The study investigated whether 15 children (ages 6-21) with IQs from 31 to 66 could spontaneously invent more efficient calculational procedures and abstract basic arithmetic relationships after individualized tutoring in computation. Experimental subjects were given training that focused on accurate computation but specific relationships and calculational shortcuts were not pointed out. As a result of the computational training, many experimental subjects spontaneously invented shortcuts for the concrete counting-all procedure they already knew including the "counting fingers strategy." More experimental than control children (who received training in nonarithmetical areas) discovered that applying the commutativity principle could shortcut their computational effort, and more experimental than control subjects mastered facts involving zero and one. (DB)