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 chi ldren of concepts about the interrelations among number, length, and crowde dness, using as stimulus materials pictures of flower beds with tulips crow ded to various amounts, planters of various lengths, and various sized bunc hes of tulips. Participants were 136 5- to 10-year-olds. The main results w ere as follows :(1) For 5- and 6-year-olds, understanding of crowdedness wa s rather difficult. (2) The direct relation between number and length was g rasped strongly even by 5-year-olds, although an excessively strong relatio n might prevent children from understanding one relation among the 3 concep ts, that is, crowdedness = number/length. (3) Though grasping the inverse r elation between length and crowdedness was the most difficult concept, 8-ye ar-olds seemed to grasp fairly well the one inverse and two direct relation s. (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.