found that some children will do this task well who cannot recognize number otherwise. Their good and fundamental idea of form and space may then be made the starting-point in instruction to develop the number concept. Comparison of different lengths and sizes leads up to quantitative concepts. The test may be varied in different ways.

Courtis Tests.-Test 4 introduces the "Courtis Tests." These have been recently so widely employed in measuring the efficiency of class work in schools that it is unnecessary to describe them here. But their application in this series of tests rests upon their adaptability to the testing of individual children. How this can be done has been instructively illustrated by the Department of Educational Investigation and Measurement of the Boston public schools. In explaining the work of the department in this direction Miss Rose A. Carrigan writes in Bulletin II:

To make sure that his ability is of a reasonably permanent nature, the pupil should measure up to the grade standard on at least three successive occasions. Whereas a single test of a thousand or more children is adequate to demonstrate the efficiency of the teaching process in general, one test is not sufficient to determine the ability of the individual. To do this last effectively, several tests are necessary; otherwise there is danger of incorrect conclusions resulting from chance scores.

This very correct statement shows the serious limitation of clinical study which is confined to one observation or test. Nevertheless, taken in connection with the other tests, one examination of a child in the addition, subtraction, multiplication, and division examples given in Series B, Arithmetic, of the Courtis Standard Research Tests will allow of helpful conclusions. The re

sults should be compared for valuation with the age and grade standards worked out by Mr. S. A. Courtis, Detroit, Mich., with such elasticity of application as will adjust them to individual types of mind.

Doctor Frank A. Ballou, director of the Boston department, has kindly given permission to use the following individual scores as illustrations of the value of this method for individual testing:

Case 68.-Fig. 18 represents the curve of a 12-yearold boy in the eighth grade of a Boston school. The solid line represents the standard for his grade, the dotted line his score. It will be seen immediately how far he left the standard, or average, behind in amount of work done, and in accuracy. He did practically double the work of the average and exceeded the standard of accuracy three times, solving all examples he attempted in subtraction, multiplication, and division.

Case 69.-Fig. 19 shows the record of another 12year-old boy, in a seventh grade at Edgerton, Wis. In the number of rights he is ahead of the standard at

all points, and his curve is more nearly a straight line than the average.

It

Case 70.-Fig. 20 is remarkable for two reasons. represents the work of an eighth-grade Boston girl of

13, who had not been thought to be in any way different from the average. Here the broken line represents her score in January, 1914; the dotted line the score three months later, in April. In both she is considerably ahead of the average standard in everything

except multiplication, where she is closer to the average. But in the three months, perhaps just because she thought that she was "good in arithmetic, anyway," she lost in amount and somewhat in accuracy, except again in multiplication, where her lower score had apparently prompted her to make some effort. This shows the value of practice, even in things which are otherwise well mastered.

Case 71. This value of practice is clearly proven by Fig. 21, exhibiting the January and April scores of an 11-year-old Boston girl in the fifth grade. At the first trial the pupil was below standard in addition and sub

traction. After three months of practice she surpassed the standard in all four of the fundamental operations. The greatest improvement was in addition, which was the operation in which she had least ability at the time of the first test.

Case 72.-Chart V exhibits the eradication of a particular weakness (in multiplication) in a 10-year-old Boston girl pupil of the fifth grade, with a corresponding loss in her former best operation (subtraction). Her

score in April is more nearly like the standard, but more even than the average.

These examples give sufficient evidence of the need of individual valuation and of attention to individual

needs. To show how very necessary individual standards are, how false ordinary school standards, and how much we are laboring under misconceptions in regard to the value of arithmetical drill as at present conducted in our schools, the author will permit himself to quote again from Doctor Ballou's paper:

The Courtis Tests have revealed great variations in the ability of pupils to add, subtract, multiply, and divide. They have shown that we have at the present time practically all grades of ability, from the fourth to the eighth in each class tested. Twenty-nine per cent of the pupils in the eighth grade could exchange places with a like number of pupils in the fourth grade without changing in the slightest the arithmetical ability in the fundamental operations of either class as a class.

The tests also show that from 35 to 50 per cent of the children tested in any one grade did not increase their ability at all in addition, subtraction, multiplication and division from the time the tests were given in January until they were given in April