The following were set as official standards, in an excellent school system, Courtis Series B being used:--
SPEED PERCENT OF GRADE. ATTEMPTS. CORRECT ANSWERS. Addition 8 12 80 7 11 80 6 10 70 5 9 70 4 8 70
Subtraction 8 12 90 7 11 90 6 10 90 5 9 80 4 7 80
Multiplication 8 11 80 7 10 80 6 9 80 5 7 70 4 6 60
Division 8 11 90 7 10 90 6 8 80 5 6 70 4 4 60
Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like those printed below, children in grade 4 got on the average less than 80 percent of correct answers. Their average speed was about 2 columns per minute. In doing division of the sort printed below children of grades 3 _B_ and 4 _A_ got less than 95 percent of correct answers, the average speed being 4 divisions per minute. In both cases the slower computers were no more accurate than the faster ones. Practice improved the speed very rapidly, but the accuracy remained substantially unchanged. Brown ['11 and '12] found a similar low status of ability and notable improvement from a moderate amount of special practice.
3 5 6 2 3 8 9 7 4 9 7 9 6 5 5 6 4 5 8 2 3 4 7 8 7 3 7 9 3 7 8 8 4 8 2 6 8 2 9 8 2 2 4 7 6 9 8 5 6 2 6 9 5 7 8 5 2 3 2 4 9 6 4 2 7 2 9 4 4 5 3 3 7 9 9 9 2 8 9 7 6 8 9 6 4 7 7 9 2 4 8 4 6 9 9 2 6 9 8 9 -- -- -- -- -- -- -- -- -- --
20 = .... 5s 56 = .... 9s and .... _r_. 30 = .... 7s and .... _r_. 89 = .... 9s and .... _r_. 20 = .... 8s and .... _r_. 56 = .... 6s and .... _r_. 31 = .... 4s and .... _r_. 86 = .... 9s and .... _r_.
It is clear that numerical work as inaccurate as this has little or no commercial or industrial value. If clerks got only six answers out of ten right as in the Courtis tests, one would need to have at least four clerks make each computation and would even then have to check many of their discrepancies by the work of still other clerks, if he wanted his accounts to show less than one error per hundred accounting units of the Courtis size.
It is also clear that the "habits of ... absolute accuracy, and satisfaction in truth as a result" which arithmetic is supposed to further must be largely mythical in pupils who get right answers only from three to nine times out of ten!
The bonds in question clearly must be made far stronger than they now are. They should in fact be strong enough to abolish errors in computation, except for those due to temporary lapses. It is much better for a child to know half of the multiplication tables, and to know that he does not know the rest, than to half-know them all; and this holds good of all the elementary bonds required for computation. Any bond should be made to work perfectly, though slowly, very soon after its formation is begun. Speed can easily be added by proper practice.