| The purpose of this study was to describe the performance of college students, within and across varying levels of conceptual understanding, as they attempted to write proofs in elementary group theory. In particular, processes, errors, correctness, and students' self-assessment were described and analyzed.; The subjects were twenty-nine college mathematics majors enrolled in three progressively more advanced abstract algebra courses. Four levels of conceptual understanding were operationally defined: pre-understanding, syntactic understanding, semantic-concrete understanding, semantic-abstract understanding; and each subject, with the exception of three "non-fitters", was placed into one of these four levels. The written work of the subjects on six selected proofs was described using precise investigator-developed coding dictionaries, the proofs were scored for correctness, and a questionnaire was administered. This generated data on processes used, errors made, correctness of proofs, and the subjects' own assessment of the proof tasks. The data were analyzed within and across the levels of conceptual understanding.; The data were discussed in the context of the conceptual analysis research paradigm, the expert/novice paradigm, "levels" models, and the proofs themselves. The major conclusions of the conceptual analysis discussion were that the performance of the subjects (i.e., specific processes, errors, correctness, and self-assessment) could be interpreted in terms of the stability of the conceptual system which they possessed concerning elementary group theory, and that the middle levels of conceptual understanding defined in this study could be characterized as unstable and transitional. Conclusions in light of the expert/novice paradigm included support of the claims made by other researchers (e.g., Silver (1985)) that there are not just experts and novices but gradations of expert and novice, and the evolution from expert to to novice tends to be a rather unstable developmental process. Concerning so-called "levels models", the implications were that a general levels model does exist but that it is very difficult to empirically describe and practically apply. Finally, conclusions concerning the proofs themselves were drawn, comparing proofs and discussing how deceptive a priori notions of "easy" and "hard" can be. The study concluded with implications for teaching proof and suggestions for further research. |
