Concepts of Interrelations among Number, Length, and Crowdedness in 5- to 10-Year-Olds

The main purpose of the present study was to examine the development in children of concepts about the interrelations among number, length, and crowdedness, using as stimulus materials pictures of flower beds with tulips crowded to various amounts, planters of various lengths, and various sized bunches of tulips. Participants were 136 5- to 10-year-olds. The main results were as follows : (1) For 5- and 6-year-olds, understanding of crowdedness was rather difficult. (2) The direct relation between number and length was grasped strongly even by 5-year-olds, although an excessively strong relation might prevent children from understanding one relation among the 3 concepts, that is, crowdedness=number/length. (3) Though grasping the inverse relation between length and crowdedness was the most difficult concept, 8-year-olds seemed to grasp fairly well the one inverse and two direct relations. (4) Integration of these 3 relations into one was very difficult. Great progress was made from when children were 8 years old to when they were 10, but even so, only 25% of 10-year-olds seemed to complete the integration.