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WHAT ARE SOME OF THE COMMON TRAITS IN THE THOUGHT PROCESSES OF UNDERGRADUATE STUDENTS CAPABLE OF CREATING PROOF?
Abstract
Mathematical proof is an important topic in mathematics education research. Many researchers have addressed various aspects of proof. One aspect that has not been addressed is what common traits are shared by those who are successful at creating proof. This research investigates the common traits in the thought processes of undergraduate students who are considered successful by their professors at creating mathematical proof. A successful proof is defined as a proof that successfully accomplishes at least one of DeVilliers (2003) six roles of proof and demonstrates adequate mathematical content, knowledge, deduction and logical reasoning abilities. This will typically be present in a proof that fits Weber’s (2004) semantic proof category, though some syntactic proofs may also qualify. Proof creation can be considered a type of problem, and Schoenfeld’s (1985) categories of resources, heuristics, control and ability are used as a framework for reporting the results. The research involved a) finding volunteers based on professorial recommendations; b) administering a proof questionnaire and conducting a video recorded interview about the results; and then c) holding a second video recorded interview where new proofs were introduced to the subjects during the interviews. The researcher used Goldin’s (2000) recommendations for making task based research scientific and made interview protocols in the style of Galbraith (1981). The interviews were transcribed and analyzed using Strauss and Corbin’s (1990) methods. The resulting codes corresponded with Schoenfeld’s four categories, so his category names were used. Resources involved the mathematical content knowledge available to the subject. Heuristics involved strategies and techniques used by the subject in creating the proof. Control involved choices in implementing resources and heuristics, planning and using time wisely. Beliefs involved the subjects’ beliefs about mathematics, proof, and their own skills. These categories are seen in other research involving proof but not all put together. The research has implications for further research possibilities in how the categories all work together and develop in successful proof creators. It also has implications for what should be taught in proofs courses to help students become successful provers.
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