### Abstract:

The focus of this study was junior-level
mathematics students' perception of proof and its
relationship to achievement. The following
problems were investigated:
1) nature of perception of proof of
undergraduate mathematics students who have
enrolled in Advanced Calculus;
2) relationship between students' perception
of selected aspects of proof and their achievement
in Advanced Calculus; and
3) relationship between measures of perception
of proof and achievement in Advanced Calculus.
Twenty versions of a questionnaire, each
containing six items, were administered randomly to
47 students in Advanced Calculus. The
questionnaire items measured selected aspects of
students' perception of proof. Student responses
to the questionnaire were evaluated and put into
response categories by three judges.
An interview script was developed based on the
results of the written questionnaires and a pilot
study involving undergraduates with a similar
backround as those in the study. The script
assessed students' subjective perception of the
nature of mathematical proof, degree to which
students enjoy proof, and amount of confidence
students have in their ability to construct proofs.
Eight follow-up interviews were conducted. They
were taped, analyzed, and categorized into an
inductively developed category system.
Achievement data were obtained from student
performance on tests and homework assignments. It
was the total number of points accumulated by each
student.
The following hypotheses were tested:
1) The correlation between total score
obtained on the written questionnaire and
achievement in Advanced Calculus for class A is not
significantly different than the same correlation
for class B.
2) There is no significant proportion of
variation in achievement that is associated with
perception of proof.
3) There is no association between achievement
in Advanced Calculus and perception of the aspects
of proof addressed by each situation on the written
questionnaire.
Data were analyzed using 2x2 contingency
tables, correlation coefficient, and qualitative
analysis of interviews. From these analyses the
following conclusions were drawn:
1) the nature and role of hypothesis in
mathematics is misunderstood by at least a large
minority of junior-level mathematics students;
2) a significant proportion of variation in
achievement is associated with perception of proof.
Several recommendations for research and practice
were discussed.