Students' cognitive performance on any mathematical concept is related with the respective self-beliefs on understanding it and on using strategies in order to overcome cognitive obstacles. The purpose of the present study was to investigate the role of various aspects of apprehension (perceptual, operative and discursive) in geometrical figure understanding and the respective students' self-beliefs about using representations as a useful tool for understanding geometrical concepts and for solving geometrical tasks. The interest was concentrated on finding the structure and the interrelations between students' self-concept beliefs, their self-efficacy beliefs about using representations and the geometrical understanding at primary and secondary education. A quantitative study was developed and data were collected from 1086 students in Grade 5 to 8. Confirmatory factor analysis provided support for the invariance of the structure for the geometrical figure understanding across the two ages groups and it affirmed the existence of a coherent model of self-beliefs dimensions about the use of representations for understanding the geometrical concepts, which becomes more stable across the educational levels. There are interesting differences concerning the interrelations among cognitive and noncognitive factors between primary and secondary education. (Contains 3 figures and 1 table.)