Jon Janelle

Date of Award

5-2014

Level of Access Assigned by Author

Campus-Only Thesis

Degree Name

Master of Science (MS)

Department

Teaching

Advisor

Natasha M. Speer

Second Committee Member

Robert Franzosa

Third Committee Member

Eric Pandiscio

Abstract

Mathematicians and mathematics education researchers have consistently asserted the crucial and multifaceted roles that deductive reasoning and proof play in mathematics. In contrast, students at many levels of education have been found to hold severely limited views of proof that may lead them to view mathematics as a rigid, formal, and largely meaningless discipline. Improving students’ understandings of and attitudes toward reasoning and proof is necessary to motivate a greater number to consider careers in STEM fields and to prevent attrition in mathematically-intensive degree programs.

This study consisted of an investigation into 59 undergraduate calculus students’ views about the nature and purposes of mathematical proof, the forms of empirical arguments they perceived as valid proofs, and the connection between their proof construction and validation practices. Previous studies of student proof conceptions have primarily focused on three groups: students in secondary geometry courses, pre-service and in-service teachers, and advanced undergraduate and graduate students who have received formal instruction in the creation of deductive proofs. However, little attention has been given to the connection between students’ proof constructions and validations or to examining students’ conceptions after the completion of a high school geometry course, but before enrollment in proof-based mathematics course. Using data obtained from written surveys and interviews, this study was designed to fill this gap in the literature.

Findings suggest that a majority, more than 80%, of college calculus students believe that the inspection of a few examples and the testing of a single extreme case are valid forms of mathematical proof. In addition, students who incorrectly validated empirical arguments as proofs were significantly more likely than their peers to construct empirical arguments when asked to verify a conjecture. Consistent with the findings of past researchers, approximately half accepted false arguments based on their proof-like surface features, for example the use of variables or formal mathematical language. While two-thirds of students were able to describe at least one purpose for proof consistent with descriptions generated by the mathematical community, many were unable to acceptably describe a single meaningful use. The pedagogical implications and limitations of these findings are discussed.

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