ABSTRACT
There are no subjects taught in the schools which give pupils as much difficulty as do the mathematical sciences. The reason for the high percentage of failures in mathematics is that the ability to think abstractly and to use symbols is far more difficult to acquire than is the ability to understand and retain the kinds of knowledge which are taught in most of the school subjects. Studies of the cases of failure in arithmetic show that the percentage of failures increases steadily as pupils pass from grade to grade. In almost all subjects it is true that pupils learn during each year new facts which supplement what was learned earlier. They even make up in the higher grades for deficiencies in their knowledge of facts taught in the lower grades. In arithmetic and the other mathematical sciences failure to master the simpler abstractions taught in the lower grades makes it impossible to understand what is taught in the upper grades. The discouraging fact with regard to pupil failures in arithmetic is that they are more numerous in the third grade than in the second, more numerous in the fourth grade than in the third, and more numerous in the fifth grade than in the fourth.
