Two experiments were carried out to examine the effect of training on children's numerical behavior in comparing 2 collections.The curriculum of the training was to enable children to grasp interrelations of number ((k+1) = (k)(1),(k-1) = (k)(1), ect.) by - associating a quantitative image to each numeral, and to use counting as a quantifying operation.(A child has a quantifying operation when assigned numerals to a set imply various relations to other numerals.) In the 1st experiment, 11 4-year-old children received 10 training sessions. Children showed a marked and statistically significant progress in the numerical judgment of 2 collections.Their performance improved significantly even when a collection had more elements than were dealt with in the training session. Furthermore, many of them also acquired number conservation, for which no direct instruction was given.Effects of the training diminished only slightly 6 months later. Different results were obtained, however, in the 2nd experiment, in which 4 5-year-old children received 5 training sessions based on almost the same curriculum as before.These subjects showed no significant improvement in comparison with the control group subjects.None of them acquired number conservation. But the training seemed successful in leading Ss to the comprehension of the nature (meaning) of counting operations and numerals assigned to collections, for the experimental Ss made successive correct responses with a suggestion to apply counting, while the controls either counted only one of collections to be compared or made a correct response when the suggestion was repeated each time. The following points were discussed based on these pieces of evidences: i) When a child comprehends inter-number relations, he can make correct responses to various tasks including numerical comparison of 2 collections and number conservation, since he can apply counting as a quantifying operation not as a mere “quotifying” (nominating) one ii) The comprehension of inter-number relations is the necessary but not sufficient condition for a child to rely exclusively on counting and to reject perceptual cues in numerical comparison.