5 | 24.2 | 30.2 | 40.6 | 36.5 | 43.6 | 16.2 | 30 |
| VI | 3 | 34.5 | 36.4 | 30.5 | 35.1 | 23.3 | 24.1 | 36.0 | 42.5 | 15.2 | 48 |
| VII | 1 | 35.2 | 37.7 | 29.1 | 32.5 | 25.1 | 27.2 | 40.5 | 45.9 | 11.7 | 42 |
| VII | 2 | 35.2 | 38.7 | 15.0 | 16.4 | 19.6 | 21.2 | 36.5 | 40.6 | 10.1 | 75 |
| VII | 3 | 27.6 | 33.7 | 8.9 | 10.1 | 11.3 | 11.3 | 25.3 | 31.5 | 19.6 | 45 |
High achievement by a system in computation went with high achievement in solving the problems, the correlation being about .50; and the system that scored high in addition or subtraction or multiplication or division usually showed closely similar excellence in the other three, the correlations being about .90.
TABLE 11
Scores Made by the Sixth-Grade Pupils of Each of Twenty-Six School Systems
| System | Score in Tests with Problems | Score in Tests in Computing |
|---|---|---|
| 23 | 356 | 1841 |
| 24 | 429 | 3513 |
| 17 | 444 | 3042 |
| 4 | 464 | 3563 |
| 25 | 464 | 2167 |
| 22 | 468 | 2311 |
| 16 | 469 | 3707 |
| 20 | 491 | 2168 |
| 18 | 509 | 3758 |
| 15 | 532 | 2779 |
| 3 | 533 | 2845 |
| 8 | 538 | 2747 |
| 6 | 550 | 3173 |
| 1 | 552 | 2935 |
| 10 | 601 | 2749 |
| 2 | 615 | 2958 |
| 21 | 627 | 2951 |
| 13 | 636 | 3049 |
| 14 | 661 | 3561 |
| 9 | 691 | 3404 |
| 7 | 734 | 3782 |
| 12 | 736 | 3410 |
| 11 | 759 | 3261 |
| 26 | 791 | 3682 |
| 19 | 848 | 4099 |
| 5 | 914 | 3569 |
Of the conditions under which arithmetical learning took place, the one most elaborately studied was the amount of time devoted to arithmetic. On the basis of replies by principals of schools to certain questions, he gave each of the twenty-six school systems a measure for the probable time spent on arithmetic up through grade 6. Leaving home study out of account, there seems to be little or no correlation between the amount of time a system devotes to arithmetic and its score in problem-solving, and not much more between time expenditure and score in computation. With home study included there is little relation to the achievement of the system in solving problems, but there is a clear effect on achievement in computation. The facts as given by Stone are:—