Candidate: Peter R. Atwood

Degree of: Doctor of Philosophy

Department: Mathematics and Statistics

Title: Learning to Construct Proofs in a First Course on Mathematical Proof

Committee:
Dr. Christian Hirsch, Chair
Dr. Dennis Pence
Dr. Eric Hart
Dr. Gary Chartrand
Dr. Robert Laing

Date: Monday, July 9, 2001, 4:00 p.m. – 6:00 p.m.
Alavi Commons, 6th Floor Everett

Abstract:
The undergraduate mathematics curriculum is comprised of lower level courses that study concepts, methods, and procedures; and upper level courses that study proofs and abstract systems. Since the 1970s, this gap in the teaching of abstract mathematical reasoning been addressed by the introduction of a transition to advanced mathematics course.

This study examined the conceptions of proof that students have upon entry to a transition course on mathematical proof; how they develop skill in planning and reporting proofs and the obstacles encountered; and the effect of instruction in proof strategies on their performance in solidifying schema in proof-planning and proof-reporting.

The subjects were n=16 sophomores and juniors in a transition course at a large mid-western university. The course was taught by one of the co-authors of the text, "Mathematical Proofs: A Guide to Understanding the Basics of Abstract Mathematics and Constructing and Writing Proofs of Your Own" (Chartrand, Polimeni, and Zhang, 1999). Assessment of students' learning to construct proofs was done through quizzes and a final exam developed by the professor with input from the researcher. These written assessments were augmented by case studies with six students from the class.

A pretest at the beginning of the course and the initial interviews indicated that understanding the distinction between a statement and its converse, using definitions of mathematical concepts within a proof, starting proofs by contradiction, and interpreting the meaning of a contradiction within a proof, were obstacles.

The six interviewees consisted of three students with A's in their previous college mathematics courses, two students with B's, and one student with C's. There were three males and three females. The researcher developed and administered five interviews, but produced valid proofs on all written assessments. One student showed all of the obstacles mentioned, and did not improve during the semester.

A comparison of the performance of the interview students with the entire class confirmed obstacles that the research literature had identified: starting proofs, the role of definitions, and the use of universal and existential quantifiers. In addition other obstacles were prominent: choosing notation and representations, complete induction, and proofs by contradiction. The interviews repeatedly sampled the students' performances on these issues in constructing proofs, inviting them to reveal their schema for how they understood this process. Three of the interview students showed that they were aware of their weaknesses, and that they worked to over come them.

The research also noted the influences of the instruction via the textbook and the professor that they expressed, either overtly or implicitly. The model of good mathematical writing, for example, was clearly present in the students' work on all of the written assessments subsequent to the pretest.

The study concluded by noting limitations of the research, suggesting some avenues for further related research, and making recommendations for practice.

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