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ERIC Number: ED550904
Record Type: Non-Journal
Publication Date: 2012
Pages: 392
Abstractor: As Provided
Reference Count: N/A
ISBN: 978-1-2679-8804-1
ISSN: N/A
Proof and Proving: Logic, Impasses, and the Relationship to Problem Solving
Savic, Milos
ProQuest LLC, Ph.D. Dissertation, New Mexico State University
Becoming a skillful prover is critical for success in advanced undergraduate and graduate mathematics courses. In this dissertation, I report my investigations of proof and the proving process in three separate studies. In the first study, I examined the amount of logic used in student-constructed proofs to help in the design of transition-to-proof courses. The study had four parts: coding 42 student-constructed proofs from a "proofs course," coding 10 student-constructed proofs from a graduate homological algebra course, interviewing three students proving a theorem, and coding the same 42 student-constructed proofs using "proof frameworks." All four parts were intended to discover the logic in proofs and in the proving process so that designers of transition-to-proof courses can take into consideration the frequency of, and the kind of, logic that occurs in constructing proofs. The second study examined how mathematicians and graduate students recover from proving impasses. I defined impasses, which are colloquially referred to as "getting stuck," as a period of time during the proving process when a prover feels or recognizes that his or her argument has not been progressing fruitfully and that he or she has no new ideas. Using the notes from the "proofs course" on semigroups, and a new data collection technique, six of nine mathematician participants and all five graduate student participants had impasses during the proving of theorems in the notes. I discuss the ways participants got "un-stuck," including incubation, a term from psychology colloquially meaning "taking a break.". Finally, researchers have noted that proving and problem solving are related. In fact, one study by Carlson and Bloom (2005) used the data that they had collected on how mathematicians solve problems to create what they called a Multidimensional Problem-Solving Framework. It occurred to me that the data I had collected in the previous study probably could be analyzed using their framework. Upon closer examination, I found that there were commonalities and differences with the phases and attributes of their framework and those of the proving process. Through these three studies, I hope to add to the existing mathematics education research literature on proof and proving. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
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Publication Type: Dissertations/Theses - Doctoral Dissertations
Education Level: Higher Education; Postsecondary Education
Audience: N/A
Language: English