# An investigation of college students' understanding of proof construction when doing mathematical analysis proofs

The purpose of this study is to identify students' approaches to analysis proofs by observing students actually doing proofs during two proof sessions and by analyzing the students' proofs required for assignments or examinations. Seven students chosen from the Analysis class at the American University in Washington, D.C. during the fall semester of 1998 are considered for this study. The researcher held office hours for ten hours a week and conducted two one-hour interviews and two one-hour proof sessions with each of the research subjects. Each proof session was audio recorded and later transcribed, and the information collected generated data on processes used, errors made and strategies used when constructing an analysis proof. The data are discussed in the context of the Concept-Understanding Scheme, which consists of concept definition, concept image, and concept usage. The importance of visualization during the proof process is discussed based on what the students wrote and said. The researcher studied the learning patterns of the students in the study to develop a model of how students in this sample do an analysis proof. Most of the subjects felt that it was important to work with others when doing proofs. Some of the subjects preferred working with other classmates, while some felt it important to work with the instructor in a one-on-one office hours setting. Most of the subjects felt that one must read the statement; write down information that comes from the given information in the problem; clearly pay attention to the given information; look up any definitions that you may need; and understand what you need to show. For most of the subjects, visual representation is not used in a majority of proofs and most of the subjects follow the analytic conventions of proof writing. The visual elements may be necessary to understand the theory or to convince the students that the theorem or statement is most likely true, but the diagrams for the most part do not influence their abstract proof writing. A phase model of how students do proofs in analysis or advanced calculus is presented based on this sample of students. The model consists of seven phases which include the convince phase, the strategy phase, the investigations phase, the resource phase, the formalization phase, the authority phase and the completion phase. Evidence of this model comes from the two proof sessions, office hours, homework and examinations. During the final interview with the research subjects, each subject described how they interact with each phase and drew diagrams to show the interplay.