A study on student perceptions of principles of logic and their application to the principle of mathematical induction

Graduation date: 2014The purpose of this article is to explore student reasoning with regard to problems in logic, particularly those related to the Principle of Mathematical Induction (PMI). The five case studies presented build off of work done by other researchers, most notably Dubinsky and Harel, who both looked at how students' schemes for logic, proof, and pattern generalization may be applied to PMI. This study examines the schemes students use for tasks involving logical implication both with and without quantifiers, modus ponens, and pattern generalization using data collected from hour-long interviews and then examines the same students' schemes when presented with tasks directly related to PMI in a second hour-long interview. We found that in addition to misrepresentations of logical statements, students also have difficulty representing ideas as mathematical statements and that this is a significant barrier to proof. Additionally, in some cases students had developed schemes which were valuable for tasks on the first interview but did not apply these schemes within the context of induction. Furthermore, it was found that students had formed false beliefs for which they had the tools to determine were false, but for which they had never been challenged or forced into a state of disequilibrium