An investigation of undergraduates' understanding of congruence of integers

Abstract

The conceptions of congruence of integers of six above-average-performing undergraduate mathematics students enrolled in a third-year introductory number theory course at a large state university in the southwestern US were examined using an exploratory case study design. Data collected include interviews, written questionnaires, and videotapes of class sessions. There were four major findings concerning these six students' conceptions of congruence. In general, the students (1) invented and used a pseudo-definition for congruence, (2) avoided working in what they called the "mod world," (3) developed an increasingly operational view of congruence (several students' relational views of congruence were heavily de-ephasized or disappeared), and (4) did not view congruences as analogous to equations. In addition, the students appropriated a strategy used to solve linear Diophantine equations in order to solve linear congruences, disregarding the method used by the instructor in class, and in the process reversed an important heuristic for solving mathematical problems. A framework was developed for analyzing the degree to which these students were employing advanced mathematical thinking. This framework represents an attempt to synthesize multiple perspectives on the nature of advanced mathematical thinking currently present in the field. In addition, the notion of classroom mathematical practices from the emergent perspective (Cobb and Yackel, 1996) was used to examine the development of the class's understanding of congruence. The development of the four interpretations described above coincided with the development of five classroom mathematical practices. The individuals' conceptions at various points in time can be viewed as "consequences" of these practices, and can be seen in turn as influencing the development of other practices. This study found many similarities between these undergraduates' conceptions of linear congruences and students' documented difficulties solving equations in algebra. The students in the study were primarily prospective secondary mathematics teachers, and since the topics studied in this type of course are closely related to those of high school mathematics, this study has implications for teacher education as well.