Karakok, Gulden

Committee Member

Oehrtman, Michael

Committee Member

Dzhamay, Anton


Educational Mathematics


University of Northern Colorado

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Greeley (Colo.)


University of Northern Colorado

Date Created





367 pages

Digital Origin

Born digital


This research is an investigation into the ideas professional mathematicians find useful in developing mathematical proofs. Specifically, this research uses the construct personal argument to describe the ideas and thoughts the individual deems relevant to making progress in proving the statement. The research looked to describe the ideas that mathematicians integrated into their personal arguments, the context surrounding the development of these ideas in terms of Dewey‘s theories of inquiry and instrumentalism, and how the mathematicians used these ideas as their arguments evolved toward a completed proof. Three research mathematicians with multiple years of experience teaching real analysis completed tasks in real analysis while thinking aloud in interview and independent settings recorded with video and Livescribe technology. Follow-up interviews were also conducted. Data were analyzed for ideas that participants found useful. Toulmin argumentation diagrams were implemented to describe the evolving arguments, and Dewey’s inquiry framework helped to describe the context surrounding the development of the ideas. Descriptive stories were written for each participant‘s work on each task documenting the argument evolution. Open, iterative coding of each idea, problem encountered, and tool was conducted. Patterns, categories, and themes across participants and tasks were identified. The mathematicians developed ideas that moved their personal arguments forward that were grouped into three categories according to their functionality: ideas that focus and configure, ideas that connect and justify, and monitoring ideas. Within these three categories were ideas in fifteen sub-types. The ideas emerged through the mathematicians’ purposeful recognition of problems to be solved as well as reflective and evaluative actions to solve them. This research implicates that using the full Toulmin model for investigating the process of creating mathematical proof since the modal qualifiers evolve to become absolute as the warrants shift to become based on deductive reasoning. In the instruction of undergraduate students, this work supports teaching content in conjunction with proof techniques and heuristic strategies for problem solving and recommends engaging students in discourse situations that would motivate moving their informal arguments into deductive proofs.

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