18 



Indiana University Studies 



ing the results of teaching with standardized tests. Without 

 doing this, a superintendent or teacher cannot know whether 

 they are securing results above or below standard, and when 

 a group of cities are given the same test for the first time 

 there has always been found a diversity of practice. 



The Achievements of Some Cities Not Uniform. The 



tables reveal striking differences in achievement among not 

 only the various cities, but also within any one system. The 

 eighth grade in City 6 ranks first in speed in division but 

 seventeenth in speed in addition. The seventh grade in City 

 8 ranks first in accuracy in division, but only sixteenth in ad- 

 dition. In the sixth grade City 2 ranks first in number of 

 division examples correctly solved, and nineteenth in addi- 

 tion examples correctly solved. Two other cities afford strik- 

 ing examples of irregularity. City 24, that ranks fourth 

 when speed alone is considered, ranks fifteenth in all points 

 and twenty-first in accuracy. This is a clear case of sacri- 

 ficing accuracy for speed. 



Again, a system may be strong in certain grades and weak 

 in others. In City 20 the averages of the rankings in all 

 points in the fifth and sixth grades are 15 and 16, whereas the 

 average for the seventh and eighth grades is 4. There are 

 many conditions which contribute to this state of affairs, some 

 of which might be differences in the amount of time devoted 

 to the subject, in the quality of the teaching, or in the extent 

 and quality of the supervision. 



This non-uniformity of achievement is evidence that the 

 product of instruction in arithmetic is not a single ability 

 which functions for all types of examples, but a group of 

 specific abilities. There may be and probably are certain 

 abilities which function in the doing of examples of different 

 types, but a pupil is not satisfactorily equipped until each of 

 the necessary specific abilities have been engendered in the 

 required degree. This makes it imperative for each teacher 

 to be conscious of the different specific abilities and definitely 

 train pupils in each of them. 



Individual Differences. The wide range of achievement in 

 both speed and accuracy is clearly seen by referring to the dis- 

 tributions given in Tables VII A to XB. These distributions 

 are the result of reducing to per cents the frequencies found 

 in the total distributions for each grade. Table VII A, for 



