Peter R. Atwood
Doctor of Philosophy
Department: Mathematics and Statistics
Title: Learning to Construct Proofs in a First Course
on Mathematical Proof
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
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 study concluded by noting limitations of the research, suggesting
some avenues for further related research, and making recommendations
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