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To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples

Bubp, Kelly M.
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1417178872

Abstract Details

2014, Doctor of Philosophy (PhD), Ohio University, Curriculum and Instruction Mathematics Education (Education).
Deciding on the truth value of mathematical statements is an essential aspect of mathematical practice in which students are rarely engaged. This study explored undergraduate students’ approaches to mathematical statements with unknown truth values. The research questions were 1. In what ways and to what extent do students use intuition and analysis to decide on the truth value of mathematical statements? 2. What are the connections between students’ process of deciding on the truth value of mathematical statements and their ability to construct associated proofs and counterexamples? 3. What types of systematic intuitive, mathematical, and logical errors do students make during the proving process, and what is the impact of these errors on the proving process? Clinical task-based interviews utilizing the think-aloud method revealed students’ reasoning processes in depth. Twelve undergraduate students each completed four mathematical tasks requiring them to decide on the truth value of a statement and prove or disprove it accordingly. Through analysis of the data, I developed a framework for distinguishing among types of reasoning based on their cognitive and mathematical properties. The framework identifies four distinct categories of reasoning – intuitive, semantic-empirical, semantic-deductive, and syntactic – each with subcategories. The students in this study used all four types of reasoning for deciding on the truth value of the statements in the tasks. Their use of semantic-deductive and syntactic reasoning mirrored mathematicians’ use of these reasoning types for decision-making. With the exception of one task, the students’ decision-making and construction processes were generally connected. Connections in which the construction process was based on decision-making process mostly facilitated proving. However, simultaneous decision-making and construction processes often led to overturned decisions. Regarding intuitive decision-making, only property-based intuitive decisions were connected to the corresponding construction process. The students in this study made numerous systematic mathematical and logical errors, but systematic intuitive errors were limited and occurred on only one task. The systematic conceptual misunderstandings surrounding the concept of function are troubling due to the centrality of this concept to mathematics. Few errors were overcome, but a certain level of uncertainty may aid students in overcoming logical errors.
Robert Klein (Committee Co-Chair)
Allyson Hallman-Thrasher (Committee Co-Chair)
271 p.
Mathematics Education
mathematical proof; intuition; dual-process theory; semantic and syntactic reasoning

Recommended Citations

Citations

  • Bubp, K. M. (2014). To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1417178872

    APA Style (7th edition)

  • Bubp, Kelly. To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples. 2014. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1417178872.

    MLA Style (8th edition)

  • Bubp, Kelly. "To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples." Doctoral dissertation, Ohio University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1417178872

    Chicago Manual of Style (17th edition)

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This open access ETD is published by Ohio University and OhioLINK.