The purpose of this study was to uncover information about and gain a greater insight into the extent to which students who are successful in a basic algebra course: l) demonstrate a reversibility of reasoning processes when solving algebraic problems; 2) demonstrate a flexibility of reasoning processes when solving algebraic problems; 3) generalize reasoning processes and algebraic operations; and 4) transfer reasoning processes and algebraic operations within the content of algebra. A Soviet "ascertaining" methodology was chosen, with this clinical approach viewed as appropriate to investigate the algebra student's thought processes. Six instruments were used to provide descriptions of the subjects. Four females were selected non-randomly for the study. Data indicated that the subjects chosen varied widely on measures of the four processes of understanding identified. Tests based on measures of instrumental understanding did not accurately reflect a student's relational understanding. Further, these students possessed definite styles of reversibility and many of the same error patterns as less successful students. It is felt that this study provides an exploration of the elusive term, "understanding" of mathematics, which mathematics educators have long noted as an important thing for students to develop. (MP)