This study aimed to investigate the informal cognitive structures regarding "point" and "straight line"--two basic and undefined terms of geometry--in children registered in preschool--the previous step before in-class formal education process. The study was conducted with the participation of 50 children enrolled in nursery, kindergarten and preschools of a total of five educational institutions--three public and two private--in a city which is in the middle of the Turkey. The qualitative research model was utilized in the study since observing, analyzing and assessing children's intuitive thinking and informal knowledge construction process would be difficult and good results would not be obtained via quantitative research methods. Data were collected through clinical interview technique. Results show that children, in general, possess major and to a large extent correct acquisitions that would be the basis of subsequent formal concept development process in children.