Date of Award
Doctor of Philosophy
Dr. Christian Hirsch
Dr. Gary Chartrand
Dr. Eric Hart
Dr. Robert Laing
This study examined the conceptions o f proof that undergraduate students have upon entry to a transition course on mathematical proof how they develop skill in planning and reporting proofs, obstacles encountered, and effects of instruction on their performance in solidifying schema in proof-planning and proof-reporting.
The subjects were sophomores and juniors (n=16) in a transition course at a large midwestern university. The course was taught by one o f the co-authors o f the text, "Mathematical Proofs" (Chartrand, Polimeni, and Zhang, 1999, in press). Assessment of learning to construct proofs was through quizzes and a final exam developed by the professor with input from the researcher. These written assessments were augmented by case studies of six students.
A pretest and initial interviews provided baseline measures o f the students' understandings. Subsequent assessments revealed how each student constructed direct proofs, proofs by contrapositive, proofs by contradiction, and proofs by mathematical induction. Half the students demonstrated that they understood the statement to be proved, and recognized definitions of the terms involved. Ten of the 16 students also showed correctly that negative results could be established by a counterexample.
The study confirmed obstacles previously identified in the literature: starting direct proofs and proofs by contrapositive, using definitions, and using universal and existential quantifiers. In addition, other obstacles were prominent: choosing mathematical notation and representations, forming induction assumptions for proofs by complete induction, and constructing proofs by contradiction.
Students' proof-constructions demonstrated habits that appropriated the presentations in the textbook and classroom. They gave clear statements of the starting assumptions, the proof strategy, and the framework of proofs by mathematical induction. The statements of starting assumptions for proofs by contradiction and the induction assumption for complete induction, however, were not successfully emulated. The study included a formulation of schema for constructing proofs that distinguished between proof concepts and mathematical concepts.
The study concluded by noting limitations of the research, suggesting avenues for further related research, and making recommendations for practice.
Atwood, Peter R., "Learning to Construct Proofs in a First Course on Mathematical Proof" (2001). Dissertations. 1350.