Haggerty: Studies in Arithmetic 



7 



Note the sixth grade division scores in cities 15 and 16. The 

 one is 8 problems correctly solved; the other is 2, or one-fourth 

 the score of the first. The multiplication rights for city 17 is 

 3.1 ; for city 12 it is more than double. The number of subtraction 

 rights of city 2 is 5.3; for city 21 it is 9. The minimum score in 

 addition is 3.4 rights for cities 1 and 7, and for city 4 it is 7.5. 

 It is the same story for all the grades. Think of such ranges of 

 efficiency as 4.4 to 9, 7 to 11.2, 6.5 to 12, and 7.0 to 12.1 for the 

 eighth grade; and 4.1 to 9, 4.2 to 8.4, and 4 to 9 for the seventh 

 grade. There are 8 eighth grades as low as the median of all 

 seventh grades in addition, and 2 as low as the median of all sixth 

 grades, and 1 whose median is but .2 above the median of all 

 fifth grades, while there are 1 sixth grade and 4 seventh grades 

 higher than the median of all eighth grades. For a correct appre- 

 ciation of these figures it must be remembered that they do not 

 represent individual children. There are in all groups half the 

 individuals below the median score and half above. How much 

 wider then must be the range between the lowest 10 per cent 

 of the lower group and the upper 10 per cent of the upper group! 



The facts represented by these numbers appear more emphatic 

 when shown in graphic form as in Figures 1 to 4. Here you see 

 at a glance the enormous amount of overlapping existing from grade 

 to grade. In general the fifth grade scores (triangles) are grouped 

 at the lower lefthand corner of the figure, and the eighth grade 

 scores (stars) are found at the upper righthand part, with the 

 sixth grade scores (squares) and the seventh grade scores (circles) 

 sprinkled between. In the center of the figure the symbols of the 

 several grades mingle in a most disorderly fashion. The amount 

 of overlapping would be simply astonishing if we had not learned 

 to expect such results from the application of standard tests. 



In Figure 5 is shown the overlapping of the fifth and seventh 

 grades in the several schools within a single city. The seventh 

 grade medians for both attempts and rights are clearly above 

 those of the fifth grade, thus showing in general a superior achieve- 

 ment for the upper grade. On the other hand, in a large number 

 of schools the fifth grade work is superior to that of many seventh 

 grade classes in the same system and in one case the more ad- 

 vanced class is inferior to the fifth grade class within the same 

 school. 



