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Abstract
This paper reports on three exploratory studies of college students decision-making behaviors when choosing proof methods in the context of proof construction. The decision strategies that students used when making proof-method decisions and the constructs influencing their decisions were explored. Nine students (three for the first study, four for the second study, and two for the third study) participated in three studies, and the students in each study were taking a transition-to-proof class with a different instructor and during a different semester while each study was being conducted. For each study, I conducted interviews with the students, observed the students transition classes, and examined their copies of class notes and homework. Based on the observations of the students decision-making activities with proof tasks, their verbal reports while making proof-method decisions, and their responses after choosing methods, I found five decision strategies and eight constructs that contributed to their proof-method decisions across the three studies. For decision strategies, the students decision acts differed depending on their familiarity with the problem statements. When a statement was familiar, the students immediately chose a method using the feature-matching strategy. When a statement was less familiar or unfamiliar, the students chose a method using one to three of the other four strategies. In particular, when they saw that more than two methods could be used to prove or disprove a statement, they used the comparison strategy to choose a method. For constructs, the students knowledge about when to use proof methods and their orientations (e.g., beliefs about proof or proof-method preferences) were the primary influences on their decision making. However, there were other types of constructs affecting their decisions. More studies with a large number of students are needed to confirm that the strategies and constructs found in the three studies are indeed the most prevalent. It is also necessary to examine how mathematicians make proof-method decisions to see what differences may exist between mathematicians (experts) and students (novice provers) with respect to decision making about proof methods and also to help students make strategic proof-method decisions in proof construction.